text:
@testset "1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p" begin
    (a, b, c, d, e, f, g, m, n, p, x, ) = @variables a b c d e f g m n p x

    #= ::Package:: =#

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(g*x)^m*(d+e*x)^n*(a+b*x+c*x^2)^p=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(g*x)^m*(d+e*x)^n*(a+b*x+c*x^2)^p*when*b=0*and*c*d^2+a*e^2=0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(d+e*x)^1*(d^2 - e^2*x^2)^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^2*(d + e*x)*(d^2 - e^2*x^2)^(1/2), x, 10, (d^3*x*sqrt(d^2 - e^2*x^2))/(8*e^2) - (d^2*(d^2 - e^2*x^2)^(3/2))/(3*e^3) - (d*x*(d^2 - e^2*x^2)^(3/2))/(4*e^2) + (d^2 - e^2*x^2)^(5/2)/(5*e^3) + (d^5*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(8*e^3)]

    @test_int [x^4*(d + e*x)*(d^2 - e^2*x^2)^(3/2), x, 8, (3*d^7*x*sqrt(d^2 - e^2*x^2))/(128*e^4) + (d^5*x*(d^2 - e^2*x^2)^(3/2))/(64*e^4) - (4*d^2*x^2*(d^2 - e^2*x^2)^(5/2))/(63*e^3) - (d*x^3*(d^2 - e^2*x^2)^(5/2))/(8*e^2) - (x^4*(d^2 - e^2*x^2)^(5/2))/(9*e) - (d^3*(128*d + 315*e*x)*(d^2 - e^2*x^2)^(5/2))/(5040*e^5) + (3*d^9*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(128*e^5)]
    @test_int [x^3*(d + e*x)*(d^2 - e^2*x^2)^(3/2), x, 7, (3*d^6*x*sqrt(d^2 - e^2*x^2))/(128*e^3) + (d^4*x*(d^2 - e^2*x^2)^(3/2))/(64*e^3) - (d*x^2*(d^2 - e^2*x^2)^(5/2))/(7*e^2) - (x^3*(d^2 - e^2*x^2)^(5/2))/(8*e) - (d^2*(32*d + 35*e*x)*(d^2 - e^2*x^2)^(5/2))/(560*e^4) + (3*d^8*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(128*e^4)]
    @test_int [x^2*(d + e*x)*(d^2 - e^2*x^2)^(3/2), x, 12, (d^5*x*sqrt(d^2 - e^2*x^2))/(16*e^2) + (d^3*x*(d^2 - e^2*x^2)^(3/2))/(24*e^2) - (d^2*(d^2 - e^2*x^2)^(5/2))/(5*e^3) - (d*x*(d^2 - e^2*x^2)^(5/2))/(6*e^2) + (d^2 - e^2*x^2)^(7/2)/(7*e^3) + (d^7*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(16*e^3)]
    @test_int [x^1*(d + e*x)*(d^2 - e^2*x^2)^(3/2), x, 5, (d^4*x*sqrt(d^2 - e^2*x^2))/(16*e) + (d^2*x*(d^2 - e^2*x^2)^(3/2))/(24*e) - ((6*d + 5*e*x)*(d^2 - e^2*x^2)^(5/2))/(30*e^2) + (d^6*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(16*e^2)]
    @test_int [x^1*(d + e*x)*(d^2 - e^2*x^2)^(3/2), x, 5, (d^4*x*sqrt(d^2 - e^2*x^2))/(16*e) + (d^2*x*(d^2 - e^2*x^2)^(3/2))/(24*e) - ((6*d + 5*e*x)*(d^2 - e^2*x^2)^(5/2))/(30*e^2) + (d^6*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(16*e^2)]
    @test_int [(d + e*x)*(d^2 - e^2*x^2)^(3/2)/x^1, x, 8, (1/8)*d^2*(8*d + 3*e*x)*sqrt(d^2 - e^2*x^2) + (1/12)*(4*d + 3*e*x)*(d^2 - e^2*x^2)^(3/2) + (3/8)*d^4*atan((e*x)/sqrt(d^2 - e^2*x^2)) - d^4*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d + e*x)*(d^2 - e^2*x^2)^(3/2)/x^2, x, 8, (1/2)*d*e*(2*d - 3*e*x)*sqrt(d^2 - e^2*x^2) - ((3*d - e*x)*(d^2 - e^2*x^2)^(3/2))/(3*x) - (3/2)*d^3*e*atan((e*x)/sqrt(d^2 - e^2*x^2)) - d^3*e*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d + e*x)*(d^2 - e^2*x^2)^(3/2)/x^3, x, 8, -((3*d*e*(d + e*x)*sqrt(d^2 - e^2*x^2))/(2*x)) - ((d - e*x)*(d^2 - e^2*x^2)^(3/2))/(2*x^2) - (3/2)*d^2*e^2*atan((e*x)/sqrt(d^2 - e^2*x^2)) + (3/2)*d^2*e^2*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d + e*x)*(d^2 - e^2*x^2)^(3/2)/x^4, x, 8, (e^2*(2*d - 3*e*x)*sqrt(d^2 - e^2*x^2))/(2*x) - ((2*d + 3*e*x)*(d^2 - e^2*x^2)^(3/2))/(6*x^3) + d*e^3*atan((e*x)/sqrt(d^2 - e^2*x^2)) + (3/2)*d*e^3*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d + e*x)*(d^2 - e^2*x^2)^(3/2)/x^5, x, 8, (e^2*(3*d + 8*e*x)*sqrt(d^2 - e^2*x^2))/(8*x^2) - ((3*d + 4*e*x)*(d^2 - e^2*x^2)^(3/2))/(12*x^4) + e^4*atan((e*x)/sqrt(d^2 - e^2*x^2)) - (3/8)*e^4*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d + e*x)*(d^2 - e^2*x^2)^(3/2)/x^6, x, 6, (3*e^3*sqrt(d^2 - e^2*x^2))/(8*x^2) - (e*(d^2 - e^2*x^2)^(3/2))/(4*x^4) - (d^2 - e^2*x^2)^(5/2)/(5*d*x^5) - (3*e^5*atanh(sqrt(d^2 - e^2*x^2)/d))/(8*d)]
    @test_int [(d + e*x)*(d^2 - e^2*x^2)^(3/2)/x^7, x, 7, (e^4*sqrt(d^2 - e^2*x^2))/(16*d*x^2) - (e^2*(d^2 - e^2*x^2)^(3/2))/(24*d*x^4) - (d^2 - e^2*x^2)^(5/2)/(6*d*x^6) - (e*(d^2 - e^2*x^2)^(5/2))/(5*d^2*x^5) - (e^6*atanh(sqrt(d^2 - e^2*x^2)/d))/(16*d^2)]
    @test_int [(d + e*x)*(d^2 - e^2*x^2)^(3/2)/x^8, x, 8, (e^5*sqrt(d^2 - e^2*x^2))/(16*d^2*x^2) - (e^3*(d^2 - e^2*x^2)^(3/2))/(24*d^2*x^4) - (d^2 - e^2*x^2)^(5/2)/(7*d*x^7) - (e*(d^2 - e^2*x^2)^(5/2))/(6*d^2*x^6) - (2*e^2*(d^2 - e^2*x^2)^(5/2))/(35*d^3*x^5) - (e^7*atanh(sqrt(d^2 - e^2*x^2)/d))/(16*d^3)]
    @test_int [(d + e*x)*(d^2 - e^2*x^2)^(3/2)/x^9, x, 9, (3*e^6*sqrt(d^2 - e^2*x^2))/(128*d^3*x^2) - (e^4*(d^2 - e^2*x^2)^(3/2))/(64*d^3*x^4) - (d^2 - e^2*x^2)^(5/2)/(8*d*x^8) - (e*(d^2 - e^2*x^2)^(5/2))/(7*d^2*x^7) - (e^2*(d^2 - e^2*x^2)^(5/2))/(16*d^3*x^6) - (2*e^3*(d^2 - e^2*x^2)^(5/2))/(35*d^4*x^5) - (3*e^8*atanh(sqrt(d^2 - e^2*x^2)/d))/(128*d^4)]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^2*(d + e*x)/(d^2 - e^2*x^2)^(1/2), x, 8, -((d^2*sqrt(d^2 - e^2*x^2))/e^3) - (d*x*sqrt(d^2 - e^2*x^2))/(2*e^2) + (d^2 - e^2*x^2)^(3/2)/(3*e^3) + (d^3*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(2*e^3)]

    @test_int [x^2*(d + e*x)/(d^2 - e^2*x^2)^(3/2), x, 5, (d*(d + e*x))/(e^3*sqrt(d^2 - e^2*x^2)) + sqrt(d^2 - e^2*x^2)/e^3 - (d*atan((e*x)/sqrt(d^2 - e^2*x^2)))/e^3]

    @test_int [x^2*(d + e*x)/(d^2 - e^2*x^2)^(5/2), x, 3, (x^2*(d + e*x))/(3*d*e*(d^2 - e^2*x^2)^(3/2)) - 2/(3*e^3*sqrt(d^2 - e^2*x^2))]

    @test_int [x^7*(d + e*x)/(d^2 - e^2*x^2)^(7/2), x, 6, (x^6*(d + e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^4*(6*d + 7*e*x))/(15*e^4*(d^2 - e^2*x^2)^(3/2)) + (x^2*(24*d + 35*e*x))/(15*e^6*sqrt(d^2 - e^2*x^2)) + ((32*d + 35*e*x)*sqrt(d^2 - e^2*x^2))/(10*e^8) - (7*d^2*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(2*e^8)]
    @test_int [x^6*(d + e*x)/(d^2 - e^2*x^2)^(7/2), x, 6, (x^5*(d + e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^3*(5*d + 6*e*x))/(15*e^4*(d^2 - e^2*x^2)^(3/2)) + (x*(5*d + 8*e*x))/(5*e^6*sqrt(d^2 - e^2*x^2)) + (16*sqrt(d^2 - e^2*x^2))/(5*e^7) - (d*atan((e*x)/sqrt(d^2 - e^2*x^2)))/e^7]
    @test_int [x^5*(d + e*x)/(d^2 - e^2*x^2)^(7/2), x, 5, (x^4*(d + e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^2*(4*d + 5*e*x))/(15*e^4*(d^2 - e^2*x^2)^(3/2)) + (8*d + 15*e*x)/(15*e^6*sqrt(d^2 - e^2*x^2)) - atan((e*x)/sqrt(d^2 - e^2*x^2))/e^6]
    @test_int [x^4*(d + e*x)/(d^2 - e^2*x^2)^(7/2), x, 4, (x^4*(d + e*x))/(5*d*e*(d^2 - e^2*x^2)^(5/2)) - (4*d^2)/(15*e^5*(d^2 - e^2*x^2)^(3/2)) + 4/(5*e^5*sqrt(d^2 - e^2*x^2))]
    @test_int [x^3*(d + e*x)/(d^2 - e^2*x^2)^(7/2), x, 3, (x^2*(d + e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (2*d + 3*e*x)/(15*e^4*(d^2 - e^2*x^2)^(3/2)) + x/(5*d^2*e^3*sqrt(d^2 - e^2*x^2))]
    @test_int [x^2*(d + e*x)/(d^2 - e^2*x^2)^(7/2), x, 3, (x^2*(d + e*x))/(5*d*e*(d^2 - e^2*x^2)^(5/2)) - (2*(d - e*x))/(15*d*e^3*(d^2 - e^2*x^2)^(3/2)) - (2*x)/(15*d^3*e^2*sqrt(d^2 - e^2*x^2))]
    @test_int [x^1*(d + e*x)/(d^2 - e^2*x^2)^(7/2), x, 3, (d + e*x)/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - x/(15*d^2*e*(d^2 - e^2*x^2)^(3/2)) - (2*x)/(15*d^4*e*sqrt(d^2 - e^2*x^2))]
    @test_int [x^0*(d + e*x)/(d^2 - e^2*x^2)^(7/2), x, 3, (d + e*x)/(5*d*e*(d^2 - e^2*x^2)^(5/2)) + (4*x)/(15*d^3*(d^2 - e^2*x^2)^(3/2)) + (8*x)/(15*d^5*sqrt(d^2 - e^2*x^2))]
    @test_int [(d + e*x)/(x^1*(d^2 - e^2*x^2)^(7/2)), x, 7, (d + e*x)/(5*d^2*(d^2 - e^2*x^2)^(5/2)) + (5*d + 4*e*x)/(15*d^4*(d^2 - e^2*x^2)^(3/2)) + (15*d + 8*e*x)/(15*d^6*sqrt(d^2 - e^2*x^2)) - atanh(sqrt(d^2 - e^2*x^2)/d)/d^6]
    @test_int [(d + e*x)/(x^2*(d^2 - e^2*x^2)^(7/2)), x, 7, (d + e*x)/(5*d^2*x*(d^2 - e^2*x^2)^(5/2)) + (6*d + 5*e*x)/(15*d^4*x*(d^2 - e^2*x^2)^(3/2)) + (8*d + 5*e*x)/(5*d^6*x*sqrt(d^2 - e^2*x^2)) - (16*sqrt(d^2 - e^2*x^2))/(5*d^7*x) - (e*atanh(sqrt(d^2 - e^2*x^2)/d))/d^7]
    @test_int [(d + e*x)/(x^3*(d^2 - e^2*x^2)^(7/2)), x, 8, (d + e*x)/(5*d^2*x^2*(d^2 - e^2*x^2)^(5/2)) + (7*d + 6*e*x)/(15*d^4*x^2*(d^2 - e^2*x^2)^(3/2)) + (35*d + 24*e*x)/(15*d^6*x^2*sqrt(d^2 - e^2*x^2)) - (7*sqrt(d^2 - e^2*x^2))/(2*d^7*x^2) - (16*e*sqrt(d^2 - e^2*x^2))/(5*d^8*x) - (7*e^2*atanh(sqrt(d^2 - e^2*x^2)/d))/(2*d^8)]

    @test_int [x^2*(d + e*x)/(d^2 - e^2*x^2)^(9/2), x, 4, (x^2*(d + e*x))/(7*d*e*(d^2 - e^2*x^2)^(7/2)) - (2*(d - 2*e*x))/(35*d*e^3*(d^2 - e^2*x^2)^(5/2)) - (4*x)/(105*d^3*e^2*(d^2 - e^2*x^2)^(3/2)) - (8*x)/(105*d^5*e^2*sqrt(d^2 - e^2*x^2))]

    @test_int [x^2*(d + e*x)/(d^2 - e^2*x^2)^(11/2), x, 5, (x^2*(d + e*x))/(9*d*e*(d^2 - e^2*x^2)^(9/2)) - (2*(d - 3*e*x))/(63*d*e^3*(d^2 - e^2*x^2)^(7/2)) - (2*x)/(105*d^3*e^2*(d^2 - e^2*x^2)^(5/2)) - (8*x)/(315*d^5*e^2*(d^2 - e^2*x^2)^(3/2)) - (16*x)/(315*d^7*e^2*sqrt(d^2 - e^2*x^2))]

    @test_int [x^2*(1 - a*x)/(1 - a^2*x^2)^(3/2), x, 4, -((1 - a*x)/(a^3*sqrt(1 - a^2*x^2))) - sqrt(1 - a^2*x^2)/a^3 - asin(a*x)/a^3]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(d+e*x)^2*(d^2 - e^2*x^2)^(p/2)=#

    #= ::Subsubsection:: =#
    #=p>0=#

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^4*(d + e*x)^2/sqrt(d^2 - e^2*x^2), x, 7, -((8*d^3*x^2*sqrt(d^2 - e^2*x^2))/(15*e^3)) - (11*d^2*x^3*sqrt(d^2 - e^2*x^2))/(24*e^2) - (2*d*x^4*sqrt(d^2 - e^2*x^2))/(5*e) - (1/6)*x^5*sqrt(d^2 - e^2*x^2) - (d^4*(256*d + 165*e*x)*sqrt(d^2 - e^2*x^2))/(240*e^5) + (11*d^6*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(16*e^5)]
    @test_int [x^3*(d + e*x)^2/sqrt(d^2 - e^2*x^2), x, 6, -((3*d^2*x^2*sqrt(d^2 - e^2*x^2))/(5*e^2)) - (d*x^3*sqrt(d^2 - e^2*x^2))/(2*e) - (1/5)*x^4*sqrt(d^2 - e^2*x^2) - (3*d^3*(8*d + 5*e*x)*sqrt(d^2 - e^2*x^2))/(20*e^4) + (3*d^5*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(4*e^4)]
    @test_int [x^2*(d + e*x)^2/sqrt(d^2 - e^2*x^2), x, 5, -((2*d*x^2*sqrt(d^2 - e^2*x^2))/(3*e)) - (1/4)*x^3*sqrt(d^2 - e^2*x^2) - (d^2*(32*d + 21*e*x)*sqrt(d^2 - e^2*x^2))/(24*e^3) + (7*d^4*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(8*e^3)]
    @test_int [x^1*(d + e*x)^2/sqrt(d^2 - e^2*x^2), x, 4, (-(1/3))*x^2*sqrt(d^2 - e^2*x^2) - (d*(5*d + 3*e*x)*sqrt(d^2 - e^2*x^2))/(3*e^2) + (d^3*atan((e*x)/sqrt(d^2 - e^2*x^2)))/e^2]
    @test_int [x^0*(d + e*x)^2/sqrt(d^2 - e^2*x^2), x, 4, -((3*d*sqrt(d^2 - e^2*x^2))/(2*e)) - ((d + e*x)*sqrt(d^2 - e^2*x^2))/(2*e) + (3*d^2*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(2*e)]
    @test_int [(d + e*x)^2/(x^1*sqrt(d^2 - e^2*x^2)), x, 7, -sqrt(d^2 - e^2*x^2) + 2*d*atan((e*x)/sqrt(d^2 - e^2*x^2)) - d*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d + e*x)^2/(x^2*sqrt(d^2 - e^2*x^2)), x, 7, -(sqrt(d^2 - e^2*x^2)/x) + e*atan((e*x)/sqrt(d^2 - e^2*x^2)) - 2*e*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d + e*x)^2/(x^3*sqrt(d^2 - e^2*x^2)), x, 5, -(sqrt(d^2 - e^2*x^2)/(2*x^2)) - (2*e*sqrt(d^2 - e^2*x^2))/(d*x) - (3*e^2*atanh(sqrt(d^2 - e^2*x^2)/d))/(2*d)]
    @test_int [(d + e*x)^2/(x^4*sqrt(d^2 - e^2*x^2)), x, 6, -(sqrt(d^2 - e^2*x^2)/(3*x^3)) - (e*sqrt(d^2 - e^2*x^2))/(d*x^2) - (5*e^2*sqrt(d^2 - e^2*x^2))/(3*d^2*x) - (e^3*atanh(sqrt(d^2 - e^2*x^2)/d))/d^2]
    @test_int [(d + e*x)^2/(x^5*sqrt(d^2 - e^2*x^2)), x, 7, -(sqrt(d^2 - e^2*x^2)/(4*x^4)) - (2*e*sqrt(d^2 - e^2*x^2))/(3*d*x^3) - (7*e^2*sqrt(d^2 - e^2*x^2))/(8*d^2*x^2) - (4*e^3*sqrt(d^2 - e^2*x^2))/(3*d^3*x) - (7*e^4*atanh(sqrt(d^2 - e^2*x^2)/d))/(8*d^3)]
    @test_int [(d + e*x)^2/(x^6*sqrt(d^2 - e^2*x^2)), x, 8, -(sqrt(d^2 - e^2*x^2)/(5*x^5)) - (e*sqrt(d^2 - e^2*x^2))/(2*d*x^4) - (3*e^2*sqrt(d^2 - e^2*x^2))/(5*d^2*x^3) - (3*e^3*sqrt(d^2 - e^2*x^2))/(4*d^3*x^2) - (6*e^4*sqrt(d^2 - e^2*x^2))/(5*d^4*x) - (3*e^5*atanh(sqrt(d^2 - e^2*x^2)/d))/(4*d^4)]

    @test_int [x^5*(d + e*x)^2/(d^2 - e^2*x^2)^(7/2), x, 6, (d^4*(d + e*x)^2)/(5*e^6*(d^2 - e^2*x^2)^(5/2)) - (22*d^3*(d + e*x))/(15*e^6*(d^2 - e^2*x^2)^(3/2)) + (2*d*(30*d + 23*e*x))/(15*e^6*sqrt(d^2 - e^2*x^2)) + sqrt(d^2 - e^2*x^2)/e^6 - (2*d*atan((e*x)/sqrt(d^2 - e^2*x^2)))/e^6]
    @test_int [x^4*(d + e*x)^2/(d^2 - e^2*x^2)^(7/2), x, 6, (d^3*(d + e*x)^2)/(5*e^5*(d^2 - e^2*x^2)^(5/2)) - (17*d^2*(d + e*x))/(15*e^5*(d^2 - e^2*x^2)^(3/2)) + (2*(15*d + 13*e*x))/(15*e^5*sqrt(d^2 - e^2*x^2)) - atan((e*x)/sqrt(d^2 - e^2*x^2))/e^5]
    @test_int [x^3*(d + e*x)^2/(d^2 - e^2*x^2)^(7/2), x, 3, (d^2*(d + e*x)^2)/(5*e^4*(d^2 - e^2*x^2)^(5/2)) - (4*d*(d + e*x))/(5*e^4*(d^2 - e^2*x^2)^(3/2)) + (5*d + 2*e*x)/(5*d*e^4*sqrt(d^2 - e^2*x^2))]
    @test_int [x^2*(d + e*x)^2/(d^2 - e^2*x^2)^(7/2), x, 3, (d*(d + e*x)^2)/(5*e^3*(d^2 - e^2*x^2)^(5/2)) - (7*(d + e*x))/(15*e^3*(d^2 - e^2*x^2)^(3/2)) + x/(15*d^2*e^2*sqrt(d^2 - e^2*x^2))]
    @test_int [x^1*(d + e*x)^2/(d^2 - e^2*x^2)^(7/2), x, 3, (d + e*x)^2/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (2*(d + e*x))/(15*d*e^2*(d^2 - e^2*x^2)^(3/2)) - (4*x)/(15*d^3*e*sqrt(d^2 - e^2*x^2))]
    @test_int [x^0*(d + e*x)^2/(d^2 - e^2*x^2)^(7/2), x, 3, (2*(d + e*x))/(5*e*(d^2 - e^2*x^2)^(5/2)) + x/(5*d^2*(d^2 - e^2*x^2)^(3/2)) + (2*x)/(5*d^4*sqrt(d^2 - e^2*x^2))]
    @test_int [(d + e*x)^2/(x^1*(d^2 - e^2*x^2)^(7/2)), x, 7, (2*(d + e*x))/(5*d*(d^2 - e^2*x^2)^(5/2)) + (5*d + 8*e*x)/(15*d^3*(d^2 - e^2*x^2)^(3/2)) + (15*d + 16*e*x)/(15*d^5*sqrt(d^2 - e^2*x^2)) - atanh(sqrt(d^2 - e^2*x^2)/d)/d^5]
    @test_int [(d + e*x)^2/(x^2*(d^2 - e^2*x^2)^(7/2)), x, 7, (2*e*(d + e*x))/(5*d^2*(d^2 - e^2*x^2)^(5/2)) + (e*(10*d + 13*e*x))/(15*d^4*(d^2 - e^2*x^2)^(3/2)) + (e*(30*d + 41*e*x))/(15*d^6*sqrt(d^2 - e^2*x^2)) - sqrt(d^2 - e^2*x^2)/(d^6*x) - (2*e*atanh(sqrt(d^2 - e^2*x^2)/d))/d^6]
    @test_int [(d + e*x)^2/(x^3*(d^2 - e^2*x^2)^(7/2)), x, 8, (2*e^2*(d + e*x))/(5*d^3*(d^2 - e^2*x^2)^(5/2)) + (e^2*(5*d + 6*e*x))/(5*d^5*(d^2 - e^2*x^2)^(3/2)) + (2*e^2*(10*d + 11*e*x))/(5*d^7*sqrt(d^2 - e^2*x^2)) - sqrt(d^2 - e^2*x^2)/(2*d^6*x^2) - (2*e*sqrt(d^2 - e^2*x^2))/(d^7*x) - (9*e^2*atanh(sqrt(d^2 - e^2*x^2)/d))/(2*d^7)]
    @test_int [(d + e*x)^2/(x^4*(d^2 - e^2*x^2)^(7/2)), x, 9, (2*e^3*(d + e*x))/(5*d^4*(d^2 - e^2*x^2)^(5/2)) + (e^3*(20*d + 23*e*x))/(15*d^6*(d^2 - e^2*x^2)^(3/2)) + (2*e^3*(45*d + 53*e*x))/(15*d^8*sqrt(d^2 - e^2*x^2)) - sqrt(d^2 - e^2*x^2)/(3*d^6*x^3) - (e*sqrt(d^2 - e^2*x^2))/(d^7*x^2) - (14*e^2*sqrt(d^2 - e^2*x^2))/(3*d^8*x) - (7*e^3*atanh(sqrt(d^2 - e^2*x^2)/d))/d^8]

    @test_int [x^3*(1 + x)^2/sqrt(1 - x^2), x, 5, (-(3/5))*x^2*sqrt(1 - x^2) - (1/2)*x^3*sqrt(1 - x^2) - (1/5)*x^4*sqrt(1 - x^2) - (3/20)*(8 + 5*x)*sqrt(1 - x^2) + (3*asin(x))/4]
    @test_int [x^2*(1 + x)^2/sqrt(1 - x^2), x, 4, (-(2/3))*x^2*sqrt(1 - x^2) - (1/4)*x^3*sqrt(1 - x^2) - (1/24)*(32 + 21*x)*sqrt(1 - x^2) + (7*asin(x))/8]
    @test_int [x^1*(1 + x)^2/sqrt(1 - x^2), x, 3, (-(1/3))*x^2*sqrt(1 - x^2) - (1/3)*(5 + 3*x)*sqrt(1 - x^2) + asin(x)]
    @test_int [x^0*(1 + x)^2/sqrt(1 - x^2), x, 3, (-(3/2))*sqrt(1 - x^2) - (1/2)*(1 + x)*sqrt(1 - x^2) + (3*asin(x))/2]
    @test_int [(1 + x)^2/(x^1*sqrt(1 - x^2)), x, 6, -sqrt(1 - x^2) + 2*asin(x) - atanh(sqrt(1 - x^2))]
    @test_int [(1 + x)^2/(x^2*sqrt(1 - x^2)), x, 6, -(sqrt(1 - x^2)/x) + asin(x) - 2*atanh(sqrt(1 - x^2))]
    @test_int [(1 + x)^2/(x^3*sqrt(1 - x^2)), x, 5, -(sqrt(1 - x^2)/(2*x^2)) - (2*sqrt(1 - x^2))/x - (3/2)*atanh(sqrt(1 - x^2))]
    @test_int [(1 + x)^2/(x^4*sqrt(1 - x^2)), x, 6, -(sqrt(1 - x^2)/(3*x^3)) - sqrt(1 - x^2)/x^2 - (5*sqrt(1 - x^2))/(3*x) - atanh(sqrt(1 - x^2))]
    @test_int [(1 + x)^2/(x^5*sqrt(1 - x^2)), x, 7, -(sqrt(1 - x^2)/(4*x^4)) - (2*sqrt(1 - x^2))/(3*x^3) - (7*sqrt(1 - x^2))/(8*x^2) - (4*sqrt(1 - x^2))/(3*x) - (7/8)*atanh(sqrt(1 - x^2))]
    @test_int [(1 + x)^2/(x^6*sqrt(1 - x^2)), x, 8, -(sqrt(1 - x^2)/(5*x^5)) - sqrt(1 - x^2)/(2*x^4) - (3*sqrt(1 - x^2))/(5*x^3) - (3*sqrt(1 - x^2))/(4*x^2) - (6*sqrt(1 - x^2))/(5*x) - (3/4)*atanh(sqrt(1 - x^2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(d+e*x)^3*(d^2 - e^2*x^2)^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [(d + e*x)^3*sqrt(d^2 - e^2*x^2)/x^5, x, 9, -((e^2*(13*d + 8*e*x)*sqrt(d^2 - e^2*x^2))/(8*x^2)) - (d*(d^2 - e^2*x^2)^(3/2))/(4*x^4) - (e*(d^2 - e^2*x^2)^(3/2))/x^3 - e^4*atan((e*x)/sqrt(d^2 - e^2*x^2)) + (13/8)*e^4*atanh(sqrt(d^2 - e^2*x^2)/d)]

    @test_int [(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)*x^5, x, 12, (35*d^12*x*sqrt(d^2 - e^2*x^2))/(2048*e^5) + (35*d^10*x*(d^2 - e^2*x^2)^(3/2))/(3072*e^5) + (7*d^8*x*(d^2 - e^2*x^2)^(5/2))/(768*e^5) - (124*d^5*x^2*(d^2 - e^2*x^2)^(7/2))/(1287*e^4) - (7*d^4*x^3*(d^2 - e^2*x^2)^(7/2))/(48*e^3) - (31*d^3*x^4*(d^2 - e^2*x^2)^(7/2))/(143*e^2) - (7*d^2*x^5*(d^2 - e^2*x^2)^(7/2))/(24*e) - (3/13)*d*x^6*(d^2 - e^2*x^2)^(7/2) - (1/14)*e*x^7*(d^2 - e^2*x^2)^(7/2) - (d^6*(31744*d + 63063*e*x)*(d^2 - e^2*x^2)^(7/2))/(1153152*e^6) + (35*d^14*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(2048*e^6)]
    @test_int [(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)*x^4, x, 11, (27*d^11*x*sqrt(d^2 - e^2*x^2))/(1024*e^4) + (9*d^9*x*(d^2 - e^2*x^2)^(3/2))/(512*e^4) + (9*d^7*x*(d^2 - e^2*x^2)^(5/2))/(640*e^4) - (20*d^4*x^2*(d^2 - e^2*x^2)^(7/2))/(143*e^3) - (9*d^3*x^3*(d^2 - e^2*x^2)^(7/2))/(40*e^2) - (45*d^2*x^4*(d^2 - e^2*x^2)^(7/2))/(143*e) - (1/4)*d*x^5*(d^2 - e^2*x^2)^(7/2) - (1/13)*e*x^6*(d^2 - e^2*x^2)^(7/2) - (d^5*(12800*d + 27027*e*x)*(d^2 - e^2*x^2)^(7/2))/(320320*e^5) + (27*d^13*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(1024*e^5)]
    @test_int [(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)*x^3, x, 10, (41*d^10*x*sqrt(d^2 - e^2*x^2))/(1024*e^3) + (41*d^8*x*(d^2 - e^2*x^2)^(3/2))/(1536*e^3) + (41*d^6*x*(d^2 - e^2*x^2)^(5/2))/(1920*e^3) - (23*d^3*x^2*(d^2 - e^2*x^2)^(7/2))/(99*e^2) - (41*d^2*x^3*(d^2 - e^2*x^2)^(7/2))/(120*e) - (3/11)*d*x^4*(d^2 - e^2*x^2)^(7/2) - (1/12)*e*x^5*(d^2 - e^2*x^2)^(7/2) - (d^4*(14720*d + 28413*e*x)*(d^2 - e^2*x^2)^(7/2))/(221760*e^4) + (41*d^12*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(1024*e^4)]
    @test_int [(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)*x^2, x, 9, (19*d^9*x*sqrt(d^2 - e^2*x^2))/(256*e^2) + (19*d^7*x*(d^2 - e^2*x^2)^(3/2))/(384*e^2) + (19*d^5*x*(d^2 - e^2*x^2)^(5/2))/(480*e^2) - (37*d^2*x^2*(d^2 - e^2*x^2)^(7/2))/(99*e) - (3/10)*d*x^3*(d^2 - e^2*x^2)^(7/2) - (1/11)*e*x^4*(d^2 - e^2*x^2)^(7/2) - (d^3*(5920*d + 13167*e*x)*(d^2 - e^2*x^2)^(7/2))/(55440*e^3) + (19*d^11*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(256*e^3)]
    @test_int [(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)*x^1, x, 9, (33*d^8*x*sqrt(d^2 - e^2*x^2))/(256*e) + (11*d^6*x*(d^2 - e^2*x^2)^(3/2))/(128*e) + (11*d^4*x*(d^2 - e^2*x^2)^(5/2))/(160*e) - (33*d^3*(d^2 - e^2*x^2)^(7/2))/(560*e^2) - (11*d^2*(d + e*x)*(d^2 - e^2*x^2)^(7/2))/(240*e^2) - (d*(d + e*x)^2*(d^2 - e^2*x^2)^(7/2))/(30*e^2) - ((d + e*x)^3*(d^2 - e^2*x^2)^(7/2))/(10*e^2) + (33*d^10*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(256*e^2)]
    @test_int [(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)*x^0, x, 8, (55/128)*d^7*x*sqrt(d^2 - e^2*x^2) + (55/192)*d^5*x*(d^2 - e^2*x^2)^(3/2) + (11/48)*d^3*x*(d^2 - e^2*x^2)^(5/2) - (11*d^2*(d^2 - e^2*x^2)^(7/2))/(56*e) - (11*d*(d + e*x)*(d^2 - e^2*x^2)^(7/2))/(72*e) - ((d + e*x)^2*(d^2 - e^2*x^2)^(7/2))/(9*e) + (55*d^9*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(128*e)]
    @test_int [(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)/x^1, x, 11, (1/128)*d^6*(128*d + 125*e*x)*sqrt(d^2 - e^2*x^2) + (1/192)*d^4*(64*d + 125*e*x)*(d^2 - e^2*x^2)^(3/2) + (1/240)*d^2*(48*d + 125*e*x)*(d^2 - e^2*x^2)^(5/2) - (3/7)*d*(d^2 - e^2*x^2)^(7/2) - (1/8)*e*x*(d^2 - e^2*x^2)^(7/2) + (125/128)*d^8*atan((e*x)/sqrt(d^2 - e^2*x^2)) - d^8*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)/x^2, x, 11, (3/16)*d^5*e*(16*d - 5*e*x)*sqrt(d^2 - e^2*x^2) + (1/8)*d^3*e*(8*d - 5*e*x)*(d^2 - e^2*x^2)^(3/2) + (1/10)*d*e*(6*d - 5*e*x)*(d^2 - e^2*x^2)^(5/2) - (1/7)*e*(d^2 - e^2*x^2)^(7/2) - (d*(d^2 - e^2*x^2)^(7/2))/x - (15/16)*d^7*e*atan((e*x)/sqrt(d^2 - e^2*x^2)) - 3*d^7*e*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)/x^3, x, 11, (1/16)*d^4*e^2*(8*d - 85*e*x)*sqrt(d^2 - e^2*x^2) + (1/24)*d^2*e^2*(4*d - 85*e*x)*(d^2 - e^2*x^2)^(3/2) + (1/30)*e^2*(3*d - 85*e*x)*(d^2 - e^2*x^2)^(5/2) - (d*(d^2 - e^2*x^2)^(7/2))/(2*x^2) - (3*e*(d^2 - e^2*x^2)^(7/2))/x - (85/16)*d^6*e^2*atan((e*x)/sqrt(d^2 - e^2*x^2)) - (1/2)*d^6*e^2*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)/x^4, x, 11, (-(1/8))*d^3*e^3*(52*d + 25*e*x)*sqrt(d^2 - e^2*x^2) - (1/12)*d*e^3*(26*d + 25*e*x)*(d^2 - e^2*x^2)^(3/2) - (e^2*(50*d + 39*e*x)*(d^2 - e^2*x^2)^(5/2))/(30*x) - (d*(d^2 - e^2*x^2)^(7/2))/(3*x^3) - (3*e*(d^2 - e^2*x^2)^(7/2))/(2*x^2) - (25/8)*d^5*e^3*atan((e*x)/sqrt(d^2 - e^2*x^2)) + (13/2)*d^5*e^3*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)/x^5, x, 11, (-(45/8))*d^2*e^4*(d - e*x)*sqrt(d^2 - e^2*x^2) + (15*d*e^3*(2*d - e*x)*(d^2 - e^2*x^2)^(3/2))/(8*x) - (3*e^2*(3*d + 2*e*x)*(d^2 - e^2*x^2)^(5/2))/(8*x^2) - (d*(d^2 - e^2*x^2)^(7/2))/(4*x^4) - (e*(d^2 - e^2*x^2)^(7/2))/x^3 + (45/8)*d^4*e^4*atan((e*x)/sqrt(d^2 - e^2*x^2)) + (45/8)*d^4*e^4*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)/x^6, x, 11, (d^2*e^4*(52*d + 25*e*x)*sqrt(d^2 - e^2*x^2))/(8*x) + (d*e^3*(25*d - 52*e*x)*(d^2 - e^2*x^2)^(3/2))/(24*x^2) - (e^2*(52*d + 25*e*x)*(d^2 - e^2*x^2)^(5/2))/(60*x^3) - (d*(d^2 - e^2*x^2)^(7/2))/(5*x^5) - (3*e*(d^2 - e^2*x^2)^(7/2))/(4*x^4) + (13/2)*d^3*e^5*atan((e*x)/sqrt(d^2 - e^2*x^2)) - (25/8)*d^3*e^5*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)/x^7, x, 11, -((d*e^5*(8*d - 85*e*x)*sqrt(d^2 - e^2*x^2))/(16*x)) + (d*e^3*(8*d + 85*e*x)*(d^2 - e^2*x^2)^(3/2))/(48*x^3) - (e^2*(85*d + 12*e*x)*(d^2 - e^2*x^2)^(5/2))/(120*x^4) - (d*(d^2 - e^2*x^2)^(7/2))/(6*x^6) - (3*e*(d^2 - e^2*x^2)^(7/2))/(5*x^5) - (1/2)*d^2*e^6*atan((e*x)/sqrt(d^2 - e^2*x^2)) - (85/16)*d^2*e^6*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)/x^8, x, 11, -((3*e^6*(16*d - 5*e*x)*sqrt(d^2 - e^2*x^2))/(16*x)) + (e^4*(16*d + 5*e*x)*(d^2 - e^2*x^2)^(3/2))/(16*x^3) - (e^2*(24*d + 5*e*x)*(d^2 - e^2*x^2)^(5/2))/(40*x^5) - (d*(d^2 - e^2*x^2)^(7/2))/(7*x^7) - (e*(d^2 - e^2*x^2)^(7/2))/(2*x^6) - 3*d*e^7*atan((e*x)/sqrt(d^2 - e^2*x^2)) - (15/16)*d*e^7*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)/x^9, x, 11, -((e^6*(125*d + 128*e*x)*sqrt(d^2 - e^2*x^2))/(128*x^2)) + (e^4*(125*d + 64*e*x)*(d^2 - e^2*x^2)^(3/2))/(192*x^4) - (e^2*(125*d + 48*e*x)*(d^2 - e^2*x^2)^(5/2))/(240*x^6) - (d*(d^2 - e^2*x^2)^(7/2))/(8*x^8) - (3*e*(d^2 - e^2*x^2)^(7/2))/(7*x^7) - e^8*atan((e*x)/sqrt(d^2 - e^2*x^2)) + (125/128)*e^8*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)/x^10, x, 9, -((55*e^7*sqrt(d^2 - e^2*x^2))/(128*x^2)) + (55*e^5*(d^2 - e^2*x^2)^(3/2))/(192*x^4) - (11*e^3*(d^2 - e^2*x^2)^(5/2))/(48*x^6) - (d*(d^2 - e^2*x^2)^(7/2))/(9*x^9) - (3*e*(d^2 - e^2*x^2)^(7/2))/(8*x^8) - (29*e^2*(d^2 - e^2*x^2)^(7/2))/(63*d*x^7) + (55*e^9*atanh(sqrt(d^2 - e^2*x^2)/d))/(128*d)]
    @test_int [(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)/x^11, x, 10, -((33*e^8*sqrt(d^2 - e^2*x^2))/(256*d*x^2)) + (11*e^6*(d^2 - e^2*x^2)^(3/2))/(128*d*x^4) - (11*e^4*(d^2 - e^2*x^2)^(5/2))/(160*d*x^6) - (d*(d^2 - e^2*x^2)^(7/2))/(10*x^10) - (e*(d^2 - e^2*x^2)^(7/2))/(3*x^9) - (33*e^2*(d^2 - e^2*x^2)^(7/2))/(80*d*x^8) - (5*e^3*(d^2 - e^2*x^2)^(7/2))/(21*d^2*x^7) + (33*e^10*atanh(sqrt(d^2 - e^2*x^2)/d))/(256*d^2)]
    @test_int [(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)/x^12, x, 11, -((19*e^9*sqrt(d^2 - e^2*x^2))/(256*d^2*x^2)) + (19*e^7*(d^2 - e^2*x^2)^(3/2))/(384*d^2*x^4) - (19*e^5*(d^2 - e^2*x^2)^(5/2))/(480*d^2*x^6) - (d*(d^2 - e^2*x^2)^(7/2))/(11*x^11) - (3*e*(d^2 - e^2*x^2)^(7/2))/(10*x^10) - (37*e^2*(d^2 - e^2*x^2)^(7/2))/(99*d*x^9) - (19*e^3*(d^2 - e^2*x^2)^(7/2))/(80*d^2*x^8) - (74*e^4*(d^2 - e^2*x^2)^(7/2))/(693*d^3*x^7) + (19*e^11*atanh(sqrt(d^2 - e^2*x^2)/d))/(256*d^3)]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^5*(d + e*x)^3/(d^2 - e^2*x^2)^(7/2), x, 7, (d^4*(d + e*x)^3)/(5*e^6*(d^2 - e^2*x^2)^(5/2)) - (23*d^3*(d + e*x)^2)/(15*e^6*(d^2 - e^2*x^2)^(3/2)) + (127*d^2*(d + e*x))/(15*e^6*sqrt(d^2 - e^2*x^2)) + (3*d*sqrt(d^2 - e^2*x^2))/e^6 + (x*sqrt(d^2 - e^2*x^2))/(2*e^5) - (13*d^2*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(2*e^6)]
    @test_int [x^4*(d + e*x)^3/(d^2 - e^2*x^2)^(7/2), x, 6, (d^3*(d + e*x)^3)/(5*e^5*(d^2 - e^2*x^2)^(5/2)) - (6*d^2*(d + e*x)^2)/(5*e^5*(d^2 - e^2*x^2)^(3/2)) + (24*d*(d + e*x))/(5*e^5*sqrt(d^2 - e^2*x^2)) + sqrt(d^2 - e^2*x^2)/e^5 - (3*d*atan((e*x)/sqrt(d^2 - e^2*x^2)))/e^5]
    @test_int [x^3*(d + e*x)^3/(d^2 - e^2*x^2)^(7/2), x, 5, (d^2*(d + e*x)^3)/(5*e^4*(d^2 - e^2*x^2)^(5/2)) - (13*d*(d + e*x)^2)/(15*e^4*(d^2 - e^2*x^2)^(3/2)) + (32*(d + e*x))/(15*e^4*sqrt(d^2 - e^2*x^2)) - atan((e*x)/sqrt(d^2 - e^2*x^2))/e^4]
    @test_int [x^2*(d + e*x)^3/(d^2 - e^2*x^2)^(7/2), x, 3, (d*(d + e*x)^3)/(5*e^3*(d^2 - e^2*x^2)^(5/2)) - (8*(d + e*x)^2)/(15*e^3*(d^2 - e^2*x^2)^(3/2)) + (7*(d + e*x))/(15*d*e^3*sqrt(d^2 - e^2*x^2))]
    @test_int [x^1*(d + e*x)^3/(d^2 - e^2*x^2)^(7/2), x, 3, (d + e*x)^3/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (2*(d + e*x))/(5*e^2*(d^2 - e^2*x^2)^(3/2)) - x/(5*d^2*e*sqrt(d^2 - e^2*x^2))]
    @test_int [x^0*(d + e*x)^3/(d^2 - e^2*x^2)^(7/2), x, 4, sqrt(d^2 - e^2*x^2)/(5*d*e*(d - e*x)^3) + (2*sqrt(d^2 - e^2*x^2))/(15*d^2*e*(d - e*x)^2) + (2*sqrt(d^2 - e^2*x^2))/(15*d^3*e*(d - e*x))]
    @test_int [(d + e*x)^3/(x^1*(d^2 - e^2*x^2)^(7/2)), x, 7, (4*(d + e*x))/(5*(d^2 - e^2*x^2)^(5/2)) + (5*d + 11*e*x)/(15*d^2*(d^2 - e^2*x^2)^(3/2)) + (15*d + 22*e*x)/(15*d^4*sqrt(d^2 - e^2*x^2)) - atanh(sqrt(d^2 - e^2*x^2)/d)/d^4]
    @test_int [(d + e*x)^3/(x^2*(d^2 - e^2*x^2)^(7/2)), x, 7, (4*e*(d + e*x))/(5*d*(d^2 - e^2*x^2)^(5/2)) + (e*(5*d + 7*e*x))/(5*d^3*(d^2 - e^2*x^2)^(3/2)) + (e*(15*d + 19*e*x))/(5*d^5*sqrt(d^2 - e^2*x^2)) - sqrt(d^2 - e^2*x^2)/(d^5*x) - (3*e*atanh(sqrt(d^2 - e^2*x^2)/d))/d^5]
    @test_int [(d + e*x)^3/(x^3*(d^2 - e^2*x^2)^(7/2)), x, 8, (4*e^2*(d + e*x))/(5*d^2*(d^2 - e^2*x^2)^(5/2)) + (e^2*(25*d + 31*e*x))/(15*d^4*(d^2 - e^2*x^2)^(3/2)) + (e^2*(90*d + 107*e*x))/(15*d^6*sqrt(d^2 - e^2*x^2)) - sqrt(d^2 - e^2*x^2)/(2*d^5*x^2) - (3*e*sqrt(d^2 - e^2*x^2))/(d^6*x) - (13*e^2*atanh(sqrt(d^2 - e^2*x^2)/d))/(2*d^6)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m / (d+e*x)^1*(d^2 - e^2*x^2)^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^4*sqrt(d^2 - e^2*x^2)/(d + e*x), x, 7, (4*d^2*x^2*sqrt(d^2 - e^2*x^2))/(15*e^3) - (d*x^3*sqrt(d^2 - e^2*x^2))/(4*e^2) + (x^4*sqrt(d^2 - e^2*x^2))/(5*e) + (d^3*(64*d - 45*e*x)*sqrt(d^2 - e^2*x^2))/(120*e^5) + (3*d^5*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(8*e^5)]
    @test_int [x^3*sqrt(d^2 - e^2*x^2)/(d + e*x), x, 6, -((d*x^2*sqrt(d^2 - e^2*x^2))/(3*e^2)) + (x^3*sqrt(d^2 - e^2*x^2))/(4*e) - (d^2*(16*d - 9*e*x)*sqrt(d^2 - e^2*x^2))/(24*e^4) - (3*d^4*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(8*e^4)]
    @test_int [x^2*sqrt(d^2 - e^2*x^2)/(d + e*x), x, 6, (d*(2*d - e*x)*sqrt(d^2 - e^2*x^2))/(2*e^3) - (d^2 - e^2*x^2)^(3/2)/(3*e^3) + (d^3*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(2*e^3)]
    @test_int [x^1*sqrt(d^2 - e^2*x^2)/(d + e*x), x, 4, -(((2*d - e*x)*sqrt(d^2 - e^2*x^2))/(2*e^2)) - (d^2*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(2*e^2)]
    @test_int [x^0*sqrt(d^2 - e^2*x^2)/(d + e*x), x, 3, sqrt(d^2 - e^2*x^2)/e + (d*atan((e*x)/sqrt(d^2 - e^2*x^2)))/e]
    @test_int [sqrt(d^2 - e^2*x^2)/(x^1*(d + e*x)), x, 7, -atan((e*x)/sqrt(d^2 - e^2*x^2)) - atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [sqrt(d^2 - e^2*x^2)/(x^2*(d + e*x)), x, 5, -(sqrt(d^2 - e^2*x^2)/(d*x)) + (e*atanh(sqrt(d^2 - e^2*x^2)/d))/d]
    @test_int [sqrt(d^2 - e^2*x^2)/(x^3*(d + e*x)), x, 6, -(sqrt(d^2 - e^2*x^2)/(2*d*x^2)) + (e*sqrt(d^2 - e^2*x^2))/(d^2*x) - (e^2*atanh(sqrt(d^2 - e^2*x^2)/d))/(2*d^2)]
    @test_int [sqrt(d^2 - e^2*x^2)/(x^4*(d + e*x)), x, 7, -(sqrt(d^2 - e^2*x^2)/(3*d*x^3)) + (e*sqrt(d^2 - e^2*x^2))/(2*d^2*x^2) - (2*e^2*sqrt(d^2 - e^2*x^2))/(3*d^3*x) + (e^3*atanh(sqrt(d^2 - e^2*x^2)/d))/(2*d^3)]
    @test_int [sqrt(d^2 - e^2*x^2)/(x^5*(d + e*x)), x, 8, -(sqrt(d^2 - e^2*x^2)/(4*d*x^4)) + (e*sqrt(d^2 - e^2*x^2))/(3*d^2*x^3) - (3*e^2*sqrt(d^2 - e^2*x^2))/(8*d^3*x^2) + (2*e^3*sqrt(d^2 - e^2*x^2))/(3*d^4*x) - (3*e^4*atanh(sqrt(d^2 - e^2*x^2)/d))/(8*d^4)]

    @test_int [x^2*(d^2 - e^2*x^2)^(3/2)/(d + e*x), x, 7, (d^3*x*sqrt(d^2 - e^2*x^2))/(8*e^2) + (d*(4*d - 3*e*x)*(d^2 - e^2*x^2)^(3/2))/(12*e^3) - (d^2 - e^2*x^2)^(5/2)/(5*e^3) + (d^5*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(8*e^3)]

    @test_int [x^4*(d^2 - e^2*x^2)^(5/2)/(d + e*x), x, 9, (3*d^7*x*sqrt(d^2 - e^2*x^2))/(128*e^4) + (d^5*x*(d^2 - e^2*x^2)^(3/2))/(64*e^4) + (4*d^2*x^2*(d^2 - e^2*x^2)^(5/2))/(63*e^3) - (d*x^3*(d^2 - e^2*x^2)^(5/2))/(8*e^2) + (x^4*(d^2 - e^2*x^2)^(5/2))/(9*e) + (d^3*(128*d - 315*e*x)*(d^2 - e^2*x^2)^(5/2))/(5040*e^5) + (3*d^9*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(128*e^5)]
    @test_int [x^3*(d^2 - e^2*x^2)^(5/2)/(d + e*x), x, 8, -((3*d^6*x*sqrt(d^2 - e^2*x^2))/(128*e^3)) - (d^4*x*(d^2 - e^2*x^2)^(3/2))/(64*e^3) - (d*x^2*(d^2 - e^2*x^2)^(5/2))/(7*e^2) + (x^3*(d^2 - e^2*x^2)^(5/2))/(8*e) - (d^2*(32*d - 35*e*x)*(d^2 - e^2*x^2)^(5/2))/(560*e^4) - (3*d^8*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(128*e^4)]
    @test_int [x^2*(d^2 - e^2*x^2)^(5/2)/(d + e*x), x, 8, (d^5*x*sqrt(d^2 - e^2*x^2))/(16*e^2) + (d^3*x*(d^2 - e^2*x^2)^(3/2))/(24*e^2) + (d*(6*d - 5*e*x)*(d^2 - e^2*x^2)^(5/2))/(30*e^3) - (d^2 - e^2*x^2)^(7/2)/(7*e^3) + (d^7*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(16*e^3)]
    @test_int [x^1*(d^2 - e^2*x^2)^(5/2)/(d + e*x), x, 6, -((d^4*x*sqrt(d^2 - e^2*x^2))/(16*e)) - (d^2*x*(d^2 - e^2*x^2)^(3/2))/(24*e) - ((6*d - 5*e*x)*(d^2 - e^2*x^2)^(5/2))/(30*e^2) - (d^6*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(16*e^2)]
    @test_int [x^0*(d^2 - e^2*x^2)^(5/2)/(d + e*x), x, 5, (3*d^3*x*sqrt(d^2 - e^2*x^2))/8 + (d*x*(d^2 - e^2*x^2)^(3/2))/4 + (d^2 - e^2*x^2)^(5/2)/(5*e) + (3*d^5*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(8*e)]
    @test_int [(d^2 - e^2*x^2)^(5/2)/(x^1*(d + e*x)), x, 9, (1/8)*d^2*(8*d - 3*e*x)*sqrt(d^2 - e^2*x^2) + (1/12)*(4*d - 3*e*x)*(d^2 - e^2*x^2)^(3/2) - (3/8)*d^4*atan((e*x)/sqrt(d^2 - e^2*x^2)) - d^4*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d^2 - e^2*x^2)^(5/2)/(x^2*(d + e*x)), x, 9, (-(1/2))*d*e*(2*d + 3*e*x)*sqrt(d^2 - e^2*x^2) - ((3*d + e*x)*(d^2 - e^2*x^2)^(3/2))/(3*x) - (3/2)*d^3*e*atan((e*x)/sqrt(d^2 - e^2*x^2)) + d^3*e*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d^2 - e^2*x^2)^(5/2)/(x^3*(d + e*x)), x, 9, (3*d*e*(d - e*x)*sqrt(d^2 - e^2*x^2))/(2*x) - ((d + e*x)*(d^2 - e^2*x^2)^(3/2))/(2*x^2) + (3/2)*d^2*e^2*atan((e*x)/sqrt(d^2 - e^2*x^2)) + (3/2)*d^2*e^2*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d^2 - e^2*x^2)^(5/2)/(x^4*(d + e*x)), x, 9, (e^2*(2*d + 3*e*x)*sqrt(d^2 - e^2*x^2))/(2*x) - ((2*d - 3*e*x)*(d^2 - e^2*x^2)^(3/2))/(6*x^3) + d*e^3*atan((e*x)/sqrt(d^2 - e^2*x^2)) - (3/2)*d*e^3*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d^2 - e^2*x^2)^(5/2)/(x^5*(d + e*x)), x, 9, (e^2*(3*d - 8*e*x)*sqrt(d^2 - e^2*x^2))/(8*x^2) - ((3*d - 4*e*x)*(d^2 - e^2*x^2)^(3/2))/(12*x^4) - e^4*atan((e*x)/sqrt(d^2 - e^2*x^2)) - (3/8)*e^4*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d^2 - e^2*x^2)^(5/2)/(x^6*(d + e*x)), x, 7, -((3*e^3*sqrt(d^2 - e^2*x^2))/(8*x^2)) + (e*(d^2 - e^2*x^2)^(3/2))/(4*x^4) - (d^2 - e^2*x^2)^(5/2)/(5*d*x^5) + (3*e^5*atanh(sqrt(d^2 - e^2*x^2)/d))/(8*d)]
    @test_int [(d^2 - e^2*x^2)^(5/2)/(x^7*(d + e*x)), x, 8, (e^4*sqrt(d^2 - e^2*x^2))/(16*d*x^2) - (e^2*(d^2 - e^2*x^2)^(3/2))/(24*d*x^4) - (d^2 - e^2*x^2)^(5/2)/(6*d*x^6) + (e*(d^2 - e^2*x^2)^(5/2))/(5*d^2*x^5) - (e^6*atanh(sqrt(d^2 - e^2*x^2)/d))/(16*d^2)]
    @test_int [(d^2 - e^2*x^2)^(5/2)/(x^8*(d + e*x)), x, 9, -((e^5*sqrt(d^2 - e^2*x^2))/(16*d^2*x^2)) + (e^3*(d^2 - e^2*x^2)^(3/2))/(24*d^2*x^4) - (d^2 - e^2*x^2)^(5/2)/(7*d*x^7) + (e*(d^2 - e^2*x^2)^(5/2))/(6*d^2*x^6) - (2*e^2*(d^2 - e^2*x^2)^(5/2))/(35*d^3*x^5) + (e^7*atanh(sqrt(d^2 - e^2*x^2)/d))/(16*d^3)]
    @test_int [(d^2 - e^2*x^2)^(5/2)/(x^9*(d + e*x)), x, 10, (3*e^6*sqrt(d^2 - e^2*x^2))/(128*d^3*x^2) - (e^4*(d^2 - e^2*x^2)^(3/2))/(64*d^3*x^4) - (d^2 - e^2*x^2)^(5/2)/(8*d*x^8) + (e*(d^2 - e^2*x^2)^(5/2))/(7*d^2*x^7) - (e^2*(d^2 - e^2*x^2)^(5/2))/(16*d^3*x^6) + (2*e^3*(d^2 - e^2*x^2)^(5/2))/(35*d^4*x^5) - (3*e^8*atanh(sqrt(d^2 - e^2*x^2)/d))/(128*d^4)]

    @test_int [(x*sqrt(1 - x^2))/(1 + x), x, 3, (-(1/2))*(2 - x)*sqrt(1 - x^2) - asin(x)/2]

    @test_int [(1 - a^2*x^2)^(3/2)/(x^2*(1 - a*x)), x, 7, -(((1 - a*x)*sqrt(1 - a^2*x^2))/x) - a*asin(a*x) - a*atanh(sqrt(1 - a^2*x^2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^4/((d + e*x)*sqrt(d^2 - e^2*x^2)), x, 6, (x^3*(d - e*x))/(e^2*sqrt(d^2 - e^2*x^2)) - (4*x^2*sqrt(d^2 - e^2*x^2))/(3*e^3) - (d*(16*d - 9*e*x)*sqrt(d^2 - e^2*x^2))/(6*e^5) - (3*d^3*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(2*e^5)]
    @test_int [x^3/((d + e*x)*sqrt(d^2 - e^2*x^2)), x, 5, (x^2*(d - e*x))/(e^2*sqrt(d^2 - e^2*x^2)) + ((4*d - 3*e*x)*sqrt(d^2 - e^2*x^2))/(2*e^4) + (3*d^2*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(2*e^4)]
    @test_int [x^2/((d + e*x)*sqrt(d^2 - e^2*x^2)), x, 5, -(sqrt(d^2 - e^2*x^2)/e^3) - (d*sqrt(d^2 - e^2*x^2))/(e^3*(d + e*x)) - (d*atan((e*x)/sqrt(d^2 - e^2*x^2)))/e^3]
    @test_int [x^1/((d + e*x)*sqrt(d^2 - e^2*x^2)), x, 3, sqrt(d^2 - e^2*x^2)/(e^2*(d + e*x)) + atan((e*x)/sqrt(d^2 - e^2*x^2))/e^2]
    @test_int [x^0/((d + e*x)*sqrt(d^2 - e^2*x^2)), x, 1, -(sqrt(d^2 - e^2*x^2)/(d*e*(d + e*x)))]
    @test_int [1/(x^1*(d + e*x)*sqrt(d^2 - e^2*x^2)), x, 5, sqrt(d^2 - e^2*x^2)/(d^2*(d + e*x)) - atanh(sqrt(d^2 - e^2*x^2)/d)/d^2]
    @test_int [1/(x^2*(d + e*x)*sqrt(d^2 - e^2*x^2)), x, 5, -((2*sqrt(d^2 - e^2*x^2))/(d^3*x)) + sqrt(d^2 - e^2*x^2)/(d^2*x*(d + e*x)) + (e*atanh(sqrt(d^2 - e^2*x^2)/d))/d^3]
    @test_int [1/(x^3*(d + e*x)*sqrt(d^2 - e^2*x^2)), x, 6, -((3*sqrt(d^2 - e^2*x^2))/(2*d^3*x^2)) + (2*e*sqrt(d^2 - e^2*x^2))/(d^4*x) + sqrt(d^2 - e^2*x^2)/(d^2*x^2*(d + e*x)) - (3*e^2*atanh(sqrt(d^2 - e^2*x^2)/d))/(2*d^4)]

    @test_int [x^5/((d + e*x)*(d^2 - e^2*x^2)^(3/2)), x, 6, (x^4*(d - e*x))/(3*e^2*(d^2 - e^2*x^2)^(3/2)) - (x^2*(4*d - 5*e*x))/(3*e^4*sqrt(d^2 - e^2*x^2)) - ((16*d - 15*e*x)*sqrt(d^2 - e^2*x^2))/(6*e^6) - (5*d^2*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(2*e^6)]
    @test_int [x^4/((d + e*x)*(d^2 - e^2*x^2)^(3/2)), x, 6, (x^3*(d - e*x))/(3*e^2*(d^2 - e^2*x^2)^(3/2)) - (x*(3*d - 4*e*x))/(3*e^4*sqrt(d^2 - e^2*x^2)) + (8*sqrt(d^2 - e^2*x^2))/(3*e^5) + (d*atan((e*x)/sqrt(d^2 - e^2*x^2)))/e^5]
    @test_int [x^3/((d + e*x)*(d^2 - e^2*x^2)^(3/2)), x, 5, (x^2*(d - e*x))/(3*e^2*(d^2 - e^2*x^2)^(3/2)) - (2*d - 3*e*x)/(3*e^4*sqrt(d^2 - e^2*x^2)) - atan((e*x)/sqrt(d^2 - e^2*x^2))/e^4]
    @test_int [x^2/((d + e*x)*(d^2 - e^2*x^2)^(3/2)), x, 3, 2/(3*e^3*sqrt(d^2 - e^2*x^2)) - x^2/(3*d*e*(d + e*x)*sqrt(d^2 - e^2*x^2))]
    @test_int [x^1/((d + e*x)*(d^2 - e^2*x^2)^(3/2)), x, 2, x/(3*d^2*e*sqrt(d^2 - e^2*x^2)) + 1/(3*e^2*(d + e*x)*sqrt(d^2 - e^2*x^2))]
    @test_int [x^0/((d + e*x)*(d^2 - e^2*x^2)^(3/2)), x, 2, (2*x)/(3*d^3*sqrt(d^2 - e^2*x^2)) - 1/(3*d*e*(d + e*x)*sqrt(d^2 - e^2*x^2))]
    @test_int [1/(x^1*(d + e*x)*(d^2 - e^2*x^2)^(3/2)), x, 6, (3*d - 2*e*x)/(3*d^4*sqrt(d^2 - e^2*x^2)) + 1/(3*d^2*(d + e*x)*sqrt(d^2 - e^2*x^2)) - atanh(sqrt(d^2 - e^2*x^2)/d)/d^4]
    @test_int [1/(x^2*(d + e*x)*(d^2 - e^2*x^2)^(3/2)), x, 6, (4*d - 3*e*x)/(3*d^4*x*sqrt(d^2 - e^2*x^2)) + 1/(3*d^2*x*(d + e*x)*sqrt(d^2 - e^2*x^2)) - (8*sqrt(d^2 - e^2*x^2))/(3*d^5*x) + (e*atanh(sqrt(d^2 - e^2*x^2)/d))/d^5]
    @test_int [1/(x^3*(d + e*x)*(d^2 - e^2*x^2)^(3/2)), x, 7, (5*d - 4*e*x)/(3*d^4*x^2*sqrt(d^2 - e^2*x^2)) + 1/(3*d^2*x^2*(d + e*x)*sqrt(d^2 - e^2*x^2)) - (5*sqrt(d^2 - e^2*x^2))/(2*d^5*x^2) + (8*e*sqrt(d^2 - e^2*x^2))/(3*d^6*x) - (5*e^2*atanh(sqrt(d^2 - e^2*x^2)/d))/(2*d^6)]

    @test_int [x^7/((d + e*x)*(d^2 - e^2*x^2)^(5/2)), x, 7, (x^6*(d - e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^4*(6*d - 7*e*x))/(15*e^4*(d^2 - e^2*x^2)^(3/2)) + (x^2*(24*d - 35*e*x))/(15*e^6*sqrt(d^2 - e^2*x^2)) + ((32*d - 35*e*x)*sqrt(d^2 - e^2*x^2))/(10*e^8) + (7*d^2*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(2*e^8)]
    @test_int [x^6/((d + e*x)*(d^2 - e^2*x^2)^(5/2)), x, 7, (x^5*(d - e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^3*(5*d - 6*e*x))/(15*e^4*(d^2 - e^2*x^2)^(3/2)) + (x*(5*d - 8*e*x))/(5*e^6*sqrt(d^2 - e^2*x^2)) - (16*sqrt(d^2 - e^2*x^2))/(5*e^7) - (d*atan((e*x)/sqrt(d^2 - e^2*x^2)))/e^7]
    @test_int [x^5/((d + e*x)*(d^2 - e^2*x^2)^(5/2)), x, 6, (x^4*(d - e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^2*(4*d - 5*e*x))/(15*e^4*(d^2 - e^2*x^2)^(3/2)) + (8*d - 15*e*x)/(15*e^6*sqrt(d^2 - e^2*x^2)) + atan((e*x)/sqrt(d^2 - e^2*x^2))/e^6]
    @test_int [x^4/((d + e*x)*(d^2 - e^2*x^2)^(5/2)), x, 5, -((x^4*(d - e*x))/(5*d*e*(d^2 - e^2*x^2)^(5/2))) + (4*d^2)/(15*e^5*(d^2 - e^2*x^2)^(3/2)) - 4/(5*e^5*sqrt(d^2 - e^2*x^2))]
    @test_int [x^3/((d + e*x)*(d^2 - e^2*x^2)^(5/2)), x, 4, (x^2*(d - e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (2*d - 3*e*x)/(15*e^4*(d^2 - e^2*x^2)^(3/2)) - x/(5*d^2*e^3*sqrt(d^2 - e^2*x^2))]
    @test_int [x^2/((d + e*x)*(d^2 - e^2*x^2)^(5/2)), x, 3, -(x^2/(5*d*e*(d + e*x)*(d^2 - e^2*x^2)^(3/2))) + (2*(d + e*x))/(15*d*e^3*(d^2 - e^2*x^2)^(3/2)) - (2*x)/(15*d^3*e^2*sqrt(d^2 - e^2*x^2))]
    @test_int [x^1/((d + e*x)*(d^2 - e^2*x^2)^(5/2)), x, 3, x/(15*d^2*e*(d^2 - e^2*x^2)^(3/2)) + 1/(5*e^2*(d + e*x)*(d^2 - e^2*x^2)^(3/2)) + (2*x)/(15*d^4*e*sqrt(d^2 - e^2*x^2))]
    @test_int [x^0/((d + e*x)*(d^2 - e^2*x^2)^(5/2)), x, 3, (4*x)/(15*d^3*(d^2 - e^2*x^2)^(3/2)) - 1/(5*d*e*(d + e*x)*(d^2 - e^2*x^2)^(3/2)) + (8*x)/(15*d^5*sqrt(d^2 - e^2*x^2))]
    @test_int [1/(x^1*(d + e*x)*(d^2 - e^2*x^2)^(5/2)), x, 7, (5*d - 4*e*x)/(15*d^4*(d^2 - e^2*x^2)^(3/2)) + 1/(5*d^2*(d + e*x)*(d^2 - e^2*x^2)^(3/2)) + (15*d - 8*e*x)/(15*d^6*sqrt(d^2 - e^2*x^2)) - atanh(sqrt(d^2 - e^2*x^2)/d)/d^6]
    @test_int [1/(x^2*(d + e*x)*(d^2 - e^2*x^2)^(5/2)), x, 7, (6*d - 5*e*x)/(15*d^4*x*(d^2 - e^2*x^2)^(3/2)) + 1/(5*d^2*x*(d + e*x)*(d^2 - e^2*x^2)^(3/2)) + (8*d - 5*e*x)/(5*d^6*x*sqrt(d^2 - e^2*x^2)) - (16*sqrt(d^2 - e^2*x^2))/(5*d^7*x) + (e*atanh(sqrt(d^2 - e^2*x^2)/d))/d^7]
    @test_int [1/(x^3*(d + e*x)*(d^2 - e^2*x^2)^(5/2)), x, 8, (7*d - 6*e*x)/(15*d^4*x^2*(d^2 - e^2*x^2)^(3/2)) + 1/(5*d^2*x^2*(d + e*x)*(d^2 - e^2*x^2)^(3/2)) + (35*d - 24*e*x)/(15*d^6*x^2*sqrt(d^2 - e^2*x^2)) - (7*sqrt(d^2 - e^2*x^2))/(2*d^7*x^2) + (16*e*sqrt(d^2 - e^2*x^2))/(5*d^8*x) - (7*e^2*atanh(sqrt(d^2 - e^2*x^2)/d))/(2*d^8)]
    @test_int [1/(x^4*(d + e*x)*(d^2 - e^2*x^2)^(5/2)), x, 9, (8*d - 7*e*x)/(15*d^4*x^3*(d^2 - e^2*x^2)^(3/2)) + 1/(5*d^2*x^3*(d + e*x)*(d^2 - e^2*x^2)^(3/2)) + (48*d - 35*e*x)/(15*d^6*x^3*sqrt(d^2 - e^2*x^2)) - (64*sqrt(d^2 - e^2*x^2))/(15*d^7*x^3) + (7*e*sqrt(d^2 - e^2*x^2))/(2*d^8*x^2) - (128*e^2*sqrt(d^2 - e^2*x^2))/(15*d^9*x) + (7*e^3*atanh(sqrt(d^2 - e^2*x^2)/d))/(2*d^9)]

    @test_int [x^3/((d + e*x)*(d^2 - e^2*x^2)^(7/2)), x, 5, (x^2*(d - e*x))/(7*e^2*(d^2 - e^2*x^2)^(7/2)) - (2*d - 3*e*x)/(35*e^4*(d^2 - e^2*x^2)^(5/2)) - x/(35*d^2*e^3*(d^2 - e^2*x^2)^(3/2)) - (2*x)/(35*d^4*e^3*sqrt(d^2 - e^2*x^2))]
    @test_int [x^2/((d + e*x)*(d^2 - e^2*x^2)^(7/2)), x, 4, -(x^2/(7*d*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2))) + (2*(d + 2*e*x))/(35*d*e^3*(d^2 - e^2*x^2)^(5/2)) - (4*x)/(105*d^3*e^2*(d^2 - e^2*x^2)^(3/2)) - (8*x)/(105*d^5*e^2*sqrt(d^2 - e^2*x^2))]

    @test_int [x^3/((1 + a*x)*sqrt(1 - a^2*x^2)), x, 4, (x^2*(1 - a*x))/(a^2*sqrt(1 - a^2*x^2)) + ((4 - 3*a*x)*sqrt(1 - a^2*x^2))/(2*a^4) + (3*asin(a*x))/(2*a^4)]
    @test_int [x^2/((1 + a*x)*sqrt(1 - a^2*x^2)), x, 4, -(sqrt(1 - a^2*x^2)/a^3) - sqrt(1 - a^2*x^2)/(a^3*(1 + a*x)) - asin(a*x)/a^3]
    @test_int [x^1/((1 + a*x)*sqrt(1 - a^2*x^2)), x, 2, sqrt(1 - a^2*x^2)/(a^2*(1 + a*x)) + asin(a*x)/a^2]
    @test_int [x^0/((1 + a*x)*sqrt(1 - a^2*x^2)), x, 1, -(sqrt(1 - a^2*x^2)/(a*(1 + a*x)))]
    @test_int [1/(x^1*(1 - a*x)*sqrt(1 - a^2*x^2)), x, 5, sqrt(1 - a^2*x^2)/(1 - a*x) - atanh(sqrt(1 - a^2*x^2))]
    @test_int [1/(x^2*(1 - a*x)*sqrt(1 - a^2*x^2)), x, 5, -((2*sqrt(1 - a^2*x^2))/x) + sqrt(1 - a^2*x^2)/(x*(1 - a*x)) - a*atanh(sqrt(1 - a^2*x^2))]
    @test_int [1/(x^3*(1 - a*x)*sqrt(1 - a^2*x^2)), x, 6, -((3*sqrt(1 - a^2*x^2))/(2*x^2)) - (2*a*sqrt(1 - a^2*x^2))/x + sqrt(1 - a^2*x^2)/(x^2*(1 - a*x)) - (3/2)*a^2*atanh(sqrt(1 - a^2*x^2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m / (d+e*x)^2*(d^2 - e^2*x^2)^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^5*(d^2 - e^2*x^2)^(5/2)/(d + e*x)^2, x, 10, -((5*d^7*x*sqrt(d^2 - e^2*x^2))/(64*e^5)) - (4*d^4*x^2*(d^2 - e^2*x^2)^(3/2))/(21*e^4) + (5*d^3*x^3*(d^2 - e^2*x^2)^(3/2))/(24*e^3) - (5*d^2*x^4*(d^2 - e^2*x^2)^(3/2))/(21*e^2) + (d*x^5*(d^2 - e^2*x^2)^(3/2))/(4*e) - (1/9)*x^6*(d^2 - e^2*x^2)^(3/2) - (d^5*(256*d - 315*e*x)*(d^2 - e^2*x^2)^(3/2))/(2016*e^6) - (5*d^9*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(64*e^6)]
    @test_int [x^4*(d^2 - e^2*x^2)^(5/2)/(d + e*x)^2, x, 9, (13*d^6*x*sqrt(d^2 - e^2*x^2))/(128*e^4) + (8*d^3*x^2*(d^2 - e^2*x^2)^(3/2))/(35*e^3) - (13*d^2*x^3*(d^2 - e^2*x^2)^(3/2))/(48*e^2) + (2*d*x^4*(d^2 - e^2*x^2)^(3/2))/(7*e) - (1/8)*x^5*(d^2 - e^2*x^2)^(3/2) + (d^4*(1024*d - 1365*e*x)*(d^2 - e^2*x^2)^(3/2))/(6720*e^5) + (13*d^8*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(128*e^5)]
    @test_int [x^3*(d^2 - e^2*x^2)^(5/2)/(d + e*x)^2, x, 8, -((d^5*x*sqrt(d^2 - e^2*x^2))/(8*e^3)) - (11*d^2*x^2*(d^2 - e^2*x^2)^(3/2))/(35*e^2) + (d*x^3*(d^2 - e^2*x^2)^(3/2))/(3*e) - (1/7)*x^4*(d^2 - e^2*x^2)^(3/2) - (d^3*(88*d - 105*e*x)*(d^2 - e^2*x^2)^(3/2))/(420*e^4) - (d^7*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(8*e^4)]
    @test_int [x^2*(d^2 - e^2*x^2)^(5/2)/(d + e*x)^2, x, 7, (3*d^4*x*sqrt(d^2 - e^2*x^2))/(16*e^2) + (2*d*x^2*(d^2 - e^2*x^2)^(3/2))/(5*e) - (1/6)*x^3*(d^2 - e^2*x^2)^(3/2) + (d^2*(32*d - 45*e*x)*(d^2 - e^2*x^2)^(3/2))/(120*e^3) + (3*d^6*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(16*e^3)]
    @test_int [x^1*(d^2 - e^2*x^2)^(5/2)/(d + e*x)^2, x, 6, -(d^3*x*sqrt(d^2 - e^2*x^2))/(4*e) - (d*x*(d^2 - e^2*x^2)^(3/2))/(6*e) - (2*(d^2 - e^2*x^2)^(5/2))/(15*e^2) - (d^2 - e^2*x^2)^(7/2)/(3*e^2*(d + e*x)^2) - (d^5*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(4*e^2)]
    @test_int [x^0*(d^2 - e^2*x^2)^(5/2)/(d + e*x)^2, x, 6, (5/8)*d^2*x*sqrt(d^2 - e^2*x^2) + (5*d*(d^2 - e^2*x^2)^(3/2))/(12*e) + ((d - e*x)*(d^2 - e^2*x^2)^(3/2))/(4*e) + (5*d^4*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(8*e)]
    @test_int [(d^2 - e^2*x^2)^(5/2)/(x^1*(d + e*x)^2), x, 9, d*(d - e*x)*sqrt(d^2 - e^2*x^2) - (1/3)*(d^2 - e^2*x^2)^(3/2) - d^3*atan((e*x)/sqrt(d^2 - e^2*x^2)) - d^3*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d^2 - e^2*x^2)^(5/2)/(x^2*(d + e*x)^2), x, 9, (-(1/2))*e*(4*d + e*x)*sqrt(d^2 - e^2*x^2) - (d^2 - e^2*x^2)^(3/2)/x - (1/2)*d^2*e*atan((e*x)/sqrt(d^2 - e^2*x^2)) + 2*d^2*e*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d^2 - e^2*x^2)^(5/2)/(x^3*(d + e*x)^2), x, 9, (e*(4*d + e*x)*sqrt(d^2 - e^2*x^2))/(2*x) - (d^2 - e^2*x^2)^(3/2)/(2*x^2) + 2*d*e^2*atan((e*x)/sqrt(d^2 - e^2*x^2)) - (1/2)*d*e^2*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d^2 - e^2*x^2)^(5/2)/(x^4*(d + e*x)^2), x, 9, (e*(d - e*x)*sqrt(d^2 - e^2*x^2))/x^2 - (d^2 - e^2*x^2)^(3/2)/(3*x^3) - e^3*atan((e*x)/sqrt(d^2 - e^2*x^2)) - e^3*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d^2 - e^2*x^2)^(5/2)/(x^5*(d + e*x)^2), x, 7, -((5*e^2*sqrt(d^2 - e^2*x^2))/(8*x^2)) - (d^2 - e^2*x^2)^(3/2)/(4*x^4) + (2*e*(d^2 - e^2*x^2)^(3/2))/(3*d*x^3) + (5*e^4*atanh(sqrt(d^2 - e^2*x^2)/d))/(8*d)]
    @test_int [(d^2 - e^2*x^2)^(5/2)/(x^6*(d + e*x)^2), x, 8, (e^3*sqrt(d^2 - e^2*x^2))/(4*d*x^2) - (d^2 - e^2*x^2)^(3/2)/(5*x^5) + (e*(d^2 - e^2*x^2)^(3/2))/(2*d*x^4) - (7*e^2*(d^2 - e^2*x^2)^(3/2))/(15*d^2*x^3) - (e^5*atanh(sqrt(d^2 - e^2*x^2)/d))/(4*d^2)]
    @test_int [(d^2 - e^2*x^2)^(5/2)/(x^7*(d + e*x)^2), x, 9, -((3*e^4*sqrt(d^2 - e^2*x^2))/(16*d^2*x^2)) - (d^2 - e^2*x^2)^(3/2)/(6*x^6) + (2*e*(d^2 - e^2*x^2)^(3/2))/(5*d*x^5) - (3*e^2*(d^2 - e^2*x^2)^(3/2))/(8*d^2*x^4) + (4*e^3*(d^2 - e^2*x^2)^(3/2))/(15*d^3*x^3) + (3*e^6*atanh(sqrt(d^2 - e^2*x^2)/d))/(16*d^3)]
    @test_int [(d^2 - e^2*x^2)^(5/2)/(x^8*(d + e*x)^2), x, 10, (e^5*sqrt(d^2 - e^2*x^2))/(8*d^3*x^2) - (d^2 - e^2*x^2)^(3/2)/(7*x^7) + (e*(d^2 - e^2*x^2)^(3/2))/(3*d*x^6) - (11*e^2*(d^2 - e^2*x^2)^(3/2))/(35*d^2*x^5) + (e^3*(d^2 - e^2*x^2)^(3/2))/(4*d^3*x^4) - (22*e^4*(d^2 - e^2*x^2)^(3/2))/(105*d^4*x^3) - (e^7*atanh(sqrt(d^2 - e^2*x^2)/d))/(8*d^4)]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^4/((d + e*x)^2*(d^2 - e^2*x^2)^(3/2)), x, 7, -((d^3*(d - e*x)^2)/(5*e^5*(d^2 - e^2*x^2)^(5/2))) + (17*d^2*(d - e*x))/(15*e^5*(d^2 - e^2*x^2)^(3/2)) - (2*(15*d - 13*e*x))/(15*e^5*sqrt(d^2 - e^2*x^2)) - atan((e*x)/sqrt(d^2 - e^2*x^2))/e^5]
    @test_int [x^3/((d + e*x)^2*(d^2 - e^2*x^2)^(3/2)), x, 4, (d^2*(d - e*x)^2)/(5*e^4*(d^2 - e^2*x^2)^(5/2)) - (4*d*(d - e*x))/(5*e^4*(d^2 - e^2*x^2)^(3/2)) + (5*d - 2*e*x)/(5*d*e^4*sqrt(d^2 - e^2*x^2))]
    @test_int [x^2/((d + e*x)^2*(d^2 - e^2*x^2)^(3/2)), x, 4, x/(15*d^2*e^2*sqrt(d^2 - e^2*x^2)) - d/(5*e^3*(d + e*x)^2*sqrt(d^2 - e^2*x^2)) + 7/(15*e^3*(d + e*x)*sqrt(d^2 - e^2*x^2)), -((d*(d - e*x)^2)/(5*e^3*(d^2 - e^2*x^2)^(5/2))) + (7*(d - e*x))/(15*e^3*(d^2 - e^2*x^2)^(3/2)) + x/(15*d^2*e^2*sqrt(d^2 - e^2*x^2))]
    @test_int [x^1/((d + e*x)^2*(d^2 - e^2*x^2)^(3/2)), x, 3, (4*x)/(15*d^3*e*sqrt(d^2 - e^2*x^2)) + 1/(5*e^2*(d + e*x)^2*sqrt(d^2 - e^2*x^2)) - 2/(15*d*e^2*(d + e*x)*sqrt(d^2 - e^2*x^2))]
    @test_int [x^0/((d + e*x)^2*(d^2 - e^2*x^2)^(3/2)), x, 3, (2*x)/(5*d^4*sqrt(d^2 - e^2*x^2)) - 1/(5*d*e*(d + e*x)^2*sqrt(d^2 - e^2*x^2)) - 1/(5*d^2*e*(d + e*x)*sqrt(d^2 - e^2*x^2))]
    @test_int [1/(x^1*(d + e*x)^2*(d^2 - e^2*x^2)^(3/2)), x, 8, (2*(d - e*x))/(5*d*(d^2 - e^2*x^2)^(5/2)) + (5*d - 8*e*x)/(15*d^3*(d^2 - e^2*x^2)^(3/2)) + (15*d - 16*e*x)/(15*d^5*sqrt(d^2 - e^2*x^2)) - atanh(sqrt(d^2 - e^2*x^2)/d)/d^5]
    @test_int [1/(x^2*(d + e*x)^2*(d^2 - e^2*x^2)^(3/2)), x, 8, -((2*e*(d - e*x))/(5*d^2*(d^2 - e^2*x^2)^(5/2))) - (e*(10*d - 13*e*x))/(15*d^4*(d^2 - e^2*x^2)^(3/2)) - (e*(30*d - 41*e*x))/(15*d^6*sqrt(d^2 - e^2*x^2)) - sqrt(d^2 - e^2*x^2)/(d^6*x) + (2*e*atanh(sqrt(d^2 - e^2*x^2)/d))/d^6]
    @test_int [1/(x^3*(d + e*x)^2*(d^2 - e^2*x^2)^(3/2)), x, 9, (2*e^2*(d - e*x))/(5*d^3*(d^2 - e^2*x^2)^(5/2)) + (e^2*(5*d - 6*e*x))/(5*d^5*(d^2 - e^2*x^2)^(3/2)) + (2*e^2*(10*d - 11*e*x))/(5*d^7*sqrt(d^2 - e^2*x^2)) - sqrt(d^2 - e^2*x^2)/(2*d^6*x^2) + (2*e*sqrt(d^2 - e^2*x^2))/(d^7*x) - (9*e^2*atanh(sqrt(d^2 - e^2*x^2)/d))/(2*d^7)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m / (d+e*x)^3*(d^2 - e^2*x^2)^(p/2)=#

    #= ::Subsubsection:: =#
    #=p>0=#

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^5/((d + e*x)^3*sqrt(d^2 - e^2*x^2)), x, 8, (d^4*(d - e*x)^3)/(5*e^6*(d^2 - e^2*x^2)^(5/2)) - (23*d^3*(d - e*x)^2)/(15*e^6*(d^2 - e^2*x^2)^(3/2)) + (127*d^2*(d - e*x))/(15*e^6*sqrt(d^2 - e^2*x^2)) + (3*d*sqrt(d^2 - e^2*x^2))/e^6 - (x*sqrt(d^2 - e^2*x^2))/(2*e^5) + (13*d^2*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(2*e^6)]
    @test_int [x^4/((d + e*x)^3*sqrt(d^2 - e^2*x^2)), x, 7, -((d^3*(d - e*x)^3)/(5*e^5*(d^2 - e^2*x^2)^(5/2))) + (6*d^2*(d - e*x)^2)/(5*e^5*(d^2 - e^2*x^2)^(3/2)) - (24*d*(d - e*x))/(5*e^5*sqrt(d^2 - e^2*x^2)) - sqrt(d^2 - e^2*x^2)/e^5 - (3*d*atan((e*x)/sqrt(d^2 - e^2*x^2)))/e^5]
    @test_int [x^3/((d + e*x)^3*sqrt(d^2 - e^2*x^2)), x, 6, (d^2*(d - e*x)^3)/(5*e^4*(d^2 - e^2*x^2)^(5/2)) - (13*d*(d - e*x)^2)/(15*e^4*(d^2 - e^2*x^2)^(3/2)) + (32*(d - e*x))/(15*e^4*sqrt(d^2 - e^2*x^2)) + atan((e*x)/sqrt(d^2 - e^2*x^2))/e^4]
    @test_int [x^2/((d + e*x)^3*sqrt(d^2 - e^2*x^2)), x, 4, -((d*sqrt(d^2 - e^2*x^2))/(5*e^3*(d + e*x)^3)) + (8*sqrt(d^2 - e^2*x^2))/(15*e^3*(d + e*x)^2) - (7*sqrt(d^2 - e^2*x^2))/(15*d*e^3*(d + e*x))]
    @test_int [x^1/((d + e*x)^3*sqrt(d^2 - e^2*x^2)), x, 3, sqrt(d^2 - e^2*x^2)/(5*e^2*(d + e*x)^3) - sqrt(d^2 - e^2*x^2)/(5*d*e^2*(d + e*x)^2) - sqrt(d^2 - e^2*x^2)/(5*d^2*e^2*(d + e*x))]
    @test_int [x^0/((d + e*x)^3*sqrt(d^2 - e^2*x^2)), x, 3, -(sqrt(d^2 - e^2*x^2)/(5*d*e*(d + e*x)^3)) - (2*sqrt(d^2 - e^2*x^2))/(15*d^2*e*(d + e*x)^2) - (2*sqrt(d^2 - e^2*x^2))/(15*d^3*e*(d + e*x))]
    @test_int [1/(x^1*(d + e*x)^3*sqrt(d^2 - e^2*x^2)), x, 8, (4*(d - e*x))/(5*(d^2 - e^2*x^2)^(5/2)) + (5*d - 11*e*x)/(15*d^2*(d^2 - e^2*x^2)^(3/2)) + (15*d - 22*e*x)/(15*d^4*sqrt(d^2 - e^2*x^2)) - atanh(sqrt(d^2 - e^2*x^2)/d)/d^4]
    @test_int [1/(x^2*(d + e*x)^3*sqrt(d^2 - e^2*x^2)), x, 8, -((4*e*(d - e*x))/(5*d*(d^2 - e^2*x^2)^(5/2))) - (e*(5*d - 7*e*x))/(5*d^3*(d^2 - e^2*x^2)^(3/2)) - (e*(15*d - 19*e*x))/(5*d^5*sqrt(d^2 - e^2*x^2)) - sqrt(d^2 - e^2*x^2)/(d^5*x) + (3*e*atanh(sqrt(d^2 - e^2*x^2)/d))/d^5]
    @test_int [1/(x^3*(d + e*x)^3*sqrt(d^2 - e^2*x^2)), x, 9, (4*e^2*(d - e*x))/(5*d^2*(d^2 - e^2*x^2)^(5/2)) + (e^2*(25*d - 31*e*x))/(15*d^4*(d^2 - e^2*x^2)^(3/2)) + (e^2*(90*d - 107*e*x))/(15*d^6*sqrt(d^2 - e^2*x^2)) - sqrt(d^2 - e^2*x^2)/(2*d^5*x^2) + (3*e*sqrt(d^2 - e^2*x^2))/(d^6*x) - (13*e^2*atanh(sqrt(d^2 - e^2*x^2)/d))/(2*d^6)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m / (d+e*x)^4*(d^2 - e^2*x^2)^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^5*sqrt(d^2 - e^2*x^2)/(d + e*x)^4, x, 9, (d^4*(d - e*x)^4)/(5*e^6*(d^2 - e^2*x^2)^(5/2)) - (8*d^3*(d - e*x)^3)/(5*e^6*(d^2 - e^2*x^2)^(3/2)) + (10*d^2*(d - e*x)^2)/(e^6*sqrt(d^2 - e^2*x^2)) + (59*d^2*sqrt(d^2 - e^2*x^2))/(3*e^6) - (2*d*x*sqrt(d^2 - e^2*x^2))/e^5 + (x^2*sqrt(d^2 - e^2*x^2))/(3*e^4) + (18*d^3*atan((e*x)/sqrt(d^2 - e^2*x^2)))/e^6]
    @test_int [x^4*sqrt(d^2 - e^2*x^2)/(d + e*x)^4, x, 7, -((d^3*(d - e*x)^4)/(5*e^5*(d^2 - e^2*x^2)^(5/2))) + (19*d^2*(d - e*x)^3)/(15*e^5*(d^2 - e^2*x^2)^(3/2)) - (6*d*(d - e*x)^2)/(e^5*sqrt(d^2 - e^2*x^2)) - ((20*d - e*x)*sqrt(d^2 - e^2*x^2))/(2*e^5) - (19*d^2*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(2*e^5)]
    @test_int [x^3*sqrt(d^2 - e^2*x^2)/(d + e*x)^4, x, 9, (8*d*sqrt(d^2 - e^2*x^2))/(e^4*(d + e*x)) + (d^2*(d^2 - e^2*x^2)^(3/2))/(5*e^4*(d + e*x)^4) - (14*d*(d^2 - e^2*x^2)^(3/2))/(15*e^4*(d + e*x)^3) - (d^2 - e^2*x^2)^(3/2)/(e^4*(d + e*x)^2) + (4*d*atan((e*x)/sqrt(d^2 - e^2*x^2)))/e^4]
    @test_int [x^2*sqrt(d^2 - e^2*x^2)/(d + e*x)^4, x, 8, -((2*sqrt(d^2 - e^2*x^2))/(e^3*(d + e*x))) - (d*(d^2 - e^2*x^2)^(3/2))/(5*e^3*(d + e*x)^4) + (3*(d^2 - e^2*x^2)^(3/2))/(5*e^3*(d + e*x)^3) - atan((e*x)/sqrt(d^2 - e^2*x^2))/e^3]
    @test_int [x^1*sqrt(d^2 - e^2*x^2)/(d + e*x)^4, x, 2, (d^2 - e^2*x^2)^(3/2)/(5*e^2*(d + e*x)^4) - (4*(d^2 - e^2*x^2)^(3/2))/(15*d*e^2*(d + e*x)^3)]
    @test_int [x^0*sqrt(d^2 - e^2*x^2)/(d + e*x)^4, x, 2, -((d^2 - e^2*x^2)^(3/2)/(5*d*e*(d + e*x)^4)) - (d^2 - e^2*x^2)^(3/2)/(15*d^2*e*(d + e*x)^3)]
    @test_int [sqrt(d^2 - e^2*x^2)/(x^1*(d + e*x)^4), x, 8, (8*d*(d - e*x))/(5*(d^2 - e^2*x^2)^(5/2)) - (4*e*x)/(5*d*(d^2 - e^2*x^2)^(3/2)) + (5*d - 8*e*x)/(5*d^3*sqrt(d^2 - e^2*x^2)) - atanh(sqrt(d^2 - e^2*x^2)/d)/d^3]
    @test_int [sqrt(d^2 - e^2*x^2)/(x^2*(d + e*x)^4), x, 8, -((8*e*(d - e*x))/(5*(d^2 - e^2*x^2)^(5/2))) - (4*e*(5*d - 8*e*x))/(15*d^2*(d^2 - e^2*x^2)^(3/2)) - (e*(60*d - 79*e*x))/(15*d^4*sqrt(d^2 - e^2*x^2)) - sqrt(d^2 - e^2*x^2)/(d^4*x) + (4*e*atanh(sqrt(d^2 - e^2*x^2)/d))/d^4]
    @test_int [sqrt(d^2 - e^2*x^2)/(x^3*(d + e*x)^4), x, 9, (8*e^2*(d - e*x))/(5*d*(d^2 - e^2*x^2)^(5/2)) + (4*e^2*(10*d - 13*e*x))/(15*d^3*(d^2 - e^2*x^2)^(3/2)) + (e^2*(135*d - 164*e*x))/(15*d^5*sqrt(d^2 - e^2*x^2)) - sqrt(d^2 - e^2*x^2)/(2*d^4*x^2) + (4*e*sqrt(d^2 - e^2*x^2))/(d^5*x) - (19*e^2*atanh(sqrt(d^2 - e^2*x^2)/d))/(2*d^5)]
    @test_int [sqrt(d^2 - e^2*x^2)/(x^4*(d + e*x)^4), x, 10, -((8*e^3*(d - e*x))/(5*d^2*(d^2 - e^2*x^2)^(5/2))) - (4*e^3*(5*d - 6*e*x))/(5*d^4*(d^2 - e^2*x^2)^(3/2)) - (e^3*(80*d - 93*e*x))/(5*d^6*sqrt(d^2 - e^2*x^2)) - sqrt(d^2 - e^2*x^2)/(3*d^4*x^3) + (2*e*sqrt(d^2 - e^2*x^2))/(d^5*x^2) - (29*e^2*sqrt(d^2 - e^2*x^2))/(3*d^6*x) + (18*e^3*atanh(sqrt(d^2 - e^2*x^2)/d))/d^6]

    @test_int [x^5*(d^2 - e^2*x^2)^(5/2)/(d + e*x)^4, x, 11, (d^4*(d - e*x)^4)/(e^6*sqrt(d^2 - e^2*x^2)) + (515*d^6*sqrt(d^2 - e^2*x^2))/(21*e^6) - (49*d^5*x*sqrt(d^2 - e^2*x^2))/(4*e^5) + (121*d^4*x^2*sqrt(d^2 - e^2*x^2))/(21*e^4) - (17*d^3*x^3*sqrt(d^2 - e^2*x^2))/(6*e^3) + (11*d^2*x^4*sqrt(d^2 - e^2*x^2))/(7*e^2) - (2*d*x^5*sqrt(d^2 - e^2*x^2))/(3*e) + (1/7)*x^6*sqrt(d^2 - e^2*x^2) + (65*d^7*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(4*e^6)]
    @test_int [x^4*(d^2 - e^2*x^2)^(5/2)/(d + e*x)^4, x, 10, -((d^3*(d - e*x)^4)/(e^5*sqrt(d^2 - e^2*x^2))) - (337*d^5*sqrt(d^2 - e^2*x^2))/(15*e^5) + (175*d^4*x*sqrt(d^2 - e^2*x^2))/(16*e^4) - (71*d^3*x^2*sqrt(d^2 - e^2*x^2))/(15*e^3) + (47*d^2*x^3*sqrt(d^2 - e^2*x^2))/(24*e^2) - (4*d*x^4*sqrt(d^2 - e^2*x^2))/(5*e) + (1/6)*x^5*sqrt(d^2 - e^2*x^2) - (239*d^6*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(16*e^5)]
    @test_int [x^3*(d^2 - e^2*x^2)^(5/2)/(d + e*x)^4, x, 9, (d^2*(d - e*x)^4)/(e^4*sqrt(d^2 - e^2*x^2)) + (101*d^4*sqrt(d^2 - e^2*x^2))/(5*e^4) - (19*d^3*x*sqrt(d^2 - e^2*x^2))/(2*e^3) + (18*d^2*x^2*sqrt(d^2 - e^2*x^2))/(5*e^2) - (d*x^3*sqrt(d^2 - e^2*x^2))/e + (1/5)*x^4*sqrt(d^2 - e^2*x^2) + (27*d^5*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(2*e^4)]
    @test_int [x^2*(d^2 - e^2*x^2)^(5/2)/(d + e*x)^4, x, 8, -((d*(d - e*x)^4)/(e^3*sqrt(d^2 - e^2*x^2))) - (95*d^3*sqrt(d^2 - e^2*x^2))/(8*e^3) - (95*d^2*(d - e*x)*sqrt(d^2 - e^2*x^2))/(24*e^3) - (19*d*(d - e*x)^2*sqrt(d^2 - e^2*x^2))/(12*e^3) - ((d - e*x)^3*sqrt(d^2 - e^2*x^2))/(4*e^3) - (95*d^4*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(8*e^3)]
    @test_int [x^1*(d^2 - e^2*x^2)^(5/2)/(d + e*x)^4, x, 6, (10*d*x*sqrt(d^2 - e^2*x^2))/e + (20*(d^2 - e^2*x^2)^(3/2))/(3*e^2) + (8*(d^2 - e^2*x^2)^(5/2))/(e^2*(d + e*x)^2) + (d^2 - e^2*x^2)^(7/2)/(e^2*(d + e*x)^4) + (10*d^3*atan((e*x)/sqrt(d^2 - e^2*x^2)))/e^2]
    @test_int [x^0*(d^2 - e^2*x^2)^(5/2)/(d + e*x)^4, x, 5, -((15*d*sqrt(d^2 - e^2*x^2))/(2*e)) - (5*(d^2 - e^2*x^2)^(3/2))/(2*e*(d + e*x)) - (2*(d^2 - e^2*x^2)^(5/2))/(e*(d + e*x)^3) - (15*d^2*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(2*e)]
    @test_int [(d^2 - e^2*x^2)^(5/2)/(x^1*(d + e*x)^4), x, 9, (8*d*(d - e*x))/sqrt(d^2 - e^2*x^2) + sqrt(d^2 - e^2*x^2) + 4*d*atan((e*x)/sqrt(d^2 - e^2*x^2)) - d*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d^2 - e^2*x^2)^(5/2)/(x^2*(d + e*x)^4), x, 9, -((8*e*(d - e*x))/sqrt(d^2 - e^2*x^2)) - sqrt(d^2 - e^2*x^2)/x - e*atan((e*x)/sqrt(d^2 - e^2*x^2)) + 4*e*atanh(sqrt(d^2 - e^2*x^2)/d)]
    @test_int [(d^2 - e^2*x^2)^(5/2)/(x^3*(d + e*x)^4), x, 7, (8*e^2*(d - e*x))/(d*sqrt(d^2 - e^2*x^2)) - sqrt(d^2 - e^2*x^2)/(2*x^2) + (4*e*sqrt(d^2 - e^2*x^2))/(d*x) - (15*e^2*atanh(sqrt(d^2 - e^2*x^2)/d))/(2*d)]
    @test_int [(d^2 - e^2*x^2)^(5/2)/(x^4*(d + e*x)^4), x, 8, -((8*e^3*(d - e*x))/(d^2*sqrt(d^2 - e^2*x^2))) - sqrt(d^2 - e^2*x^2)/(3*x^3) + (2*e*sqrt(d^2 - e^2*x^2))/(d*x^2) - (23*e^2*sqrt(d^2 - e^2*x^2))/(3*d^2*x) + (10*e^3*atanh(sqrt(d^2 - e^2*x^2)/d))/d^2]
    @test_int [(d^2 - e^2*x^2)^(5/2)/(x^5*(d + e*x)^4), x, 9, (8*e^4*(d - e*x))/(d^3*sqrt(d^2 - e^2*x^2)) - sqrt(d^2 - e^2*x^2)/(4*x^4) + (4*e*sqrt(d^2 - e^2*x^2))/(3*d*x^3) - (31*e^2*sqrt(d^2 - e^2*x^2))/(8*d^2*x^2) + (32*e^3*sqrt(d^2 - e^2*x^2))/(3*d^3*x) - (95*e^4*atanh(sqrt(d^2 - e^2*x^2)/d))/(8*d^3)]
    @test_int [(d^2 - e^2*x^2)^(5/2)/(x^6*(d + e*x)^4), x, 10, -((8*e^5*(d - e*x))/(d^4*sqrt(d^2 - e^2*x^2))) - sqrt(d^2 - e^2*x^2)/(5*x^5) + (e*sqrt(d^2 - e^2*x^2))/(d*x^4) - (13*e^2*sqrt(d^2 - e^2*x^2))/(5*d^2*x^3) + (11*e^3*sqrt(d^2 - e^2*x^2))/(2*d^3*x^2) - (66*e^4*sqrt(d^2 - e^2*x^2))/(5*d^4*x) + (27*e^5*atanh(sqrt(d^2 - e^2*x^2)/d))/(2*d^4)]

    @test_int [x^2*sqrt(1 - a^2*x^2)/(1 - a*x)^4, x, 7, (2*sqrt(1 - a^2*x^2))/(a^3*(1 - a*x)) + (1 - a^2*x^2)^(3/2)/(5*a^3*(1 - a*x)^4) - (3*(1 - a^2*x^2)^(3/2))/(5*a^3*(1 - a*x)^3) - asin(a*x)/a^3]
    @test_int [x^2*sqrt(1 - a^2*x^2)/(1 - a*x)^5, x, 4, (1 - a^2*x^2)^(3/2)/(7*a^3*(1 - a*x)^5) - (12*(1 - a^2*x^2)^(3/2))/(35*a^3*(1 - a*x)^4) + (23*(1 - a^2*x^2)^(3/2))/(105*a^3*(1 - a*x)^3)]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^3/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)), x, 9, -((24*x)/(5005*d^3*e^3*(d^2 - e^2*x^2)^(5/2))) + d^2/(13*e^4*(d + e*x)^4*(d^2 - e^2*x^2)^(5/2)) - (30*d)/(143*e^4*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)) + 21/(143*e^4*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2)) + 4/(1001*d*e^4*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) - (32*x)/(5005*d^5*e^3*(d^2 - e^2*x^2)^(3/2)) - (64*x)/(5005*d^7*e^3*sqrt(d^2 - e^2*x^2))]
    @test_int [x^2/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)), x, 8, (14*x)/(2145*d^4*e^2*(d^2 - e^2*x^2)^(5/2)) - d/(13*e^3*(d + e*x)^4*(d^2 - e^2*x^2)^(5/2)) + 17/(143*e^3*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)) - 7/(1287*d*e^3*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2)) - 7/(1287*d^2*e^3*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) + (56*x)/(6435*d^6*e^2*(d^2 - e^2*x^2)^(3/2)) + (112*x)/(6435*d^8*e^2*sqrt(d^2 - e^2*x^2))]
    @test_int [x^1/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)), x, 7, (64*x)/(2145*d^5*e*(d^2 - e^2*x^2)^(5/2)) + 1/(13*e^2*(d + e*x)^4*(d^2 - e^2*x^2)^(5/2)) - 4/(143*d*e^2*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)) - 32/(1287*d^2*e^2*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2)) - 32/(1287*d^3*e^2*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) + (256*x)/(6435*d^7*e*(d^2 - e^2*x^2)^(3/2)) + (512*x)/(6435*d^9*e*sqrt(d^2 - e^2*x^2))]
    @test_int [x^0/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)), x, 7, (48*x)/(715*d^6*(d^2 - e^2*x^2)^(5/2)) - 1/(13*d*e*(d + e*x)^4*(d^2 - e^2*x^2)^(5/2)) - 9/(143*d^2*e*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)) - 8/(143*d^3*e*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2)) - 8/(143*d^4*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) + (64*x)/(715*d^8*(d^2 - e^2*x^2)^(3/2)) + (128*x)/(715*d^10*sqrt(d^2 - e^2*x^2))]
    @test_int [1/(x^1*(d + e*x)^4*(d^2 - e^2*x^2)^(7/2)), x, 12, (8*d*(d - e*x))/(13*(d^2 - e^2*x^2)^(13/2)) - (4*e*x)/(13*d*(d^2 - e^2*x^2)^(11/2)) + (13*d - 40*e*x)/(117*d^3*(d^2 - e^2*x^2)^(9/2)) + (117*d - 320*e*x)/(819*d^5*(d^2 - e^2*x^2)^(7/2)) + (273*d - 640*e*x)/(1365*d^7*(d^2 - e^2*x^2)^(5/2)) + (273*d - 512*e*x)/(819*d^9*(d^2 - e^2*x^2)^(3/2)) + (819*d - 1024*e*x)/(819*d^11*sqrt(d^2 - e^2*x^2)) - atanh(sqrt(d^2 - e^2*x^2)/d)/d^11]
    @test_int [1/(x^2*(d + e*x)^4*(d^2 - e^2*x^2)^(7/2)), x, 12, -((8*e*(d - e*x))/(13*(d^2 - e^2*x^2)^(13/2))) - (4*e*(13*d - 24*e*x))/(143*d^2*(d^2 - e^2*x^2)^(11/2)) - (e*(572*d - 1103*e*x))/(1287*d^4*(d^2 - e^2*x^2)^(9/2)) - (e*(5148*d - 10111*e*x))/(9009*d^6*(d^2 - e^2*x^2)^(7/2)) - (e*(12012*d - 23225*e*x))/(15015*d^8*(d^2 - e^2*x^2)^(5/2)) - (e*(12012*d - 21583*e*x))/(9009*d^10*(d^2 - e^2*x^2)^(3/2)) - (e*(36036*d - 52175*e*x))/(9009*d^12*sqrt(d^2 - e^2*x^2)) - sqrt(d^2 - e^2*x^2)/(d^12*x) + (4*e*atanh(sqrt(d^2 - e^2*x^2)/d))/d^12]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(d+e*x)^(n/2)*(d^2 - e^2*x^2)^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [sqrt(c - a*c*x)*sqrt(1 - a^2*x^2)/x^2, x, 4, -((a*c*sqrt(1 - a^2*x^2))/sqrt(c - a*c*x)) - (c^2*(1 - a^2*x^2)^(3/2))/(x*(c - a*c*x)^(3/2)) + a*sqrt(c)*atanh((sqrt(c)*sqrt(1 - a^2*x^2))/sqrt(c - a*c*x))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [sqrt(c - a*c*x)/(x*sqrt(1 - a^2*x^2)), x, 2, -2*sqrt(c)*atanh((sqrt(c)*sqrt(1 - a^2*x^2))/sqrt(c - a*c*x))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(g*x)^(m/2)*(d+e*x)^(n/2)*(d^2 - e^2*x^2)^(p/2)=#

    #= Following*pairs*of*integrands*are*equal. =#
    @test_int [sqrt(1 - a*x)/sqrt(x), x, 3, sqrt(x)*sqrt(1 - a*x) + asin(sqrt(a)*sqrt(x))/sqrt(a)]
    @test_int [sqrt(1 - a^2*x^2)/(sqrt(x)*sqrt(1 + a*x)), x, 4, sqrt(x)*sqrt(1 - a*x) + asin(sqrt(a)*sqrt(x))/sqrt(a)]

    @test_int [sqrt(1 + a*x)/sqrt(x), x, 3, sqrt(x)*sqrt(1 + a*x) + asinh(sqrt(a)*sqrt(x))/sqrt(a)]
    @test_int [sqrt(1 - a^2*x^2)/(sqrt(x)*sqrt(1 - a*x)), x, 4, sqrt(x)*sqrt(1 + a*x) + asinh(sqrt(a)*sqrt(x))/sqrt(a)]

    @test_int [sqrt(x)*sqrt(1 - a*x), x, 4, -((sqrt(x)*sqrt(1 - a*x))/(4*a)) + (1/2)*x^(3/2)*sqrt(1 - a*x) + asin(sqrt(a)*sqrt(x))/(4*a^(3/2))]
    @test_int [sqrt(x)*sqrt(1 - a^2*x^2)/sqrt(1 + a*x), x, 5, -((sqrt(x)*sqrt(1 - a*x))/(4*a)) + (1/2)*x^(3/2)*sqrt(1 - a*x) + asin(sqrt(a)*sqrt(x))/(4*a^(3/2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(g*x)^m*(d+e*x)^n*(d^2 - e^2*x^2)^(p/2)*when*m*symbolic=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [(g*x)^m*(d^2 - e^2*x^2)^(5/2)*(d + e*x)^3, x, 7, -((3*d*(g*x)^(1 + m)*(d^2 - e^2*x^2)^(7/2))/(g*(8 + m))) - (e*(g*x)^(2 + m)*(d^2 - e^2*x^2)^(7/2))/(g^2*(9 + m)) + (d^7*(11 + 4*m)*(g*x)^(1 + m)*sqrt(d^2 - e^2*x^2)*HypergeometricFunctions._₂F₁(-(5/2), (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2))/(g*(1 + m)*(8 + m)*sqrt(1 - (e^2*x^2)/d^2)) + (d^6*e*(29 + 4*m)*(g*x)^(2 + m)*sqrt(d^2 - e^2*x^2)*HypergeometricFunctions._₂F₁(-(5/2), (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2))/(g^2*(2 + m)*(9 + m)*sqrt(1 - (e^2*x^2)/d^2))]
    @test_int [(g*x)^m*(d^2 - e^2*x^2)^(5/2)*(d + e*x)^2, x, 6, -(((g*x)^(1 + m)*(d^2 - e^2*x^2)^(7/2))/(g*(8 + m))) + (d^6*(9 + 2*m)*(g*x)^(1 + m)*sqrt(d^2 - e^2*x^2)*HypergeometricFunctions._₂F₁(-(5/2), (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2))/(g*(1 + m)*(8 + m)*sqrt(1 - (e^2*x^2)/d^2)) + (2*d^5*e*(g*x)^(2 + m)*sqrt(d^2 - e^2*x^2)*HypergeometricFunctions._₂F₁(-(5/2), (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2))/(g^2*(2 + m)*sqrt(1 - (e^2*x^2)/d^2))]
    @test_int [(g*x)^m*(d^2 - e^2*x^2)^(5/2)*(d + e*x)^1, x, 5, (d^5*(g*x)^(1 + m)*sqrt(d^2 - e^2*x^2)*HypergeometricFunctions._₂F₁(-(5/2), (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2))/(g*(1 + m)*sqrt(1 - (e^2*x^2)/d^2)) + (d^4*e*(g*x)^(2 + m)*sqrt(d^2 - e^2*x^2)*HypergeometricFunctions._₂F₁(-(5/2), (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2))/(g^2*(2 + m)*sqrt(1 - (e^2*x^2)/d^2))]
    @test_int [(g*x)^m*(d^2 - e^2*x^2)^(5/2)*(d + e*x)^0, x, 2, (d^4*(g*x)^(1 + m)*sqrt(d^2 - e^2*x^2)*HypergeometricFunctions._₂F₁(-(5/2), (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2))/(g*(1 + m)*sqrt(1 - (e^2*x^2)/d^2))]
    @test_int [(g*x)^m*(d^2 - e^2*x^2)^(5/2)/(d + e*x)^1, x, 8, (d^3*(g*x)^(1 + m)*sqrt(d^2 - e^2*x^2)*HypergeometricFunctions._₂F₁(-(3/2), (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2))/(g*(1 + m)*sqrt(1 - (e^2*x^2)/d^2)) - (d^2*e*(g*x)^(2 + m)*sqrt(d^2 - e^2*x^2)*HypergeometricFunctions._₂F₁(-(3/2), (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2))/(g^2*(2 + m)*sqrt(1 - (e^2*x^2)/d^2))]
    @test_int [(g*x)^m*(d^2 - e^2*x^2)^(5/2)/(d + e*x)^2, x, 7, -(((g*x)^(1 + m)*(d^2 - e^2*x^2)^(3/2))/(g*(4 + m))) + (d^2*(5 + 2*m)*(g*x)^(1 + m)*sqrt(d^2 - e^2*x^2)*HypergeometricFunctions._₂F₁(-(1/2), (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2))/(g*(1 + m)*(4 + m)*sqrt(1 - (e^2*x^2)/d^2)) - (2*d*e*(g*x)^(2 + m)*sqrt(d^2 - e^2*x^2)*HypergeometricFunctions._₂F₁(-(1/2), (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2))/(g^2*(2 + m)*sqrt(1 - (e^2*x^2)/d^2))]
    @test_int [(g*x)^m*(d^2 - e^2*x^2)^(5/2)/(d + e*x)^3, x, 8, -((3*d*(g*x)^(1 + m)*sqrt(d^2 - e^2*x^2))/(g*(2 + m))) + (e*(g*x)^(2 + m)*sqrt(d^2 - e^2*x^2))/(g^2*(3 + m)) + (d^3*(5 + 4*m)*(g*x)^(1 + m)*sqrt(1 - (e^2*x^2)/d^2)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2))/(g*(1 + m)*(2 + m)*sqrt(d^2 - e^2*x^2)) - (d^2*e*(11 + 4*m)*(g*x)^(2 + m)*sqrt(1 - (e^2*x^2)/d^2)*HypergeometricFunctions._₂F₁(1/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2))/(g^2*(2 + m)*(3 + m)*sqrt(d^2 - e^2*x^2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(g*x)^m*(d + e*x)^3/(d^2 - e^2*x^2)^(7/2), x, 6, (4*(g*x)^(1 + m)*(d + e*x))/(5*g*(d^2 - e^2*x^2)^(5/2)) + ((1 - 4*m)*(g*x)^(1 + m)*sqrt(1 - (e^2*x^2)/d^2)*HypergeometricFunctions._₂F₁(5/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2))/(5*d^3*g*(1 + m)*sqrt(d^2 - e^2*x^2)) + (e*(7 - 4*m)*(g*x)^(2 + m)*sqrt(1 - (e^2*x^2)/d^2)*HypergeometricFunctions._₂F₁(5/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2))/(5*d^4*g^2*(2 + m)*sqrt(d^2 - e^2*x^2))]
    @test_int [(g*x)^m*(d + e*x)^2/(d^2 - e^2*x^2)^(7/2), x, 6, (2*(g*x)^(1 + m)*(d + e*x))/(5*d*g*(d^2 - e^2*x^2)^(5/2)) + ((3 - 2*m)*(g*x)^(1 + m)*sqrt(1 - (e^2*x^2)/d^2)*HypergeometricFunctions._₂F₁(5/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2))/(5*d^4*g*(1 + m)*sqrt(d^2 - e^2*x^2)) + (2*e*(3 - m)*(g*x)^(2 + m)*sqrt(1 - (e^2*x^2)/d^2)*HypergeometricFunctions._₂F₁(5/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2))/(5*d^5*g^2*(2 + m)*sqrt(d^2 - e^2*x^2))]
    @test_int [(g*x)^m*(d + e*x)^1/(d^2 - e^2*x^2)^(7/2), x, 5, ((g*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, (1/2)*(-4 + m), (3 + m)/2, (e^2*x^2)/d^2))/(d*g*(1 + m)*(d^2 - e^2*x^2)^(5/2)) + (e*(g*x)^(2 + m)*HypergeometricFunctions._₂F₁(1, (1/2)*(-3 + m), (4 + m)/2, (e^2*x^2)/d^2))/(d^2*g^2*(2 + m)*(d^2 - e^2*x^2)^(5/2)), ((g*x)^(1 + m)*sqrt(1 - (e^2*x^2)/d^2)*HypergeometricFunctions._₂F₁(7/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2))/(d^5*g*(1 + m)*sqrt(d^2 - e^2*x^2)) + (e*(g*x)^(2 + m)*sqrt(1 - (e^2*x^2)/d^2)*HypergeometricFunctions._₂F₁(7/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2))/(d^6*g^2*(2 + m)*sqrt(d^2 - e^2*x^2))]
    @test_int [(g*x)^m*(d + e*x)^0/(d^2 - e^2*x^2)^(7/2), x, 2, ((g*x)^(1 + m)*sqrt(1 - (e^2*x^2)/d^2)*HypergeometricFunctions._₂F₁(7/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2))/(d^6*g*(1 + m)*sqrt(d^2 - e^2*x^2))]
    @test_int [(g*x)^m/((d + e*x)^1*(d^2 - e^2*x^2)^(7/2)), x, 8, ((g*x)^(1 + m)*sqrt(1 - (e^2*x^2)/d^2)*HypergeometricFunctions._₂F₁(9/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2))/(d^7*g*(1 + m)*sqrt(d^2 - e^2*x^2)) - (e*(g*x)^(2 + m)*sqrt(1 - (e^2*x^2)/d^2)*HypergeometricFunctions._₂F₁(9/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2))/(d^8*g^2*(2 + m)*sqrt(d^2 - e^2*x^2))]
    @test_int [(g*x)^m/((d + e*x)^2*(d^2 - e^2*x^2)^(7/2)), x, 7, (2*(g*x)^(1 + m)*(d - e*x))/(9*d*g*(d^2 - e^2*x^2)^(9/2)) + ((7 - 2*m)*(g*x)^(1 + m)*sqrt(1 - (e^2*x^2)/d^2)*HypergeometricFunctions._₂F₁(9/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2))/(9*d^8*g*(1 + m)*sqrt(d^2 - e^2*x^2)) - (2*e*(7 - m)*(g*x)^(2 + m)*sqrt(1 - (e^2*x^2)/d^2)*HypergeometricFunctions._₂F₁(9/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2))/(9*d^9*g^2*(2 + m)*sqrt(d^2 - e^2*x^2))]
    @test_int [(g*x)^m/((d + e*x)^3*(d^2 - e^2*x^2)^(7/2)), x, 7, (4*(g*x)^(1 + m)*(d - e*x))/(11*g*(d^2 - e^2*x^2)^(11/2)) + ((7 - 4*m)*(g*x)^(1 + m)*sqrt(1 - (e^2*x^2)/d^2)*HypergeometricFunctions._₂F₁(11/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2))/(11*d^9*g*(1 + m)*sqrt(d^2 - e^2*x^2)) - (e*(25 - 4*m)*(g*x)^(2 + m)*sqrt(1 - (e^2*x^2)/d^2)*HypergeometricFunctions._₂F₁(11/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2))/(11*d^10*g^2*(2 + m)*sqrt(d^2 - e^2*x^2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(d+e*x)^n*(d^2 - e^2*x^2)^p*when*p*symbolic=#

    #= ::Subsubsection::Closed:: =#
    #=n>0=#

    @test_int [x^5*(d + e*x)*(d^2 - e^2*x^2)^p, x, 6, -((d^5*(d^2 - e^2*x^2)^(1 + p))/(2*e^6*(1 + p))) + (d^3*(d^2 - e^2*x^2)^(2 + p))/(e^6*(2 + p)) - (d*(d^2 - e^2*x^2)^(3 + p))/(2*e^6*(3 + p)) + ((1/7)*e*x^7*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(7/2, -p, 9/2, (e^2*x^2)/d^2))/(1 - (e^2*x^2)/d^2)^p]
    @test_int [x^4*(d + e*x)*(d^2 - e^2*x^2)^p, x, 6, -((d^4*(d^2 - e^2*x^2)^(1 + p))/(2*e^5*(1 + p))) + (d^2*(d^2 - e^2*x^2)^(2 + p))/(e^5*(2 + p)) - (d^2 - e^2*x^2)^(3 + p)/(2*e^5*(3 + p)) + ((1/5)*d*x^5*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(5/2, -p, 7/2, (e^2*x^2)/d^2))/(1 - (e^2*x^2)/d^2)^p]
    @test_int [x^3*(d + e*x)*(d^2 - e^2*x^2)^p, x, 6, -((d^3*(d^2 - e^2*x^2)^(1 + p))/(2*e^4*(1 + p))) + (d*(d^2 - e^2*x^2)^(2 + p))/(2*e^4*(2 + p)) + ((1/5)*e*x^5*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(5/2, -p, 7/2, (e^2*x^2)/d^2))/(1 - (e^2*x^2)/d^2)^p]
    @test_int [x^2*(d + e*x)*(d^2 - e^2*x^2)^p, x, 6, -((d^2*(d^2 - e^2*x^2)^(1 + p))/(2*e^3*(1 + p))) + (d^2 - e^2*x^2)^(2 + p)/(2*e^3*(2 + p)) + ((1/3)*d*x^3*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(3/2, -p, 5/2, (e^2*x^2)/d^2))/(1 - (e^2*x^2)/d^2)^p]
    @test_int [x^1*(d + e*x)*(d^2 - e^2*x^2)^p, x, 4, -((d*(d^2 - e^2*x^2)^(1 + p))/(2*e^2*(1 + p))) + ((1/3)*e*x^3*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(3/2, -p, 5/2, (e^2*x^2)/d^2))/(1 - (e^2*x^2)/d^2)^p]
    @test_int [x^0*(d + e*x)*(d^2 - e^2*x^2)^p, x, 3, -((d^2 - e^2*x^2)^(1 + p)/(2*e*(1 + p))) + (d*x*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, (e^2*x^2)/d^2))/(1 - (e^2*x^2)/d^2)^p]
    @test_int [(d + e*x)*(d^2 - e^2*x^2)^p/x^1, x, 5, (e*x*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, (e^2*x^2)/d^2))/(1 - (e^2*x^2)/d^2)^p - ((d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2))/(2*d*(1 + p))]
    @test_int [(d + e*x)*(d^2 - e^2*x^2)^p/x^2, x, 5, -((d*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(-(1/2), -p, 1/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*x)) - (e*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2))/(2*d^2*(1 + p))]
    @test_int [(d + e*x)*(d^2 - e^2*x^2)^p/x^3, x, 5, -((e*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(-(1/2), -p, 1/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*x)) - (e^2*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(2, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2))/(2*d^3*(1 + p))]

    @test_int [x^5*(d + e*x)^2*(d^2 - e^2*x^2)^p, x, 7, -((d^6*(d^2 - e^2*x^2)^(1 + p))/(e^6*(1 + p))) + (5*d^4*(d^2 - e^2*x^2)^(2 + p))/(2*e^6*(2 + p)) - (2*d^2*(d^2 - e^2*x^2)^(3 + p))/(e^6*(3 + p)) + (d^2 - e^2*x^2)^(4 + p)/(2*e^6*(4 + p)) + ((2/7)*d*e*x^7*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(7/2, -p, 9/2, (e^2*x^2)/d^2))/(1 - (e^2*x^2)/d^2)^p]
    @test_int [x^4*(d + e*x)^2*(d^2 - e^2*x^2)^p, x, 8, -((d^5*(d^2 - e^2*x^2)^(1 + p))/(e^5*(1 + p))) - (x^5*(d^2 - e^2*x^2)^(1 + p))/(7 + 2*p) + (2*d^3*(d^2 - e^2*x^2)^(2 + p))/(e^5*(2 + p)) - (d*(d^2 - e^2*x^2)^(3 + p))/(e^5*(3 + p)) + (2*d^2*(6 + p)*x^5*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(5/2, -p, 7/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(5*(7 + 2*p)))]
    @test_int [x^3*(d + e*x)^2*(d^2 - e^2*x^2)^p, x, 7, -((d^4*(d^2 - e^2*x^2)^(1 + p))/(e^4*(1 + p))) + (3*d^2*(d^2 - e^2*x^2)^(2 + p))/(2*e^4*(2 + p)) - (d^2 - e^2*x^2)^(3 + p)/(2*e^4*(3 + p)) + ((2/5)*d*e*x^5*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(5/2, -p, 7/2, (e^2*x^2)/d^2))/(1 - (e^2*x^2)/d^2)^p]
    @test_int [x^2*(d + e*x)^2*(d^2 - e^2*x^2)^p, x, 8, -((d^3*(d^2 - e^2*x^2)^(1 + p))/(e^3*(1 + p))) - (x^3*(d^2 - e^2*x^2)^(1 + p))/(5 + 2*p) + (d*(d^2 - e^2*x^2)^(2 + p))/(e^3*(2 + p)) + (2*d^2*(4 + p)*x^3*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(3/2, -p, 5/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(3*(5 + 2*p)))]
    @test_int [x^1*(d + e*x)^2*(d^2 - e^2*x^2)^p, x, 7, -((d^2*(d^2 - e^2*x^2)^(1 + p))/(e^2*(1 + p))) + (d^2 - e^2*x^2)^(2 + p)/(2*e^2*(2 + p)) + ((2/3)*d*e*x^3*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(3/2, -p, 5/2, (e^2*x^2)/d^2))/(1 - (e^2*x^2)/d^2)^p]
    @test_int [x^0*(d + e*x)^2*(d^2 - e^2*x^2)^p, x, 2, -((2^(2 + p)*d*(1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(-2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)))/(e*(1 + p)))]
    @test_int [(d + e*x)^2*(d^2 - e^2*x^2)^p/x^1, x, 7, -((d^2 - e^2*x^2)^(1 + p)/(2*(1 + p))) + (2*d*e*x*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, (e^2*x^2)/d^2))/(1 - (e^2*x^2)/d^2)^p - ((d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2))/(2*(1 + p))]
    @test_int [(d + e*x)^2*(d^2 - e^2*x^2)^p/x^2, x, 6, -((d^2 - e^2*x^2)^(1 + p)/x) - (2*e^2*p*x*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, (e^2*x^2)/d^2))/(1 - (e^2*x^2)/d^2)^p - (e*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2))/(d*(1 + p))]
    @test_int [(d + e*x)^2*(d^2 - e^2*x^2)^p/x^3, x, 6, -((d^2 - e^2*x^2)^(1 + p)/(2*x^2)) - (2*d*e*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(-(1/2), -p, 1/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*x) - (e^2*(1 - p)*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2))/(2*d^2*(1 + p))]

    @test_int [x^5*(d + e*x)^3*(d^2 - e^2*x^2)^p, x, 7, -((2*d^7*(d^2 - e^2*x^2)^(1 + p))/(e^6*(1 + p))) - (e*x^7*(d^2 - e^2*x^2)^(1 + p))/(9 + 2*p) + (11*d^5*(d^2 - e^2*x^2)^(2 + p))/(2*e^6*(2 + p)) - (5*d^3*(d^2 - e^2*x^2)^(3 + p))/(e^6*(3 + p)) + (3*d*(d^2 - e^2*x^2)^(4 + p))/(2*e^6*(4 + p)) + (2*d^2*e*(17 + 3*p)*x^7*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(7/2, -p, 9/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(7*(9 + 2*p)))]
    @test_int [x^4*(d + e*x)^3*(d^2 - e^2*x^2)^p, x, 7, -((2*d^6*(d^2 - e^2*x^2)^(1 + p))/(e^5*(1 + p))) - (3*d*x^5*(d^2 - e^2*x^2)^(1 + p))/(7 + 2*p) + (9*d^4*(d^2 - e^2*x^2)^(2 + p))/(2*e^5*(2 + p)) - (3*d^2*(d^2 - e^2*x^2)^(3 + p))/(e^5*(3 + p)) + (d^2 - e^2*x^2)^(4 + p)/(2*e^5*(4 + p)) + (2*d^3*(11 + p)*x^5*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(5/2, -p, 7/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(5*(7 + 2*p)))]
    @test_int [x^3*(d + e*x)^3*(d^2 - e^2*x^2)^p, x, 7, -((2*d^5*(d^2 - e^2*x^2)^(1 + p))/(e^4*(1 + p))) - (e*x^5*(d^2 - e^2*x^2)^(1 + p))/(7 + 2*p) + (7*d^3*(d^2 - e^2*x^2)^(2 + p))/(2*e^4*(2 + p)) - (3*d*(d^2 - e^2*x^2)^(3 + p))/(2*e^4*(3 + p)) + (2*d^2*e*(13 + 3*p)*x^5*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(5/2, -p, 7/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(5*(7 + 2*p)))]
    @test_int [x^2*(d + e*x)^3*(d^2 - e^2*x^2)^p, x, 7, -((2*d^4*(d^2 - e^2*x^2)^(1 + p))/(e^3*(1 + p))) - (3*d*x^3*(d^2 - e^2*x^2)^(1 + p))/(5 + 2*p) + (5*d^2*(d^2 - e^2*x^2)^(2 + p))/(2*e^3*(2 + p)) - (d^2 - e^2*x^2)^(3 + p)/(2*e^3*(3 + p)) + (2*d^3*(7 + p)*x^3*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(3/2, -p, 5/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(3*(5 + 2*p)))]
    @test_int [x^1*(d + e*x)^3*(d^2 - e^2*x^2)^p, x, 3, If(13>=8, -(((d + e*x)^3*(d^2 - e^2*x^2)^(1 + p))/(e^2*(5 + 2*p))) - (3*2^(3 + p)*d^3*(1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(-3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)))/(e^2*(1 + p)*(5 + 2*p)), -(((d + e*x)^3*(d^2 - e^2*x^2)^(1 + p))/(e^2*(5 + 2*p))) - (3*2^(3 + p)*d^3*(1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(-3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)))/(e^2*(5 + 7*p + 2*p^2)))]
    @test_int [x^0*(d + e*x)^3*(d^2 - e^2*x^2)^p, x, 2, -((2^(3 + p)*d^2*(1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(-3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)))/(e*(1 + p)))]
    @test_int [(d + e*x)^3*(d^2 - e^2*x^2)^p/x^1, x, 7, -((3*d*(d^2 - e^2*x^2)^(1 + p))/(2*(1 + p))) - (e*x*(d^2 - e^2*x^2)^(1 + p))/(3 + 2*p) + (2*d^2*e*(5 + 3*p)*x*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(3 + 2*p)) - (d*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2))/(2*(1 + p))]
    @test_int [(d + e*x)^3*(d^2 - e^2*x^2)^p/x^2, x, 8, -((e*(d^2 - e^2*x^2)^(1 + p))/(2*(1 + p))) - (d*(d^2 - e^2*x^2)^(1 + p))/x + (2*d*e^2*(1 - p)*x*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, (e^2*x^2)/d^2))/(1 - (e^2*x^2)/d^2)^p - (3*e*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2))/(2*(1 + p))]
    @test_int [(d + e*x)^3*(d^2 - e^2*x^2)^p/x^3, x, 7, -((d*(d^2 - e^2*x^2)^(1 + p))/(2*x^2)) - (3*e*(d^2 - e^2*x^2)^(1 + p))/x - (2*e^3*(1 + 3*p)*x*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, (e^2*x^2)/d^2))/(1 - (e^2*x^2)/d^2)^p - (e^2*(3 - p)*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, 1 - (e^2*x^2)/d^2))/(2*d*(1 + p))]

    #= ::Subsubsection::Closed:: =#
    #=n<0=#

    @test_int [x^4*(d^2 - e^2*x^2)^p/(d + e*x), x, 7, (d^4*(d^2 - e^2*x^2)^p)/(2*e^5*p) - (d^2*(d^2 - e^2*x^2)^(1 + p))/(e^5*(1 + p)) + (d^2 - e^2*x^2)^(2 + p)/(2*e^5*(2 + p)) + (x^5*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(5/2, 1 - p, 7/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(5*d))]
    @test_int [x^3*(d^2 - e^2*x^2)^p/(d + e*x), x, 7, -((d^3*(d^2 - e^2*x^2)^p)/(2*e^4*p)) + (d*(d^2 - e^2*x^2)^(1 + p))/(2*e^4*(1 + p)) - (e*x^5*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(5/2, 1 - p, 7/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(5*d^2))]
    @test_int [x^2*(d^2 - e^2*x^2)^p/(d + e*x), x, 7, (d^2*(d^2 - e^2*x^2)^p)/(2*e^3*p) - (d^2 - e^2*x^2)^(1 + p)/(2*e^3*(1 + p)) + (x^3*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(3/2, 1 - p, 5/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(3*d))]
    @test_int [x^1*(d^2 - e^2*x^2)^p/(d + e*x), x, 5, -((d*(d^2 - e^2*x^2)^p)/(2*e^2*p)) - (e*x^3*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(3/2, 1 - p, 5/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(3*d^2))]
    @test_int [x^0*(d^2 - e^2*x^2)^p/(d + e*x), x, 2, -((2^(-1 + p)*(1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)))/(d^2*e*(1 + p)))]
    @test_int [(d^2 - e^2*x^2)^p/(x^1*(d + e*x)), x, 6, -((e*x*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(1/2, 1 - p, 3/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*d^2)) - ((d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(1, p, 1 + p, 1 - (e^2*x^2)/d^2))/(2*d*p)]
    @test_int [(d^2 - e^2*x^2)^p/(x^2*(d + e*x)), x, 6, -(((d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(-(1/2), 1 - p, 1/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(d*x))) + (e*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(1, p, 1 + p, 1 - (e^2*x^2)/d^2))/(2*d^2*p)]
    @test_int [(d^2 - e^2*x^2)^p/(x^3*(d + e*x)), x, 6, (e*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(-(1/2), 1 - p, 1/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(d^2*x)) - (e^2*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(2, p, 1 + p, 1 - (e^2*x^2)/d^2))/(2*d^3*p)]

    @test_int [x^5*(d^2 - e^2*x^2)^p/(d + e*x)^2, x, 8, (d^6*(d^2 - e^2*x^2)^(-1 + p))/(e^6*(1 - p)) + (5*d^4*(d^2 - e^2*x^2)^p)/(2*e^6*p) - (2*d^2*(d^2 - e^2*x^2)^(1 + p))/(e^6*(1 + p)) + (d^2 - e^2*x^2)^(2 + p)/(2*e^6*(2 + p)) - (2*e*x^7*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(7/2, 2 - p, 9/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(7*d^3))]
    @test_int [x^4*(d^2 - e^2*x^2)^p/(d + e*x)^2, x, 9, -((d^5*(d^2 - e^2*x^2)^(-1 + p))/(e^5*(1 - p))) - (x^5*(d^2 - e^2*x^2)^(-1 + p))/(3 + 2*p) - (2*d^3*(d^2 - e^2*x^2)^p)/(e^5*p) + (d*(d^2 - e^2*x^2)^(1 + p))/(e^5*(1 + p)) + (2*(4 + p)*x^5*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(5/2, 2 - p, 7/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(5*d^2*(3 + 2*p)))]
    @test_int [x^3*(d^2 - e^2*x^2)^p/(d + e*x)^2, x, 8, (d^4*(d^2 - e^2*x^2)^(-1 + p))/(e^4*(1 - p)) + (3*d^2*(d^2 - e^2*x^2)^p)/(2*e^4*p) - (d^2 - e^2*x^2)^(1 + p)/(2*e^4*(1 + p)) - (2*e*x^5*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(5/2, 2 - p, 7/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(5*d^3))]
    @test_int [x^2*(d^2 - e^2*x^2)^p/(d + e*x)^2, x, 9, -((d^3*(d^2 - e^2*x^2)^(-1 + p))/(e^3*(1 - p))) - (x^3*(d^2 - e^2*x^2)^(-1 + p))/(1 + 2*p) - (d*(d^2 - e^2*x^2)^p)/(e^3*p) + (2*(2 + p)*x^3*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(3/2, 2 - p, 5/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(3*d^2*(1 + 2*p)))]
    @test_int [x^1*(d^2 - e^2*x^2)^p/(d + e*x)^2, x, 3, (d^2 - e^2*x^2)^(1 + p)/(2*e^2*(1 - p)*(d + e*x)^2) - (2^(-1 + p)*(1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)))/(d^2*e^2*(1 - p^2))]
    @test_int [x^0*(d^2 - e^2*x^2)^p/(d + e*x)^2, x, 2, -((2^(-2 + p)*(1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)))/(d^3*e*(1 + p)))]
    @test_int [(d^2 - e^2*x^2)^p/(x^1*(d + e*x)^2), x, 8, (d^2 - e^2*x^2)^(-1 + p)/(1 - p) - (2*e*x*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(1/2, 2 - p, 3/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*d^3) - ((d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(1, p, 1 + p, 1 - (e^2*x^2)/d^2))/(2*d^2*p)]
    @test_int [(d^2 - e^2*x^2)^p/(x^2*(d + e*x)^2), x, 7, -((d^2 - e^2*x^2)^(-1 + p)/x) + (2*e^2*(2 - p)*x*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(1/2, 2 - p, 3/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*d^4) - (e*(d^2 - e^2*x^2)^(-1 + p)*HypergeometricFunctions._₂F₁(1, -1 + p, p, 1 - (e^2*x^2)/d^2))/(d*(1 - p))]
    @test_int [(d^2 - e^2*x^2)^p/(x^3*(d + e*x)^2), x, 7, -((d^2 - e^2*x^2)^(-1 + p)/(2*x^2)) + (2*e*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(-(1/2), 2 - p, 1/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(d^3*x)) + (e^2*(3 - p)*(d^2 - e^2*x^2)^(-1 + p)*HypergeometricFunctions._₂F₁(1, -1 + p, p, 1 - (e^2*x^2)/d^2))/(2*d^2*(1 - p))]
    @test_int [(d^2 - e^2*x^2)^p/(x^4*(d + e*x)^2), x, 7, -((d^2 - e^2*x^2)^(-1 + p)/(3*x^3)) - (2*e^2*(4 - p)*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(-(1/2), 2 - p, 1/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(3*d^4*x)) - (e^3*(d^2 - e^2*x^2)^(-1 + p)*HypergeometricFunctions._₂F₁(2, -1 + p, p, 1 - (e^2*x^2)/d^2))/(d^3*(1 - p))]
    @test_int [(d^2 - e^2*x^2)^p/(x^5*(d + e*x)^2), x, 7, -((d^2 - e^2*x^2)^(-1 + p)/(4*x^4)) + (2*e*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(-(3/2), 2 - p, -(1/2), (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(3*d^3*x^3)) + (e^4*(5 - p)*(d^2 - e^2*x^2)^(-1 + p)*HypergeometricFunctions._₂F₁(2, -1 + p, p, 1 - (e^2*x^2)/d^2))/(4*d^4*(1 - p))]

    @test_int [x^4*(d^2 - e^2*x^2)^p/(d + e*x)^3, x, 8, -((2*d^6*(d^2 - e^2*x^2)^(-2 + p))/(e^5*(2 - p))) - (3*d*x^5*(d^2 - e^2*x^2)^(-2 + p))/(1 + 2*p) + (9*d^4*(d^2 - e^2*x^2)^(-1 + p))/(2*e^5*(1 - p)) + (3*d^2*(d^2 - e^2*x^2)^p)/(e^5*p) - (d^2 - e^2*x^2)^(1 + p)/(2*e^5*(1 + p)) + (2*(8 + p)*x^5*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(5/2, 3 - p, 7/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(5*d^3*(1 + 2*p)))]
    @test_int [x^3*(d^2 - e^2*x^2)^p/(d + e*x)^3, x, 8, (2*d^5*(d^2 - e^2*x^2)^(-2 + p))/(e^4*(2 - p)) + (e*x^5*(d^2 - e^2*x^2)^(-2 + p))/(1 + 2*p) - (7*d^3*(d^2 - e^2*x^2)^(-1 + p))/(2*e^4*(1 - p)) - (3*d*(d^2 - e^2*x^2)^p)/(2*e^4*p) - (2*e*(4 + 3*p)*x^5*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(5/2, 3 - p, 7/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(5*d^4*(1 + 2*p)))]
    @test_int [x^2*(d^2 - e^2*x^2)^p/(d + e*x)^3, x, 4, If(13>=8, -((d*(d^2 - e^2*x^2)^(1 + p))/(2*e^3*(2 - p)*(d + e*x)^3)) - (d^2 - e^2*x^2)^(1 + p)/(2*e^3*p*(d + e*x)^2) + (2^(-3 + p)*(4 + p)*(1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)))/(d^2*e^3*(2 - p)*p*(1 + p)), -((d*(d^2 - e^2*x^2)^(1 + p))/(2*e^3*(2 - p)*(d + e*x)^3)) - (d^2 - e^2*x^2)^(1 + p)/(2*e^3*p*(d + e*x)^2) + (2^(-3 + p)*(4 + p)*(1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)))/(d^2*e^3*p*(2 + p - p^2)))]
    @test_int [x^1*(d^2 - e^2*x^2)^p/(d + e*x)^3, x, 3, If(13>=8, (d^2 - e^2*x^2)^(1 + p)/(2*e^2*(2 - p)*(d + e*x)^3) - (3*2^(-3 + p)*(1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)))/(d^3*e^2*(2 - p)*(1 + p)), (d^2 - e^2*x^2)^(1 + p)/(2*e^2*(2 - p)*(d + e*x)^3) - (3*2^(-3 + p)*(1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)))/(d^3*e^2*(2 + p - p^2)))]
    @test_int [x^0*(d^2 - e^2*x^2)^p/(d + e*x)^3, x, 2, -((2^(-3 + p)*(1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)))/(d^4*e*(1 + p)))]
    @test_int [(d^2 - e^2*x^2)^p/(x^1*(d + e*x)^3), x, 8, (2*d*(d^2 - e^2*x^2)^(-2 + p))/(2 - p) - (e*x*(d^2 - e^2*x^2)^(-2 + p))/(3 - 2*p) - (2*e*(4 - 3*p)*x*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(1/2, 3 - p, 3/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(d^4*(3 - 2*p))) + ((d^2 - e^2*x^2)^(-1 + p)*HypergeometricFunctions._₂F₁(1, -1 + p, p, 1 - (e^2*x^2)/d^2))/(2*d*(1 - p))]
    @test_int [(d^2 - e^2*x^2)^p/(x^2*(d + e*x)^3), x, 9, -((2*e*(d^2 - e^2*x^2)^(-2 + p))/(2 - p)) - (d*(d^2 - e^2*x^2)^(-2 + p))/x + (2*e^2*(4 - p)*x*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(1/2, 3 - p, 3/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*d^5) - (3*e*(d^2 - e^2*x^2)^(-1 + p)*HypergeometricFunctions._₂F₁(1, -1 + p, p, 1 - (e^2*x^2)/d^2))/(2*d^2*(1 - p))]
    @test_int [(d^2 - e^2*x^2)^p/(x^3*(d + e*x)^3), x, 8, -((d*(d^2 - e^2*x^2)^(-2 + p))/(2*x^2)) + (3*e*(d^2 - e^2*x^2)^(-2 + p))/x - (2*e^3*(8 - 3*p)*x*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(1/2, 3 - p, 3/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*d^6) + (e^2*(6 - p)*(d^2 - e^2*x^2)^(-2 + p)*HypergeometricFunctions._₂F₁(1, -2 + p, -1 + p, 1 - (e^2*x^2)/d^2))/(2*d*(2 - p))]
    @test_int [(d^2 - e^2*x^2)^p/(x^4*(d + e*x)^3), x, 8, -((d*(d^2 - e^2*x^2)^(-2 + p))/(3*x^3)) + (3*e*(d^2 - e^2*x^2)^(-2 + p))/(2*x^2) - (2*e^2*(8 - p)*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(-(1/2), 3 - p, 1/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(3*d^5*x)) - (e^3*(10 - 3*p)*(d^2 - e^2*x^2)^(-2 + p)*HypergeometricFunctions._₂F₁(1, -2 + p, -1 + p, 1 - (e^2*x^2)/d^2))/(2*d^2*(2 - p))]
    @test_int [(d^2 - e^2*x^2)^p/(x^5*(d + e*x)^3), x, 8, -((d*(d^2 - e^2*x^2)^(-2 + p))/(4*x^4)) + (e*(d^2 - e^2*x^2)^(-2 + p))/x^3 + (2*e^3*(4 - p)*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(-(1/2), 3 - p, 1/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(d^6*x)) + (e^4*(10 - p)*(d^2 - e^2*x^2)^(-2 + p)*HypergeometricFunctions._₂F₁(2, -2 + p, -1 + p, 1 - (e^2*x^2)/d^2))/(4*d^3*(2 - p))]

    @test_int [x^4*(d^2 - e^2*x^2)^p/(d + e*x)^4, x, 9, -((4*d^7*(d^2 - e^2*x^2)^(-3 + p))/(e^5*(3 - p))) + (d^2*(13 + 12*p)*x^5*(d^2 - e^2*x^2)^(-3 + p))/(1 - 4*p^2) - (e^2*x^7*(d^2 - e^2*x^2)^(-3 + p))/(1 + 2*p) + (10*d^5*(d^2 - e^2*x^2)^(-2 + p))/(e^5*(2 - p)) - (8*d^3*(d^2 - e^2*x^2)^(-1 + p))/(e^5*(1 - p)) - (2*d*(d^2 - e^2*x^2)^p)/(e^5*p) - (4*(16 + 15*p + p^2)*x^5*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(5/2, 4 - p, 7/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(5*d^4*(1 - 4*p^2)))]
    @test_int [x^3*(d^2 - e^2*x^2)^p/(d + e*x)^4, x, 5, If(13>=8, (d^2*(d^2 - e^2*x^2)^(1 + p))/(2*e^4*(3 - p)*(d + e*x)^4) - (d*(1 + 2*p)*(d^2 - e^2*x^2)^(1 + p))/(e^4*(1 - 2*p)*p*(d + e*x)^3) - (d^2 - e^2*x^2)^(1 + p)/(2*e^4*p*(d + e*x)^2) + (3*2^(-2 + p)*(2 + p)*(1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)))/(d^2*e^4*(1 - 2*p)*(3 - p)*p*(1 + p)), (d^2*(d^2 - e^2*x^2)^(1 + p))/(2*e^4*(3 - p)*(d + e*x)^4) - (d*(1 + 2*p)*(d^2 - e^2*x^2)^(1 + p))/(e^4*(1 - 2*p)*p*(d + e*x)^3) - (d^2 - e^2*x^2)^(1 + p)/(2*e^4*p*(d + e*x)^2) + (3*2^(-2 + p)*(2 + p)*(1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)))/(d^2*e^4*p*(3 - 4*p - 5*p^2 + 2*p^3)))]
    @test_int [x^2*(d^2 - e^2*x^2)^p/(d + e*x)^4, x, 4, If(13>=8, -((d*(d^2 - e^2*x^2)^(1 + p))/(2*e^3*(3 - p)*(d + e*x)^4)) + (d^2 - e^2*x^2)^(1 + p)/(e^3*(1 - 2*p)*(d + e*x)^3) - (2^(-3 + p)*(7 + p)*(1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)))/(d^3*e^3*(1 - 2*p)*(3 - p)*(1 + p)), -((d*(d^2 - e^2*x^2)^(1 + p))/(2*e^3*(3 - p)*(d + e*x)^4)) + (d^2 - e^2*x^2)^(1 + p)/(e^3*(1 - 2*p)*(d + e*x)^3) - (2^(-3 + p)*(7 + p)*(1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)))/(d^3*e^3*(3 - 4*p - 5*p^2 + 2*p^3)))]
    @test_int [x^1*(d^2 - e^2*x^2)^p/(d + e*x)^4, x, 3, If(13>=8, (d^2 - e^2*x^2)^(1 + p)/(2*e^2*(3 - p)*(d + e*x)^4) - (2^(-2 + p)*(1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)))/(d^4*e^2*(3 - p)*(1 + p)), (d^2 - e^2*x^2)^(1 + p)/(2*e^2*(3 - p)*(d + e*x)^4) - (2^(-2 + p)*(1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(3 - p, 1 + p, 2 + p, (d - e*x)/(2*d)))/(d^4*e^2*(3 + 2*p - p^2)))]
    @test_int [x^0*(d^2 - e^2*x^2)^p/(d + e*x)^4, x, 2, -((2^(-4 + p)*(1 + (e*x)/d)^(-1 - p)*(d^2 - e^2*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(4 - p, 1 + p, 2 + p, (d - e*x)/(2*d)))/(d^5*e*(1 + p)))]
    @test_int [(d^2 - e^2*x^2)^p/(x^1*(d + e*x)^4), x, 9, (4*d^2*(d^2 - e^2*x^2)^(-3 + p))/(3 - p) - (4*d*e*x*(d^2 - e^2*x^2)^(-3 + p))/(5 - 2*p) - (d^2 - e^2*x^2)^(-2 + p)/(2*(2 - p)) - (8*e*(2 - p)*x*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(1/2, 4 - p, 3/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(d^5*(5 - 2*p))) + ((d^2 - e^2*x^2)^(-2 + p)*HypergeometricFunctions._₂F₁(1, -2 + p, -1 + p, 1 - (e^2*x^2)/d^2))/(2*(2 - p))]
    @test_int [(d^2 - e^2*x^2)^p/(x^2*(d + e*x)^4), x, 9, -((4*d*e*(d^2 - e^2*x^2)^(-3 + p))/(3 - p)) - (d^2*(d^2 - e^2*x^2)^(-3 + p))/x + (e^2*x*(d^2 - e^2*x^2)^(-3 + p))/(5 - 2*p) + (4*e^2*(16 - 9*p + p^2)*x*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(1/2, 4 - p, 3/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(d^6*(5 - 2*p))) - (2*e*(d^2 - e^2*x^2)^(-2 + p)*HypergeometricFunctions._₂F₁(1, -2 + p, -1 + p, 1 - (e^2*x^2)/d^2))/(d*(2 - p))]
    @test_int [(d^2 - e^2*x^2)^p/(x^3*(d + e*x)^4), x, 10, (e^2*(11 - p)*(d^2 - e^2*x^2)^(-3 + p))/(2*(3 - p)) - (d^2*(d^2 - e^2*x^2)^(-3 + p))/(2*x^2) + (4*d*e*(d^2 - e^2*x^2)^(-3 + p))/x - (8*e^3*(4 - p)*x*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(1/2, 4 - p, 3/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*d^7) + (e^2*(10 - p)*(d^2 - e^2*x^2)^(-2 + p)*HypergeometricFunctions._₂F₁(1, -2 + p, -1 + p, 1 - (e^2*x^2)/d^2))/(2*d^2*(2 - p))]
    @test_int [(d^2 - e^2*x^2)^p/(x^4*(d + e*x)^4), x, 9, -((d^2*(d^2 - e^2*x^2)^(-3 + p))/(3*x^3)) + (2*d*e*(d^2 - e^2*x^2)^(-3 + p))/x^2 - (e^2*(27 - 2*p)*(d^2 - e^2*x^2)^(-3 + p))/(3*x) + (4*e^4*(48 - 17*p + p^2)*x*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(1/2, 4 - p, 3/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(3*d^8)) - (2*e^3*(5 - p)*(d^2 - e^2*x^2)^(-3 + p)*HypergeometricFunctions._₂F₁(1, -3 + p, -2 + p, 1 - (e^2*x^2)/d^2))/(d*(3 - p))]
    @test_int [(d^2 - e^2*x^2)^p/(x^5*(d + e*x)^4), x, 9, -((d^2*(d^2 - e^2*x^2)^(-3 + p))/(4*x^4)) + (4*d*e*(d^2 - e^2*x^2)^(-3 + p))/(3*x^3) - (e^2*(17 - p)*(d^2 - e^2*x^2)^(-3 + p))/(4*x^2) + (8*e^3*(6 - p)*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁(-(1/2), 4 - p, 1/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(3*d^7*x)) + (e^4*(70 - 21*p + p^2)*(d^2 - e^2*x^2)^(-3 + p)*HypergeometricFunctions._₂F₁(1, -3 + p, -2 + p, 1 - (e^2*x^2)/d^2))/(4*d^2*(3 - p))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(g*x)^m*(d+e*x)^n*(d^2 - e^2*x^2)^p*when*m*and*p*symbolic=#

    @test_int [(g*x)^m*(d^2 - e^2*x^2)^p*(d + e*x)^3, x, 7, -((3*d*(g*x)^(1 + m)*(d^2 - e^2*x^2)^(1 + p))/(g*(3 + m + 2*p))) - (e*(g*x)^(2 + m)*(d^2 - e^2*x^2)^(1 + p))/(g^2*(4 + m + 2*p)) + (2*d^3*(3 + 2*m + p)*(g*x)^(1 + m)*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁((1 + m)/2, -p, (3 + m)/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(g*(1 + m)*(3 + m + 2*p))) + (2*d^2*e*(7 + 2*m + 3*p)*(g*x)^(2 + m)*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁((2 + m)/2, -p, (4 + m)/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(g^2*(2 + m)*(4 + m + 2*p)))]
    @test_int [(g*x)^m*(d^2 - e^2*x^2)^p*(d + e*x)^2, x, 6, -(((g*x)^(1 + m)*(d^2 - e^2*x^2)^(1 + p))/(g*(3 + m + 2*p))) + (2*d^2*(2 + m + p)*(g*x)^(1 + m)*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁((1 + m)/2, -p, (3 + m)/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(g*(1 + m)*(3 + m + 2*p))) + (2*d*e*(g*x)^(2 + m)*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁((2 + m)/2, -p, (4 + m)/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(g^2*(2 + m)))]
    @test_int [(g*x)^m*(d^2 - e^2*x^2)^p*(d + e*x)^1, x, 5, (d*(g*x)^(1 + m)*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁((1 + m)/2, -p, (3 + m)/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(g*(1 + m))) + (e*(g*x)^(2 + m)*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁((2 + m)/2, -p, (4 + m)/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(g^2*(2 + m)))]
    @test_int [(g*x)^m*(d^2 - e^2*x^2)^p*(d + e*x)^0, x, 2, ((g*x)^(1 + m)*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁((1 + m)/2, -p, (3 + m)/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(g*(1 + m)))]
    @test_int [(g*x)^m*(d^2 - e^2*x^2)^p/(d + e*x)^1, x, 8, ((g*x)^(1 + m)*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁((1 + m)/2, 1 - p, (3 + m)/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(d*g*(1 + m))) - (e*(g*x)^(2 + m)*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁((2 + m)/2, 1 - p, (4 + m)/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(d^2*g^2*(2 + m)))]
    @test_int [(g*x)^m*(d^2 - e^2*x^2)^p/(d + e*x)^2, x, 7, ((g*x)^(1 + m)*(d^2 - e^2*x^2)^(-1 + p))/(g*(1 - m - 2*p)) - (2*(m + p)*(g*x)^(1 + m)*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁((1 + m)/2, 2 - p, (3 + m)/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(d^2*g*(1 + m)*(1 - m - 2*p))) - (2*e*(g*x)^(2 + m)*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁((2 + m)/2, 2 - p, (4 + m)/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(d^3*g^2*(2 + m)))]
    @test_int [(g*x)^m*(d^2 - e^2*x^2)^p/(d + e*x)^3, x, 8, (3*d*(g*x)^(1 + m)*(d^2 - e^2*x^2)^(-2 + p))/(g*(3 - m - 2*p)) - (e*(g*x)^(2 + m)*(d^2 - e^2*x^2)^(-2 + p))/(g^2*(2 - m - 2*p)) - (2*(2*m + p)*(g*x)^(1 + m)*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁((1 + m)/2, 3 - p, (3 + m)/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(d^3*g*(1 + m)*(3 - m - 2*p))) - (2*e*(2 - 2*m - 3*p)*(g*x)^(2 + m)*(d^2 - e^2*x^2)^p*HypergeometricFunctions._₂F₁((2 + m)/2, 3 - p, (4 + m)/2, (e^2*x^2)/d^2))/((1 - (e^2*x^2)/d^2)^p*(d^4*g^2*(2 + m)*(2 - m - 2*p)))]

    @test_int [(g*x)^m*(1 - a^2*x^2)^p/(1 + a*x), x, 6, ((g*x)^(1 + m)*HypergeometricFunctions._₂F₁((1 + m)/2, 1 - p, (3 + m)/2, a^2*x^2))/(g*(1 + m)) - (a*(g*x)^(2 + m)*HypergeometricFunctions._₂F₁((2 + m)/2, 1 - p, (4 + m)/2, a^2*x^2))/(g^2*(2 + m))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(g*x)^m*(d+e*x)^n*(d^2 - e^2*x^2)^p*when*n*and*p*symbolic=#

    @test_int [(g*x)^m*(d + e*x)^n*(d^2 - e^2*x^2)^p, x, 4, ((g*x)^(1 + m)*(d + e*x)^n*(1 + (e*x)/d)^(-n - p)*(d^2 - e^2*x^2)^p*AppellF1(1 + m, -p, -n - p, 2 + m, (e*x)/d, -((e*x)/d)))/((1 - (e*x)/d)^p*(g*(1 + m)))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(g*x)^m*(d+e*x)^n*(a+b*x+c*x^2)^p*when*b=0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(d+e*x)^(n/2)*(a+c*x^2)^p=#

    @test_int [x*sqrt(1 + x)/(1 + x^2), x, 11, 2*sqrt(1 + x) + atan((sqrt(2*(1 + sqrt(2))) - 2*sqrt(1 + x))/sqrt(2*(-1 + sqrt(2))))/sqrt(2*(1 + sqrt(2))) - atan((sqrt(2*(1 + sqrt(2))) + 2*sqrt(1 + x))/sqrt(2*(-1 + sqrt(2))))/sqrt(2*(1 + sqrt(2))) + (1/2)*sqrt((1/2)*(1 + sqrt(2)))*log(1 + sqrt(2) + x - sqrt(2*(1 + sqrt(2)))*sqrt(1 + x)) - (1/2)*sqrt((1/2)*(1 + sqrt(2)))*log(1 + sqrt(2) + x + sqrt(2*(1 + sqrt(2)))*sqrt(1 + x))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m / (d+e*x)^1*(a+c*x^2)^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^4*sqrt(a + c*x^2)/(d + e*x), x, 9, (d*(8*c*d^3 - e*(4*c*d^2 - a*e^2)*x)*sqrt(a + c*x^2))/(8*c*e^5) + ((47*c*d^2 - 8*a*e^2)*(a + c*x^2)^(3/2))/(60*c^2*e^3) - (13*d*(d + e*x)*(a + c*x^2)^(3/2))/(20*c*e^3) + ((d + e*x)^2*(a + c*x^2)^(3/2))/(5*c*e^3) - (d*(8*c^2*d^4 + 4*a*c*d^2*e^2 - a^2*e^4)*atanh((sqrt(c)*x)/sqrt(a + c*x^2)))/(8*c^(3/2)*e^6) - (d^4*sqrt(c*d^2 + a*e^2)*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/e^6]
    @test_int [x^3*sqrt(a + c*x^2)/(d + e*x), x, 8, -(((8*c*d^3 - e*(4*c*d^2 - a*e^2)*x)*sqrt(a + c*x^2))/(8*c*e^4)) - (7*d*(a + c*x^2)^(3/2))/(12*c*e^2) + ((d + e*x)*(a + c*x^2)^(3/2))/(4*c*e^2) + ((8*c^2*d^4 + 4*a*c*d^2*e^2 - a^2*e^4)*atanh((sqrt(c)*x)/sqrt(a + c*x^2)))/(8*c^(3/2)*e^5) + (d^3*sqrt(c*d^2 + a*e^2)*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/e^5]
    @test_int [x^2*sqrt(a + c*x^2)/(d + e*x), x, 8, (d*(2*d - e*x)*sqrt(a + c*x^2))/(2*e^3) + (a + c*x^2)^(3/2)/(3*c*e) - (d*(2*c*d^2 + a*e^2)*atanh((sqrt(c)*x)/sqrt(a + c*x^2)))/(2*sqrt(c)*e^4) - (d^2*sqrt(c*d^2 + a*e^2)*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/e^4]
    @test_int [x^1*sqrt(a + c*x^2)/(d + e*x), x, 6, -(((2*d - e*x)*sqrt(a + c*x^2))/(2*e^2)) + ((2*c*d^2 + a*e^2)*atanh((sqrt(c)*x)/sqrt(a + c*x^2)))/(2*sqrt(c)*e^3) + (d*sqrt(c*d^2 + a*e^2)*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/e^3]
    @test_int [x^0*sqrt(a + c*x^2)/(d + e*x), x, 6, sqrt(a + c*x^2)/e - (sqrt(c)*d*atanh((sqrt(c)*x)/sqrt(a + c*x^2)))/e^2 - (sqrt(c*d^2 + a*e^2)*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/e^2]
    @test_int [sqrt(a + c*x^2)/(x^1*(d + e*x)), x, 9, (sqrt(c)*atanh((sqrt(c)*x)/sqrt(a + c*x^2)))/e + (sqrt(c*d^2 + a*e^2)*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(d*e) - (sqrt(a)*atanh(sqrt(a + c*x^2)/sqrt(a)))/d]
    @test_int [sqrt(a + c*x^2)/(x^2*(d + e*x)), x, 15, -(sqrt(a + c*x^2)/(d*x)) - (sqrt(c*d^2 + a*e^2)*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/d^2 + (sqrt(a)*e*atanh(sqrt(a + c*x^2)/sqrt(a)))/d^2]
    @test_int [sqrt(a + c*x^2)/(x^3*(d + e*x)), x, 19, -(sqrt(a + c*x^2)/(2*d*x^2)) + (e*sqrt(a + c*x^2))/(d^2*x) + (e*sqrt(c*d^2 + a*e^2)*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/d^3 - (c*atanh(sqrt(a + c*x^2)/sqrt(a)))/(2*sqrt(a)*d) - (sqrt(a)*e^2*atanh(sqrt(a + c*x^2)/sqrt(a)))/d^3]
    @test_int [sqrt(a + c*x^2)/(x^4*(d + e*x)), x, 20, (e*sqrt(a + c*x^2))/(2*d^2*x^2) - (e^2*sqrt(a + c*x^2))/(d^3*x) - (a + c*x^2)^(3/2)/(3*a*d*x^3) - (e^2*sqrt(c*d^2 + a*e^2)*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/d^4 + (c*e*atanh(sqrt(a + c*x^2)/sqrt(a)))/(2*sqrt(a)*d^2) + (sqrt(a)*e^3*atanh(sqrt(a + c*x^2)/sqrt(a)))/d^4]
    @test_int [sqrt(a + c*x^2)/(x^5*(d + e*x)), x, 25, -(sqrt(a + c*x^2)/(4*d*x^4)) - (c*sqrt(a + c*x^2))/(8*a*d*x^2) - (e^2*sqrt(a + c*x^2))/(2*d^3*x^2) + (e^3*sqrt(a + c*x^2))/(d^4*x) + (e*(a + c*x^2)^(3/2))/(3*a*d^2*x^3) + (e^3*sqrt(c*d^2 + a*e^2)*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/d^5 + (c^2*atanh(sqrt(a + c*x^2)/sqrt(a)))/(8*a^(3/2)*d) - (c*e^2*atanh(sqrt(a + c*x^2)/sqrt(a)))/(2*sqrt(a)*d^3) - (sqrt(a)*e^4*atanh(sqrt(a + c*x^2)/sqrt(a)))/d^5]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^4/((d + e*x)*sqrt(a + c*x^2)), x, 8, ((11*c*d^2 - 4*a*e^2)*sqrt(a + c*x^2))/(6*c^2*e^3) - (7*d*(d + e*x)*sqrt(a + c*x^2))/(6*c*e^3) + ((d + e*x)^2*sqrt(a + c*x^2))/(3*c*e^3) - (d*(2*c*d^2 - a*e^2)*atanh((sqrt(c)*x)/sqrt(a + c*x^2)))/(2*c^(3/2)*e^4) - (d^4*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(e^4*sqrt(c*d^2 + a*e^2))]
    @test_int [x^3/((d + e*x)*sqrt(a + c*x^2)), x, 7, -((3*d*sqrt(a + c*x^2))/(2*c*e^2)) + ((d + e*x)*sqrt(a + c*x^2))/(2*c*e^2) + ((2*c*d^2 - a*e^2)*atanh((sqrt(c)*x)/sqrt(a + c*x^2)))/(2*c^(3/2)*e^3) + (d^3*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(e^3*sqrt(c*d^2 + a*e^2))]
    @test_int [x^2/((d + e*x)*sqrt(a + c*x^2)), x, 7, sqrt(a + c*x^2)/(c*e) - (d*atanh((sqrt(c)*x)/sqrt(a + c*x^2)))/(sqrt(c)*e^2) - (d^2*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(e^2*sqrt(c*d^2 + a*e^2))]
    @test_int [x^1/((d + e*x)*sqrt(a + c*x^2)), x, 5, atanh((sqrt(c)*x)/sqrt(a + c*x^2))/(sqrt(c)*e) + (d*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(e*sqrt(c*d^2 + a*e^2))]
    @test_int [x^0/((d + e*x)*sqrt(a + c*x^2)), x, 2, -(atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2)))/sqrt(c*d^2 + a*e^2))]
    @test_int [1/(x^1*(d + e*x)*sqrt(a + c*x^2)), x, 7, (e*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(d*sqrt(c*d^2 + a*e^2)) - atanh(sqrt(a + c*x^2)/sqrt(a))/(sqrt(a)*d)]
    @test_int [1/(x^2*(d + e*x)*sqrt(a + c*x^2)), x, 8, -(sqrt(a + c*x^2)/(a*d*x)) - (e^2*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(d^2*sqrt(c*d^2 + a*e^2)) + (e*atanh(sqrt(a + c*x^2)/sqrt(a)))/(sqrt(a)*d^2)]
    @test_int [1/(x^3*(d + e*x)*sqrt(a + c*x^2)), x, 12, -(sqrt(a + c*x^2)/(2*a*d*x^2)) + (e*sqrt(a + c*x^2))/(a*d^2*x) + (e^3*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(d^3*sqrt(c*d^2 + a*e^2)) + (c*atanh(sqrt(a + c*x^2)/sqrt(a)))/(2*a^(3/2)*d) - (e^2*atanh(sqrt(a + c*x^2)/sqrt(a)))/(sqrt(a)*d^3)]

    @test_int [x^4/((d + e*x)*(a + c*x^2)^(3/2)), x, 7, (a*(a*e + c*d*x))/(c^2*(c*d^2 + a*e^2)*sqrt(a + c*x^2)) + sqrt(a + c*x^2)/(c^2*e) - (d*atanh((sqrt(c)*x)/sqrt(a + c*x^2)))/(c^(3/2)*e^2) - (d^4*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(e^2*(c*d^2 + a*e^2)^(3/2))]
    @test_int [x^3/((d + e*x)*(a + c*x^2)^(3/2)), x, 6, (a*(d - e*x))/(c*(c*d^2 + a*e^2)*sqrt(a + c*x^2)) + atanh((sqrt(c)*x)/sqrt(a + c*x^2))/(c^(3/2)*e) + (d^3*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(e*(c*d^2 + a*e^2)^(3/2))]
    @test_int [x^2/((d + e*x)*(a + c*x^2)^(3/2)), x, 4, -((a*e + c*d*x)/(c*(c*d^2 + a*e^2)*sqrt(a + c*x^2))) - (d^2*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(c*d^2 + a*e^2)^(3/2)]
    @test_int [x^1/((d + e*x)*(a + c*x^2)^(3/2)), x, 4, -((d - e*x)/((c*d^2 + a*e^2)*sqrt(a + c*x^2))) + (d*e*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(c*d^2 + a*e^2)^(3/2)]
    @test_int [x^0/((d + e*x)*(a + c*x^2)^(3/2)), x, 4, (a*e + c*d*x)/(a*(c*d^2 + a*e^2)*sqrt(a + c*x^2)) - (e^2*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(c*d^2 + a*e^2)^(3/2)]
    @test_int [1/(x^1*(d + e*x)*(a + c*x^2)^(3/2)), x, 10, 1/(a*d*sqrt(a + c*x^2)) - (e*(a*e + c*d*x))/(a*d*(c*d^2 + a*e^2)*sqrt(a + c*x^2)) + (e^3*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(d*(c*d^2 + a*e^2)^(3/2)) - atanh(sqrt(a + c*x^2)/sqrt(a))/(a^(3/2)*d)]
    @test_int [1/(x^2*(d + e*x)*(a + c*x^2)^(3/2)), x, 12, -(e/(a*d^2*sqrt(a + c*x^2))) - 1/(a*d*x*sqrt(a + c*x^2)) - (2*c*x)/(a^2*d*sqrt(a + c*x^2)) + (e^2*(a*e + c*d*x))/(a*d^2*(c*d^2 + a*e^2)*sqrt(a + c*x^2)) - (e^4*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(d^2*(c*d^2 + a*e^2)^(3/2)) + (e*atanh(sqrt(a + c*x^2)/sqrt(a)))/(a^(3/2)*d^2)]
    @test_int [1/(x^3*(d + e*x)*(a + c*x^2)^(3/2)), x, 17, -((3*c)/(2*a^2*d*sqrt(a + c*x^2))) + e^2/(a*d^3*sqrt(a + c*x^2)) - 1/(2*a*d*x^2*sqrt(a + c*x^2)) + e/(a*d^2*x*sqrt(a + c*x^2)) + (2*c*e*x)/(a^2*d^2*sqrt(a + c*x^2)) - (e^3*(a*e + c*d*x))/(a*d^3*(c*d^2 + a*e^2)*sqrt(a + c*x^2)) + (e^5*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(d^3*(c*d^2 + a*e^2)^(3/2)) + (3*c*atanh(sqrt(a + c*x^2)/sqrt(a)))/(2*a^(5/2)*d) - (e^2*atanh(sqrt(a + c*x^2)/sqrt(a)))/(a^(3/2)*d^3)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m / (d+e*x)^2*(a+c*x^2)^(p/2)=#

    #= ::Subsubsection:: =#
    #=p>0=#

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^5/((d + e*x)^2*sqrt(a + c*x^2)), x, 9, ((13*c*d^2 - 2*a*e^2)*sqrt(a + c*x^2))/(3*c^2*e^4) + (d^5*sqrt(a + c*x^2))/(e^4*(c*d^2 + a*e^2)*(d + e*x)) - (5*d*(d + e*x)*sqrt(a + c*x^2))/(3*c*e^4) + ((d + e*x)^2*sqrt(a + c*x^2))/(3*c*e^4) - (d*(4*c*d^2 - a*e^2)*atanh((sqrt(c)*x)/sqrt(a + c*x^2)))/(c^(3/2)*e^5) - (d^4*(4*c*d^2 + 5*a*e^2)*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(e^5*(c*d^2 + a*e^2)^(3/2))]
    @test_int [x^4/((d + e*x)^2*sqrt(a + c*x^2)), x, 8, -((5*d*sqrt(a + c*x^2))/(2*c*e^3)) - (d^4*sqrt(a + c*x^2))/(e^3*(c*d^2 + a*e^2)*(d + e*x)) + ((d + e*x)*sqrt(a + c*x^2))/(2*c*e^3) + ((6*c*d^2 - a*e^2)*atanh((sqrt(c)*x)/sqrt(a + c*x^2)))/(2*c^(3/2)*e^4) + (d^3*(3*c*d^2 + 4*a*e^2)*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(e^4*(c*d^2 + a*e^2)^(3/2))]
    @test_int [x^3/((d + e*x)^2*sqrt(a + c*x^2)), x, 7, sqrt(a + c*x^2)/(c*e^2) + (d^3*sqrt(a + c*x^2))/(e^2*(c*d^2 + a*e^2)*(d + e*x)) - (2*d*atanh((sqrt(c)*x)/sqrt(a + c*x^2)))/(sqrt(c)*e^3) - (d^2*(2*c*d^2 + 3*a*e^2)*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(e^3*(c*d^2 + a*e^2)^(3/2))]
    @test_int [x^2/((d + e*x)^2*sqrt(a + c*x^2)), x, 6, -((d^2*sqrt(a + c*x^2))/(e*(c*d^2 + a*e^2)*(d + e*x))) + atanh((sqrt(c)*x)/sqrt(a + c*x^2))/(sqrt(c)*e^2) + (d*(c*d^2 + 2*a*e^2)*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(e^2*(c*d^2 + a*e^2)^(3/2))]
    @test_int [x^1/((d + e*x)^2*sqrt(a + c*x^2)), x, 3, (d*sqrt(a + c*x^2))/((c*d^2 + a*e^2)*(d + e*x)) - (a*e*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(c*d^2 + a*e^2)^(3/2)]
    @test_int [x^0/((d + e*x)^2*sqrt(a + c*x^2)), x, 3, -((e*sqrt(a + c*x^2))/((c*d^2 + a*e^2)*(d + e*x))) - (c*d*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(c*d^2 + a*e^2)^(3/2)]
    @test_int [1/(x^1*(d + e*x)^2*sqrt(a + c*x^2)), x, 10, (e^2*sqrt(a + c*x^2))/(d*(c*d^2 + a*e^2)*(d + e*x)) + (c*e*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(c*d^2 + a*e^2)^(3/2) + (e*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(d^2*sqrt(c*d^2 + a*e^2)) - atanh(sqrt(a + c*x^2)/sqrt(a))/(sqrt(a)*d^2)]
    @test_int [1/(x^2*(d + e*x)^2*sqrt(a + c*x^2)), x, 11, -(sqrt(a + c*x^2)/(a*d^2*x)) - (e^3*sqrt(a + c*x^2))/(d^2*(c*d^2 + a*e^2)*(d + e*x)) - (c*e^2*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(d*(c*d^2 + a*e^2)^(3/2)) - (2*e^2*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(d^3*sqrt(c*d^2 + a*e^2)) + (2*e*atanh(sqrt(a + c*x^2)/sqrt(a)))/(sqrt(a)*d^3)]
    @test_int [1/(x^3*(d + e*x)^2*sqrt(a + c*x^2)), x, 15, -(sqrt(a + c*x^2)/(2*a*d^2*x^2)) + (2*e*sqrt(a + c*x^2))/(a*d^3*x) + (e^4*sqrt(a + c*x^2))/(d^3*(c*d^2 + a*e^2)*(d + e*x)) + (c*e^3*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(d^2*(c*d^2 + a*e^2)^(3/2)) + (3*e^3*atanh((a*e - c*d*x)/(sqrt(c*d^2 + a*e^2)*sqrt(a + c*x^2))))/(d^4*sqrt(c*d^2 + a*e^2)) + (c*atanh(sqrt(a + c*x^2)/sqrt(a)))/(2*a^(3/2)*d^2) - (3*e^2*atanh(sqrt(a + c*x^2)/sqrt(a)))/(sqrt(a)*d^4)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(d+e*x)^n*(a+c*x^2)^p*when*n*symbolic=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^2*(a + b*x)^n*(c + d*x^2), x, 2, (a^2*(b^2*c + a^2*d)*(a + b*x)^(1 + n))/(b^5*(1 + n)) - (2*a*(b^2*c + 2*a^2*d)*(a + b*x)^(2 + n))/(b^5*(2 + n)) + ((b^2*c + 6*a^2*d)*(a + b*x)^(3 + n))/(b^5*(3 + n)) - (4*a*d*(a + b*x)^(4 + n))/(b^5*(4 + n)) + (d*(a + b*x)^(5 + n))/(b^5*(5 + n))]
    @test_int [x^1*(a + b*x)^n*(c + d*x^2), x, 2, -((a*(b^2*c + a^2*d)*(a + b*x)^(1 + n))/(b^4*(1 + n))) + ((b^2*c + 3*a^2*d)*(a + b*x)^(2 + n))/(b^4*(2 + n)) - (3*a*d*(a + b*x)^(3 + n))/(b^4*(3 + n)) + (d*(a + b*x)^(4 + n))/(b^4*(4 + n))]
    @test_int [x^0*(a + b*x)^n*(c + d*x^2), x, 2, ((b^2*c + a^2*d)*(a + b*x)^(1 + n))/(b^3*(1 + n)) - (2*a*d*(a + b*x)^(2 + n))/(b^3*(2 + n)) + (d*(a + b*x)^(3 + n))/(b^3*(3 + n))]
    @test_int [(a + b*x)^n*(c + d*x^2)/x^1, x, 3, -((a*d*(a + b*x)^(1 + n))/(b^2*(1 + n))) + (d*(a + b*x)^(2 + n))/(b^2*(2 + n)) - (c*(a + b*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, 1 + (b*x)/a))/(a*(1 + n))]

    @test_int [x^2*(a + b*x)^n*(c + d*x^2)^2, x, 2, (a^2*(b^2*c + a^2*d)^2*(a + b*x)^(1 + n))/(b^7*(1 + n)) - (2*a*(b^2*c + a^2*d)*(b^2*c + 3*a^2*d)*(a + b*x)^(2 + n))/(b^7*(2 + n)) + ((b^4*c^2 + 12*a^2*b^2*c*d + 15*a^4*d^2)*(a + b*x)^(3 + n))/(b^7*(3 + n)) - (4*a*d*(2*b^2*c + 5*a^2*d)*(a + b*x)^(4 + n))/(b^7*(4 + n)) + (d*(2*b^2*c + 15*a^2*d)*(a + b*x)^(5 + n))/(b^7*(5 + n)) - (6*a*d^2*(a + b*x)^(6 + n))/(b^7*(6 + n)) + (d^2*(a + b*x)^(7 + n))/(b^7*(7 + n))]
    @test_int [x^1*(a + b*x)^n*(c + d*x^2)^2, x, 2, -((a*(b^2*c + a^2*d)^2*(a + b*x)^(1 + n))/(b^6*(1 + n))) + ((b^2*c + a^2*d)*(b^2*c + 5*a^2*d)*(a + b*x)^(2 + n))/(b^6*(2 + n)) - (2*a*d*(3*b^2*c + 5*a^2*d)*(a + b*x)^(3 + n))/(b^6*(3 + n)) + (2*d*(b^2*c + 5*a^2*d)*(a + b*x)^(4 + n))/(b^6*(4 + n)) - (5*a*d^2*(a + b*x)^(5 + n))/(b^6*(5 + n)) + (d^2*(a + b*x)^(6 + n))/(b^6*(6 + n))]
    @test_int [x^0*(a + b*x)^n*(c + d*x^2)^2, x, 2, ((b^2*c + a^2*d)^2*(a + b*x)^(1 + n))/(b^5*(1 + n)) - (4*a*d*(b^2*c + a^2*d)*(a + b*x)^(2 + n))/(b^5*(2 + n)) + (2*d*(b^2*c + 3*a^2*d)*(a + b*x)^(3 + n))/(b^5*(3 + n)) - (4*a*d^2*(a + b*x)^(4 + n))/(b^5*(4 + n)) + (d^2*(a + b*x)^(5 + n))/(b^5*(5 + n))]
    @test_int [(a + b*x)^n*(c + d*x^2)^2/x^1, x, 4, -((a*d*(2*b^2*c + a^2*d)*(a + b*x)^(1 + n))/(b^4*(1 + n))) + (d*(2*b^2*c + 3*a^2*d)*(a + b*x)^(2 + n))/(b^4*(2 + n)) - (3*a*d^2*(a + b*x)^(3 + n))/(b^4*(3 + n)) + (d^2*(a + b*x)^(4 + n))/(b^4*(4 + n)) - (c^2*(a + b*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, 1 + (b*x)/a))/(a*(1 + n))]

    @test_int [x^2*(a + b*x)^n*(c + d*x^2)^3, x, 2, (a^2*(b^2*c + a^2*d)^3*(a + b*x)^(1 + n))/(b^9*(1 + n)) - (2*a*(b^2*c + a^2*d)^2*(b^2*c + 4*a^2*d)*(a + b*x)^(2 + n))/(b^9*(2 + n)) + ((b^2*c + a^2*d)*(b^4*c^2 + 17*a^2*b^2*c*d + 28*a^4*d^2)*(a + b*x)^(3 + n))/(b^9*(3 + n)) - (4*a*d*(3*b^4*c^2 + 15*a^2*b^2*c*d + 14*a^4*d^2)*(a + b*x)^(4 + n))/(b^9*(4 + n)) + (d*(3*b^4*c^2 + 45*a^2*b^2*c*d + 70*a^4*d^2)*(a + b*x)^(5 + n))/(b^9*(5 + n)) - (2*a*d^2*(9*b^2*c + 28*a^2*d)*(a + b*x)^(6 + n))/(b^9*(6 + n)) + (d^2*(3*b^2*c + 28*a^2*d)*(a + b*x)^(7 + n))/(b^9*(7 + n)) - (8*a*d^3*(a + b*x)^(8 + n))/(b^9*(8 + n)) + (d^3*(a + b*x)^(9 + n))/(b^9*(9 + n))]
    @test_int [x^1*(a + b*x)^n*(c + d*x^2)^3, x, 2, -((a*(b^2*c + a^2*d)^3*(a + b*x)^(1 + n))/(b^8*(1 + n))) + ((b^2*c + a^2*d)^2*(b^2*c + 7*a^2*d)*(a + b*x)^(2 + n))/(b^8*(2 + n)) - (3*a*d*(b^2*c + a^2*d)*(3*b^2*c + 7*a^2*d)*(a + b*x)^(3 + n))/(b^8*(3 + n)) + (d*(3*b^4*c^2 + 30*a^2*b^2*c*d + 35*a^4*d^2)*(a + b*x)^(4 + n))/(b^8*(4 + n)) - (5*a*d^2*(3*b^2*c + 7*a^2*d)*(a + b*x)^(5 + n))/(b^8*(5 + n)) + (3*d^2*(b^2*c + 7*a^2*d)*(a + b*x)^(6 + n))/(b^8*(6 + n)) - (7*a*d^3*(a + b*x)^(7 + n))/(b^8*(7 + n)) + (d^3*(a + b*x)^(8 + n))/(b^8*(8 + n))]
    @test_int [x^0*(a + b*x)^n*(c + d*x^2)^3, x, 2, ((b^2*c + a^2*d)^3*(a + b*x)^(1 + n))/(b^7*(1 + n)) - (6*a*d*(b^2*c + a^2*d)^2*(a + b*x)^(2 + n))/(b^7*(2 + n)) + (3*d*(b^2*c + a^2*d)*(b^2*c + 5*a^2*d)*(a + b*x)^(3 + n))/(b^7*(3 + n)) - (4*a*d^2*(3*b^2*c + 5*a^2*d)*(a + b*x)^(4 + n))/(b^7*(4 + n)) + (3*d^2*(b^2*c + 5*a^2*d)*(a + b*x)^(5 + n))/(b^7*(5 + n)) - (6*a*d^3*(a + b*x)^(6 + n))/(b^7*(6 + n)) + (d^3*(a + b*x)^(7 + n))/(b^7*(7 + n))]
    @test_int [(a + b*x)^n*(c + d*x^2)^3/x^1, x, 4, -((a*d*(3*b^4*c^2 + 3*a^2*b^2*c*d + a^4*d^2)*(a + b*x)^(1 + n))/(b^6*(1 + n))) + (d*(3*b^4*c^2 + 9*a^2*b^2*c*d + 5*a^4*d^2)*(a + b*x)^(2 + n))/(b^6*(2 + n)) - (a*d^2*(9*b^2*c + 10*a^2*d)*(a + b*x)^(3 + n))/(b^6*(3 + n)) + (d^2*(3*b^2*c + 10*a^2*d)*(a + b*x)^(4 + n))/(b^6*(4 + n)) - (5*a*d^3*(a + b*x)^(5 + n))/(b^6*(5 + n)) + (d^3*(a + b*x)^(6 + n))/(b^6*(6 + n)) - (c^3*(a + b*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, 1 + (b*x)/a))/(a*(1 + n))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^4*(d + e*x)^n/(a + c*x^2), x, 6, ((c*d^2 - a*e^2)*(d + e*x)^(1 + n))/(c^2*e^3*(1 + n)) - (2*d*(d + e*x)^(2 + n))/(c*e^3*(2 + n)) + (d + e*x)^(3 + n)/(c*e^3*(3 + n)) + ((-a)^(3/2)*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d - sqrt(-a)*e)))/(2*c^2*(sqrt(c)*d - sqrt(-a)*e)*(1 + n)) - ((-a)^(3/2)*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d + sqrt(-a)*e)))/(2*c^2*(sqrt(c)*d + sqrt(-a)*e)*(1 + n))]
    @test_int [x^3*(d + e*x)^n/(a + c*x^2), x, 6, -((d*(d + e*x)^(1 + n))/(c*e^2*(1 + n))) + (d + e*x)^(2 + n)/(c*e^2*(2 + n)) + (a*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d - sqrt(-a)*e)))/(2*c^(3/2)*(sqrt(c)*d - sqrt(-a)*e)*(1 + n)) + (a*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d + sqrt(-a)*e)))/(2*c^(3/2)*(sqrt(c)*d + sqrt(-a)*e)*(1 + n))]
    @test_int [x^2*(d + e*x)^n/(a + c*x^2), x, 6, (d + e*x)^(1 + n)/(c*e*(1 + n)) + (sqrt(-a)*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d - sqrt(-a)*e)))/(2*c*(sqrt(c)*d - sqrt(-a)*e)*(1 + n)) - (sqrt(-a)*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d + sqrt(-a)*e)))/(2*c*(sqrt(c)*d + sqrt(-a)*e)*(1 + n))]
    @test_int [x^1*(d + e*x)^n/(a + c*x^2), x, 4, -(((d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d - sqrt(-a)*e)))/(2*sqrt(c)*(sqrt(c)*d - sqrt(-a)*e)*(1 + n))) - ((d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d + sqrt(-a)*e)))/(2*sqrt(c)*(sqrt(c)*d + sqrt(-a)*e)*(1 + n))]
    @test_int [x^0*(d + e*x)^n/(a + c*x^2), x, 4, ((d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d - sqrt(-a)*e)))/(2*sqrt(-a)*(sqrt(c)*d - sqrt(-a)*e)*(1 + n)) - ((d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d + sqrt(-a)*e)))/(2*sqrt(-a)*(sqrt(c)*d + sqrt(-a)*e)*(1 + n))]
    @test_int [(d + e*x)^n/(x^1*(a + c*x^2)), x, 7, (sqrt(c)*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d - sqrt(-a)*e)))/(2*a*(sqrt(c)*d - sqrt(-a)*e)*(1 + n)) + (sqrt(c)*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d + sqrt(-a)*e)))/(2*a*(sqrt(c)*d + sqrt(-a)*e)*(1 + n)) - ((d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, 1 + (e*x)/d))/(a*d*(1 + n))]
    @test_int [(d + e*x)^n/(x^2*(a + c*x^2)), x, 7, (c*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d - sqrt(-a)*e)))/(2*(-a)^(3/2)*(sqrt(c)*d - sqrt(-a)*e)*(1 + n)) - (c*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d + sqrt(-a)*e)))/(2*(-a)^(3/2)*(sqrt(c)*d + sqrt(-a)*e)*(1 + n)) + (e*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(2, 1 + n, 2 + n, 1 + (e*x)/d))/(a*d^2*(1 + n))]

    @test_int [x^4*(d + e*x)^n/(a + c*x^2)^2, x, 5, (d + e*x)^(1 + n)/(c^2*e*(1 + n)) + (a*(a*e + c*d*x)*(d + e*x)^(1 + n))/(2*c^2*(c*d^2 + a*e^2)*(a + c*x^2)) + ((3*sqrt(-a)*c*d^2 + a*sqrt(c)*d*e*n + sqrt(-a)*a*e^2*(3 + n))*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d - sqrt(-a)*e)))/(4*c^2*(sqrt(c)*d - sqrt(-a)*e)*(c*d^2 + a*e^2)*(1 + n)) - ((3*sqrt(-a)*c*d^2 - a*sqrt(c)*d*e*n + sqrt(-a)*a*e^2*(3 + n))*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d + sqrt(-a)*e)))/(4*c^2*(sqrt(c)*d + sqrt(-a)*e)*(c*d^2 + a*e^2)*(1 + n))]
    @test_int [x^3*(d + e*x)^n/(a + c*x^2)^2, x, 5, (a*(d - e*x)*(d + e*x)^(1 + n))/(2*c*(c*d^2 + a*e^2)*(a + c*x^2)) + ((sqrt(-a)*d*e*n - (2*c*d^2 + a*e^2*(2 + n))/sqrt(c))*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d - sqrt(-a)*e)))/(4*c*(sqrt(c)*d - sqrt(-a)*e)*(c*d^2 + a*e^2)*(1 + n)) - ((2*c*d^2 + sqrt(-a)*sqrt(c)*d*e*n + a*e^2*(2 + n))*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d + sqrt(-a)*e)))/(4*c^(3/2)*(sqrt(c)*d + sqrt(-a)*e)*(c*d^2 + a*e^2)*(1 + n))]
    @test_int [x^2*(d + e*x)^n/(a + c*x^2)^2, x, 5, -(((a*e + c*d*x)*(d + e*x)^(1 + n))/(2*c*(c*d^2 + a*e^2)*(a + c*x^2))) + ((c*d^2 - sqrt(-a)*sqrt(c)*d*e*n + a*e^2*(1 + n))*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d - sqrt(-a)*e)))/(4*sqrt(-a)*c*(sqrt(c)*d - sqrt(-a)*e)*(c*d^2 + a*e^2)*(1 + n)) - ((c*d^2 + sqrt(-a)*sqrt(c)*d*e*n + a*e^2*(1 + n))*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d + sqrt(-a)*e)))/(4*sqrt(-a)*c*(sqrt(c)*d + sqrt(-a)*e)*(c*d^2 + a*e^2)*(1 + n))]
    @test_int [x^1*(d + e*x)^n/(a + c*x^2)^2, x, 5, -(((d - e*x)*(d + e*x)^(1 + n))/(2*(c*d^2 + a*e^2)*(a + c*x^2))) + (e*(sqrt(c)*d + sqrt(-a)*e)*n*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d - sqrt(-a)*e)))/(4*sqrt(-a)*sqrt(c)*(sqrt(c)*d - sqrt(-a)*e)*(c*d^2 + a*e^2)*(1 + n)) + (e*(sqrt(-a)*sqrt(c)*d + a*e)*n*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d + sqrt(-a)*e)))/(4*a*sqrt(c)*(sqrt(c)*d + sqrt(-a)*e)*(c*d^2 + a*e^2)*(1 + n))]
    @test_int [x^0*(d + e*x)^n/(a + c*x^2)^2, x, 5, ((a*e + c*d*x)*(d + e*x)^(1 + n))/(2*a*(c*d^2 + a*e^2)*(a + c*x^2)) - ((c*d^2 + a*e^2*(1 - n) + sqrt(-a)*sqrt(c)*d*e*n)*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d - sqrt(-a)*e)))/(4*(-a)^(3/2)*(sqrt(c)*d - sqrt(-a)*e)*(c*d^2 + a*e^2)*(1 + n)) + ((c*d^2 + a*e^2*(1 - n) - sqrt(-a)*sqrt(c)*d*e*n)*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d + sqrt(-a)*e)))/(4*(-a)^(3/2)*(sqrt(c)*d + sqrt(-a)*e)*(c*d^2 + a*e^2)*(1 + n))]
    @test_int [(d + e*x)^n/(x^1*(a + c*x^2)^2), x, 12, (c*(d - e*x)*(d + e*x)^(1 + n))/(2*a*(c*d^2 + a*e^2)*(a + c*x^2)) + (sqrt(c)*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d - sqrt(-a)*e)))/(2*a^2*(sqrt(c)*d - sqrt(-a)*e)*(1 + n)) + (sqrt(c)*e*(sqrt(c)*d + sqrt(-a)*e)*n*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d - sqrt(-a)*e)))/(4*(-a)^(3/2)*(sqrt(c)*d - sqrt(-a)*e)*(c*d^2 + a*e^2)*(1 + n)) + (sqrt(c)*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d + sqrt(-a)*e)))/(2*a^2*(sqrt(c)*d + sqrt(-a)*e)*(1 + n)) - (sqrt(c)*e*(sqrt(-a)*sqrt(c)*d + a*e)*n*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d + sqrt(-a)*e)))/(4*a^2*(sqrt(c)*d + sqrt(-a)*e)*(c*d^2 + a*e^2)*(1 + n)) - ((d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, 1 + (e*x)/d))/(a^2*d*(1 + n))]
    @test_int [(d + e*x)^n/(x^2*(a + c*x^2)^2), x, 12, -((c*(a*e + c*d*x)*(d + e*x)^(1 + n))/(2*a^2*(c*d^2 + a*e^2)*(a + c*x^2))) - (c*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d - sqrt(-a)*e)))/(2*(-a)^(5/2)*(sqrt(c)*d - sqrt(-a)*e)*(1 + n)) - (c*(c*d^2 + a*e^2*(1 - n) + sqrt(-a)*sqrt(c)*d*e*n)*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d - sqrt(-a)*e)))/(4*(-a)^(5/2)*(sqrt(c)*d - sqrt(-a)*e)*(c*d^2 + a*e^2)*(1 + n)) + (c*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d + sqrt(-a)*e)))/(2*(-a)^(5/2)*(sqrt(c)*d + sqrt(-a)*e)*(1 + n)) + (c*(c*d^2 + a*e^2*(1 - n) - sqrt(-a)*sqrt(c)*d*e*n)*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (sqrt(c)*(d + e*x))/(sqrt(c)*d + sqrt(-a)*e)))/(4*(-a)^(5/2)*(sqrt(c)*d + sqrt(-a)*e)*(c*d^2 + a*e^2)*(1 + n)) + (e*(d + e*x)^(1 + n)*HypergeometricFunctions._₂F₁(2, 1 + n, 2 + n, 1 + (e*x)/d))/(a^2*d^2*(1 + n))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(g*x)^m*(d+e*x)^n*(a+c*x^2)^p*when*m*and*n*symbolic=#

    @test_int [(g*x)^m*(d + e*x)^n*(a + c*x^2)^2, x, 6, -((c*d*(2 + m)*(c*d^2*(12 + 7*m + m^2) + 2*a*e^2*(20 + m^2 + 9*n + n^2 + m*(9 + 2*n)))*(g*x)^(1 + m)*(d + e*x)^(1 + n))/(e^4*g*(2 + m + n)*(3 + m + n)*(4 + m + n)*(5 + m + n))) + (c*(c*d^2*(12 + 7*m + m^2) + 2*a*e^2*(20 + m^2 + 9*n + n^2 + m*(9 + 2*n)))*(g*x)^(2 + m)*(d + e*x)^(1 + n))/(e^3*g^2*(3 + m + n)*(4 + m + n)*(5 + m + n)) - (c^2*d*(4 + m)*(g*x)^(3 + m)*(d + e*x)^(1 + n))/(e^2*g^3*(4 + m + n)*(5 + m + n)) + (c^2*(g*x)^(4 + m)*(d + e*x)^(1 + n))/(e*g^4*(5 + m + n)) + ((a^2*e^4*(2 + m + n)*(3 + m + n)*(4 + m + n)*(5 + m + n) + c*d^2*(1 + m)*(2 + m)*(c*d^2*(12 + 7*m + m^2) + 2*a*e^2*(20 + m^2 + 9*n + n^2 + m*(9 + 2*n))))*(g*x)^(1 + m)*(d + e*x)^n*HypergeometricFunctions._₂F₁(1 + m, -n, 2 + m, -((e*x)/d)))/(1 + (e*x)/d)^n/(e^4*g*(1 + m)*(2 + m + n)*(3 + m + n)*(4 + m + n)*(5 + m + n))]
    @test_int [(g*x)^m*(d + e*x)^n*(a + c*x^2)^1, x, 4, -((c*d*(2 + m)*(g*x)^(1 + m)*(d + e*x)^(1 + n))/(e^2*g*(2 + m + n)*(3 + m + n))) + (c*(g*x)^(2 + m)*(d + e*x)^(1 + n))/(e*g^2*(3 + m + n)) + ((c*d^2*(1 + m)*(2 + m) + a*e^2*(2 + m + n)*(3 + m + n))*(g*x)^(1 + m)*(d + e*x)^n*HypergeometricFunctions._₂F₁(1 + m, -n, 2 + m, -((e*x)/d)))/((1 + (e*x)/d)^n*(e^2*g*(1 + m)*(2 + m + n)*(3 + m + n)))]
    @test_int [(g*x)^m*(d + e*x)^n/(a + c*x^2)^1, x, 6, ((g*x)^(1 + m)*(d + e*x)^n*AppellF1(1 + m, -n, 1, 2 + m, -((e*x)/d), -((sqrt(c)*x)/sqrt(-a))))/((1 + (e*x)/d)^n*(2*a*g*(1 + m))) + ((g*x)^(1 + m)*(d + e*x)^n*AppellF1(1 + m, -n, 1, 2 + m, -((e*x)/d), (sqrt(c)*x)/sqrt(-a)))/((1 + (e*x)/d)^n*(2*a*g*(1 + m)))]
    @test_int [(g*x)^m*(d + e*x)^n/(a + c*x^2)^2, x, 12, ((g*x)^(1 + m)*(d + e*x)^n*AppellF1(1 + m, -n, 1, 2 + m, -((e*x)/d), -((sqrt(c)*x)/sqrt(-a))))/((1 + (e*x)/d)^n*(4*a^2*g*(1 + m))) + ((g*x)^(1 + m)*(d + e*x)^n*AppellF1(1 + m, -n, 1, 2 + m, -((e*x)/d), (sqrt(c)*x)/sqrt(-a)))/((1 + (e*x)/d)^n*(4*a^2*g*(1 + m))) + ((g*x)^(1 + m)*(d + e*x)^n*AppellF1(1 + m, -n, 2, 2 + m, -((e*x)/d), -((sqrt(c)*x)/sqrt(-a))))/((1 + (e*x)/d)^n*(4*a^2*g*(1 + m))) + ((g*x)^(1 + m)*(d + e*x)^n*AppellF1(1 + m, -n, 2, 2 + m, -((e*x)/d), (sqrt(c)*x)/sqrt(-a)))/((1 + (e*x)/d)^n*(4*a^2*g*(1 + m)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(d+e*x)^n*(a+c*x^2)^p*when*p*symbolic=#

    #= ::Subsubsection::Closed:: =#
    #=n>0=#

    @test_int [x^5*(d + e*x)*(a + b*x^2)^p, x, 6, (a^2*d*(a + b*x^2)^(1 + p))/(2*b^3*(1 + p)) - (a*d*(a + b*x^2)^(2 + p))/(b^3*(2 + p)) + (d*(a + b*x^2)^(3 + p))/(2*b^3*(3 + p)) + (e*x^7*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(7/2, -p, 9/2, -((b*x^2)/a)))/(7*(1 + (b*x^2)/a)^p)]
    @test_int [x^4*(d + e*x)*(a + b*x^2)^p, x, 6, (a^2*e*(a + b*x^2)^(1 + p))/(2*b^3*(1 + p)) - (a*e*(a + b*x^2)^(2 + p))/(b^3*(2 + p)) + (e*(a + b*x^2)^(3 + p))/(2*b^3*(3 + p)) + (d*x^5*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(5/2, -p, 7/2, -((b*x^2)/a)))/(5*(1 + (b*x^2)/a)^p)]
    @test_int [x^3*(d + e*x)*(a + b*x^2)^p, x, 6, -(a*d*(a + b*x^2)^(1 + p))/(2*b^2*(1 + p)) + (d*(a + b*x^2)^(2 + p))/(2*b^2*(2 + p)) + (e*x^5*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(5/2, -p, 7/2, -((b*x^2)/a)))/(5*(1 + (b*x^2)/a)^p)]
    @test_int [x^2*(d + e*x)*(a + b*x^2)^p, x, 6, -(a*e*(a + b*x^2)^(1 + p))/(2*b^2*(1 + p)) + (e*(a + b*x^2)^(2 + p))/(2*b^2*(2 + p)) + (d*x^3*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(3/2, -p, 5/2, -((b*x^2)/a)))/(3*(1 + (b*x^2)/a)^p)]
    @test_int [x^1*(d + e*x)*(a + b*x^2)^p, x, 4, (d*(a + b*x^2)^(1 + p))/(2*b*(1 + p)) + (e*x^3*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(3/2, -p, 5/2, -((b*x^2)/a)))/(3*(1 + (b*x^2)/a)^p)]
    @test_int [x^0*(d + e*x)*(a + b*x^2)^p, x, 3, (e*(a + b*x^2)^(1 + p))/(2*b*(1 + p)) + (d*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/(1 + (b*x^2)/a)^p]
    @test_int [((d + e*x)*(a + b*x^2)^p)/x^1, x, 5, (e*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/(1 + (b*x^2)/a)^p - (d*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, 1 + (b*x^2)/a))/(2*a*(1 + p))]
    @test_int [((d + e*x)*(a + b*x^2)^p)/x^2, x, 5, -((d*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(-1/2, -p, 1/2, -((b*x^2)/a)))/(x*(1 + (b*x^2)/a)^p)) - (e*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, 1 + (b*x^2)/a))/(2*a*(1 + p))]
    @test_int [((d + e*x)*(a + b*x^2)^p)/x^3, x, 5, -((e*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(-1/2, -p, 1/2, -((b*x^2)/a)))/(x*(1 + (b*x^2)/a)^p)) + (b*d*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(2, 1 + p, 2 + p, 1 + (b*x^2)/a))/(2*a^2*(1 + p))]

    @test_int [x^5*(d + e*x)^2*(a + b*x^2)^p, x, 7, (a^2*(b*d^2 - a*e^2)*(a + b*x^2)^(1 + p))/(2*b^4*(1 + p)) - (a*(2*b*d^2 - 3*a*e^2)*(a + b*x^2)^(2 + p))/(2*b^4*(2 + p)) + ((b*d^2 - 3*a*e^2)*(a + b*x^2)^(3 + p))/(2*b^4*(3 + p)) + (e^2*(a + b*x^2)^(4 + p))/(2*b^4*(4 + p)) + (2*d*e*x^7*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(7/2, -p, 9/2, -((b*x^2)/a)))/(7*(1 + (b*x^2)/a)^p)]
    @test_int [x^4*(d + e*x)^2*(a + b*x^2)^p, x, 8, (a^2*d*e*(a + b*x^2)^(1 + p))/(b^3*(1 + p)) + (e^2*x^5*(a + b*x^2)^(1 + p))/(b*(7 + 2*p)) - (2*a*d*e*(a + b*x^2)^(2 + p))/(b^3*(2 + p)) + (d*e*(a + b*x^2)^(3 + p))/(b^3*(3 + p)) - ((5*a*e^2 - b*d^2*(7 + 2*p))*x^5*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(5/2, -p, 7/2, -((b*x^2)/a)))/(5*b*(7 + 2*p)*(1 + (b*x^2)/a)^p)]
    @test_int [x^3*(d + e*x)^2*(a + b*x^2)^p, x, 7, -(a*(b*d^2 - a*e^2)*(a + b*x^2)^(1 + p))/(2*b^3*(1 + p)) + ((b*d^2 - 2*a*e^2)*(a + b*x^2)^(2 + p))/(2*b^3*(2 + p)) + (e^2*(a + b*x^2)^(3 + p))/(2*b^3*(3 + p)) + (2*d*e*x^5*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(5/2, -p, 7/2, -((b*x^2)/a)))/(5*(1 + (b*x^2)/a)^p)]
    @test_int [x^2*(d + e*x)^2*(a + b*x^2)^p, x, 8, -((a*d*e*(a + b*x^2)^(1 + p))/(b^2*(1 + p))) + (e^2*x^3*(a + b*x^2)^(1 + p))/(b*(5 + 2*p)) + (d*e*(a + b*x^2)^(2 + p))/(b^2*(2 + p)) - ((3*a*e^2 - b*d^2*(5 + 2*p))*x^3*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(3/2, -p, 5/2, -((b*x^2)/a)))/(3*b*(5 + 2*p)*(1 + (b*x^2)/a)^p)]
    @test_int [x^1*(d + e*x)^2*(a + b*x^2)^p, x, 7, ((b*d^2 - a*e^2)*(a + b*x^2)^(1 + p))/(2*b^2*(1 + p)) + (e^2*(a + b*x^2)^(2 + p))/(2*b^2*(2 + p)) + (2*d*e*x^3*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(3/2, -p, 5/2, -((b*x^2)/a)))/(3*(1 + (b*x^2)/a)^p)]
    @test_int [x^0*(d + e*x)^2*(a + b*x^2)^p, x, 4, If(13>=8, (d*e*(2 + p)*(a + b*x^2)^(1 + p))/(b*(1 + p)*(3 + 2*p)) + (e*(d + e*x)*(a + b*x^2)^(1 + p))/(b*(3 + 2*p)) - ((a*e^2 - b*d^2*(3 + 2*p))*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/((1 + (b*x^2)/a)^p*(b*(3 + 2*p))), (d*e*(2 + p)*(a + b*x^2)^(1 + p))/(b*(3 + 5*p + 2*p^2)) + (e*(d + e*x)*(a + b*x^2)^(1 + p))/(b*(3 + 2*p)) - ((a*e^2 - b*d^2*(3 + 2*p))*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/((1 + (b*x^2)/a)^p*(b*(3 + 2*p))))]
    @test_int [((d + e*x)^2*(a + b*x^2)^p)/x^1, x, 7, (e^2*(a + b*x^2)^(1 + p))/(2*b*(1 + p)) + (2*d*e*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/(1 + (b*x^2)/a)^p - (d^2*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, 1 + (b*x^2)/a))/(2*a*(1 + p))]
    @test_int [((d + e*x)^2*(a + b*x^2)^p)/x^2, x, 6, -((d^2*(a + b*x^2)^(1 + p))/(a*x)) + ((a*e^2 + b*d^2*(1 + 2*p))*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/(a*(1 + (b*x^2)/a)^p) - (d*e*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, 1 + (b*x^2)/a))/(a*(1 + p))]
    @test_int [((d + e*x)^2*(a + b*x^2)^p)/x^3, x, 6, -(d^2*(a + b*x^2)^(1 + p))/(2*a*x^2) - (2*d*e*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(-1/2, -p, 1/2, -((b*x^2)/a)))/(x*(1 + (b*x^2)/a)^p) - ((a*e^2 + b*d^2*p)*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, 1 + (b*x^2)/a))/(2*a^2*(1 + p))]

    @test_int [x^5*(d + e*x)^3*(a + b*x^2)^p, x, 7, (a^2*d*(b*d^2 - 3*a*e^2)*(a + b*x^2)^(1 + p))/(2*b^4*(1 + p)) + (e^3*x^7*(a + b*x^2)^(1 + p))/(b*(9 + 2*p)) - (a*d*(2*b*d^2 - 9*a*e^2)*(a + b*x^2)^(2 + p))/(2*b^4*(2 + p)) + (d*(b*d^2 - 9*a*e^2)*(a + b*x^2)^(3 + p))/(2*b^4*(3 + p)) + (3*d*e^2*(a + b*x^2)^(4 + p))/(2*b^4*(4 + p)) - (e*(7*a*e^2 - 3*b*d^2*(9 + 2*p))*x^7*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(7/2, -p, 9/2, -((b*x^2)/a)))/(7*b*(9 + 2*p)*(1 + (b*x^2)/a)^p)]
    @test_int [x^4*(d + e*x)^3*(a + b*x^2)^p, x, 7, (a^2*e*(3*b*d^2 - a*e^2)*(a + b*x^2)^(1 + p))/(2*b^4*(1 + p)) + (3*d*e^2*x^5*(a + b*x^2)^(1 + p))/(b*(7 + 2*p)) - (3*a*e*(2*b*d^2 - a*e^2)*(a + b*x^2)^(2 + p))/(2*b^4*(2 + p)) + (3*e*(b*d^2 - a*e^2)*(a + b*x^2)^(3 + p))/(2*b^4*(3 + p)) + (e^3*(a + b*x^2)^(4 + p))/(2*b^4*(4 + p)) - (d*(15*a*e^2 - b*d^2*(7 + 2*p))*x^5*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(5/2, -p, 7/2, -((b*x^2)/a)))/(5*b*(7 + 2*p)*(1 + (b*x^2)/a)^p)]
    @test_int [x^3*(d + e*x)^3*(a + b*x^2)^p, x, 7, -(a*d*(b*d^2 - 3*a*e^2)*(a + b*x^2)^(1 + p))/(2*b^3*(1 + p)) + (e^3*x^5*(a + b*x^2)^(1 + p))/(b*(7 + 2*p)) + (d*(b*d^2 - 6*a*e^2)*(a + b*x^2)^(2 + p))/(2*b^3*(2 + p)) + (3*d*e^2*(a + b*x^2)^(3 + p))/(2*b^3*(3 + p)) - (e*(5*a*e^2 - 3*b*d^2*(7 + 2*p))*x^5*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(5/2, -p, 7/2, -((b*x^2)/a)))/(5*b*(7 + 2*p)*(1 + (b*x^2)/a)^p)]
    @test_int [x^2*(d + e*x)^3*(a + b*x^2)^p, x, 7, -(a*e*(3*b*d^2 - a*e^2)*(a + b*x^2)^(1 + p))/(2*b^3*(1 + p)) + (3*d*e^2*x^3*(a + b*x^2)^(1 + p))/(b*(5 + 2*p)) + (e*(3*b*d^2 - 2*a*e^2)*(a + b*x^2)^(2 + p))/(2*b^3*(2 + p)) + (e^3*(a + b*x^2)^(3 + p))/(2*b^3*(3 + p)) - (d*(9*a*e^2 - b*d^2*(5 + 2*p))*x^3*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(3/2, -p, 5/2, -((b*x^2)/a)))/(3*b*(5 + 2*p)*(1 + (b*x^2)/a)^p)]
    @test_int [x^1*(d + e*x)^3*(a + b*x^2)^p, x, 7, (d*(b*d^2 - 3*a*e^2)*(a + b*x^2)^(1 + p))/(2*b^2*(1 + p)) + (e^3*x^3*(a + b*x^2)^(1 + p))/(b*(5 + 2*p)) + (3*d*e^2*(a + b*x^2)^(2 + p))/(2*b^2*(2 + p)) - (e*(a*e^2 - b*d^2*(5 + 2*p))*x^3*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(3/2, -p, 5/2, -((b*x^2)/a)))/(b*(5 + 2*p)*(1 + (b*x^2)/a)^p)]
    @test_int [x^0*(d + e*x)^3*(a + b*x^2)^p, x, 4, -((e*(a*e^2 - 3*b*d^2*(2 + p))*(a + b*x^2)^(1 + p))/(2*b^2*(1 + p)*(2 + p))) + (3*d*e^2*x*(a + b*x^2)^(1 + p))/(b*(3 + 2*p)) + (e^3*x^2*(a + b*x^2)^(1 + p))/(2*b*(2 + p)) - (d*(3*a*e^2 - b*d^2*(3 + 2*p))*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/((1 + (b*x^2)/a)^p*(b*(3 + 2*p))), (e*(d + e*x)^2*(a + b*x^2)^(1 + p))/(2*b*(2 + p)) - (e*((3 + 2*p)*(a*e^2 - b*d^2*(5 + 2*p)) - 2*b*d*e*(1 + p)*(3 + p)*x)*(a + b*x^2)^(1 + p))/(2*b^2*(2 + p)*(3 + 5*p + 2*p^2)) - (d*(3*a*e^2 - b*d^2*(3 + 2*p))*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/((1 + (b*x^2)/a)^p*(b*(3 + 2*p)))]
    @test_int [((d + e*x)^3*(a + b*x^2)^p)/x^1, x, 7, (3*d*e^2*(a + b*x^2)^(1 + p))/(2*b*(1 + p)) + (e^3*x*(a + b*x^2)^(1 + p))/(b*(3 + 2*p)) - (e*(a*e^2 - 3*b*d^2*(3 + 2*p))*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/(b*(3 + 2*p)*(1 + (b*x^2)/a)^p) - (d^3*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, 1 + (b*x^2)/a))/(2*a*(1 + p))]
    @test_int [((d + e*x)^3*(a + b*x^2)^p)/x^2, x, 8, (e^3*(a + b*x^2)^(1 + p))/(2*b*(1 + p)) - (d^3*(a + b*x^2)^(1 + p))/(a*x) + (d*(3*a*e^2 + b*d^2*(1 + 2*p))*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/(a*(1 + (b*x^2)/a)^p) - (3*d^2*e*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, 1 + (b*x^2)/a))/(2*a*(1 + p))]
    @test_int [((d + e*x)^3*(a + b*x^2)^p)/x^3, x, 7, -(d^3*(a + b*x^2)^(1 + p))/(2*a*x^2) - (3*d^2*e*(a + b*x^2)^(1 + p))/(a*x) + (e*(a*e^2 + 3*b*d^2*(1 + 2*p))*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/(a*(1 + (b*x^2)/a)^p) - (d*(3*a*e^2 + b*d^2*p)*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, 1 + (b*x^2)/a))/(2*a^2*(1 + p))]

    #= ::Subsubsection::Closed:: =#
    #=n<0=#

    @test_int [x^4*(a + b*x^2)^p/(d + e*x), x, 7, ((b*d^2 - a*e^2)*(a + b*x^2)^(1 + p))/(2*b^2*e^3*(1 + p)) + (a + b*x^2)^(2 + p)/(2*b^2*e*(2 + p)) + (x^5*(a + b*x^2)^p*AppellF1(5/2, -p, 1, 7/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*(5*d)) - (d^4*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(2*e^3*(b*d^2 + a*e^2)*(1 + p))]
    @test_int [x^3*(a + b*x^2)^p/(d + e*x), x, 6, -((d*(a + b*x^2)^(1 + p))/(2*b*e^2*(1 + p))) - (e*x^5*(a + b*x^2)^p*AppellF1(5/2, -p, 1, 7/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*(5*d^2)) + (d^3*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(2*e^2*(b*d^2 + a*e^2)*(1 + p))]
    @test_int [x^2*(a + b*x^2)^p/(d + e*x), x, 6, (a + b*x^2)^(1 + p)/(2*b*e*(1 + p)) + (x^3*(a + b*x^2)^p*AppellF1(3/2, -p, 1, 5/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*(3*d)) - (d^2*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(2*e*(b*d^2 + a*e^2)*(1 + p))]
    @test_int [x^1*(a + b*x^2)^p/(d + e*x), x, 9, -((x*(a + b*x^2)^p*AppellF1(1/2, -p, 1, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*e)) + (x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/((1 + (b*x^2)/a)^p*e) + (d*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(2*(b*d^2 + a*e^2)*(1 + p))]
    @test_int [x^0*(a + b*x^2)^p/(d + e*x), x, 6, (x*(a + b*x^2)^p*AppellF1(1/2, -p, 1, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*d) - (e*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(2*(b*d^2 + a*e^2)*(1 + p))]
    @test_int [(a + b*x^2)^p/(x^1*(d + e*x)), x, 7, -((e*x*(a + b*x^2)^p*AppellF1(1/2, -p, 1, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*d^2)) + (e^2*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(2*d*(b*d^2 + a*e^2)*(1 + p)) - ((a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, 1 + (b*x^2)/a))/(2*a*d*(1 + p))]
    @test_int [(a + b*x^2)^p/(x^2*(d + e*x)), x, 7, -(((a + b*x^2)^p*AppellF1(-(1/2), -p, 1, 1/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*(d*x))) - (e^3*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(2*d^2*(b*d^2 + a*e^2)*(1 + p)) + (e*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, 1 + (b*x^2)/a))/(2*a*d^2*(1 + p))]
    @test_int [(a + b*x^2)^p/(x^3*(d + e*x)), x, 8, -((a + b*x^2)^(1 + p)/(2*a*d*x^2)) + (e*(a + b*x^2)^p*AppellF1(-(1/2), -p, 1, 1/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*(d^2*x)) + (e^4*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(2*d^3*(b*d^2 + a*e^2)*(1 + p)) - ((a*e^2 + b*d^2*p)*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, 1 + (b*x^2)/a))/(2*a^2*d^3*(1 + p))]

    @test_int [x^4*(a + b*x^2)^p/(d + e*x)^2, x, 12, -((d*(4 + 3*p)*(a + b*x^2)^(1 + p))/(b*e^3*(1 + p)*(3 + 2*p))) - (d^4*(a + b*x^2)^(1 + p))/(e^3*(b*d^2 + a*e^2)*(d + e*x)) + ((d + e*x)*(a + b*x^2)^(1 + p))/(b*e^3*(3 + 2*p)) - (2*d^2*(2*a*e^2 + b*d^2*(2 + p))*x*(a + b*x^2)^p*AppellF1(1/2, -p, 1, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*(e^4*(b*d^2 + a*e^2))) - ((a^2*e^4 - 2*a*b*d^2*e^2*(4 + 3*p) - 2*b^2*d^4*(6 + 7*p + 2*p^2))*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/((1 + (b*x^2)/a)^p*(b*e^4*(b*d^2 + a*e^2)*(3 + 2*p))) + (d^3*(2*a*e^2 + b*d^2*(2 + p))*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(e^3*(b*d^2 + a*e^2)^2*(1 + p))]
    @test_int [x^3*(a + b*x^2)^p/(d + e*x)^2, x, 11, (a + b*x^2)^(1 + p)/(2*b*e^2*(1 + p)) + (d^3*(a + b*x^2)^(1 + p))/(e^2*(b*d^2 + a*e^2)*(d + e*x)) + (d*(3*a*e^2 + b*d^2*(3 + 2*p))*x*(a + b*x^2)^p*AppellF1(1/2, -p, 1, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*(e^3*(b*d^2 + a*e^2))) - (d*(2*a*e^2 + b*d^2*(3 + 2*p))*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/((1 + (b*x^2)/a)^p*(e^3*(b*d^2 + a*e^2))) - (d^2*(3*a*e^2 + b*d^2*(3 + 2*p))*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(2*e^2*(b*d^2 + a*e^2)^2*(1 + p))]
    @test_int [x^2*(a + b*x^2)^p/(d + e*x)^2, x, 10, -((d^2*(a + b*x^2)^(1 + p))/(e*(b*d^2 + a*e^2)*(d + e*x))) - (2*(a*e^2 + b*d^2*(1 + p))*x*(a + b*x^2)^p*AppellF1(1/2, -p, 1, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*(e^2*(b*d^2 + a*e^2))) + ((a*e^2 + 2*b*d^2*(1 + p))*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/((1 + (b*x^2)/a)^p*(e^2*(b*d^2 + a*e^2))) + (d*(a*e^2 + b*d^2*(1 + p))*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(e*(b*d^2 + a*e^2)^2*(1 + p))]
    @test_int [x^1*(a + b*x^2)^p/(d + e*x)^2, x, 10, (d*(a + b*x^2)^(1 + p))/((b*d^2 + a*e^2)*(d + e*x)) + ((a*e^2 + b*d^2*(1 + 2*p))*x*(a + b*x^2)^p*AppellF1(1/2, -p, 1, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*(d*e*(b*d^2 + a*e^2))) - (b*d*(1 + 2*p)*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/((1 + (b*x^2)/a)^p*(e*(b*d^2 + a*e^2))) - ((a*e^2 + b*d^2*(1 + 2*p))*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(2*(b*d^2 + a*e^2)^2*(1 + p))]
    @test_int [x^0*(a + b*x^2)^p/(d + e*x)^2, x, 8, (e^2*x*(a + b*x^2)^(1 + p))/((b*d^2 + a*e^2)*(d^2 - e^2*x^2)) - (2*b*p*x*(a + b*x^2)^p*AppellF1(1/2, -p, 1, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*(b*d^2 + a*e^2)) + (b*(1 + 2*p)*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/((1 + (b*x^2)/a)^p*(b*d^2 + a*e^2)) - (b*d*e*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(2, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/((b*d^2 + a*e^2)^2*(1 + p)), (x*(a + b*x^2)^p*AppellF1(1/2, -p, 2, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*d^2) + (e^2*x^3*(a + b*x^2)^p*AppellF1(3/2, -p, 2, 5/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*(3*d^4)) - (b*d*e*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(2, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/((b*d^2 + a*e^2)^2*(1 + p))]
    @test_int [(a + b*x^2)^p/(x^1*(d + e*x)^2), x, 18, -((e*x*(a + b*x^2)^p*AppellF1(1/2, -p, 1, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*d^3)) - (e*x*(a + b*x^2)^p*AppellF1(1/2, -p, 2, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*d^3) - (e^3*x^3*(a + b*x^2)^p*AppellF1(3/2, -p, 2, 5/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*(3*d^5)) + (e^2*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(2*d^2*(b*d^2 + a*e^2)*(1 + p)) - ((a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, 1 + (b*x^2)/a))/(2*a*d^2*(1 + p)) + (b*e^2*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(2, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/((b*d^2 + a*e^2)^2*(1 + p))]
    @test_int [(a + b*x^2)^p/(x^2*(d + e*x)^2), x, 20, (2*e^2*x*(a + b*x^2)^p*AppellF1(1/2, -p, 1, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*d^4) + (e^2*x*(a + b*x^2)^p*AppellF1(1/2, -p, 2, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*d^4) + (e^4*x^3*(a + b*x^2)^p*AppellF1(3/2, -p, 2, 5/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*(3*d^6)) - ((a + b*x^2)^p*HypergeometricFunctions._₂F₁(-(1/2), -p, 1/2, -((b*x^2)/a)))/((1 + (b*x^2)/a)^p*(d^2*x)) - (e^3*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(d^3*(b*d^2 + a*e^2)*(1 + p)) + (e*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, 1 + (b*x^2)/a))/(a*d^3*(1 + p)) - (b*e^3*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(2, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(d*(b*d^2 + a*e^2)^2*(1 + p))]

    @test_int [x^4*(a + b*x^2)^p/(d + e*x)^3, x, 12, (a + b*x^2)^(1 + p)/(2*b*e^3*(1 + p)) - (d^4*(a + b*x^2)^(1 + p))/(2*e^3*(b*d^2 + a*e^2)*(d + e*x)^2) + (d^3*(4*a*e^2 + b*d^2*(3 + p))*(a + b*x^2)^(1 + p))/(e^3*(b*d^2 + a*e^2)^2*(d + e*x)) + (d*(6*a^2*e^4 + 3*a*b*d^2*e^2*(4 + 3*p) + b^2*d^4*(6 + 7*p + 2*p^2))*x*(a + b*x^2)^p*AppellF1(1/2, -p, 1, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*(e^4*(b*d^2 + a*e^2)^2)) - (d*(3*a^2*e^4 + 2*a*b*d^2*e^2*(5 + 4*p) + b^2*d^4*(6 + 7*p + 2*p^2))*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/((1 + (b*x^2)/a)^p*(e^4*(b*d^2 + a*e^2)^2)) - (d^2*(6*a^2*e^4 + 3*a*b*d^2*e^2*(4 + 3*p) + b^2*d^4*(6 + 7*p + 2*p^2))*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(2*e^3*(b*d^2 + a*e^2)^3*(1 + p))]
    @test_int [x^3*(a + b*x^2)^p/(d + e*x)^3, x, 11, (d^3*(a + b*x^2)^(1 + p))/(2*e^2*(b*d^2 + a*e^2)*(d + e*x)^2) - (d^2*(3*a*e^2 + b*d^2*(2 + p))*(a + b*x^2)^(1 + p))/(e^2*(b*d^2 + a*e^2)^2*(d + e*x)) - ((3*a^2*e^4 + a*b*d^2*e^2*(6 + 7*p) + b^2*d^4*(3 + 5*p + 2*p^2))*x*(a + b*x^2)^p*AppellF1(1/2, -p, 1, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*(e^3*(b*d^2 + a*e^2)^2)) + ((a^2*e^4 + a*b*d^2*e^2*(5 + 6*p) + b^2*d^4*(3 + 5*p + 2*p^2))*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/((1 + (b*x^2)/a)^p*(e^3*(b*d^2 + a*e^2)^2)) + (d*(3*a^2*e^4 + a*b*d^2*e^2*(6 + 7*p) + b^2*d^4*(3 + 5*p + 2*p^2))*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(2*e^2*(b*d^2 + a*e^2)^3*(1 + p))]
    @test_int [x^2*(a + b*x^2)^p/(d + e*x)^3, x, 11, -((d^2*(a + b*x^2)^(1 + p))/(2*e*(b*d^2 + a*e^2)*(d + e*x)^2)) + (d*(2*a*e^2 + b*d^2*(1 + p))*(a + b*x^2)^(1 + p))/(e*(b*d^2 + a*e^2)^2*(d + e*x)) + ((a^2*e^4 + a*b*d^2*e^2*(2 + 5*p) + b^2*d^4*(1 + 3*p + 2*p^2))*x*(a + b*x^2)^p*AppellF1(1/2, -p, 1, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*(d*e^2*(b*d^2 + a*e^2)^2)) - (b*d*(1 + 2*p)*(2*a*e^2 + b*d^2*(1 + p))*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/((1 + (b*x^2)/a)^p*(e^2*(b*d^2 + a*e^2)^2)) - ((a^2*e^4 + a*b*d^2*e^2*(2 + 5*p) + b^2*d^4*(1 + 3*p + 2*p^2))*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(2*e*(b*d^2 + a*e^2)^3*(1 + p))]
    @test_int [x^1*(a + b*x^2)^p/(d + e*x)^3, x, 11, (d*(a + b*x^2)^(1 + p))/(2*(b*d^2 + a*e^2)*(d + e*x)^2) - ((a*e^2 + b*d^2*p)*(a + b*x^2)^(1 + p))/((b*d^2 + a*e^2)^2*(d + e*x)) - (b*p*(3*a*e^2 + b*d^2*(1 + 2*p))*x*(a + b*x^2)^p*AppellF1(1/2, -p, 1, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*(e*(b*d^2 + a*e^2)^2)) + (b*(1 + 2*p)*(a*e^2 + b*d^2*p)*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/((1 + (b*x^2)/a)^p*(e*(b*d^2 + a*e^2)^2)) + (b*d*p*(3*a*e^2 + b*d^2*(1 + 2*p))*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(2*(b*d^2 + a*e^2)^3*(1 + p))]
    @test_int [x^0*(a + b*x^2)^p/(d + e*x)^3, x, 11, -((d^2*e*(a + b*x^2)^(1 + p))/(4*(b*d^2 + a*e^2)*(d^2 - e^2*x^2)^2)) + (x*(a + b*x^2)^p*AppellF1(1/2, -p, 3, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*d^3) + (e^2*x^3*(a + b*x^2)^p*AppellF1(3/2, -p, 3, 5/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*d^5) + (b*e*(2*a*e^2 + b*d^2*(1 + p))*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(2, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(4*(b*d^2 + a*e^2)^3*(1 + p)) - (3*b^2*d^2*e*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(3, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(2*(b*d^2 + a*e^2)^3*(1 + p))]
    @test_int [(a + b*x^2)^p/(x^1*(d + e*x)^3), x, 29, (d*e^2*(a + b*x^2)^(1 + p))/(4*(b*d^2 + a*e^2)*(d^2 - e^2*x^2)^2) - (e*x*(a + b*x^2)^p*AppellF1(1/2, -p, 1, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*d^4) - (e*x*(a + b*x^2)^p*AppellF1(1/2, -p, 2, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*d^4) - (e*x*(a + b*x^2)^p*AppellF1(1/2, -p, 3, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*d^4) - (e^3*x^3*(a + b*x^2)^p*AppellF1(3/2, -p, 2, 5/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*(3*d^6)) - (e^3*x^3*(a + b*x^2)^p*AppellF1(3/2, -p, 3, 5/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*d^6) + (e^2*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(2*d^3*(b*d^2 + a*e^2)*(1 + p)) - ((a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, 1 + (b*x^2)/a))/(2*a*d^3*(1 + p)) + (b*e^2*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(2, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(d*(b*d^2 + a*e^2)^2*(1 + p)) - (b*e^2*(2*a*e^2 + b*d^2*(1 + p))*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(2, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(4*d*(b*d^2 + a*e^2)^3*(1 + p)) + (3*b^2*d*e^2*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(3, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(2*(b*d^2 + a*e^2)^3*(1 + p))]
    @test_int [(a + b*x^2)^p/(x^2*(d + e*x)^3), x, 31, -((e^3*(a + b*x^2)^(1 + p))/(4*(b*d^2 + a*e^2)*(d^2 - e^2*x^2)^2)) + (3*e^2*x*(a + b*x^2)^p*AppellF1(1/2, -p, 1, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*d^5) + (2*e^2*x*(a + b*x^2)^p*AppellF1(1/2, -p, 2, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*d^5) + (e^2*x*(a + b*x^2)^p*AppellF1(1/2, -p, 3, 3/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*d^5) + (2*e^4*x^3*(a + b*x^2)^p*AppellF1(3/2, -p, 2, 5/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*(3*d^7)) + (e^4*x^3*(a + b*x^2)^p*AppellF1(3/2, -p, 3, 5/2, -((b*x^2)/a), (e^2*x^2)/d^2))/((1 + (b*x^2)/a)^p*d^7) - ((a + b*x^2)^p*HypergeometricFunctions._₂F₁(-(1/2), -p, 1/2, -((b*x^2)/a)))/((1 + (b*x^2)/a)^p*(d^3*x)) - (3*e^3*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(2*d^4*(b*d^2 + a*e^2)*(1 + p)) + (3*e*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, 1 + (b*x^2)/a))/(2*a*d^4*(1 + p)) - (2*b*e^3*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(2, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(d^2*(b*d^2 + a*e^2)^2*(1 + p)) + (b*e^3*(2*a*e^2 + b*d^2*(1 + p))*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(2, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(4*d^2*(b*d^2 + a*e^2)^3*(1 + p)) - (3*b^2*e^3*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(3, 1 + p, 2 + p, (e^2*(a + b*x^2))/(b*d^2 + a*e^2)))/(2*(b*d^2 + a*e^2)^3*(1 + p))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(g*x)^m*(d+e*x)^n*(a+c*x^2)^p*when*m*and*p*symbolic=#

    @test_int [(g*x)^m*(d + e*x)^3*(a + c*x^2)^p, x, 7, (3*d*e^2*(g*x)^(1 + m)*(a + c*x^2)^(1 + p))/(c*g*(3 + m + 2*p)) + (e^3*(g*x)^(2 + m)*(a + c*x^2)^(1 + p))/(c*g^2*(4 + m + 2*p)) - (d*(3*a*e^2*(1 + m) - c*d^2*(3 + m + 2*p))*(g*x)^(1 + m)*(a + c*x^2)^p*HypergeometricFunctions._₂F₁((1 + m)/2, -p, (3 + m)/2, -((c*x^2)/a)))/((1 + (c*x^2)/a)^p*(c*g*(1 + m)*(3 + m + 2*p))) - (e*(a*e^2*(2 + m) - 3*c*d^2*(4 + m + 2*p))*(g*x)^(2 + m)*(a + c*x^2)^p*HypergeometricFunctions._₂F₁((2 + m)/2, -p, (4 + m)/2, -((c*x^2)/a)))/((1 + (c*x^2)/a)^p*(c*g^2*(2 + m)*(4 + m + 2*p)))]
    @test_int [(g*x)^m*(d + e*x)^2*(a + c*x^2)^p, x, 6, (e^2*(g*x)^(1 + m)*(a + c*x^2)^(1 + p))/(c*g*(3 + m + 2*p)) - ((a*e^2*(1 + m) - c*d^2*(3 + m + 2*p))*(g*x)^(1 + m)*(a + c*x^2)^p*HypergeometricFunctions._₂F₁((1 + m)/2, -p, (3 + m)/2, -((c*x^2)/a)))/((1 + (c*x^2)/a)^p*(c*g*(1 + m)*(3 + m + 2*p))) + (2*d*e*(g*x)^(2 + m)*(a + c*x^2)^p*HypergeometricFunctions._₂F₁((2 + m)/2, -p, (4 + m)/2, -((c*x^2)/a)))/((1 + (c*x^2)/a)^p*(g^2*(2 + m)))]
    @test_int [(g*x)^m*(d + e*x)^1*(a + c*x^2)^p, x, 5, (d*(g*x)^(1 + m)*(a + c*x^2)^p*HypergeometricFunctions._₂F₁((1 + m)/2, -p, (3 + m)/2, -((c*x^2)/a)))/((1 + (c*x^2)/a)^p*(g*(1 + m))) + (e*(g*x)^(2 + m)*(a + c*x^2)^p*HypergeometricFunctions._₂F₁((2 + m)/2, -p, (4 + m)/2, -((c*x^2)/a)))/((1 + (c*x^2)/a)^p*(g^2*(2 + m)))]
    @test_int [(g*x)^m*(d + e*x)^0*(a + c*x^2)^p, x, 2, ((g*x)^(1 + m)*(a + c*x^2)^p*HypergeometricFunctions._₂F₁((1 + m)/2, -p, (3 + m)/2, -((c*x^2)/a)))/((1 + (c*x^2)/a)^p*(g*(1 + m)))]
    @test_int [(g*x)^m/(d + e*x)^1*(a + c*x^2)^p, x, 5, (x*(g*x)^m*(a + c*x^2)^p*AppellF1((1 + m)/2, -p, 1, (3 + m)/2, -((c*x^2)/a), (e^2*x^2)/d^2))/((1 + (c*x^2)/a)^p*(d*(1 + m))) - (e*x^2*(g*x)^m*(a + c*x^2)^p*AppellF1((2 + m)/2, -p, 1, (4 + m)/2, -((c*x^2)/a), (e^2*x^2)/d^2))/((1 + (c*x^2)/a)^p*(d^2*(2 + m)))]
    @test_int [(g*x)^m/(d + e*x)^2*(a + c*x^2)^p, x, 8, (x*(g*x)^m*(a + c*x^2)^p*AppellF1((1 + m)/2, -p, 2, (3 + m)/2, -((c*x^2)/a), (e^2*x^2)/d^2))/((1 + (c*x^2)/a)^p*(d^2*(1 + m))) - (2*e*x^2*(g*x)^m*(a + c*x^2)^p*AppellF1((2 + m)/2, -p, 2, (4 + m)/2, -((c*x^2)/a), (e^2*x^2)/d^2))/((1 + (c*x^2)/a)^p*(d^3*(2 + m))) + (e^2*x^3*(g*x)^m*(a + c*x^2)^p*AppellF1((3 + m)/2, -p, 2, (5 + m)/2, -((c*x^2)/a), (e^2*x^2)/d^2))/((1 + (c*x^2)/a)^p*(d^4*(3 + m)))]
    @test_int [(g*x)^m/(d + e*x)^3*(a + c*x^2)^p, x, 10, (x*(g*x)^m*(a + c*x^2)^p*AppellF1((1 + m)/2, -p, 3, (3 + m)/2, -((c*x^2)/a), (e^2*x^2)/d^2))/((1 + (c*x^2)/a)^p*(d^3*(1 + m))) - (3*e*x^2*(g*x)^m*(a + c*x^2)^p*AppellF1((2 + m)/2, -p, 3, (4 + m)/2, -((c*x^2)/a), (e^2*x^2)/d^2))/((1 + (c*x^2)/a)^p*(d^4*(2 + m))) + (3*e^2*x^3*(g*x)^m*(a + c*x^2)^p*AppellF1((3 + m)/2, -p, 3, (5 + m)/2, -((c*x^2)/a), (e^2*x^2)/d^2))/((1 + (c*x^2)/a)^p*(d^5*(3 + m))) - (e^3*x^4*(g*x)^m*(a + c*x^2)^p*AppellF1((4 + m)/2, -p, 3, (6 + m)/2, -((c*x^2)/a), (e^2*x^2)/d^2))/((1 + (c*x^2)/a)^p*(d^6*(4 + m)))]

    #= ::Section:: =#
    #=Integrands*of*the*form*x^m*(d+e*x)^n*(a+b*x+c*x^2)^p*when*b^2-4*a*c=0=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*x^m*(d+e*x)^n*(a+b*x+c*x^2)^p*when*c*d^2-b*d*e+a*e^2=0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^2 / (d+e*x)*(a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^3*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x), x, 6, (1/24)*(a/(c*d) - (7*d)/e^2)*x^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2) + (x^3*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(4*e) - ((105*c^3*d^6 - 25*a*c^2*d^4*e^2 - 17*a^2*c*d^2*e^4 - 15*a^3*e^6 - 2*c*d*e*(35*c^2*d^4 - 6*a*c*d^2*e^2 - 5*a^2*e^4)*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(192*c^3*d^3*e^4) + ((c*d^2 - a*e^2)*(35*c^3*d^6 + 15*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 + 5*a^3*e^6)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(128*c^(7/2)*d^(7/2)*e^(9/2))]
    @test_int [x^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x), x, 5, (x^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3*e) + (((5*c*d^2 - 3*a*e^2)*(3*c*d^2 + a*e^2) - 2*c*d*e*(5*c*d^2 - a*e^2)*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(24*c^2*d^2*e^3) - ((c*d^2 - a*e^2)*(5*c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(16*c^(5/2)*d^(5/2)*e^(7/2))]
    @test_int [x^1*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x), x, 4, (-(1/4))*(a/(c*d) + (3*d)/e^2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2) + (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(2*c*d*e*(d + e*x)) + ((c*d^2 - a*e^2)*(3*c*d^2 + a*e^2)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(8*c^(3/2)*d^(3/2)*e^(5/2))]
    @test_int [x^0*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x), x, 3, sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/e - ((c*d^2 - a*e^2)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(2*sqrt(c)*sqrt(d)*e^(3/2))]
    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(x^1*(d + e*x)), x, 6, (sqrt(c)*sqrt(d)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/sqrt(e) - (sqrt(a)*sqrt(e)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/sqrt(d)]
    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(x^2*(d + e*x)), x, 4, -(sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d*x)) - ((c*d^2 - a*e^2)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(2*sqrt(a)*d^(3/2)*sqrt(e))]
    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(x^3*(d + e*x)), x, 5, -(sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(2*d*x^2)) - ((c/(a*e) - (3*e)/d^2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(4*x) + ((c*d^2 - a*e^2)*(c*d^2 + 3*a*e^2)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(8*a^(3/2)*d^(5/2)*e^(3/2))]
    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(x^4*(d + e*x)), x, 6, -(sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(3*d*x^3)) - ((c/(a*e) - (5*e)/d^2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(12*x^2) + ((3*c*d^2 - 5*a*e^2)*(c*d^2 + 3*a*e^2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(24*a^2*d^3*e^2*x) - ((c*d^2 - a*e^2)*(c^2*d^4 + 2*a*c*d^2*e^2 + 5*a^2*e^4)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(16*a^(5/2)*d^(7/2)*e^(5/2))]
    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(x^5*(d + e*x)), x, 7, -(sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(4*d*x^4)) - ((c/(a*e) - (7*e)/d^2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(24*x^3) + ((5*c^2*d^4 + 6*a*c*d^2*e^2 - 35*a^2*e^4)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(96*a^2*d^3*e^2*x^2) - ((15*c^3*d^6 + 17*a*c^2*d^4*e^2 + 25*a^2*c*d^2*e^4 - 105*a^3*e^6)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(192*a^3*d^4*e^3*x) + ((c*d^2 - a*e^2)*(5*c^3*d^6 + 9*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 + 35*a^3*e^6)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(128*a^(7/2)*d^(9/2)*e^(7/2))]

    @test_int [x^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x), x, 7, ((21*c^4*d^8 - 6*a^2*c^2*d^4*e^4 - 8*a^3*c*d^2*e^6 - 7*a^4*e^8)*(c*d^2 + a*e^2 + 2*c*d*e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(512*c^4*d^4*e^5) + (1/20)*(a/(c*d) - (3*d)/e^2)*x^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2) + (x^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(6*e) - ((105*c^3*d^6 - 21*a*c^2*d^4*e^2 - 33*a^2*c*d^2*e^4 - 35*a^3*e^6 - 6*c*d*e*(21*c^2*d^4 - 6*a*c*d^2*e^2 - 7*a^2*e^4)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(960*c^3*d^3*e^4) - ((c*d^2 - a*e^2)^3*(21*c^3*d^6 + 21*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 + 7*a^3*e^6)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(1024*c^(9/2)*d^(9/2)*e^(11/2))]
    @test_int [x^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x), x, 6, -(((c*d^2 - a*e^2)*(7*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*(c*d^2 + a*e^2 + 2*c*d*e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(128*c^3*d^3*e^4)) + (x^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(5*e) + ((35*c^2*d^4 - 12*a*c*d^2*e^2 - 15*a^2*e^4 - 6*c*d*e*(7*c*d^2 - 3*a*e^2)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(240*c^2*d^2*e^3) + ((c*d^2 - a*e^2)^3*(7*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(256*c^(7/2)*d^(7/2)*e^(9/2))]
    @test_int [x^1*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x), x, 5, ((c*d^2 - a*e^2)*(5*c*d^2 + 3*a*e^2)*(c*d^2 + a*e^2 + 2*c*d*e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(64*c^2*d^2*e^3) - (1/24)*((3*a)/(c*d) + (5*d)/e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2) + (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(4*c*d*e*(d + e*x)) - ((c*d^2 - a*e^2)^3*(5*c*d^2 + 3*a*e^2)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(128*c^(5/2)*d^(5/2)*e^(7/2))]
    @test_int [x^0*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x), x, 4, (1/8)*(a/(c*d) - d/e^2)*(c*d^2 + a*e^2 + 2*c*d*e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2) + (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(3*e) + ((c*d^2 - a*e^2)^3*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(16*c^(3/2)*d^(3/2)*e^(5/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^1*(d + e*x)), x, 7, ((c*d^2 + 5*a*e^2 + 2*c*d*e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(4*e) - ((c^2*d^4 - 6*a*c*d^2*e^2 - 3*a^2*e^4)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(8*sqrt(c)*sqrt(d)*e^(3/2)) - a^(3/2)*sqrt(d)*e^(3/2)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^2*(d + e*x)), x, 7, -(((a*e - c*d*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/x) + (sqrt(c)*sqrt(d)*(c*d^2 + 3*a*e^2)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(2*sqrt(e)) - (sqrt(a)*sqrt(e)*(3*c*d^2 + a*e^2)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(2*sqrt(d))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^3*(d + e*x)), x, 7, -(((2*a*d*e + (5*c*d^2 + a*e^2)*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(4*d*x^2)) + c^(3/2)*d^(3/2)*sqrt(e)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))) - ((3*c^2*d^4 + 6*a*c*d^2*e^2 - a^2*e^4)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(8*sqrt(a)*d^(3/2)*sqrt(e))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^4*(d + e*x)), x, 5, -(((c/(a*e) - e/d^2)*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(8*x^2)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(3*d*x^3) + ((c*d^2 - a*e^2)^3*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(16*a^(3/2)*d^(5/2)*e^(3/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^5*(d + e*x)), x, 6, ((c*d^2 - a*e^2)*(3*c*d^2 + 5*a*e^2)*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(64*a^2*d^3*e^2*x^2) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(4*d*x^4) - (((3*c)/(a*e) - (5*e)/d^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(24*x^3) - ((c*d^2 - a*e^2)^3*(3*c*d^2 + 5*a*e^2)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(128*a^(5/2)*d^(7/2)*e^(5/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^6*(d + e*x)), x, 7, -(((c*d^2 - a*e^2)*(3*c^2*d^4 + 6*a*c*d^2*e^2 + 7*a^2*e^4)*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(128*a^3*d^4*e^3*x^2)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(5*d*x^5) - (((3*c)/(a*e) - (7*e)/d^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(40*x^4) + ((15*c^2*d^4 + 12*a*c*d^2*e^2 - 35*a^2*e^4)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(240*a^2*d^3*e^2*x^3) + ((c*d^2 - a*e^2)^3*(3*c^2*d^4 + 6*a*c*d^2*e^2 + 7*a^2*e^4)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(256*a^(7/2)*d^(9/2)*e^(7/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^7*(d + e*x)), x, 8, ((7*c^4*d^8 + 8*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 21*a^4*e^8)*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(512*a^4*d^5*e^4*x^2) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(6*d*x^6) - ((c/(a*e) - (3*e)/d^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(20*x^5) + ((7*c^2*d^4 + 6*a*c*d^2*e^2 - 21*a^2*e^4)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(160*a^2*d^3*e^2*x^4) - ((35*c^3*d^6 + 33*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 - 105*a^3*e^6)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(960*a^3*d^4*e^3*x^3) - ((c*d^2 - a*e^2)^3*(7*c^3*d^6 + 15*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 + 21*a^3*e^6)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(1024*a^(9/2)*d^(11/2)*e^(9/2))]

    @test_int [x^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x), x, 8, -((3*(c*d^2 - a*e^2)^3*(33*c^3*d^6 + 45*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 + 15*a^3*e^6)*(c*d^2 + a*e^2 + 2*c*d*e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(16384*c^5*d^5*e^6)) + ((c*d^2 - a*e^2)*(33*c^3*d^6 + 45*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 + 15*a^3*e^6)*(c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(2048*c^4*d^4*e^5) + (1/112)*((5*a)/(c*d) - (11*d)/e^2)*x^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2) + (x^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(8*e) - ((231*c^3*d^6 - 15*a*c^2*d^4*e^2 - 95*a^2*c*d^2*e^4 - 105*a^3*e^6 - 10*c*d*e*(33*c^2*d^4 - 10*a*c*d^2*e^2 - 15*a^2*e^4)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(4480*c^3*d^3*e^4) + (3*(c*d^2 - a*e^2)^5*(33*c^3*d^6 + 45*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 + 15*a^3*e^6)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(32768*c^(11/2)*d^(11/2)*e^(13/2))]
    @test_int [x^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x), x, 7, ((c*d^2 - a*e^2)^3*(9*c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*(c*d^2 + a*e^2 + 2*c*d*e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(1024*c^4*d^4*e^5) - ((c*d^2 - a*e^2)*(9*c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*(c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(384*c^3*d^3*e^4) + (x^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(7*e) + ((63*c^2*d^4 - 20*a*c*d^2*e^2 - 35*a^2*e^4 - 10*c*d*e*(9*c*d^2 - 5*a*e^2)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(840*c^2*d^2*e^3) - ((c*d^2 - a*e^2)^5*(9*c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(2048*c^(9/2)*d^(9/2)*e^(11/2))]
    @test_int [x^1*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x), x, 6, -(((c*d^2 - a*e^2)^3*(7*c*d^2 + 5*a*e^2)*(c*d^2 + a*e^2 + 2*c*d*e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(512*c^3*d^3*e^4)) + ((c*d^2 - a*e^2)*(7*c*d^2 + 5*a*e^2)*(c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(192*c^2*d^2*e^3) - (1/60)*((5*a)/(c*d) + (7*d)/e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2) + (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(6*c*d*e*(d + e*x)) + ((c*d^2 - a*e^2)^5*(7*c*d^2 + 5*a*e^2)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(1024*c^(7/2)*d^(7/2)*e^(9/2))]
    @test_int [x^0*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x), x, 5, (3*(c*d^2 - a*e^2)^3*(c*d^2 + a*e^2 + 2*c*d*e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(128*c^2*d^2*e^3) + (1/16)*(a/(c*d) - d/e^2)*(c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2) + (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(5*e) - (3*(c*d^2 - a*e^2)^5*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(256*c^(5/2)*d^(5/2)*e^(7/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^1*(d + e*x)), x, 8, -(((3*c^3*d^6 - 11*a*c^2*d^4*e^2 - 83*a^2*c*d^2*e^4 - 5*a^3*e^6 + 2*c*d*e*(c*d^2 - 5*a*e^2)*(3*c*d^2 + a*e^2)*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(64*c*d*e^2)) + ((3*c*d^2 + 11*a*e^2 + 6*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(24*e) + ((3*c^4*d^8 - 20*a*c^3*d^6*e^2 + 90*a^2*c^2*d^4*e^4 + 60*a^3*c*d^2*e^6 - 5*a^4*e^8)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(128*c^(3/2)*d^(3/2)*e^(5/2)) - a^(5/2)*d^(3/2)*e^(5/2)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^2*(d + e*x)), x, 8, ((c^2*d^4 + 28*a*c*d^2*e^2 + 19*a^2*e^4 + 2*c*d*e*(c*d^2 + 7*a*e^2)*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(8*e) - ((3*a*e - c*d*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*x) - ((c^3*d^6 - 15*a*c^2*d^4*e^2 - 45*a^2*c*d^2*e^4 - 5*a^3*e^6)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(16*sqrt(c)*sqrt(d)*e^(3/2)) - (1/2)*a^(3/2)*sqrt(d)*e^(3/2)*(5*c*d^2 + 3*a*e^2)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^3*(d + e*x)), x, 8, -((3*(a*e*(3*c*d^2 + a*e^2) - c*d*(c*d^2 + 3*a*e^2)*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(4*x)) - ((a*e - c*d*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(2*x^2) + (3*sqrt(c)*sqrt(d)*(c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(8*sqrt(e)) - (3*sqrt(a)*sqrt(e)*(5*c^2*d^4 + 10*a*c*d^2*e^2 + a^2*e^4)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(8*sqrt(d))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^4*(d + e*x)), x, 8, -(((5*c^2*d^4 + 12*a*c*d^2*e^2 - a^2*e^4 - 2*c*d*e*(7*c*d^2 + a*e^2)*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(8*d*x)) - ((4*a*d*e + 3*(3*c*d^2 + a*e^2)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(12*d*x^3) + (1/2)*c^(3/2)*d^(3/2)*sqrt(e)*(3*c*d^2 + 5*a*e^2)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))) - ((5*c^3*d^6 + 45*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - a^3*e^6)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(16*sqrt(a)*d^(3/2)*sqrt(e))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^5*(d + e*x)), x, 8, -(((2*a*d*e*(5*c*d^2 - a*e^2)*(c*d^2 + 3*a*e^2) + (5*c^3*d^6 + 83*a*c^2*d^4*e^2 + 11*a^2*c*d^2*e^4 - 3*a^3*e^6)*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(64*a*d^2*e*x^2)) - ((6*a*d*e + (11*c*d^2 + 3*a*e^2)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(24*d*x^4) + c^(5/2)*d^(5/2)*e^(3/2)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))) + ((5*c^4*d^8 - 60*a*c^3*d^6*e^2 - 90*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 3*a^4*e^8)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(128*a^(3/2)*d^(5/2)*e^(3/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^6*(d + e*x)), x, 6, (3*(c*d^2 - a*e^2)^3*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(128*a^2*d^3*e^2*x^2) - ((c/(a*e) - e/d^2)*(2*a*d*e + (c*d^2 + a*e^2)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(16*x^4) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(5*d*x^5) - (3*(c*d^2 - a*e^2)^5*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(256*a^(5/2)*d^(7/2)*e^(5/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^7*(d + e*x)), x, 7, -(((c*d^2 - a*e^2)^3*(5*c*d^2 + 7*a*e^2)*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(512*a^3*d^4*e^3*x^2)) + ((c*d^2 - a*e^2)*(5*c*d^2 + 7*a*e^2)*(2*a*d*e + (c*d^2 + a*e^2)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(192*a^2*d^3*e^2*x^4) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(6*d*x^6) - (((5*c)/(a*e) - (7*e)/d^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(60*x^5) + ((c*d^2 - a*e^2)^5*(5*c*d^2 + 7*a*e^2)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(1024*a^(7/2)*d^(9/2)*e^(7/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^8*(d + e*x)), x, 8, ((c*d^2 - a*e^2)^3*(5*c^2*d^4 + 10*a*c*d^2*e^2 + 9*a^2*e^4)*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(1024*a^4*d^5*e^4*x^2) - ((c*d^2 - a*e^2)*(5*c^2*d^4 + 10*a*c*d^2*e^2 + 9*a^2*e^4)*(2*a*d*e + (c*d^2 + a*e^2)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(384*a^3*d^4*e^3*x^4) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(7*d*x^7) - (((5*c)/(a*e) - (9*e)/d^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(84*x^6) + ((35*c^2*d^4 + 20*a*c*d^2*e^2 - 63*a^2*e^4)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(840*a^2*d^3*e^2*x^5) - ((c*d^2 - a*e^2)^5*(5*c^2*d^4 + 10*a*c*d^2*e^2 + 9*a^2*e^4)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(2048*a^(9/2)*d^(11/2)*e^(9/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^9*(d + e*x)), x, 9, -((3*(c*d^2 - a*e^2)^3*(15*c^3*d^6 + 35*a*c^2*d^4*e^2 + 45*a^2*c*d^2*e^4 + 33*a^3*e^6)*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(16384*a^5*d^6*e^5*x^2)) + ((c*d^2 - a*e^2)*(15*c^3*d^6 + 35*a*c^2*d^4*e^2 + 45*a^2*c*d^2*e^4 + 33*a^3*e^6)*(2*a*d*e + (c*d^2 + a*e^2)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(2048*a^4*d^5*e^4*x^4) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(8*d*x^8) - (((5*c)/(a*e) - (11*e)/d^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(112*x^7) + ((15*c^2*d^4 + 10*a*c*d^2*e^2 - 33*a^2*e^4)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(448*a^2*d^3*e^2*x^6) - ((105*c^3*d^6 + 95*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - 231*a^3*e^6)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(4480*a^3*d^4*e^3*x^5) + (3*(c*d^2 - a*e^2)^5*(15*c^3*d^6 + 35*a*c^2*d^4*e^2 + 45*a^2*c*d^2*e^4 + 33*a^3*e^6)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(32768*a^(11/2)*d^(13/2)*e^(11/2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^3/((d + e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)), x, 5, -((3*(3*c*d^2 + a*e^2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(4*c^2*d^2*e^3)) - (2*d^3*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(e^3*(c*d^2 - a*e^2)*(d + e*x)) + ((d + e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(2*c*d*e^3) + (3*(5*c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(8*c^(5/2)*d^(5/2)*e^(7/2)), -((2*d*x^2*(a*e*(c*d^2 - a*e^2) + c*d*(c*d^2 - a*e^2)*x))/(e*(c*d^2 - a*e^2)^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))) - (((5*c*d^2 - 3*a*e^2)*(3*c*d^2 + a*e^2) - 2*c*d*e*(5*c*d^2 - a*e^2)*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(4*c^2*d^2*e^3*(c*d^2 - a*e^2)) + (3*(5*c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(8*c^(5/2)*d^(5/2)*e^(7/2))]
    @test_int [x^2/((d + e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)), x, 4, sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(c*d*e^2) + (2*d^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(e^2*(c*d^2 - a*e^2)*(d + e*x)) - ((3*c*d^2 + a*e^2)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(2*c^(3/2)*d^(3/2)*e^(5/2))]
    @test_int [x^1/((d + e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)), x, 3, -((2*d*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(e*(c*d^2 - a*e^2)*(d + e*x))) + atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)))/(sqrt(c)*sqrt(d)*e^(3/2))]
    @test_int [x^0/((d + e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)), x, 1, (2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/((c*d^2 - a*e^2)*(d + e*x))]
    @test_int [1/(x^1*(d + e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)), x, 5, -((2*e*(a*e + c*d*x))/(d*(c*d^2 - a*e^2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))) - atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)))/(sqrt(a)*d^(3/2)*sqrt(e))]
    @test_int [1/(x^2*(d + e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)), x, 5, -((2*e*(a*e + c*d*x))/(d*(c*d^2 - a*e^2)*x*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))) - ((c*d^2 - 3*a*e^2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(a*d^2*e*(c*d^2 - a*e^2)*x) + ((c*d^2 + 3*a*e^2)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(2*a^(3/2)*d^(5/2)*e^(3/2))]
    @test_int [1/(x^3*(d + e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)), x, 6, -((2*e*(a*e + c*d*x))/(d*(c*d^2 - a*e^2)*x^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))) - ((c*d^2 - 5*a*e^2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(2*a*d^2*e*(c*d^2 - a*e^2)*x^2) + ((3*c*d^2 - 5*a*e^2)*(c*d^2 + 3*a*e^2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(4*a^2*d^3*e^2*(c*d^2 - a*e^2)*x) - (3*(c^2*d^4 + 2*a*c*d^2*e^2 + 5*a^2*e^4)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(8*a^(5/2)*d^(7/2)*e^(5/2))]

    @test_int [x^5/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)), x, 6, -((2*d*x^4*(a*e*(c*d^2 - a*e^2) + c*d*(c*d^2 - a*e^2)*x))/(3*e*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))) - (2*x^2*(a*d*e*(c*d^2 - a*e^2)*(7*c^2*d^4 - 12*a*c*d^2*e^2 - 3*a^2*e^4) + (c*d^2 - a*e^2)*(7*c^3*d^6 - 11*a*c^2*d^4*e^2 - a^2*c*d^2*e^4 - 3*a^3*e^6)*x))/(3*c*d*e^2*(c*d^2 - a*e^2)^4*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)) - ((105*c^4*d^8 - 190*a*c^3*d^6*e^2 + 36*a^2*c^2*d^4*e^4 + 30*a^3*c*d^2*e^6 - 45*a^4*e^8 - 2*c*d*e*(35*c^3*d^6 - 61*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 - 15*a^3*e^6)*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(12*c^3*d^3*e^4*(c*d^2 - a*e^2)^3) + (5*(7*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(8*c^(7/2)*d^(7/2)*e^(9/2))]
    @test_int [x^4/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)), x, 6, -((2*d*x^3*(a*e*(c*d^2 - a*e^2) + c*d*(c*d^2 - a*e^2)*x))/(3*e*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))) - (2*x*(a*d*e*(c*d^2 - a*e^2)*(5*c^2*d^4 - 10*a*c*d^2*e^2 - 3*a^2*e^4) + (c*d^2 - a*e^2)*(5*c^3*d^6 - 9*a*c^2*d^4*e^2 - a^2*c*d^2*e^4 - 3*a^3*e^6)*x))/(3*c*d*e^2*(c*d^2 - a*e^2)^4*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)) + ((15*c^3*d^6 - 31*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 - 9*a^3*e^6)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3*c^2*d^2*e^3*(c*d^2 - a*e^2)^3) - ((5*c*d^2 + 3*a*e^2)*atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(2*c^(5/2)*d^(5/2)*e^(7/2))]
    @test_int [x^3/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)), x, 5, -((2*d*x^2*(a*e*(c*d^2 - a*e^2) + c*d*(c*d^2 - a*e^2)*x))/(3*e*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))) - (2*(a*d*e*(c*d^2 - 3*a*e^2)*(3*c*d^2 + a*e^2) + (3*c^3*d^6 - 7*a*c^2*d^4*e^2 - a^2*c*d^2*e^4 - 3*a^3*e^6)*x))/(3*c*d*e^2*(c*d^2 - a*e^2)^3*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)) + atanh((c*d^2 + a*e^2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)))/(c^(3/2)*d^(3/2)*e^(5/2))]
    @test_int [x^2/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)), x, 3, (2*x^2)/(3*(c*d^2 - a*e^2)*(d + e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)) - (8*a*e*(2*a*d*e + (c*d^2 + a*e^2)*x))/(3*(c*d^2 - a*e^2)^3*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [x^1/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)), x, 2, -((2*d)/(3*e*(c*d^2 - a*e^2)*(d + e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))) + (2*(c*d^2 + 3*a*e^2)*(c*d^2 + a*e^2 + 2*c*d*e*x))/(3*e*(c*d^2 - a*e^2)^3*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [x^0/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)), x, 2, 2/(3*(c*d^2 - a*e^2)*(d + e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)) - (8*c*d*(c*d^2 + a*e^2 + 2*c*d*e*x))/(3*(c*d^2 - a*e^2)^3*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [1/(x^1*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)), x, 6, -((2*e*(a*e + c*d*x))/(3*d*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))) + (2*(3*c^3*d^6 + a*c^2*d^4*e^2 + 7*a^2*c*d^2*e^4 - 3*a^3*e^6 + c*d*e*(3*c*d^2 - a*e^2)*(c*d^2 + 3*a*e^2)*x))/(3*a*d^2*e*(c*d^2 - a*e^2)^3*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)) - atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)))/(a^(3/2)*d^(5/2)*e^(3/2))]
    @test_int [1/(x^2*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)), x, 6, -((2*e*(a*e + c*d*x))/(3*d*(c*d^2 - a*e^2)*x*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))) + (2*(3*c^3*d^6 + a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 - 5*a^3*e^6 + c*d*e*(3*c^2*d^4 + 10*a*c*d^2*e^2 - 5*a^2*e^4)*x))/(3*a*d^2*e*(c*d^2 - a*e^2)^3*x*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)) - ((9*c^3*d^6 - 9*a*c^2*d^4*e^2 + 31*a^2*c*d^2*e^4 - 15*a^3*e^6)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3*a^2*d^3*e^2*(c*d^2 - a*e^2)^3*x) + ((3*c*d^2 + 5*a*e^2)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(2*a^(5/2)*d^(7/2)*e^(5/2))]
    @test_int [1/(x^3*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)), x, 7, -((2*e*(a*e + c*d*x))/(3*d*(c*d^2 - a*e^2)*x^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))) + (2*(3*c^3*d^6 + a*c^2*d^4*e^2 + 11*a^2*c*d^2*e^4 - 7*a^3*e^6 + c*d*e*(3*c^2*d^4 + 12*a*c*d^2*e^2 - 7*a^2*e^4)*x))/(3*a*d^2*e*(c*d^2 - a*e^2)^3*x^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)) - ((15*c^3*d^6 - 9*a*c^2*d^4*e^2 + 61*a^2*c*d^2*e^4 - 35*a^3*e^6)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(6*a^2*d^3*e^2*(c*d^2 - a*e^2)^3*x^2) + ((45*c^4*d^8 - 30*a*c^3*d^6*e^2 - 36*a^2*c^2*d^4*e^4 + 190*a^3*c*d^2*e^6 - 105*a^4*e^8)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(12*a^3*d^4*e^3*(c*d^2 - a*e^2)^3*x) - (5*(3*c^2*d^4 + 6*a*c*d^2*e^2 + 7*a^2*e^4)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(8*a^(7/2)*d^(9/2)*e^(7/2))]
    @test_int [1/(x^4*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)), x, 8, -((2*e*(a*e + c*d*x))/(3*d*(c*d^2 - a*e^2)*x^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))) + (2*(3*c^3*d^6 + a*c^2*d^4*e^2 + 13*a^2*c*d^2*e^4 - 9*a^3*e^6 + c*d*e*(3*c^2*d^4 + 14*a*c*d^2*e^2 - 9*a^2*e^4)*x))/(3*a*d^2*e*(c*d^2 - a*e^2)^3*x^3*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)) - ((7*c^3*d^6 - 3*a*c^2*d^4*e^2 + 33*a^2*c*d^2*e^4 - 21*a^3*e^6)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3*a^2*d^3*e^2*(c*d^2 - a*e^2)^3*x^3) + ((35*c^4*d^8 - 16*a*c^3*d^6*e^2 - 18*a^2*c^2*d^4*e^4 + 168*a^3*c*d^2*e^6 - 105*a^4*e^8)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(12*a^3*d^4*e^3*(c*d^2 - a*e^2)^3*x^2) - ((105*c^5*d^10 - 55*a*c^4*d^8*e^2 - 54*a^2*c^3*d^6*e^4 - 78*a^3*c^2*d^4*e^6 + 525*a^4*c*d^2*e^8 - 315*a^5*e^10)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(24*a^4*d^5*e^4*(c*d^2 - a*e^2)^3*x) + (5*(7*c^3*d^6 + 15*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 + 21*a^3*e^6)*atanh((2*a*d*e + (c*d^2 + a*e^2)*x)/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))/(16*a^(9/2)*d^(11/2)*e^(9/2))]

    @test_int [x^2/(d + e*x)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2), x, 3, (2*x^2)/(5*(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (8*(a*d*e*(c*d^2 - a*e^2)*(c*d^2 + 3*a*e^2) + (c^3*d^6 + a^2*c*d^2*e^4 - 2*a^3*e^6)*x))/(15*e*(c*d^2 - a*e^2)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (8*(c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*(c*d^2 + a*e^2 + 2*c*d*e*x))/(15*e*(c*d^2 - a*e^2)^5*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]

    @test_int [x^2/(d + e*x)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2), x, 4, (2*x^2)/(7*(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)) - (8*(2*a*d*e*(c*d^2 + 2*a*e^2) + (2*c^2*d^4 + a*c*d^2*e^2 + 3*a^2*e^4)*x))/(35*e*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)) + (16*(3*c^2*d^4 + 14*a*c*d^2*e^2 + 7*a^2*e^4)*(c*d^2 + a*e^2 + 2*c*d*e*x))/(105*e*(c*d^2 - a*e^2)^5*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (128*c*d*(3*c^2*d^4 + 14*a*c*d^2*e^2 + 7*a^2*e^4)*(c*d^2 + a*e^2 + 2*c*d*e*x))/(105*(c*d^2 - a*e^2)^7*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*x^m*(d+e*x)^p*(a+b*x+c*x^2)^p*when*b*d+a*e=0*and*c*d+b*e=0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(d+e*x)^(p/2)*(a+b*x+c*x^2)^(p/2)*when*b*d+a*e=0*and*c*d+b*e=0=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^3*sqrt(1 + x)*sqrt(1 - x + x^2), x, 4, (6*x*sqrt(1 + x)*sqrt(1 - x + x^2))/55 + (2*x^4*sqrt(1 + x)*sqrt(1 - x + x^2))/11 - (4*3^(3/4)*sqrt(2 + sqrt(3))*(1 + x)^(3/2)*sqrt(1 - x + x^2)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(55*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*(1 + x^3))]
    @test_int [x^2*sqrt(1 + x)*sqrt(1 - x + x^2), x, 1, (2/9)*(1 + x)^(3/2)*(1 - x + x^2)^(3/2)]
    @test_int [x*sqrt(1 + x)*sqrt(1 - x + x^2), x, 5, (2*x^2*sqrt(1 + x)*sqrt(1 - x + x^2))/7 + (6*sqrt(1 + x)*sqrt(1 - x + x^2))/(7*(1 + sqrt(3) + x)) - (3*3^(1/4)*sqrt(2 - sqrt(3))*(1 + x)^(3/2)*sqrt(1 - x + x^2)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.E(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(7*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*(1 + x^3)) + (2*sqrt(2)*3^(3/4)*(1 + x)^(3/2)*sqrt(1 - x + x^2)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(7*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*(1 + x^3))]
    @test_int [sqrt(1 + x)*sqrt(1 - x + x^2), x, 3, (2*x*sqrt(1 + x)*sqrt(1 - x + x^2))/5 + (2*3^(3/4)*sqrt(2 + sqrt(3))*(1 + x)^(3/2)*sqrt(1 - x + x^2)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(5*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*(1 + x^3))]
    @test_int [(sqrt(1 + x)*sqrt(1 - x + x^2))/x, x, 5, (2*sqrt(1 + x)*sqrt(1 - x + x^2))/3 - (2*sqrt(1 + x)*sqrt(1 - x + x^2)*atanh(sqrt(1 + x^3)))/(3*sqrt(1 + x^3))]
    @test_int [(sqrt(1 + x)*sqrt(1 - x + x^2))/x^2, x, 5, -((sqrt(1 + x)*sqrt(1 - x + x^2))/x) + (3*sqrt(1 + x)*sqrt(1 - x + x^2))/(1 + sqrt(3) + x) - (3*3^(1/4)*sqrt(2 - sqrt(3))*(1 + x)^(3/2)*sqrt(1 - x + x^2)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.E(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(2*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*(1 + x^3)) + (sqrt(2)*3^(3/4)*(1 + x)^(3/2)*sqrt(1 - x + x^2)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(sqrt((1 + x)/(1 + sqrt(3) + x)^2)*(1 + x^3))]
    @test_int [(sqrt(1 + x)*sqrt(1 - x + x^2))/x^3, x, 3, -(sqrt(1 + x)*sqrt(1 - x + x^2))/(2*x^2) + (3^(3/4)*sqrt(2 + sqrt(3))*(1 + x)^(3/2)*sqrt(1 - x + x^2)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(2*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*(1 + x^3))]

    @test_int [x^3*(1 + x)^(3/2)*(1 - x + x^2)^(3/2), x, 5, (54*x*sqrt(1 + x)*sqrt(1 - x + x^2))/935 + (18*x^4*sqrt(1 + x)*sqrt(1 - x + x^2))/187 + (2*x^4*sqrt(1 + x)*sqrt(1 - x + x^2)*(1 + x^3))/17 - (36*3^(3/4)*sqrt(2 + sqrt(3))*(1 + x)^(3/2)*sqrt(1 - x + x^2)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(935*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*(1 + x^3))]
    @test_int [x^2*(1 + x)^(3/2)*(1 - x + x^2)^(3/2), x, 1, (2/15)*(1 + x)^(5/2)*(1 - x + x^2)^(5/2)]
    @test_int [x*(1 + x)^(3/2)*(1 - x + x^2)^(3/2), x, 6, (18*x^2*sqrt(1 + x)*sqrt(1 - x + x^2))/91 + (54*sqrt(1 + x)*sqrt(1 - x + x^2))/(91*(1 + sqrt(3) + x)) + (2*x^2*sqrt(1 + x)*sqrt(1 - x + x^2)*(1 + x^3))/13 - (27*3^(1/4)*sqrt(2 - sqrt(3))*(1 + x)^(3/2)*sqrt(1 - x + x^2)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.E(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(91*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*(1 + x^3)) + (18*sqrt(2)*3^(3/4)*(1 + x)^(3/2)*sqrt(1 - x + x^2)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(91*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*(1 + x^3))]
    @test_int [(1 + x)^(3/2)*(1 - x + x^2)^(3/2), x, 4, (18*x*sqrt(1 + x)*sqrt(1 - x + x^2))/55 + (2*x*sqrt(1 + x)*sqrt(1 - x + x^2)*(1 + x^3))/11 + (18*3^(3/4)*sqrt(2 + sqrt(3))*(1 + x)^(3/2)*sqrt(1 - x + x^2)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(55*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*(1 + x^3))]
    @test_int [((1 + x)^(3/2)*(1 - x + x^2)^(3/2))/x, x, 6, (2*sqrt(1 + x)*sqrt(1 - x + x^2))/3 + (2*sqrt(1 + x)*sqrt(1 - x + x^2)*(1 + x^3))/9 - (2*sqrt(1 + x)*sqrt(1 - x + x^2)*atanh(sqrt(1 + x^3)))/(3*sqrt(1 + x^3))]
    @test_int [((1 + x)^(3/2)*(1 - x + x^2)^(3/2))/x^2, x, 6, (9*x^2*sqrt(1 + x)*sqrt(1 - x + x^2))/7 + (27*sqrt(1 + x)*sqrt(1 - x + x^2))/(7*(1 + sqrt(3) + x)) - (sqrt(1 + x)*sqrt(1 - x + x^2)*(1 + x^3))/x - (27*3^(1/4)*sqrt(2 - sqrt(3))*(1 + x)^(3/2)*sqrt(1 - x + x^2)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.E(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(14*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*(1 + x^3)) + (9*sqrt(2)*3^(3/4)*(1 + x)^(3/2)*sqrt(1 - x + x^2)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(7*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*(1 + x^3))]
    @test_int [((1 + x)^(3/2)*(1 - x + x^2)^(3/2))/x^3, x, 4, (9*x*sqrt(1 + x)*sqrt(1 - x + x^2))/10 - (sqrt(1 + x)*sqrt(1 - x + x^2)*(1 + x^3))/(2*x^2) + (9*3^(3/4)*sqrt(2 + sqrt(3))*(1 + x)^(3/2)*sqrt(1 - x + x^2)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(10*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*(1 + x^3))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^3/(sqrt(1 + x)*sqrt(1 - x + x^2)), x, 3, (2*x*(1 + x^3))/(5*sqrt(1 + x)*sqrt(1 - x + x^2)) - (4*sqrt(2 + sqrt(3))/3^(1/4)*sqrt(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(5*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 - x + x^2))]
    @test_int [x^2/(sqrt(1 + x)*sqrt(1 - x + x^2)), x, 1, (2/3)*sqrt(1 + x)*sqrt(1 - x + x^2)]
    @test_int [x/(sqrt(1 + x)*sqrt(1 - x + x^2)), x, 4, (2*(1 + x^3))/(sqrt(1 + x)*(1 + sqrt(3) + x)*sqrt(1 - x + x^2)) - (3^(1/4)*sqrt(2 - sqrt(3))*sqrt(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.E(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 - x + x^2)) + (2*sqrt(2)*sqrt(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3^(1/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 - x + x^2))]
    @test_int [1/(sqrt(1 + x)*sqrt(1 - x + x^2)), x, 2, (2*sqrt(2 + sqrt(3))/3^(1/4)*sqrt(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 - x + x^2))]
    @test_int [1/(x*sqrt(1 + x)*sqrt(1 - x + x^2)), x, 4, (-2*sqrt(1 + x^3)*atanh(sqrt(1 + x^3)))/(3*sqrt(1 + x)*sqrt(1 - x + x^2))]
    @test_int [1/(x^2*sqrt(1 + x)*sqrt(1 - x + x^2)), x, 5, -((1 + x^3)/(x*sqrt(1 + x)*sqrt(1 - x + x^2))) + (1 + x^3)/(sqrt(1 + x)*(1 + sqrt(3) + x)*sqrt(1 - x + x^2)) - (3^(1/4)*sqrt(2 - sqrt(3))*sqrt(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.E(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(2*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 - x + x^2)) + (sqrt(2)*sqrt(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3^(1/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 - x + x^2))]
    @test_int [1/(x^3*sqrt(1 + x)*sqrt(1 - x + x^2)), x, 3, -(1 + x^3)/(2*x^2*sqrt(1 + x)*sqrt(1 - x + x^2)) - (sqrt(2 + sqrt(3))/3^(1/4)*sqrt(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(2*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 - x + x^2))]

    @test_int [x^3/((1 + x)^(3/2)*(1 - x + x^2)^(3/2)), x, 3, (-2*x)/(3*sqrt(1 + x)*sqrt(1 - x + x^2)) + (4*sqrt(2 + sqrt(3))/3^(1/4)*sqrt(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 - x + x^2))]
    @test_int [x^2/((1 + x)^(3/2)*(1 - x + x^2)^(3/2)), x, 1, -2/(3*sqrt(1 + x)*sqrt(1 - x + x^2))]
    @test_int [x/((1 + x)^(3/2)*(1 - x + x^2)^(3/2)), x, 5, (2*x^2)/(3*sqrt(1 + x)*sqrt(1 - x + x^2)) - (2*(1 + x^3))/(3*sqrt(1 + x)*(1 + sqrt(3) + x)*sqrt(1 - x + x^2)) + (3^(1/4)*sqrt(2 - sqrt(3))*sqrt(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.E(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 - x + x^2)) - (2*sqrt(2)*sqrt(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 - x + x^2))]
    @test_int [1/((1 + x)^(3/2)*(1 - x + x^2)^(3/2)), x, 3, (2*x)/(3*sqrt(1 + x)*sqrt(1 - x + x^2)) + (2*sqrt(2 + sqrt(3))/3^(1/4)*sqrt(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 - x + x^2))]
    @test_int [1/(x*(1 + x)^(3/2)*(1 - x + x^2)^(3/2)), x, 5, 2/(3*sqrt(1 + x)*sqrt(1 - x + x^2)) - (2*sqrt(1 + x^3)*atanh(sqrt(1 + x^3)))/(3*sqrt(1 + x)*sqrt(1 - x + x^2))]
    @test_int [1/(x^2*(1 + x)^(3/2)*(1 - x + x^2)^(3/2)), x, 6, 2/(3*x*sqrt(1 + x)*sqrt(1 - x + x^2)) - (5*(1 + x^3))/(3*x*sqrt(1 + x)*sqrt(1 - x + x^2)) + (5*(1 + x^3))/(3*sqrt(1 + x)*(1 + sqrt(3) + x)*sqrt(1 - x + x^2)) - (5*3^(1/4)*sqrt(2 - sqrt(3))*sqrt(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.E(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(6*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 - x + x^2)) + (5*sqrt(2)*sqrt(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 - x + x^2))]
    @test_int [1/(x^3*(1 + x)^(3/2)*(1 - x + x^2)^(3/2)), x, 4, 2/(3*x^2*sqrt(1 + x)*sqrt(1 - x + x^2)) - (7*(1 + x^3))/(6*x^2*sqrt(1 + x)*sqrt(1 - x + x^2)) - (7*sqrt(2 + sqrt(3))/3^(1/4)*sqrt(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(6*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 - x + x^2))]

    @test_int [x^3/((1 + x)^(5/2)*(1 - x + x^2)^(5/2)), x, 4, (4*x)/(27*sqrt(1 + x)*sqrt(1 - x + x^2)) - (2*x)/(9*sqrt(1 + x)*sqrt(1 - x + x^2)*(1 + x^3)) + (4*sqrt(2 + sqrt(3))*sqrt(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(27*3^(1/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 - x + x^2))]
    @test_int [x^2/((1 + x)^(5/2)*(1 - x + x^2)^(5/2)), x, 1, -(2/(9*(1 + x)^(3/2)*(1 - x + x^2)^(3/2)))]
    @test_int [x/((1 + x)^(5/2)*(1 - x + x^2)^(5/2)), x, 6, (10*x^2)/(27*sqrt(1 + x)*sqrt(1 - x + x^2)) + (2*x^2)/(9*sqrt(1 + x)*sqrt(1 - x + x^2)*(1 + x^3)) - (10*(1 + x^3))/(27*sqrt(1 + x)*(1 + sqrt(3) + x)*sqrt(1 - x + x^2)) + (5*3^(1/4)*sqrt(2 - sqrt(3))*sqrt(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.E(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(27*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 - x + x^2)) - (10*sqrt(2)*sqrt(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(27*3^(1/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 - x + x^2))]
    @test_int [1/((1 + x)^(5/2)*(1 - x + x^2)^(5/2)), x, 4, (14*x)/(27*sqrt(1 + x)*sqrt(1 - x + x^2)) + (2*x)/(9*sqrt(1 + x)*sqrt(1 - x + x^2)*(1 + x^3)) + (14*sqrt(2 + sqrt(3))/3^(1/4)*sqrt(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(27*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 - x + x^2))]
    @test_int [1/(x*(1 + x)^(5/2)*(1 - x + x^2)^(5/2)), x, 6, 2/(3*sqrt(1 + x)*sqrt(1 - x + x^2)) + 2/(9*sqrt(1 + x)*sqrt(1 - x + x^2)*(1 + x^3)) - (2*sqrt(1 + x^3)*atanh(sqrt(1 + x^3)))/(3*sqrt(1 + x)*sqrt(1 - x + x^2))]
    @test_int [1/(x^2*(1 + x)^(5/2)*(1 - x + x^2)^(5/2)), x, 7, 22/(27*x*sqrt(1 + x)*sqrt(1 - x + x^2)) + 2/(9*x*sqrt(1 + x)*sqrt(1 - x + x^2)*(1 + x^3)) - (55*(1 + x^3))/(27*x*sqrt(1 + x)*sqrt(1 - x + x^2)) + (55*(1 + x^3))/(27*sqrt(1 + x)*(1 + sqrt(3) + x)*sqrt(1 - x + x^2)) - (55*3^(1/4)*sqrt(2 - sqrt(3))*sqrt(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.E(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(54*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 - x + x^2)) + (55*sqrt(2)*sqrt(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(27*3^(1/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 - x + x^2))]
    @test_int [1/(x^3*(1 + x)^(5/2)*(1 - x + x^2)^(5/2)), x, 5, 26/(27*x^2*sqrt(1 + x)*sqrt(1 - x + x^2)) + 2/(9*x^2*sqrt(1 + x)*sqrt(1 - x + x^2)*(1 + x^3)) - (91*(1 + x^3))/(54*x^2*sqrt(1 + x)*sqrt(1 - x + x^2)) - (91*sqrt(2 + sqrt(3))/3^(1/4)*sqrt(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(54*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 - x + x^2))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*x^m*(d+e*x)^n*(a+b*x+c*x^2)^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(d+e*x)^n*(a+b*x+c*x^2)^p=#

    #= ::Subsubsection:: =#
    #=p>0=#

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x/((-1 + x)^3*(3 + 5*x + 4*x^2)^2), x, 7, -(21/(736*(1 - x)^2)) - 97/(4416*(1 - x)) + (39 + 44*x)/(276*(1 - x)^2*(3 + 5*x + 4*x^2)) + (6023*atan((5 + 8*x)/sqrt(23)))/(52992*sqrt(23)) + (11*log(1 - x))/2304 - (11*log(3 + 5*x + 4*x^2))/4608]

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^m*(d+e*x)^n*(a+b*x+c*x^2)^(p/2)=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(d+e*x)^(n/2)*(a+b*x+c*x^2)^p=#

    #= ::Subsubsection:: =#
    #=p>0=#

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^4*sqrt(d + e*x)/(a + b*x + c*x^2), x, 6, -((2*b*(b^2 - 2*a*c)*sqrt(d + e*x))/c^4) + (2*(c^2*d^2 + b^2*e^2 + c*e*(b*d - a*e))*(d + e*x)^(3/2))/(3*c^3*e^3) - (2*(2*c*d + b*e)*(d + e*x)^(5/2))/(5*c^2*e^3) + (2*(d + e*x)^(7/2))/(7*c*e^3) + (sqrt(2)*(b^3*c*d - 2*a*b*c^2*d - b^4*e + 3*a*b^2*c*e - a^2*c^2*e - (b^4*c*d - 4*a*b^2*c^2*d + 2*a^2*c^3*d - b^5*e + 5*a*b^3*c*e - 5*a^2*b*c^2*e)/sqrt(b^2 - 4*a*c))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(c^(9/2)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) + (sqrt(2)*(b^3*c*d - 2*a*b*c^2*d - b^4*e + 3*a*b^2*c*e - a^2*c^2*e + (b^4*c*d - 4*a*b^2*c^2*d + 2*a^2*c^3*d - b^5*e + 5*a*b^3*c*e - 5*a^2*b*c^2*e)/sqrt(b^2 - 4*a*c))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(c^(9/2)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]
    @test_int [x^3*sqrt(d + e*x)/(a + b*x + c*x^2), x, 6, (2*(b^2 - a*c)*sqrt(d + e*x))/c^3 - (2*(c*d + b*e)*(d + e*x)^(3/2))/(3*c^2*e^2) + (2*(d + e*x)^(5/2))/(5*c*e^2) + ((b^3 - 3*a*b*c - sqrt(b^2 - 4*a*c)*(b^2 - a*c))*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(sqrt(2)*c^(7/2)*sqrt(b^2 - 4*a*c)) - ((b^3 - 3*a*b*c + sqrt(b^2 - 4*a*c)*(b^2 - a*c))*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(sqrt(2)*c^(7/2)*sqrt(b^2 - 4*a*c)), (2*(b^2 - a*c)*sqrt(d + e*x))/c^3 - (2*(c*d + b*e)*(d + e*x)^(3/2))/(3*c^2*e^2) + (2*(d + e*x)^(5/2))/(5*c*e^2) - (sqrt(2)*(b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c*e - (b^3*c*d - 3*a*b*c^2*d - b^4*e + 4*a*b^2*c*e - 2*a^2*c^2*e)/sqrt(b^2 - 4*a*c))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(c^(7/2)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) - (sqrt(2)*(b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c*e + (b^3*c*d - 3*a*b*c^2*d - b^4*e + 4*a*b^2*c*e - 2*a^2*c^2*e)/sqrt(b^2 - 4*a*c))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(c^(7/2)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]
    @test_int [x^2*sqrt(d + e*x)/(a + b*x + c*x^2), x, 6, -((2*b*sqrt(d + e*x))/c^2) + (2*(d + e*x)^(3/2))/(3*c*e) + (sqrt(2)*(b*c*d - b^2*e + a*c*e - (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/sqrt(b^2 - 4*a*c))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(c^(5/2)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) + (sqrt(2)*(b*c*d - b^2*e + a*c*e + (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/sqrt(b^2 - 4*a*c))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(c^(5/2)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]
    @test_int [x^1*sqrt(d + e*x)/(a + b*x + c*x^2), x, 5, (2*sqrt(d + e*x))/c + (sqrt(2)*(b*c*d - b^2*e + 2*a*c*e - sqrt(b^2 - 4*a*c)*(c*d - b*e))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(c^(3/2)*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) - (sqrt(2)*(b*c*d - b^2*e + 2*a*c*e + sqrt(b^2 - 4*a*c)*(c*d - b*e))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(c^(3/2)*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]
    @test_int [x^0*sqrt(d + e*x)/(a + b*x + c*x^2), x, 4, -((sqrt(2)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(sqrt(c)*sqrt(b^2 - 4*a*c))) + (sqrt(2)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(sqrt(c)*sqrt(b^2 - 4*a*c))]
    @test_int [sqrt(d + e*x)/(x^1*(a + b*x + c*x^2)), x, 7, -((2*sqrt(d)*atanh(sqrt(d + e*x)/sqrt(d)))/a) + (sqrt(2)*sqrt(c)*(b*d + sqrt(b^2 - 4*a*c)*d - 2*a*e)*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(a*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) - (sqrt(2)*sqrt(c)*(b*d - sqrt(b^2 - 4*a*c)*d - 2*a*e)*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(a*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]
    @test_int [sqrt(d + e*x)/(x^2*(a + b*x + c*x^2)), x, 9, If(13>=8, -(sqrt(d + e*x)/(a*x)) + (e*atanh(sqrt(d + e*x)/sqrt(d)))/(a*sqrt(d)) + (2*(b*d - a*e)*atanh(sqrt(d + e*x)/sqrt(d)))/(a^2*sqrt(d)) - (sqrt(2)*sqrt(c)*(b^2*d - 2*a*c*d - a*b*e + sqrt(b^2 - 4*a*c)*(b*d - a*e))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(a^2*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) + (sqrt(2)*sqrt(c)*(b^2*d - b*(sqrt(b^2 - 4*a*c)*d + a*e) - a*(2*c*d - sqrt(b^2 - 4*a*c)*e))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(a^2*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)), -(sqrt(d + e*x)/(a*x)) + (e*atanh(sqrt(d + e*x)/sqrt(d)))/(a*sqrt(d)) + (2*(b*d - a*e)*atanh(sqrt(d + e*x)/sqrt(d)))/(a^2*sqrt(d)) - (sqrt(2)*sqrt(c)*(b^2*d - 2*a*c*d - a*b*e + sqrt(b^2 - 4*a*c)*(b*d - a*e))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(a^2*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) + (sqrt(2)*sqrt(c)*(b^2*d - 2*a*c*d - a*b*e - sqrt(b^2 - 4*a*c)*(b*d - a*e))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(a^2*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))]
    @test_int [sqrt(d + e*x)/(x^3*(a + b*x + c*x^2)), x, 12, -(sqrt(d + e*x)/(2*a*x^2)) + (3*e*sqrt(d + e*x))/(4*a*d*x) + ((b*d - a*e)*sqrt(d + e*x))/(a^2*d*x) - (3*e^2*atanh(sqrt(d + e*x)/sqrt(d)))/(4*a*d^(3/2)) - (e*(b*d - a*e)*atanh(sqrt(d + e*x)/sqrt(d)))/(a^2*d^(3/2)) - (2*(b^2*d - a*c*d - a*b*e)*atanh(sqrt(d + e*x)/sqrt(d)))/(a^3*sqrt(d)) + (sqrt(2)*sqrt(c)*(b^3*d - a*c*(sqrt(b^2 - 4*a*c)*d - 2*a*e) + b^2*(sqrt(b^2 - 4*a*c)*d - a*e) - a*b*(3*c*d + sqrt(b^2 - 4*a*c)*e))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(a^3*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) - (sqrt(2)*sqrt(c)*(b^3*d - b^2*(sqrt(b^2 - 4*a*c)*d + a*e) + a*c*(sqrt(b^2 - 4*a*c)*d + 2*a*e) - a*b*(3*c*d - sqrt(b^2 - 4*a*c)*e))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(a^3*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]

    @test_int [x^4*(d + e*x)^(3/2)/(a + b*x + c*x^2), x, 6, -((2*(b^3*c*d - 2*a*b*c^2*d - b^4*e + 3*a*b^2*c*e - a^2*c^2*e)*sqrt(d + e*x))/c^5) - (2*b*(b^2 - 2*a*c)*(d + e*x)^(3/2))/(3*c^4) + (2*(c^2*d^2 + b^2*e^2 + c*e*(b*d - a*e))*(d + e*x)^(5/2))/(5*c^3*e^3) - (2*(2*c*d + b*e)*(d + e*x)^(7/2))/(7*c^2*e^3) + (2*(d + e*x)^(9/2))/(9*c*e^3) + (sqrt(2)*((b*c*d - b^2*e + a*c*e)*(b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e) + (2*b^5*c*d*e - 10*a*b^3*c^2*d*e + 10*a^2*b*c^3*d*e - b^6*e^2 + a*b^2*c^2*(4*c*d^2 - 9*a*e^2) - b^4*c*(c*d^2 - 6*a*e^2) - 2*a^2*c^3*(c*d^2 - a*e^2))/sqrt(b^2 - 4*a*c))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(c^(11/2)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) + (sqrt(2)*((b*c*d - b^2*e + a*c*e)*(b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e) - (2*b^5*c*d*e - 10*a*b^3*c^2*d*e + 10*a^2*b*c^3*d*e - b^6*e^2 + a*b^2*c^2*(4*c*d^2 - 9*a*e^2) - b^4*c*(c*d^2 - 6*a*e^2) - 2*a^2*c^3*(c*d^2 - a*e^2))/sqrt(b^2 - 4*a*c))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(c^(11/2)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]
    @test_int [x^3*(d + e*x)^(3/2)/(a + b*x + c*x^2), x, 6, (2*(b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c*e)*sqrt(d + e*x))/c^4 + (2*(b^2 - a*c)*(d + e*x)^(3/2))/(3*c^3) - (2*(c*d + b*e)*(d + e*x)^(5/2))/(5*c^2*e^2) + (2*(d + e*x)^(7/2))/(7*c*e^2) + (sqrt(2)*(2*b^3*c*d*e - 4*a*b*c^2*d*e - b^4*e^2 - b^2*c*(c*d^2 - 3*a*e^2) + a*c^2*(c*d^2 - a*e^2) - (2*b^4*c*d*e - 8*a*b^2*c^2*d*e + 4*a^2*c^3*d*e - b^5*e^2 - b^3*c*(c*d^2 - 5*a*e^2) + a*b*c^2*(3*c*d^2 - 5*a*e^2))/sqrt(b^2 - 4*a*c))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(c^(9/2)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) + (sqrt(2)*(2*b^3*c*d*e - 4*a*b*c^2*d*e - b^4*e^2 - b^2*c*(c*d^2 - 3*a*e^2) + a*c^2*(c*d^2 - a*e^2) + (2*b^4*c*d*e - 8*a*b^2*c^2*d*e + 4*a^2*c^3*d*e - b^5*e^2 - b^3*c*(c*d^2 - 5*a*e^2) + a*b*c^2*(3*c*d^2 - 5*a*e^2))/sqrt(b^2 - 4*a*c))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(c^(9/2)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]
    @test_int [x^2*(d + e*x)^(3/2)/(a + b*x + c*x^2), x, 6, -((2*(b*c*d - b^2*e + a*c*e)*sqrt(d + e*x))/c^3) - (2*b*(d + e*x)^(3/2))/(3*c^2) + (2*(d + e*x)^(5/2))/(5*c*e) + (sqrt(2)*((c*d - b*e)*(b*c*d - b^2*e + 2*a*c*e) + (2*b^3*c*d*e - 6*a*b*c^2*d*e - b^4*e^2 - b^2*c*(c*d^2 - 4*a*e^2) + 2*a*c^2*(c*d^2 - a*e^2))/sqrt(b^2 - 4*a*c))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(c^(7/2)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) + (sqrt(2)*((c*d - b*e)*(b*c*d - b^2*e + 2*a*c*e) - (2*b^3*c*d*e - 6*a*b*c^2*d*e - b^4*e^2 - b^2*c*(c*d^2 - 4*a*e^2) + 2*a*c^2*(c*d^2 - a*e^2))/sqrt(b^2 - 4*a*c))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(c^(7/2)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]
    @test_int [x^1*(d + e*x)^(3/2)/(a + b*x + c*x^2), x, 6, (2*(c*d - b*e)*sqrt(d + e*x))/c^2 + (2*(d + e*x)^(3/2))/(3*c) + (sqrt(2)*(b^3*e^2 - b^2*e*(2*c*d + sqrt(b^2 - 4*a*c)*e) + c*(a*sqrt(b^2 - 4*a*c)*e^2 - c*d*(sqrt(b^2 - 4*a*c)*d - 4*a*e)) + b*c*(c*d^2 + e*(2*sqrt(b^2 - 4*a*c)*d - 3*a*e)))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(c^(5/2)*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) - (sqrt(2)*(b^3*e^2 - b^2*e*(2*c*d - sqrt(b^2 - 4*a*c)*e) + b*c*(c*d^2 - e*(2*sqrt(b^2 - 4*a*c)*d + 3*a*e)) - c*(a*sqrt(b^2 - 4*a*c)*e^2 - c*d*(sqrt(b^2 - 4*a*c)*d + 4*a*e)))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(c^(5/2)*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]
    @test_int [x^0*(d + e*x)^(3/2)/(a + b*x + c*x^2), x, 5, (2*e*sqrt(d + e*x))/c - (sqrt(2)*(2*c^2*d^2 + b*(b - sqrt(b^2 - 4*a*c))*e^2 - 2*c*e*(b*d - sqrt(b^2 - 4*a*c)*d + a*e))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(c^(3/2)*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) + (sqrt(2)*(2*c^2*d^2 + b*(b + sqrt(b^2 - 4*a*c))*e^2 - 2*c*e*(b*d + sqrt(b^2 - 4*a*c)*d + a*e))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(c^(3/2)*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]
    @test_int [(d + e*x)^(3/2)/(x^1*(a + b*x + c*x^2)), x, 7, -((2*d^(3/2)*atanh(sqrt(d + e*x)/sqrt(d)))/a) - (sqrt(2)*(a*sqrt(b^2 - 4*a*c)*e^2 - c*d*(sqrt(b^2 - 4*a*c)*d - 4*a*e) - b*(c*d^2 + a*e^2))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(a*sqrt(c)*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) - (sqrt(2)*(a*sqrt(b^2 - 4*a*c)*e^2 - c*d*(sqrt(b^2 - 4*a*c)*d + 4*a*e) + b*(c*d^2 + a*e^2))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(a*sqrt(c)*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]
    @test_int [(d + e*x)^(3/2)/(x^2*(a + b*x + c*x^2)), x, 9, -((d*sqrt(d + e*x))/(a*x)) + (sqrt(d)*e*atanh(sqrt(d + e*x)/sqrt(d)))/a + (2*sqrt(d)*(b*d - 2*a*e)*atanh(sqrt(d + e*x)/sqrt(d)))/a^2 - (sqrt(2)*sqrt(c)*(b^2*d^2 + b*d*(sqrt(b^2 - 4*a*c)*d - 2*a*e) - 2*a*(c*d^2 + e*(sqrt(b^2 - 4*a*c)*d - a*e)))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(a^2*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) + (sqrt(2)*sqrt(c)*(b^2*d^2 - b*d*(sqrt(b^2 - 4*a*c)*d + 2*a*e) - 2*a*(c*d^2 - e*(sqrt(b^2 - 4*a*c)*d + a*e)))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(a^2*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]
    @test_int [(d + e*x)^(3/2)/(x^3*(a + b*x + c*x^2)), x, 12, -((d*sqrt(d + e*x))/(2*a*x^2)) + (3*e*sqrt(d + e*x))/(4*a*x) + ((b*d - 2*a*e)*sqrt(d + e*x))/(a^2*x) - (3*e^2*atanh(sqrt(d + e*x)/sqrt(d)))/(4*a*sqrt(d)) - (e*(b*d - 2*a*e)*atanh(sqrt(d + e*x)/sqrt(d)))/(a^2*sqrt(d)) - (2*(b^2*d^2 - 2*a*b*d*e - a*(c*d^2 - a*e^2))*atanh(sqrt(d + e*x)/sqrt(d)))/(a^3*sqrt(d)) + (sqrt(2)*sqrt(c)*(b^3*d^2 + b^2*d*(sqrt(b^2 - 4*a*c)*d - 2*a*e) + a*(a*sqrt(b^2 - 4*a*c)*e^2 - c*d*(sqrt(b^2 - 4*a*c)*d - 4*a*e)) - a*b*(3*c*d^2 + e*(2*sqrt(b^2 - 4*a*c)*d - a*e)))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)))/(a^3*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)) - (sqrt(2)*sqrt(c)*(b^3*d^2 - b^2*d*(sqrt(b^2 - 4*a*c)*d + 2*a*e) - a*b*(3*c*d^2 - e*(2*sqrt(b^2 - 4*a*c)*d + a*e)) - a*(a*sqrt(b^2 - 4*a*c)*e^2 - c*d*(sqrt(b^2 - 4*a*c)*d + 4*a*e)))*atanh((sqrt(2)*sqrt(c)*sqrt(d + e*x))/sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(a^3*sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(d+e*x)^n*(a+b*x+c*x^2)^p*when*n*symbolic=#

    #= ::Subsubsection:: =#
    #=p>0=#

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^m*(e + f*x)^n/(a + b*x + c*x^2), x, 6, (2*c*x^(1 + m)*(e + f*x)^n*AppellF1(1 + m, -n, 1, 2 + m, -((f*x)/e), -((2*c*x)/(b - sqrt(b^2 - 4*a*c)))))/((1 + (f*x)/e)^n*(sqrt(b^2 - 4*a*c)*(b - sqrt(b^2 - 4*a*c))*(1 + m))) - (2*c*x^(1 + m)*(e + f*x)^n*AppellF1(1 + m, -n, 1, 2 + m, -((f*x)/e), -((2*c*x)/(b + sqrt(b^2 - 4*a*c)))))/((1 + (f*x)/e)^n*(sqrt(b^2 - 4*a*c)*(b + sqrt(b^2 - 4*a*c))*(1 + m)))]

    @test_int [x^3*(e + f*x)^n/(a + b*x + c*x^2), x, 4, -(((c*e + b*f)*(e + f*x)^(1 + n))/(c^2*f^2*(1 + n))) + (e + f*x)^(2 + n)/(c*f^2*(2 + n)) + ((a - b^2/c + (b*(b^2 - 3*a*c))/(c*sqrt(b^2 - 4*a*c)))*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b - sqrt(b^2 - 4*a*c))*f)))/(c*(2*c*e - (b - sqrt(b^2 - 4*a*c))*f)*(1 + n)) + ((a - b^2/c - (b*(b^2 - 3*a*c))/(c*sqrt(b^2 - 4*a*c)))*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b + sqrt(b^2 - 4*a*c))*f)))/(c*(2*c*e - (b + sqrt(b^2 - 4*a*c))*f)*(1 + n))]
    @test_int [x^2*(e + f*x)^n/(a + b*x + c*x^2), x, 4, (e + f*x)^(1 + n)/(c*f*(1 + n)) + ((b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b - sqrt(b^2 - 4*a*c))*f)))/(c*(2*c*e - (b - sqrt(b^2 - 4*a*c))*f)*(1 + n)) + ((b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b + sqrt(b^2 - 4*a*c))*f)))/(c*(2*c*e - (b + sqrt(b^2 - 4*a*c))*f)*(1 + n))]
    @test_int [x^1*(e + f*x)^n/(a + b*x + c*x^2), x, 4, -(((1 - b/sqrt(b^2 - 4*a*c))*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b - sqrt(b^2 - 4*a*c))*f)))/((2*c*e - (b - sqrt(b^2 - 4*a*c))*f)*(1 + n))) - ((1 + b/sqrt(b^2 - 4*a*c))*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b + sqrt(b^2 - 4*a*c))*f)))/((2*c*e - (b + sqrt(b^2 - 4*a*c))*f)*(1 + n))]
    @test_int [x^0*(e + f*x)^n/(a + b*x + c*x^2), x, 4, -((2*c*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b - sqrt(b^2 - 4*a*c))*f)))/(sqrt(b^2 - 4*a*c)*(2*c*e - (b - sqrt(b^2 - 4*a*c))*f)*(1 + n))) + (2*c*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b + sqrt(b^2 - 4*a*c))*f)))/(sqrt(b^2 - 4*a*c)*(2*c*e - (b + sqrt(b^2 - 4*a*c))*f)*(1 + n))]
    @test_int [(e + f*x)^n/(x^1*(a + b*x + c*x^2)), x, 7, (c*(1 + b/sqrt(b^2 - 4*a*c))*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b - sqrt(b^2 - 4*a*c))*f)))/(a*(2*c*e - (b - sqrt(b^2 - 4*a*c))*f)*(1 + n)) + (c*(1 - b/sqrt(b^2 - 4*a*c))*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b + sqrt(b^2 - 4*a*c))*f)))/(a*(2*c*e - (b + sqrt(b^2 - 4*a*c))*f)*(1 + n)) - ((e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, 1 + (f*x)/e))/(a*e*(1 + n))]
    @test_int [(e + f*x)^n/(x^2*(a + b*x + c*x^2)), x, 8, -((c*(b + (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b - sqrt(b^2 - 4*a*c))*f)))/(a^2*(2*c*e - (b - sqrt(b^2 - 4*a*c))*f)*(1 + n))) - (c*(b - (b^2 - 2*a*c)/sqrt(b^2 - 4*a*c))*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b + sqrt(b^2 - 4*a*c))*f)))/(a^2*(2*c*e - (b + sqrt(b^2 - 4*a*c))*f)*(1 + n)) + (b*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, 1 + (f*x)/e))/(a^2*e*(1 + n)) + (f*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(2, 1 + n, 2 + n, 1 + (f*x)/e))/(a*e^2*(1 + n))]

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(f+g*x)^n*(a+b*x+c*x^2)^p=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(f+g*x)^n*(a+b*x+c*x^2)^p*when*b=0*and*c*d^2+a*e^2=0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(f+g*x)^2*(d^2-e^2*x^2)^p=#

    #= ::Subsubsection:: =#
    #=p>0=#

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(d + e*x)^4*(f + g*x)^2/(d^2 - e^2*x^2), x, 3, -((d^2*(7*e^2*f^2 + 16*d*e*f*g + 8*d^2*g^2)*x)/e^2) - (d*(2*e^2*f^2 + 7*d*e*f*g + 4*d^2*g^2)*x^2)/e - (1/3)*(e*f + d*g)*(e*f + 7*d*g)*x^3 - (1/2)*e*g*(e*f + 2*d*g)*x^4 - (1/5)*e^2*g^2*x^5 - (8*d^3*(e*f + d*g)^2*log(d - e*x))/e^3]
    @test_int [(d + e*x)^3*(f + g*x)^2/(d^2 - e^2*x^2), x, 3, -((d*(e*f + 2*d*g)*(3*e*f + 2*d*g)*x)/e^2) - ((e^2*f^2 + 6*d*e*f*g + 4*d^2*g^2)*x^2)/(2*e) - (1/3)*g*(2*e*f + 3*d*g)*x^3 - (1/4)*e*g^2*x^4 - (4*d^2*(e*f + d*g)^2*log(d - e*x))/e^3]
    @test_int [(d + e*x)^2*(f + g*x)^2/(d^2 - e^2*x^2), x, 3, -((2*d*g*(e*f + d*g)*x)/e^2) - (d*(f + g*x)^2)/e - (f + g*x)^3/(3*g) - (2*d*(e*f + d*g)^2*log(d - e*x))/e^3]
    @test_int [(d + e*x)^1*(f + g*x)^2/(d^2 - e^2*x^2), x, 3, -((g*(e*f + d*g)*x)/e^2) - (f + g*x)^2/(2*e) - ((e*f + d*g)^2*log(d - e*x))/e^3]
    @test_int [(d + e*x)^0*(f + g*x)^2/(d^2 - e^2*x^2), x, 5, -((g^2*x)/e^2) - ((e*f + d*g)^2*log(d - e*x))/(2*d*e^3) + ((e*f - d*g)^2*log(d + e*x))/(2*d*e^3)]
    @test_int [(f + g*x)^2/((d + e*x)^1*(d^2 - e^2*x^2)), x, 3, -((e*f - d*g)^2/(2*d*e^3*(d + e*x))) - ((e*f + d*g)^2*log(d - e*x))/(4*d^2*e^3) + ((e*f - d*g)*(e*f + 3*d*g)*log(d + e*x))/(4*d^2*e^3)]
    @test_int [(f + g*x)^2/((d + e*x)^2*(d^2 - e^2*x^2)), x, 4, -((e*f - d*g)^2/(4*d*e^3*(d + e*x)^2)) - ((e*f - d*g)*(e*f + 3*d*g))/(4*d^2*e^3*(d + e*x)) + ((e*f + d*g)^2*atanh((e*x)/d))/(4*d^3*e^3)]
    @test_int [(f + g*x)^2/((d + e*x)^3*(d^2 - e^2*x^2)), x, 4, -((e*f - d*g)^2/(6*d*e^3*(d + e*x)^3)) - ((e*f - d*g)*(e*f + 3*d*g))/(8*d^2*e^3*(d + e*x)^2) - (e*f + d*g)^2/(8*d^3*e^3*(d + e*x)) + ((e*f + d*g)^2*atanh((e*x)/d))/(8*d^4*e^3)]
    @test_int [(f + g*x)^2/((d + e*x)^4*(d^2 - e^2*x^2)), x, 4, -((e*f - d*g)^2/(8*d*e^3*(d + e*x)^4)) - ((e*f - d*g)*(e*f + 3*d*g))/(12*d^2*e^3*(d + e*x)^3) - (e*f + d*g)^2/(16*d^3*e^3*(d + e*x)^2) - (e*f + d*g)^2/(16*d^4*e^3*(d + e*x)) + ((e*f + d*g)^2*atanh((e*x)/d))/(16*d^5*e^3)]

    @test_int [(d + e*x)^7*(f + g*x)^2/(d^2 - e^2*x^2)^2, x, 3, (d^3*(49*e^2*f^2 + 160*d*e*f*g + 112*d^2*g^2)*x)/e^2 + (d^2*(23*e^2*f^2 + 98*d*e*f*g + 80*d^2*g^2)*x^2)/(2*e) + (1/3)*d*(7*e^2*f^2 + 46*d*e*f*g + 49*d^2*g^2)*x^3 + (1/4)*e*(e^2*f^2 + 14*d*e*f*g + 23*d^2*g^2)*x^4 + (1/5)*e^2*g*(2*e*f + 7*d*g)*x^5 + (1/6)*e^3*g^2*x^6 + (32*d^5*(e*f + d*g)^2)/(e^3*(d - e*x)) + (16*d^4*(e*f + d*g)*(5*e*f + 9*d*g)*log(d - e*x))/e^3]
    @test_int [(d + e*x)^6*(f + g*x)^2/(d^2 - e^2*x^2)^2, x, 3, (d^2*(17*e^2*f^2 + 64*d*e*f*g + 48*d^2*g^2)*x)/e^2 + (d*(3*e^2*f^2 + 17*d*e*f*g + 16*d^2*g^2)*x^2)/e + (1/3)*(e^2*f^2 + 12*d*e*f*g + 17*d^2*g^2)*x^3 + (1/2)*e*g*(e*f + 3*d*g)*x^4 + (1/5)*e^2*g^2*x^5 + (16*d^4*(e*f + d*g)^2)/(e^3*(d - e*x)) + (32*d^3*(e*f + d*g)*(e*f + 2*d*g)*log(d - e*x))/e^3]
    @test_int [(d + e*x)^5*(f + g*x)^2/(d^2 - e^2*x^2)^2, x, 3, (d*(5*e^2*f^2 + 24*d*e*f*g + 20*d^2*g^2)*x)/e^2 + ((e^2*f^2 + 10*d*e*f*g + 12*d^2*g^2)*x^2)/(2*e) + (1/3)*g*(2*e*f + 5*d*g)*x^3 + (1/4)*e*g^2*x^4 + (8*d^3*(e*f + d*g)^2)/(e^3*(d - e*x)) + (4*d^2*(e*f + d*g)*(3*e*f + 7*d*g)*log(d - e*x))/e^3]
    @test_int [(d + e*x)^4*(f + g*x)^2/(d^2 - e^2*x^2)^2, x, 3, ((e^2*f^2 + 8*d*e*f*g + 8*d^2*g^2)*x)/e^2 + (g*(e*f + 2*d*g)*x^2)/e + (g^2*x^3)/3 + (4*d^2*(e*f + d*g)^2)/(e^3*(d - e*x)) + (4*d*(e*f + d*g)*(e*f + 3*d*g)*log(d - e*x))/e^3]
    @test_int [(d + e*x)^3*(f + g*x)^2/(d^2 - e^2*x^2)^2, x, 3, (g*(2*e*f + 3*d*g)*x)/e^2 + (g^2*x^2)/(2*e) + (2*d*(e*f + d*g)^2)/(e^3*(d - e*x)) + ((e*f + d*g)*(e*f + 5*d*g)*log(d - e*x))/e^3]
    @test_int [(d + e*x)^2*(f + g*x)^2/(d^2 - e^2*x^2)^2, x, 3, (g^2*x)/e^2 + (e*f + d*g)^2/(e^3*(d - e*x)) + (2*g*(e*f + d*g)*log(d - e*x))/e^3]
    @test_int [(d + e*x)^1*(f + g*x)^2/(d^2 - e^2*x^2)^2, x, 3, (e*f + d*g)^2/(2*d*e^3*(d - e*x)) - ((e*f - 3*d*g)*(e*f + d*g)*log(d - e*x))/(4*d^2*e^3) + ((e*f - d*g)^2*log(d + e*x))/(4*d^2*e^3)]
    @test_int [(d + e*x)^0*(f + g*x)^2/(d^2 - e^2*x^2)^2, x, 2, ((d^2*g + e^2*f*x)*(f + g*x))/(2*d^2*e^2*(d^2 - e^2*x^2)) + ((e*f - d*g)*(e*f + d*g)*atanh((e*x)/d))/(2*d^3*e^3)]
    @test_int [(f + g*x)^2/((d + e*x)^1*(d^2 - e^2*x^2)^2), x, 4, (e*f + d*g)^2/(8*d^3*e^3*(d - e*x)) - (e*f - d*g)^2/(8*d^2*e^3*(d + e*x)^2) - (e^2*f^2 - d^2*g^2)/(4*d^3*e^3*(d + e*x)) + ((3*e*f - d*g)*(e*f + d*g)*atanh((e*x)/d))/(8*d^4*e^3)]
    @test_int [(f + g*x)^2/((d + e*x)^2*(d^2 - e^2*x^2)^2), x, 4, (e*f + d*g)^2/(16*d^4*e^3*(d - e*x)) - (e*f - d*g)^2/(12*d^2*e^3*(d + e*x)^3) - (e^2*f^2 - d^2*g^2)/(8*d^3*e^3*(d + e*x)^2) - ((3*e*f - d*g)*(e*f + d*g))/(16*d^4*e^3*(d + e*x)) + (f*(e*f + d*g)*atanh((e*x)/d))/(4*d^5*e^2)]
    @test_int [(f + g*x)^2/((d + e*x)^3*(d^2 - e^2*x^2)^2), x, 4, (e*f + d*g)^2/(32*d^5*e^3*(d - e*x)) - (e*f - d*g)^2/(16*d^2*e^3*(d + e*x)^4) - (e^2*f^2 - d^2*g^2)/(12*d^3*e^3*(d + e*x)^3) - ((3*e*f - d*g)*(e*f + d*g))/(32*d^4*e^3*(d + e*x)^2) - (f*(e*f + d*g))/(8*d^5*e^2*(d + e*x)) + ((e*f + d*g)*(5*e*f + d*g)*atanh((e*x)/d))/(32*d^6*e^3)]
    @test_int [(f + g*x)^2/((d + e*x)^4*(d^2 - e^2*x^2)^2), x, 4, (e*f + d*g)^2/(64*d^6*e^3*(d - e*x)) - (e*f - d*g)^2/(20*d^2*e^3*(d + e*x)^5) - (e^2*f^2 - d^2*g^2)/(16*d^3*e^3*(d + e*x)^4) - ((3*e*f - d*g)*(e*f + d*g))/(48*d^4*e^3*(d + e*x)^3) - (f*(e*f + d*g))/(16*d^5*e^2*(d + e*x)^2) - ((e*f + d*g)*(5*e*f + d*g))/(64*d^6*e^3*(d + e*x)) + ((e*f + d*g)*(3*e*f + d*g)*atanh((e*x)/d))/(32*d^7*e^3)]

    @test_int [(d + e*x)^7*(f + g*x)^2/(d^2 - e^2*x^2)^3, x, 3, -((d*(7*e^2*f^2 + 48*d*e*f*g + 56*d^2*g^2)*x)/e^2) - ((e*f + 2*d*g)*(e*f + 12*d*g)*x^2)/(2*e) - (1/3)*g*(2*e*f + 7*d*g)*x^3 - (1/4)*e*g^2*x^4 + (8*d^4*(e*f + d*g)^2)/(e^3*(d - e*x)^2) - (32*d^3*(e*f + d*g)*(e*f + 2*d*g))/(e^3*(d - e*x)) - (8*d^2*(3*e^2*f^2 + 14*d*e*f*g + 13*d^2*g^2)*log(d - e*x))/e^3]
    @test_int [(d + e*x)^6*(f + g*x)^2/(d^2 - e^2*x^2)^3, x, 3, -(((e^2*f^2 + 12*d*e*f*g + 18*d^2*g^2)*x)/e^2) - (g*(e*f + 3*d*g)*x^2)/e - (g^2*x^3)/3 + (4*d^3*(e*f + d*g)^2)/(e^3*(d - e*x)^2) - (4*d^2*(e*f + d*g)*(3*e*f + 7*d*g))/(e^3*(d - e*x)) - (2*d*(3*e^2*f^2 + 18*d*e*f*g + 19*d^2*g^2)*log(d - e*x))/e^3]
    @test_int [(d + e*x)^5*(f + g*x)^2/(d^2 - e^2*x^2)^3, x, 3, -((g*(2*e*f + 5*d*g)*x)/e^2) - (g^2*x^2)/(2*e) + (2*d^2*(e*f + d*g)^2)/(e^3*(d - e*x)^2) - (4*d*(e*f + d*g)*(e*f + 3*d*g))/(e^3*(d - e*x)) - ((e^2*f^2 + 10*d*e*f*g + 13*d^2*g^2)*log(d - e*x))/e^3]
    @test_int [(d + e*x)^4*(f + g*x)^2/(d^2 - e^2*x^2)^3, x, 3, -((g^2*x)/e^2) + (d*(e*f + d*g)^2)/(e^3*(d - e*x)^2) - ((e*f + d*g)*(e*f + 5*d*g))/(e^3*(d - e*x)) - (2*g*(e*f + 2*d*g)*log(d - e*x))/e^3]
    @test_int [(d + e*x)^3*(f + g*x)^2/(d^2 - e^2*x^2)^3, x, 3, (e*f + d*g)^2/(2*e^3*(d - e*x)^2) - (2*g*(e*f + d*g))/(e^3*(d - e*x)) - (g^2*log(d - e*x))/e^3]
    @test_int [(d + e*x)^2*(f + g*x)^2/(d^2 - e^2*x^2)^3, x, 4, (e*f + d*g)^2/(4*d*e^3*(d - e*x)^2) + ((e*f - 3*d*g)*(e*f + d*g))/(4*d^2*e^3*(d - e*x)) + ((e*f - d*g)^2*atanh((e*x)/d))/(4*d^3*e^3)]
    @test_int [(d + e*x)^1*(f + g*x)^2/(d^2 - e^2*x^2)^3, x, 4, (e*f + d*g)^2/(8*d^2*e^3*(d - e*x)^2) + (e^2*f^2 - d^2*g^2)/(4*d^3*e^3*(d - e*x)) - (e*f - d*g)^2/(8*d^3*e^3*(d + e*x)) + ((e*f - d*g)*(3*e*f + d*g)*atanh((e*x)/d))/(8*d^4*e^3)]
    @test_int [(d + e*x)^0*(f + g*x)^2/(d^2 - e^2*x^2)^3, x, 3, ((d^2*g + e^2*f*x)*(f + g*x))/(4*d^2*e^2*(d^2 - e^2*x^2)^2) + (2*d^2*f*g + (3*e^2*f^2 - d^2*g^2)*x)/(8*d^4*e^2*(d^2 - e^2*x^2)) + ((3*e^2*f^2 - d^2*g^2)*atanh((e*x)/d))/(8*d^5*e^3)]
    @test_int [(f + g*x)^2/((d + e*x)^1*(d^2 - e^2*x^2)^3), x, 4, (e*f + d*g)^2/(32*d^4*e^3*(d - e*x)^2) + (f*(e*f + d*g))/(8*d^5*e^2*(d - e*x)) - (e*f - d*g)^2/(24*d^3*e^3*(d + e*x)^3) - ((e*f - d*g)*(3*e*f + d*g))/(32*d^4*e^3*(d + e*x)^2) - (3*e^2*f^2 - d^2*g^2)/(16*d^5*e^3*(d + e*x)) + ((5*e^2*f^2 + 2*d*e*f*g - d^2*g^2)*atanh((e*x)/d))/(16*d^6*e^3)]
    @test_int [(f + g*x)^2/((d + e*x)^2*(d^2 - e^2*x^2)^3), x, 4, (e*f + d*g)^2/(64*d^5*e^3*(d - e*x)^2) + ((e*f + d*g)*(5*e*f + d*g))/(64*d^6*e^3*(d - e*x)) - (e*f - d*g)^2/(32*d^3*e^3*(d + e*x)^4) - ((e*f - d*g)*(3*e*f + d*g))/(48*d^4*e^3*(d + e*x)^3) - (3*e^2*f^2 - d^2*g^2)/(32*d^5*e^3*(d + e*x)^2) - (5*e^2*f^2 + 2*d*e*f*g - d^2*g^2)/(32*d^6*e^3*(d + e*x)) + ((15*e^2*f^2 + 10*d*e*f*g - d^2*g^2)*atanh((e*x)/d))/(64*d^7*e^3)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(f+g*x)^n*(d^2-e^2*x^2)^(p/2)=#

    @test_int [(d + e*x)^3*(f + g*x)^5/(d^2 - e^2*x^2)^(7/2), x, 7, ((e*f + d*g)^5*(d + e*x)^3)/(5*d*e^6*(d^2 - e^2*x^2)^(5/2)) + ((2*e*f - 23*d*g)*(e*f + d*g)^4*(d + e*x)^2)/(15*d^2*e^6*(d^2 - e^2*x^2)^(3/2)) + ((e*f + d*g)^3*(2*e^2*f^2 - 21*d*e*f*g + 127*d^2*g^2)*(d + e*x))/(15*d^3*e^6*sqrt(d^2 - e^2*x^2)) + (g^4*(5*e*f + 3*d*g)*sqrt(d^2 - e^2*x^2))/e^6 + (g^5*x*sqrt(d^2 - e^2*x^2))/(2*e^5) - (g^3*(20*e^2*f^2 + 30*d*e*f*g + 13*d^2*g^2)*atan((e*x)/sqrt(d^2 - e^2*x^2)))/(2*e^6)]
    @test_int [(d + e*x)^3*(f + g*x)^4/(d^2 - e^2*x^2)^(7/2), x, 6, ((e*f + d*g)^4*(d + e*x)^3)/(5*d*e^5*(d^2 - e^2*x^2)^(5/2)) + (2*(e*f - 9*d*g)*(e*f + d*g)^3*(d + e*x)^2)/(15*d^2*e^5*(d^2 - e^2*x^2)^(3/2)) + (2*(e*f + d*g)^2*(e^2*f^2 - 8*d*e*f*g + 36*d^2*g^2)*(d + e*x))/(15*d^3*e^5*sqrt(d^2 - e^2*x^2)) + (g^4*sqrt(d^2 - e^2*x^2))/e^5 - (g^3*(4*e*f + 3*d*g)*atan((e*x)/sqrt(d^2 - e^2*x^2)))/e^5]
    @test_int [(d + e*x)^3*(f + g*x)^3/(d^2 - e^2*x^2)^(7/2), x, 5, ((e*f + d*g)^3*(d + e*x)^3)/(5*d*e^4*(d^2 - e^2*x^2)^(5/2)) + ((2*e*f - 13*d*g)*(e*f + d*g)^2*(d + e*x)^2)/(15*d^2*e^4*(d^2 - e^2*x^2)^(3/2)) + ((e*f + d*g)*(2*e^2*f^2 - 11*d*e*f*g + 32*d^2*g^2)*(d + e*x))/(15*d^3*e^4*sqrt(d^2 - e^2*x^2)) - (g^3*atan((e*x)/sqrt(d^2 - e^2*x^2)))/e^4]
    @test_int [(d + e*x)^3*(f + g*x)^2/(d^2 - e^2*x^2)^(7/2), x, 3, ((e*f + d*g)^2*(d + e*x)^3)/(5*d*e^3*(d^2 - e^2*x^2)^(5/2)) + (2*(e*f - 4*d*g)*(e*f + d*g)*(d + e*x)^2)/(15*d^2*e^3*(d^2 - e^2*x^2)^(3/2)) + ((2*e^2*f^2 - 6*d*e*f*g + 7*d^2*g^2)*(d + e*x))/(15*d^3*e^3*sqrt(d^2 - e^2*x^2))]
    @test_int [(d + e*x)^3*(f + g*x)^1/(d^2 - e^2*x^2)^(7/2), x, 3, ((e*f + d*g)*(d + e*x)^3)/(5*d*e^2*(d^2 - e^2*x^2)^(5/2)) + (2*(2*e*f - 3*d*g)*(d + e*x))/(15*d*e^2*(d^2 - e^2*x^2)^(3/2)) + ((2*e*f - 3*d*g)*x)/(15*d^3*e*sqrt(d^2 - e^2*x^2))]
    @test_int [(d + e*x)^3*(f + g*x)^0/(d^2 - e^2*x^2)^(7/2), x, 4, sqrt(d^2 - e^2*x^2)/(5*d*e*(d - e*x)^3) + (2*sqrt(d^2 - e^2*x^2))/(15*d^2*e*(d - e*x)^2) + (2*sqrt(d^2 - e^2*x^2))/(15*d^3*e*(d - e*x))]
    @test_int [(d + e*x)^3/((f + g*x)^1*(d^2 - e^2*x^2)^(7/2)), x, 6, (4*d*(d + e*x))/(5*(e*f + d*g)*(d^2 - e^2*x^2)^(5/2)) - (5*d*(e*f - d*g) - e*(e*f + 11*d*g)*x)/(15*d*(e*f + d*g)^2*(d^2 - e^2*x^2)^(3/2)) + (15*d^3*g^2 + e*(2*e^2*f^2 + 9*d*e*f*g + 22*d^2*g^2)*x)/(15*d^3*(e*f + d*g)^3*sqrt(d^2 - e^2*x^2)) + (g^3*atan((d^2*g + e^2*f*x)/(sqrt(e^2*f^2 - d^2*g^2)*sqrt(d^2 - e^2*x^2))))/((e*f + d*g)^3*sqrt(e^2*f^2 - d^2*g^2))]
    @test_int [(d + e*x)^3/((f + g*x)^2*(d^2 - e^2*x^2)^(7/2)), x, 6, (4*d*e*(d + e*x))/(5*(e*f + d*g)^2*(d^2 - e^2*x^2)^(5/2)) - (e*(5*d*(e*f - 3*d*g) - e*(e*f + 21*d*g)*x))/(15*d*(e*f + d*g)^3*(d^2 - e^2*x^2)^(3/2)) + (e*(45*d^3*g^2 + e*(2*e^2*f^2 + 14*d*e*f*g + 57*d^2*g^2)*x))/(15*d^3*(e*f + d*g)^4*sqrt(d^2 - e^2*x^2)) + (g^4*sqrt(d^2 - e^2*x^2))/((e*f - d*g)*(e*f + d*g)^4*(f + g*x)) + (e*g^3*(4*e*f - 3*d*g)*atan((d^2*g + e^2*f*x)/(sqrt(e^2*f^2 - d^2*g^2)*sqrt(d^2 - e^2*x^2))))/((e*f - d*g)*(e*f + d*g)^4*sqrt(e^2*f^2 - d^2*g^2))]
    @test_int [(d + e*x)^3/((f + g*x)^3*(d^2 - e^2*x^2)^(7/2)), x, 7, (4*d*e^2*(d + e*x))/(5*(e*f + d*g)^3*(d^2 - e^2*x^2)^(5/2)) - (e^2*(5*d*(e*f - 5*d*g) - e*(e*f + 31*d*g)*x))/(15*d*(e*f + d*g)^4*(d^2 - e^2*x^2)^(3/2)) + (e^2*(90*d^3*g^2 + e*(2*e^2*f^2 + 19*d*e*f*g + 107*d^2*g^2)*x))/(15*d^3*(e*f + d*g)^5*sqrt(d^2 - e^2*x^2)) + (g^4*sqrt(d^2 - e^2*x^2))/(2*(e*f - d*g)*(e*f + d*g)^4*(f + g*x)^2) + (3*e*g^4*(3*e*f - 2*d*g)*sqrt(d^2 - e^2*x^2))/(2*(e*f - d*g)^2*(e*f + d*g)^5*(f + g*x)) + (e^2*g^3*(20*e^2*f^2 - 30*d*e*f*g + 13*d^2*g^2)*atan((d^2*g + e^2*f*x)/(sqrt(e^2*f^2 - d^2*g^2)*sqrt(d^2 - e^2*x^2))))/(2*(e*f - d*g)^2*(e*f + d*g)^5*sqrt(e^2*f^2 - d^2*g^2))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(f+g*x)^n*(a+b*x+c*x^2)^p*when*b=0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(f+g*x)^(n/2)*(a+c*x^2)^p=#

    #= ::Subsubsection:: =#
    #=p>0 & n>0=#

    #= ::Subsubsection::Closed:: =#
    #=p>0 & n<0=#

    @test_int [(a + c*x^2)/((d + e*x)^(3/2)*(f + g*x)), x, 4, -((2*(c*d^2 + a*e^2))/(e^2*(e*f - d*g)*sqrt(d + e*x))) + (2*c*sqrt(d + e*x))/(e^2*g) - (2*(c*f^2 + a*g^2)*atan((sqrt(g)*sqrt(d + e*x))/sqrt(e*f - d*g)))/(g^(3/2)*(e*f - d*g)^(3/2))]

    @test_int [(d + e*x)^3*(a + c*x^2)/sqrt(f + g*x), x, 3, -((2*(e*f - d*g)^3*(c*f^2 + a*g^2)*sqrt(f + g*x))/g^6) + (2*(e*f - d*g)^2*(3*a*e*g^2 + c*f*(5*e*f - 2*d*g))*(f + g*x)^(3/2))/(3*g^6) - (2*(e*f - d*g)*(3*a*e^2*g^2 + c*(10*e^2*f^2 - 8*d*e*f*g + d^2*g^2))*(f + g*x)^(5/2))/(5*g^6) + (2*e*(a*e^2*g^2 + c*(10*e^2*f^2 - 12*d*e*f*g + 3*d^2*g^2))*(f + g*x)^(7/2))/(7*g^6) - (2*c*e^2*(5*e*f - 3*d*g)*(f + g*x)^(9/2))/(9*g^6) + (2*c*e^3*(f + g*x)^(11/2))/(11*g^6)]
    @test_int [(d + e*x)^2*(a + c*x^2)/sqrt(f + g*x), x, 3, (2*(e*f - d*g)^2*(c*f^2 + a*g^2)*sqrt(f + g*x))/g^5 - (4*(e*f - d*g)*(a*e*g^2 + c*f*(2*e*f - d*g))*(f + g*x)^(3/2))/(3*g^5) + (2*(a*e^2*g^2 + c*(6*e^2*f^2 - 6*d*e*f*g + d^2*g^2))*(f + g*x)^(5/2))/(5*g^5) - (4*c*e*(2*e*f - d*g)*(f + g*x)^(7/2))/(7*g^5) + (2*c*e^2*(f + g*x)^(9/2))/(9*g^5)]
    @test_int [(d + e*x)^1*(a + c*x^2)/sqrt(f + g*x), x, 2, -((2*(e*f - d*g)*(c*f^2 + a*g^2)*sqrt(f + g*x))/g^4) + (2*(a*e*g^2 + c*f*(3*e*f - 2*d*g))*(f + g*x)^(3/2))/(3*g^4) - (2*c*(3*e*f - d*g)*(f + g*x)^(5/2))/(5*g^4) + (2*c*e*(f + g*x)^(7/2))/(7*g^4)]
    @test_int [(d + e*x)^0*(a + c*x^2)/sqrt(f + g*x), x, 2, (2*(c*f^2 + a*g^2)*sqrt(f + g*x))/g^3 - (4*c*f*(f + g*x)^(3/2))/(3*g^3) + (2*c*(f + g*x)^(5/2))/(5*g^3)]
    @test_int [(a + c*x^2)/((d + e*x)^1*sqrt(f + g*x)), x, 4, -((2*c*(e*f + d*g)*sqrt(f + g*x))/(e^2*g^2)) + (2*c*(f + g*x)^(3/2))/(3*e*g^2) - (2*(c*d^2 + a*e^2)*atanh((sqrt(e)*sqrt(f + g*x))/sqrt(e*f - d*g)))/(e^(5/2)*sqrt(e*f - d*g))]
    @test_int [(a + c*x^2)/((d + e*x)^2*sqrt(f + g*x)), x, 4, (2*c*sqrt(f + g*x))/(e^2*g) - ((a + (c*d^2)/e^2)*sqrt(f + g*x))/((e*f - d*g)*(d + e*x)) + ((a*e^2*g + c*d*(4*e*f - 3*d*g))*atanh((sqrt(e)*sqrt(f + g*x))/sqrt(e*f - d*g)))/(e^(5/2)*(e*f - d*g)^(3/2))]
    @test_int [(a + c*x^2)/((d + e*x)^3*sqrt(f + g*x)), x, 4, -(((a + (c*d^2)/e^2)*sqrt(f + g*x))/(2*(e*f - d*g)*(d + e*x)^2)) + ((3*a*e^2*g + c*d*(8*e*f - 5*d*g))*sqrt(f + g*x))/(4*e^2*(e*f - d*g)^2*(d + e*x)) - ((3*a*e^2*g^2 + c*(8*e^2*f^2 - 8*d*e*f*g + 3*d^2*g^2))*atanh((sqrt(e)*sqrt(f + g*x))/sqrt(e*f - d*g)))/(4*e^(5/2)*(e*f - d*g)^(5/2))]

    @test_int [(d + e*x)^3*(a + c*x^2)/(f + g*x)^(3/2), x, 3, (2*(e*f - d*g)^3*(c*f^2 + a*g^2))/(g^6*sqrt(f + g*x)) + (2*(e*f - d*g)^2*(3*a*e*g^2 + c*f*(5*e*f - 2*d*g))*sqrt(f + g*x))/g^6 - (2*(e*f - d*g)*(3*a*e^2*g^2 + c*(10*e^2*f^2 - 8*d*e*f*g + d^2*g^2))*(f + g*x)^(3/2))/(3*g^6) + (2*e*(a*e^2*g^2 + c*(10*e^2*f^2 - 12*d*e*f*g + 3*d^2*g^2))*(f + g*x)^(5/2))/(5*g^6) - (2*c*e^2*(5*e*f - 3*d*g)*(f + g*x)^(7/2))/(7*g^6) + (2*c*e^3*(f + g*x)^(9/2))/(9*g^6)]
    @test_int [(d + e*x)^2*(a + c*x^2)/(f + g*x)^(3/2), x, 3, -((2*(e*f - d*g)^2*(c*f^2 + a*g^2))/(g^5*sqrt(f + g*x))) - (4*(e*f - d*g)*(a*e*g^2 + c*f*(2*e*f - d*g))*sqrt(f + g*x))/g^5 + (2*(a*e^2*g^2 + c*(6*e^2*f^2 - 6*d*e*f*g + d^2*g^2))*(f + g*x)^(3/2))/(3*g^5) - (4*c*e*(2*e*f - d*g)*(f + g*x)^(5/2))/(5*g^5) + (2*c*e^2*(f + g*x)^(7/2))/(7*g^5)]
    @test_int [(d + e*x)^1*(a + c*x^2)/(f + g*x)^(3/2), x, 2, (2*(e*f - d*g)*(c*f^2 + a*g^2))/(g^4*sqrt(f + g*x)) + (2*(a*e*g^2 + c*f*(3*e*f - 2*d*g))*sqrt(f + g*x))/g^4 - (2*c*(3*e*f - d*g)*(f + g*x)^(3/2))/(3*g^4) + (2*c*e*(f + g*x)^(5/2))/(5*g^4)]
    @test_int [(d + e*x)^0*(a + c*x^2)/(f + g*x)^(3/2), x, 2, -((2*(c*f^2 + a*g^2))/(g^3*sqrt(f + g*x))) - (4*c*f*sqrt(f + g*x))/g^3 + (2*c*(f + g*x)^(3/2))/(3*g^3)]
    @test_int [(a + c*x^2)/((d + e*x)^1*(f + g*x)^(3/2)), x, 4, (2*(c*f^2 + a*g^2))/(g^2*(e*f - d*g)*sqrt(f + g*x)) + (2*c*sqrt(f + g*x))/(e*g^2) - (2*(c*d^2 + a*e^2)*atanh((sqrt(e)*sqrt(f + g*x))/sqrt(e*f - d*g)))/(e^(3/2)*(e*f - d*g)^(3/2))]
    @test_int [(a + c*x^2)/((d + e*x)^2*(f + g*x)^(3/2)), x, 4, -((2*(c*f^2 + a*g^2))/(g*(e*f - d*g)^2*sqrt(f + g*x))) - ((c*d^2 + a*e^2)*sqrt(f + g*x))/(e*(e*f - d*g)^2*(d + e*x)) + ((3*a*e^2*g + c*d*(4*e*f - d*g))*atanh((sqrt(e)*sqrt(f + g*x))/sqrt(e*f - d*g)))/(e^(3/2)*(e*f - d*g)^(5/2))]
    @test_int [(a + c*x^2)/((d + e*x)^3*(f + g*x)^(3/2)), x, 5, (2*(c*f^2 + a*g^2))/((e*f - d*g)^3*sqrt(f + g*x)) - ((c*d^2 + a*e^2)*sqrt(f + g*x))/(2*e*(e*f - d*g)^2*(d + e*x)^2) + ((7*a*e^2*g + c*d*(8*e*f - d*g))*sqrt(f + g*x))/(4*e*(e*f - d*g)^3*(d + e*x)) - ((15*a*e^2*g^2 + c*(8*e^2*f^2 + 8*d*e*f*g - d^2*g^2))*atanh((sqrt(e)*sqrt(f + g*x))/sqrt(e*f - d*g)))/(4*e^(3/2)*(e*f - d*g)^(7/2))]

    #= ::Subsubsection:: =#
    #=p<0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^(m/2)*(f+g*x)^(n/2)*(a+c*x^2)^p=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [(a + c*x^2)/(sqrt(d + e*x)*sqrt(f + g*x)), x, 5, -((c*(3*e*f + 5*d*g)*sqrt(d + e*x)*sqrt(f + g*x))/(4*e^2*g^2)) + (c*(d + e*x)^(3/2)*sqrt(f + g*x))/(2*e^2*g) + ((8*a*e^2*g^2 + c*(3*e^2*f^2 + 2*d*e*f*g + 3*d^2*g^2))*atanh((sqrt(g)*sqrt(d + e*x))/(sqrt(e)*sqrt(f + g*x))))/(4*e^(5/2)*g^(5/2))]

    @test_int [(-1 + 2*x^2)/(sqrt(-1 + x)*sqrt(1 + x)), x, 1, sqrt(-1 + x)*x*sqrt(1 + x)]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(d + e*x)^(3/2)*sqrt(f + g*x)/(a + c*x^2), x, 11, (e*sqrt(d + e*x)*sqrt(f + g*x))/c + (sqrt(e)*(e*f + 3*d*g)*atanh((sqrt(g)*sqrt(d + e*x))/(sqrt(e)*sqrt(f + g*x))))/(c*sqrt(g)) + (((a*(a*e^2*g - c*d*(2*e*f + d*g)))/sqrt(c) - sqrt(-a)*(c*d^2*f - a*e*(e*f + 2*d*g)))*atanh((sqrt(sqrt(c)*f - sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d - sqrt(-a)*e)*sqrt(f + g*x))))/(a*c*sqrt(sqrt(c)*d - sqrt(-a)*e)*sqrt(sqrt(c)*f - sqrt(-a)*g)) + (((a*(a*e^2*g - c*d*(2*e*f + d*g)))/sqrt(c) + sqrt(-a)*(c*d^2*f - a*e*(e*f + 2*d*g)))*atanh((sqrt(sqrt(c)*f + sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d + sqrt(-a)*e)*sqrt(f + g*x))))/(a*c*sqrt(sqrt(c)*d + sqrt(-a)*e)*sqrt(sqrt(c)*f + sqrt(-a)*g))]
    @test_int [(d + e*x)^(1/2)*sqrt(f + g*x)/(a + c*x^2), x, 10, (2*sqrt(e)*sqrt(g)*atanh((sqrt(g)*sqrt(d + e*x))/(sqrt(e)*sqrt(f + g*x))))/c + ((c*d*f - a*e*g - sqrt(-a)*sqrt(c)*(e*f + d*g))*atanh((sqrt(sqrt(c)*f - sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d - sqrt(-a)*e)*sqrt(f + g*x))))/(sqrt(-a)*c*sqrt(sqrt(c)*d - sqrt(-a)*e)*sqrt(sqrt(c)*f - sqrt(-a)*g)) - ((c*d*f - a*e*g + sqrt(-a)*sqrt(c)*(e*f + d*g))*atanh((sqrt(sqrt(c)*f + sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d + sqrt(-a)*e)*sqrt(f + g*x))))/(sqrt(-a)*c*sqrt(sqrt(c)*d + sqrt(-a)*e)*sqrt(sqrt(c)*f + sqrt(-a)*g))]
    @test_int [sqrt(f + g*x)/((d + e*x)^(1/2)*(a + c*x^2)), x, 6, (sqrt(sqrt(c)*f - sqrt(-a)*g)*atanh((sqrt(sqrt(c)*f - sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d - sqrt(-a)*e)*sqrt(f + g*x))))/(sqrt(-a)*sqrt(c)*sqrt(sqrt(c)*d - sqrt(-a)*e)) - (sqrt(sqrt(c)*f + sqrt(-a)*g)*atanh((sqrt(sqrt(c)*f + sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d + sqrt(-a)*e)*sqrt(f + g*x))))/(sqrt(-a)*sqrt(c)*sqrt(sqrt(c)*d + sqrt(-a)*e))]
    @test_int [sqrt(f + g*x)/((d + e*x)^(3/2)*(a + c*x^2)), x, 8, -((2*e*sqrt(f + g*x))/((c*d^2 + a*e^2)*sqrt(d + e*x))) + ((c*d*f + a*e*g + sqrt(-a)*sqrt(c)*(e*f - d*g))*atanh((sqrt(sqrt(c)*f - sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d - sqrt(-a)*e)*sqrt(f + g*x))))/(sqrt(-a)*sqrt(sqrt(c)*d - sqrt(-a)*e)*(c*d^2 + a*e^2)*sqrt(sqrt(c)*f - sqrt(-a)*g)) - ((c*d*f + a*e*g - sqrt(-a)*sqrt(c)*(e*f - d*g))*atanh((sqrt(sqrt(c)*f + sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d + sqrt(-a)*e)*sqrt(f + g*x))))/(sqrt(-a)*sqrt(sqrt(c)*d + sqrt(-a)*e)*(c*d^2 + a*e^2)*sqrt(sqrt(c)*f + sqrt(-a)*g))]
    @test_int [sqrt(f + g*x)/((d + e*x)^(5/2)*(a + c*x^2)), x, 11, -((2*e*sqrt(f + g*x))/(3*(c*d^2 + a*e^2)*(d + e*x)^(3/2))) + (4*e*g*sqrt(f + g*x))/(3*(c*d^2 + a*e^2)*(e*f - d*g)*sqrt(d + e*x)) + (e*(c*d*f + a*e*g - sqrt(-a)*sqrt(c)*(e*f - d*g))*sqrt(f + g*x))/(sqrt(-a)*(sqrt(c)*d + sqrt(-a)*e)*(c*d^2 + a*e^2)*(e*f - d*g)*sqrt(d + e*x)) - (e*(c*d*f + a*e*g + sqrt(-a)*sqrt(c)*(e*f - d*g))*sqrt(f + g*x))/(sqrt(-a)*(sqrt(c)*d - sqrt(-a)*e)*(c*d^2 + a*e^2)*(e*f - d*g)*sqrt(d + e*x)) + (sqrt(c)*(c*d*f + a*e*g + sqrt(-a)*sqrt(c)*(e*f - d*g))*atanh((sqrt(sqrt(c)*f - sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d - sqrt(-a)*e)*sqrt(f + g*x))))/(sqrt(-a)*(sqrt(c)*d - sqrt(-a)*e)^(3/2)*(c*d^2 + a*e^2)*sqrt(sqrt(c)*f - sqrt(-a)*g)) + (sqrt(c)*(sqrt(-a)*c*d*f + sqrt(-a)*a*e*g + a*sqrt(c)*(e*f - d*g))*atanh((sqrt(sqrt(c)*f + sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d + sqrt(-a)*e)*sqrt(f + g*x))))/(a*(sqrt(c)*d + sqrt(-a)*e)^(3/2)*(c*d^2 + a*e^2)*sqrt(sqrt(c)*f + sqrt(-a)*g))]

    @test_int [(d + e*x)^(3/2)/(sqrt(f + g*x)*(a + c*x^2)), x, 11, (2*e^(3/2)*atanh((sqrt(g)*sqrt(d + e*x))/(sqrt(e)*sqrt(f + g*x))))/(c*sqrt(g)) + ((c*d^2 - 2*sqrt(-a)*sqrt(c)*d*e - a*e^2)*atanh((sqrt(sqrt(c)*f - sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d - sqrt(-a)*e)*sqrt(f + g*x))))/(sqrt(-a)*c*sqrt(sqrt(c)*d - sqrt(-a)*e)*sqrt(sqrt(c)*f - sqrt(-a)*g)) - ((c*d^2 + 2*sqrt(-a)*sqrt(c)*d*e - a*e^2)*atanh((sqrt(sqrt(c)*f + sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d + sqrt(-a)*e)*sqrt(f + g*x))))/(sqrt(-a)*c*sqrt(sqrt(c)*d + sqrt(-a)*e)*sqrt(sqrt(c)*f + sqrt(-a)*g))]
    @test_int [(d + e*x)^(1/2)/(sqrt(f + g*x)*(a + c*x^2)), x, 6, (sqrt(sqrt(c)*d - sqrt(-a)*e)*atanh((sqrt(sqrt(c)*f - sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d - sqrt(-a)*e)*sqrt(f + g*x))))/(sqrt(-a)*sqrt(c)*sqrt(sqrt(c)*f - sqrt(-a)*g)) - (sqrt(sqrt(c)*d + sqrt(-a)*e)*atanh((sqrt(sqrt(c)*f + sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d + sqrt(-a)*e)*sqrt(f + g*x))))/(sqrt(-a)*sqrt(c)*sqrt(sqrt(c)*f + sqrt(-a)*g))]
    @test_int [1/((d + e*x)^(1/2)*sqrt(f + g*x)*(a + c*x^2)), x, 6, atanh((sqrt(sqrt(c)*f - sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d - sqrt(-a)*e)*sqrt(f + g*x)))/(sqrt(-a)*sqrt(sqrt(c)*d - sqrt(-a)*e)*sqrt(sqrt(c)*f - sqrt(-a)*g)) - atanh((sqrt(sqrt(c)*f + sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d + sqrt(-a)*e)*sqrt(f + g*x)))/(sqrt(-a)*sqrt(sqrt(c)*d + sqrt(-a)*e)*sqrt(sqrt(c)*f + sqrt(-a)*g))]
    @test_int [1/((d + e*x)^(3/2)*sqrt(f + g*x)*(a + c*x^2)), x, 8, -((e*sqrt(f + g*x))/(sqrt(-a)*(sqrt(c)*d - sqrt(-a)*e)*(e*f - d*g)*sqrt(d + e*x))) + (e*sqrt(f + g*x))/(sqrt(-a)*(sqrt(c)*d + sqrt(-a)*e)*(e*f - d*g)*sqrt(d + e*x)) + (sqrt(c)*atanh((sqrt(sqrt(c)*f - sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d - sqrt(-a)*e)*sqrt(f + g*x))))/(sqrt(-a)*(sqrt(c)*d - sqrt(-a)*e)^(3/2)*sqrt(sqrt(c)*f - sqrt(-a)*g)) - (sqrt(c)*atanh((sqrt(sqrt(c)*f + sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d + sqrt(-a)*e)*sqrt(f + g*x))))/(sqrt(-a)*(sqrt(c)*d + sqrt(-a)*e)^(3/2)*sqrt(sqrt(c)*f + sqrt(-a)*g))]

    @test_int [(d + e*x)^(3/2)/((f + g*x)^(3/2)*(a + c*x^2)), x, 21, (2*(e*f - d*g)*sqrt(d + e*x))/((c*f^2 + a*g^2)*sqrt(f + g*x)) - (2*sqrt(e)*(e*f - d*g)*atanh((sqrt(g)*sqrt(d + e*x))/(sqrt(e)*sqrt(f + g*x))))/(sqrt(g)*(c*f^2 + a*g^2)) - (sqrt(e)*(c*d*f + a*e*g - sqrt(-a)*sqrt(c)*(e*f - d*g))*atanh((sqrt(g)*sqrt(d + e*x))/(sqrt(e)*sqrt(f + g*x))))/(sqrt(-a)*sqrt(c)*sqrt(g)*(c*f^2 + a*g^2)) + (sqrt(e)*(c*d*f + a*e*g + sqrt(-a)*sqrt(c)*(e*f - d*g))*atanh((sqrt(g)*sqrt(d + e*x))/(sqrt(e)*sqrt(f + g*x))))/(sqrt(-a)*sqrt(c)*sqrt(g)*(c*f^2 + a*g^2)) + (sqrt(sqrt(c)*d - sqrt(-a)*e)*(c*d*f + a*e*g - sqrt(-a)*sqrt(c)*(e*f - d*g))*atanh((sqrt(sqrt(c)*f - sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d - sqrt(-a)*e)*sqrt(f + g*x))))/(sqrt(-a)*sqrt(c)*sqrt(sqrt(c)*f - sqrt(-a)*g)*(c*f^2 + a*g^2)) - (sqrt(sqrt(c)*d + sqrt(-a)*e)*(c*d*f + a*e*g + sqrt(-a)*sqrt(c)*(e*f - d*g))*atanh((sqrt(sqrt(c)*f + sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d + sqrt(-a)*e)*sqrt(f + g*x))))/(sqrt(-a)*sqrt(c)*sqrt(sqrt(c)*f + sqrt(-a)*g)*(c*f^2 + a*g^2))]
    @test_int [(d + e*x)^(1/2)/((f + g*x)^(3/2)*(a + c*x^2)), x, 8, -((2*g*sqrt(d + e*x))/((c*f^2 + a*g^2)*sqrt(f + g*x))) + ((c*d*f + a*e*g - sqrt(-a)*sqrt(c)*(e*f - d*g))*atanh((sqrt(sqrt(c)*f - sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d - sqrt(-a)*e)*sqrt(f + g*x))))/(sqrt(-a)*sqrt(sqrt(c)*d - sqrt(-a)*e)*sqrt(sqrt(c)*f - sqrt(-a)*g)*(c*f^2 + a*g^2)) - ((c*d*f + a*e*g + sqrt(-a)*sqrt(c)*(e*f - d*g))*atanh((sqrt(sqrt(c)*f + sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d + sqrt(-a)*e)*sqrt(f + g*x))))/(sqrt(-a)*sqrt(sqrt(c)*d + sqrt(-a)*e)*sqrt(sqrt(c)*f + sqrt(-a)*g)*(c*f^2 + a*g^2))]
    @test_int [1/((d + e*x)^(1/2)*(f + g*x)^(3/2)*(a + c*x^2)), x, 8, (g*sqrt(d + e*x))/(sqrt(-a)*(sqrt(c)*f - sqrt(-a)*g)*(e*f - d*g)*sqrt(f + g*x)) - (g*sqrt(d + e*x))/(sqrt(-a)*(sqrt(c)*f + sqrt(-a)*g)*(e*f - d*g)*sqrt(f + g*x)) + (sqrt(c)*atanh((sqrt(sqrt(c)*f - sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d - sqrt(-a)*e)*sqrt(f + g*x))))/(sqrt(-a)*sqrt(sqrt(c)*d - sqrt(-a)*e)*(sqrt(c)*f - sqrt(-a)*g)^(3/2)) - (sqrt(c)*atanh((sqrt(sqrt(c)*f + sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d + sqrt(-a)*e)*sqrt(f + g*x))))/(sqrt(-a)*sqrt(sqrt(c)*d + sqrt(-a)*e)*(sqrt(c)*f + sqrt(-a)*g)^(3/2))]
    @test_int [1/((d + e*x)^(3/2)*(f + g*x)^(3/2)*(a + c*x^2)), x, 12, -(e/(sqrt(-a)*(sqrt(c)*d - sqrt(-a)*e)*(e*f - d*g)*sqrt(d + e*x)*sqrt(f + g*x))) + e/(sqrt(-a)*(sqrt(c)*d + sqrt(-a)*e)*(e*f - d*g)*sqrt(d + e*x)*sqrt(f + g*x)) + (g*(2*sqrt(-a)*e*g - sqrt(c)*(e*f + d*g))*sqrt(d + e*x))/(sqrt(-a)*(sqrt(c)*d - sqrt(-a)*e)*(sqrt(c)*f - sqrt(-a)*g)*(e*f - d*g)^2*sqrt(f + g*x)) + (g*(2*sqrt(-a)*e*g + sqrt(c)*(e*f + d*g))*sqrt(d + e*x))/(sqrt(-a)*(sqrt(c)*d + sqrt(-a)*e)*(sqrt(c)*f + sqrt(-a)*g)*(e*f - d*g)^2*sqrt(f + g*x)) + (c*atanh((sqrt(sqrt(c)*f - sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d - sqrt(-a)*e)*sqrt(f + g*x))))/(sqrt(-a)*(sqrt(c)*d - sqrt(-a)*e)^(3/2)*(sqrt(c)*f - sqrt(-a)*g)^(3/2)) - (c*atanh((sqrt(sqrt(c)*f + sqrt(-a)*g)*sqrt(d + e*x))/(sqrt(sqrt(c)*d + sqrt(-a)*e)*sqrt(f + g*x))))/(sqrt(-a)*(sqrt(c)*d + sqrt(-a)*e)^(3/2)*(sqrt(c)*f + sqrt(-a)*g)^(3/2))]

    @test_int [sqrt(x)/(sqrt(1 + x)*(1 + x^2)), x, 6, (-(1/2))*(1 - im)^(3/2)*atanh((sqrt(1 - im)*sqrt(x))/sqrt(1 + x)) - (1/2)*(1 + im)^(3/2)*atanh((sqrt(1 + im)*sqrt(x))/sqrt(1 + x))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(f+g*x)^n*(a+c*x^2)^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [(f + g*x)^2*sqrt(1 - x^2)/(1 - x)^4, x, 5, ((f + g)^2*(1 + x)^4)/(5*(1 - x^2)^(5/2)) + ((f - 9*g)*(f + g)*(1 + x)^3)/(15*(1 - x^2)^(3/2)) + (2*g^2*(1 + x))/sqrt(1 - x^2) - g^2*asin(x)]

    @test_int [(1 - a^2*x^2)^(3/2)/((c + d*x)*(1 - a*x)^2), x, 6, -(sqrt(1 - a^2*x^2)/d) - ((a*c - 2*d)*asin(a*x))/d^2 + ((a*c - d)^2*atan((d + a^2*c*x)/(sqrt(a^2*c^2 - d^2)*sqrt(1 - a^2*x^2))))/(d^2*sqrt(a^2*c^2 - d^2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(1 + a*x)^2/((c + d*x)*sqrt(1 - a^2*x^2)), x, 5, -(sqrt(1 - a^2*x^2)/d) - ((a*c - 2*d)*asin(a*x))/d^2 + ((a*c - d)^2*atan((d + a^2*c*x)/(sqrt(a^2*c^2 - d^2)*sqrt(1 - a^2*x^2))))/(d^2*sqrt(a^2*c^2 - d^2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(f+g*x)^(n/2)*(a+c*x^2)^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0 & n>0=#

    @test_int [(d + e*x)^3*sqrt(f + g*x)*sqrt(a + c*x^2), x, 10, (-2*(150*a^2*e^4*g^4 - 6*a*c*e^2*g^2*(2*e^2*f^2 - 33*d*e*f*g + 165*d^2*g^2) + c^2*(187*e^4*f^4 - 732*d*e^3*f^3*g + 1098*d^2*e^2*f^2*g^2 - 798*d^3*e*f*g^3 + 315*d^4*g^4))*sqrt(f + g*x)*sqrt(a + c*x^2))/(3465*c^2*e*g^4) + (2*(d + e*x)^4*sqrt(f + g*x)*sqrt(a + c*x^2))/(11*e) - (2*(2*a*e^2*g^2*(74*e*f - 231*d*g) - c*(233*e^3*f^3 - 843*d*e^2*f^2*g + 1107*d^2*e*f*g^2 - 567*d^3*g^3))*(f + g*x)^(3/2)*sqrt(a + c*x^2))/(3465*c*g^4) + (2*e*(18*a*e^2*g^2 - c*(29*e^2*f^2 - 96*d*e*f*g + 81*d^2*g^2))*(f + g*x)^(5/2)*sqrt(a + c*x^2))/(693*c*g^4) + (2*e^2*(e*f - 3*d*g)*(f + g*x)^(7/2)*sqrt(a + c*x^2))/(99*g^4) + (4*sqrt(-a)*(3*a^2*e^2*g^4*(26*e*f + 231*d*g) - c^2*f^2*(64*e^3*f^3 - 264*d*e^2*f^2*g + 396*d^2*e*f*g^2 - 231*d^3*g^3) - 9*a*c*g^2*(6*e^3*f^3 - 33*d*e^2*f^2*g + 88*d^2*e*f*g^2 + 77*d^3*g^3))*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(3465*c^(3/2)*g^5*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) - (4*sqrt(-a)*(c*f^2 + a*g^2)*(75*a^2*e^3*g^4 - 3*a*c*e*g^2*(2*e^2*f^2 - 33*d*e*f*g + 165*d^2*g^2) - c^2*f*(64*e^3*f^3 - 264*d*e^2*f^2*g + 396*d^2*e*f*g^2 - 231*d^3*g^3))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(3465*c^(5/2)*g^5*sqrt(f + g*x)*sqrt(a + c*x^2))]
    @test_int [(d + e*x)^2*sqrt(f + g*x)*sqrt(a + c*x^2), x, 9, (-2*(6*a*e^2*g^2*(e*f - 10*d*g) - c*(19*e^3*f^3 - 57*d*e^2*f^2*g + 63*d^2*e*f*g^2 - 35*d^3*g^3))*sqrt(f + g*x)*sqrt(a + c*x^2))/(315*c*e*g^3) + (2*(d + e*x)^3*sqrt(f + g*x)*sqrt(a + c*x^2))/(9*e) + (4*(7*a*e^2*g^2 - c*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2))*(f + g*x)^(3/2)*sqrt(a + c*x^2))/(315*c*g^3) + (2*e*(e*f - 3*d*g)*(f + g*x)^(5/2)*sqrt(a + c*x^2))/(63*g^3) + (4*sqrt(-a)*(21*a^2*e^2*g^4 + 3*a*c*g^2*(3*e^2*f^2 - 16*d*e*f*g - 21*d^2*g^2) + c^2*f^2*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2))*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(315*c^(3/2)*g^4*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) - (4*sqrt(-a)*(c*f^2 + a*g^2)*(3*a*e*g^2*(e*f - 10*d*g) + c*f*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(315*c^(3/2)*g^4*sqrt(f + g*x)*sqrt(a + c*x^2))]
    @test_int [(d + e*x)^1*sqrt(f + g*x)*sqrt(a + c*x^2), x, 7, (-2*sqrt(f + g*x)*(5*a*e*g^2 + c*f*(4*e*f - 7*d*g) - 3*c*g*(e*f + 7*d*g)*x)*sqrt(a + c*x^2))/(105*c*g^2) + (2*e*sqrt(f + g*x)*(a + c*x^2)^(3/2))/(7*c) - (4*sqrt(-a)*(c*f^2*(4*e*f - 7*d*g) + a*g^2*(8*e*f + 21*d*g))*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(105*sqrt(c)*g^3*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) + (4*sqrt(-a)*(c*f^2 + a*g^2)*(5*a*e*g^2 + c*f*(4*e*f - 7*d*g))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(105*c^(3/2)*g^3*sqrt(f + g*x)*sqrt(a + c*x^2))]
    @test_int [(d + e*x)^0*sqrt(f + g*x)*sqrt(a + c*x^2), x, 7, (-4*f*sqrt(f + g*x)*sqrt(a + c*x^2))/(15*g) + (2*(f + g*x)^(3/2)*sqrt(a + c*x^2))/(5*g) + (4*sqrt(-a)*(c*f^2 - 3*a*g^2)*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(15*sqrt(c)*g^2*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) - (4*sqrt(-a)*f*(c*f^2 + a*g^2)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(15*sqrt(c)*g^2*sqrt(f + g*x)*sqrt(a + c*x^2))]
    @test_int [sqrt(f + g*x)*sqrt(a + c*x^2)/(d + e*x)^1, x, 14, (2*sqrt(f + g*x)*sqrt(a + c*x^2))/(3*e) - (2*sqrt(-a)*sqrt(c)*(e*f - 3*d*g)*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(3*e^2*g*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) + (2*sqrt(-a)*sqrt(c)*f*(e*f - 3*d*g)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(3*e^2*g*sqrt(f + g*x)*sqrt(a + c*x^2)) - (2*sqrt(-a)*(2*a*e^2*g - 3*c*d*(e*f - d*g))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(3*sqrt(c)*e^3*sqrt(f + g*x)*sqrt(a + c*x^2)) - (2*(c*d^2 + a*e^2)*(e*f - d*g)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.Pi((2*e)/((sqrt(c)*d)/sqrt(-a) + e), asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (2*sqrt(-a)*g)/(sqrt(c)*f + sqrt(-a)*g)))/(e^3*((sqrt(c)*d)/sqrt(-a) + e)*sqrt(f + g*x)*sqrt(a + c*x^2))]
    @test_int [sqrt(f + g*x)*sqrt(a + c*x^2)/(d + e*x)^2, x, 14, -((sqrt(f + g*x)*sqrt(a + c*x^2))/(e*(d + e*x))) - (3*sqrt(-a)*sqrt(c)*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(e^2*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) + (3*sqrt(-a)*sqrt(c)*f*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(e^2*sqrt(f + g*x)*sqrt(a + c*x^2)) - (sqrt(-a)*sqrt(c)*(2*e*f - 3*d*g)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(e^3*sqrt(f + g*x)*sqrt(a + c*x^2)) - ((a*e^2*g - c*d*(2*e*f - 3*d*g))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.Pi((2*e)/((sqrt(c)*d)/sqrt(-a) + e), asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (2*sqrt(-a)*g)/(sqrt(c)*f + sqrt(-a)*g)))/(e^3*((sqrt(c)*d)/sqrt(-a) + e)*sqrt(f + g*x)*sqrt(a + c*x^2))]
    @test_int [sqrt(f + g*x)*sqrt(a + c*x^2)/(d + e*x)^3, x, 23, -((sqrt(f + g*x)*sqrt(a + c*x^2))/(2*e*(d + e*x)^2)) - ((a*e^2*g - c*d*(2*e*f - 3*d*g))*sqrt(f + g*x)*sqrt(a + c*x^2))/(4*e*(c*d^2 + a*e^2)*(e*f - d*g)*(d + e*x)) - (sqrt(-a)*sqrt(c)*(a*e^2*g - c*d*(2*e*f - 3*d*g))*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(4*e^2*(c*d^2 + a*e^2)*(e*f - d*g)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) - (3*sqrt(-a)*sqrt(c)*g*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(2*e^3*sqrt(f + g*x)*sqrt(a + c*x^2)) + (sqrt(-a)*sqrt(c)*f*(a*e^2*g - c*d*(2*e*f - 3*d*g))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(4*e^2*(c*d^2 + a*e^2)*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + c*x^2)) - (sqrt(-a)*sqrt(c)*d*g*(a*e^2*g - c*d*(2*e*f - 3*d*g))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(4*e^3*(c*d^2 + a*e^2)*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + c*x^2)) - (c*(e*f - 3*d*g)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.Pi((2*e)/((sqrt(c)*d)/sqrt(-a) + e), asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (2*sqrt(-a)*g)/(sqrt(c)*f + sqrt(-a)*g)))/(e^3*((sqrt(c)*d)/sqrt(-a) + e)*sqrt(f + g*x)*sqrt(a + c*x^2)) + ((a*e^2*g - c*d*(2*e*f - 3*d*g))^2*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.Pi((2*e)/((sqrt(c)*d)/sqrt(-a) + e), asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (2*sqrt(-a)*g)/(sqrt(c)*f + sqrt(-a)*g)))/(4*e^3*((sqrt(c)*d)/sqrt(-a) + e)*(c*d^2 + a*e^2)*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + c*x^2))]

    #= ::Subsubsection::Closed:: =#
    #=p>0 & n<0=#

    @test_int [(d + e*x)^3*sqrt(a + c*x^2)/sqrt(f + g*x), x, 9, (-4*(9*a*e^2*g^2*(2*e*f - 5*d*g) + c*(76*e^3*f^3 - 204*d*e^2*f^2*g + 168*d^2*e*f*g^2 - 35*d^3*g^3))*sqrt(f + g*x)*sqrt(a + c*x^2))/(315*c*g^4) + (2*(d + e*x)^3*sqrt(f + g*x)*sqrt(a + c*x^2))/(9*g) + (4*e*(7*a*e^2*g^2 + c*(64*e^2*f^2 - 111*d*e*f*g + 42*d^2*g^2))*(f + g*x)^(3/2)*sqrt(a + c*x^2))/(315*c*g^4) - (4*e^2*(4*e*f - 3*d*g)*(f + g*x)^(5/2)*sqrt(a + c*x^2))/(63*g^4) + (4*sqrt(-a)*(21*a^2*e^3*g^4 - 3*a*c*e*g^2*(10*e^2*f^2 - 39*d*e*f*g + 63*d^2*g^2) - c^2*f*(64*e^3*f^3 - 216*d*e^2*f^2*g + 252*d^2*e*f*g^2 - 105*d^3*g^3))*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(315*c^(3/2)*g^5*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) - (4*sqrt(-a)*(c*f^2 + a*g^2)*(9*a*e^2*g^2*(2*e*f - 5*d*g) - c*(64*e^3*f^3 - 216*d*e^2*f^2*g + 252*d^2*e*f*g^2 - 105*d^3*g^3))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(315*c^(3/2)*g^5*sqrt(f + g*x)*sqrt(a + c*x^2))]
    @test_int [(d + e*x)^2*sqrt(a + c*x^2)/sqrt(f + g*x), x, 8, (4*(5*a*e^2*g^2 + c*(21*e^2*f^2 - 34*d*e*f*g + 10*d^2*g^2))*sqrt(f + g*x)*sqrt(a + c*x^2))/(105*c*g^3) + (2*(d + e*x)^2*sqrt(f + g*x)*sqrt(a + c*x^2))/(7*g) - (4*e*(3*e*f - 2*d*g)*(f + g*x)^(3/2)*sqrt(a + c*x^2))/(35*g^3) + (4*sqrt(-a)*(a*e*g^2*(13*e*f - 42*d*g) + c*f*(24*e^2*f^2 - 56*d*e*f*g + 35*d^2*g^2))*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(105*sqrt(c)*g^4*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) + (4*sqrt(-a)*(c*f^2 + a*g^2)*(5*a*e^2*g^2 - c*(24*e^2*f^2 - 56*d*e*f*g + 35*d^2*g^2))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(105*c^(3/2)*g^4*sqrt(f + g*x)*sqrt(a + c*x^2))]
    @test_int [(d + e*x)^1*sqrt(a + c*x^2)/sqrt(f + g*x), x, 6, (-2*sqrt(f + g*x)*(4*e*f - 5*d*g - 3*e*g*x)*sqrt(a + c*x^2))/(15*g^2) - (4*sqrt(-a)*(3*a*e*g^2 + c*f*(4*e*f - 5*d*g))*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(15*sqrt(c)*g^3*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) + (4*sqrt(-a)*(4*e*f - 5*d*g)*(c*f^2 + a*g^2)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(15*sqrt(c)*g^3*sqrt(f + g*x)*sqrt(a + c*x^2))]
    @test_int [(d + e*x)^0*sqrt(a + c*x^2)/sqrt(f + g*x), x, 6, (2*sqrt(f + g*x)*sqrt(a + c*x^2))/(3*g) + (4*sqrt(-a)*sqrt(c)*f*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(3*g^2*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) - (4*sqrt(-a)*(c*f^2 + a*g^2)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(3*sqrt(c)*g^2*sqrt(f + g*x)*sqrt(a + c*x^2))]
    @test_int [sqrt(a + c*x^2)/((d + e*x)^1*sqrt(f + g*x)), x, 10, (-2*sqrt(-a)*sqrt(c)*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(e*g*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) + (2*sqrt(-a)*sqrt(c)*(e*f + d*g)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(e^2*g*sqrt(f + g*x)*sqrt(a + c*x^2)) - (2*(c*d^2 + a*e^2)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.Pi((2*e)/((sqrt(c)*d)/sqrt(-a) + e), asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (2*sqrt(-a)*g)/(sqrt(c)*f + sqrt(-a)*g)))/(e^2*((sqrt(c)*d)/sqrt(-a) + e)*sqrt(f + g*x)*sqrt(a + c*x^2))]
    @test_int [sqrt(a + c*x^2)/((d + e*x)^2*sqrt(f + g*x)), x, 14, -((sqrt(f + g*x)*sqrt(a + c*x^2))/((e*f - d*g)*(d + e*x))) - (sqrt(-a)*sqrt(c)*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(e*(e*f - d*g)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) + (sqrt(-a)*sqrt(c)*f*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(e*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + c*x^2)) - (sqrt(-a)*sqrt(c)*(2*e*f - d*g)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(e^2*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + c*x^2)) + ((a*e^2*g + c*d*(2*e*f - d*g))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.Pi((2*e)/((sqrt(c)*d)/sqrt(-a) + e), asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (2*sqrt(-a)*g)/(sqrt(c)*f + sqrt(-a)*g)))/(e^2*((sqrt(c)*d)/sqrt(-a) + e)*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + c*x^2))]
    @test_int [sqrt(a + c*x^2)/((d + e*x)^3*sqrt(f + g*x)), x, 23, -((sqrt(f + g*x)*sqrt(a + c*x^2))/(2*(e*f - d*g)*(d + e*x)^2)) + ((3*a*e^2*g + c*d*(2*e*f + d*g))*sqrt(f + g*x)*sqrt(a + c*x^2))/(4*(c*d^2 + a*e^2)*(e*f - d*g)^2*(d + e*x)) + (sqrt(-a)*sqrt(c)*(3*a*e^2*g + c*d*(2*e*f + d*g))*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(4*e*(c*d^2 + a*e^2)*(e*f - d*g)^2*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) + (sqrt(-a)*sqrt(c)*g*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(2*e^2*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + c*x^2)) - (sqrt(-a)*sqrt(c)*f*(3*a*e^2*g + c*d*(2*e*f + d*g))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(4*e*(c*d^2 + a*e^2)*(e*f - d*g)^2*sqrt(f + g*x)*sqrt(a + c*x^2)) + (sqrt(-a)*sqrt(c)*d*g*(3*a*e^2*g + c*d*(2*e*f + d*g))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(4*e^2*(c*d^2 + a*e^2)*(e*f - d*g)^2*sqrt(f + g*x)*sqrt(a + c*x^2)) - (c*(e*f + d*g)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.Pi((2*e)/((sqrt(c)*d)/sqrt(-a) + e), asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (2*sqrt(-a)*g)/(sqrt(c)*f + sqrt(-a)*g)))/(e^2*((sqrt(c)*d)/sqrt(-a) + e)*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + c*x^2)) - ((a*e^2*g - c*d*(2*e*f - 3*d*g))*(3*a*e^2*g + c*d*(2*e*f + d*g))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.Pi((2*e)/((sqrt(c)*d)/sqrt(-a) + e), asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (2*sqrt(-a)*g)/(sqrt(c)*f + sqrt(-a)*g)))/(4*e^2*((sqrt(c)*d)/sqrt(-a) + e)*(c*d^2 + a*e^2)*(e*f - d*g)^2*sqrt(f + g*x)*sqrt(a + c*x^2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0 & n>0=#

    @test_int [(d + e*x)^3*sqrt(f + g*x)/sqrt(a + c*x^2), x, 8, (-2*e*(25*a*e^2*g^2 + c*(7*e^2*f^2 + 12*d*e*f*g - 90*d^2*g^2))*sqrt(f + g*x)*sqrt(a + c*x^2))/(105*c^2*g^2) + (2*e*(d + e*x)^2*sqrt(f + g*x)*sqrt(a + c*x^2))/(7*c) + (2*e^2*(e*f + 11*d*g)*(f + g*x)^(3/2)*sqrt(a + c*x^2))/(35*c*g^2) + (2*sqrt(-a)*(a*e^2*g^2*(19*e*f + 189*d*g) - c*(8*e^3*f^3 - 42*d*e^2*f^2*g + 105*d^2*e*f*g^2 + 105*d^3*g^3))*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(105*c^(3/2)*g^3*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) - (2*sqrt(-a)*e*(c*f^2 + a*g^2)*(25*a*e^2*g^2 - c*(8*e^2*f^2 - 42*d*e*f*g + 105*d^2*g^2))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(105*c^(5/2)*g^3*sqrt(f + g*x)*sqrt(a + c*x^2))]
    @test_int [(d + e*x)^2*sqrt(f + g*x)/sqrt(a + c*x^2), x, 7, (2*e*(e*f + 7*d*g)*sqrt(f + g*x)*sqrt(a + c*x^2))/(15*c*g) + (2*e*(d + e*x)*sqrt(f + g*x)*sqrt(a + c*x^2))/(5*c) + (2*sqrt(-a)*(9*a*e^2*g^2 + c*(2*e^2*f^2 - 10*d*e*f*g - 15*d^2*g^2))*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(15*c^(3/2)*g^2*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) - (4*sqrt(-a)*e*(e*f - 5*d*g)*(c*f^2 + a*g^2)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(15*c^(3/2)*g^2*sqrt(f + g*x)*sqrt(a + c*x^2))]
    @test_int [(d + e*x)^1*sqrt(f + g*x)/sqrt(a + c*x^2), x, 6, (2*e*sqrt(f + g*x)*sqrt(a + c*x^2))/(3*c) - (2*sqrt(-a)*(e*f + 3*d*g)*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(3*sqrt(c)*g*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) + (2*sqrt(-a)*e*(c*f^2 + a*g^2)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(3*c^(3/2)*g*sqrt(f + g*x)*sqrt(a + c*x^2))]
    @test_int [(d + e*x)^0*sqrt(f + g*x)/sqrt(a + c*x^2), x, 2, (-2*sqrt(-a)*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(sqrt(c)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2))]
    @test_int [sqrt(f + g*x)/((d + e*x)^1*sqrt(a + c*x^2)), x, 7, (-2*sqrt(-a)*g*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(sqrt(c)*e*sqrt(f + g*x)*sqrt(a + c*x^2)) - (2*(e*f - d*g)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.Pi((2*e)/((sqrt(c)*d)/sqrt(-a) + e), asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (2*sqrt(-a)*g)/(sqrt(c)*f + sqrt(-a)*g)))/(e*((sqrt(c)*d)/sqrt(-a) + e)*sqrt(f + g*x)*sqrt(a + c*x^2))]
    @test_int [sqrt(f + g*x)/((d + e*x)^2*sqrt(a + c*x^2)), x, 14, -((e*sqrt(f + g*x)*sqrt(a + c*x^2))/((c*d^2 + a*e^2)*(d + e*x))) - (sqrt(-a)*sqrt(c)*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/((c*d^2 + a*e^2)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) + (sqrt(-a)*sqrt(c)*f*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/((c*d^2 + a*e^2)*sqrt(f + g*x)*sqrt(a + c*x^2)) - (sqrt(-a)*sqrt(c)*d*g*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(e*(c*d^2 + a*e^2)*sqrt(f + g*x)*sqrt(a + c*x^2)) - ((a*e^2*g + c*d*(2*e*f - d*g))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.Pi((2*e)/((sqrt(c)*d)/sqrt(-a) + e), asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (2*sqrt(-a)*g)/(sqrt(c)*f + sqrt(-a)*g)))/(e*((sqrt(c)*d)/sqrt(-a) + e)*(c*d^2 + a*e^2)*sqrt(f + g*x)*sqrt(a + c*x^2))]
    @test_int [sqrt(f + g*x)/((d + e*x)^3*sqrt(a + c*x^2)), x, 23, -(e*sqrt(f + g*x)*sqrt(a + c*x^2))/(2*(c*d^2 + a*e^2)*(d + e*x)^2) - (e*(a*e^2*g + c*d*(6*e*f - 5*d*g))*sqrt(f + g*x)*sqrt(a + c*x^2))/(4*(c*d^2 + a*e^2)^2*(e*f - d*g)*(d + e*x)) - (sqrt(-a)*sqrt(c)*(a*e^2*g + c*d*(6*e*f - 5*d*g))*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(4*(c*d^2 + a*e^2)^2*(e*f - d*g)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) + (sqrt(-a)*sqrt(c)*g*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(2*e*(c*d^2 + a*e^2)*sqrt(f + g*x)*sqrt(a + c*x^2)) + (sqrt(-a)*sqrt(c)*f*(a*e^2*g + c*d*(6*e*f - 5*d*g))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(4*(c*d^2 + a*e^2)^2*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + c*x^2)) - (sqrt(-a)*sqrt(c)*d*g*(a*e^2*g + c*d*(6*e*f - 5*d*g))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(4*e*(c*d^2 + a*e^2)^2*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + c*x^2)) + (c*(e*f - 3*d*g)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.Pi((2*e)/((sqrt(c)*d)/sqrt(-a) + e), asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (2*sqrt(-a)*g)/(sqrt(c)*f + sqrt(-a)*g)))/(e*((sqrt(c)*d)/sqrt(-a) + e)*(c*d^2 + a*e^2)*sqrt(f + g*x)*sqrt(a + c*x^2)) + ((a*e^2*g + c*d*(6*e*f - 5*d*g))*(a*e^2*g - c*d*(2*e*f - 3*d*g))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.Pi((2*e)/((sqrt(c)*d)/sqrt(-a) + e), asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (2*sqrt(-a)*g)/(sqrt(c)*f + sqrt(-a)*g)))/(4*e*((sqrt(c)*d)/sqrt(-a) + e)*(c*d^2 + a*e^2)^2*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + c*x^2))]

    @test_int [(f + g*x)^(5/2)/((d + e*x)*sqrt(a + c*x^2)), x, 16, (2*g^2*sqrt(f + g*x)*sqrt(a + c*x^2))/(3*c*e) - (2*sqrt(-a)*g*(7*e*f - 3*d*g)*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 + (a*sqrt(c)*x)/(-a)^(3/2))/sqrt(2)), (2*a*g)/((-sqrt(-a))*sqrt(c)*f + a*g)))/(3*sqrt(c)*e^2*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) + (2*sqrt(-a)*g*(a*e^2*g^2 + c*(-2*e^2*f^2 + 6*d*e*f*g - 3*d^2*g^2))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 + (a*sqrt(c)*x)/(-a)^(3/2))/sqrt(2)), (2*a*g)/((-sqrt(-a))*sqrt(c)*f + a*g)))/(3*c^(3/2)*e^3*sqrt(f + g*x)*sqrt(a + c*x^2)) - (2*(e*f - d*g)^2*sqrt((g*(sqrt(-a) - sqrt(c)*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(-((g*(sqrt(-a) + sqrt(c)*x))/(sqrt(c)*f - sqrt(-a)*g)))*Elliptic.Pi((e*(f + (sqrt(-a)*g)/sqrt(c)))/(e*f - d*g), asin(sqrt(c/(c*f + sqrt(-a)*sqrt(c)*g))*sqrt(f + g*x)), (sqrt(c)*f + sqrt(-a)*g)/(sqrt(c)*f - sqrt(-a)*g)))/(e^3*sqrt(c/(c*f + sqrt(-a)*sqrt(c)*g))*sqrt(a + c*x^2)), (2*g^2*sqrt(f + g*x)*sqrt(a + c*x^2))/(3*c*e) - (8*sqrt(-a)*f*g*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(3*sqrt(c)*e*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) - (2*sqrt(-a)*g*(e*f - d*g)*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(sqrt(c)*e^2*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) - (2*sqrt(-a)*g*(e*f - d*g)^2*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(sqrt(c)*e^3*sqrt(f + g*x)*sqrt(a + c*x^2)) + (2*sqrt(-a)*g*(c*f^2 + a*g^2)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(3*c^(3/2)*e*sqrt(f + g*x)*sqrt(a + c*x^2)) - (2*(e*f - d*g)^3*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.Pi((2*e)/((sqrt(c)*d)/sqrt(-a) + e), asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (2*sqrt(-a)*g)/(sqrt(c)*f + sqrt(-a)*g)))/(e^3*((sqrt(c)*d)/sqrt(-a) + e)*sqrt(f + g*x)*sqrt(a + c*x^2))]

    @test_int [(f + g*x)^(3/2)/((d + e*x)*sqrt(a + c*x^2)), x, 10, -((2*sqrt(-a)*g*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(sqrt(c)*e*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2))) - (2*sqrt(-a)*g*(e*f - d*g)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(sqrt(c)*e^2*sqrt(f + g*x)*sqrt(a + c*x^2)) - (2*(e*f - d*g)^2*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.Pi((2*e)/((sqrt(c)*d)/sqrt(-a) + e), asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (2*sqrt(-a)*g)/(sqrt(c)*f + sqrt(-a)*g)))/(e^2*((sqrt(c)*d)/sqrt(-a) + e)*sqrt(f + g*x)*sqrt(a + c*x^2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0 & n<0=#

    @test_int [(d + e*x)^3/(sqrt(f + g*x)*sqrt(a + c*x^2)), x, 7, (-8*e^2*(e*f - 3*d*g)*sqrt(f + g*x)*sqrt(a + c*x^2))/(15*c*g^2) + (2*e^2*(d + e*x)*sqrt(f + g*x)*sqrt(a + c*x^2))/(5*c*g) + (2*sqrt(-a)*e*(9*a*e^2*g^2 - c*(8*e^2*f^2 - 30*d*e*f*g + 45*d^2*g^2))*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(15*c^(3/2)*g^3*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) - (2*sqrt(-a)*(a*e^2*g^2*(7*e*f - 15*d*g) - c*(8*e^3*f^3 - 30*d*e^2*f^2*g + 45*d^2*e*f*g^2 - 15*d^3*g^3))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(15*c^(3/2)*g^3*sqrt(f + g*x)*sqrt(a + c*x^2))]
    @test_int [(d + e*x)^2/(sqrt(f + g*x)*sqrt(a + c*x^2)), x, 7, (2*e^2*sqrt(f + g*x)*sqrt(a + c*x^2))/(3*c*g) + (4*sqrt(-a)*e*(e*f - 3*d*g)*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(3*sqrt(c)*g^2*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) - (2*sqrt(-a)*((3*c*d^2 - a*e^2)*g^2 + 2*c*e*f*(e*f - 3*d*g))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(3*c^(3/2)*g^2*sqrt(f + g*x)*sqrt(a + c*x^2))]
    @test_int [(d + e*x)^1/(sqrt(f + g*x)*sqrt(a + c*x^2)), x, 5, (-2*sqrt(-a)*e*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(sqrt(c)*g*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) + (2*sqrt(-a)*(e*f - d*g)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(sqrt(c)*g*sqrt(f + g*x)*sqrt(a + c*x^2))]
    @test_int [(d + e*x)^0/(sqrt(f + g*x)*sqrt(a + c*x^2)), x, 2, (-2*sqrt(-a)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/(sqrt(c)*sqrt(f + g*x)*sqrt(a + c*x^2))]
    @test_int [1/((d + e*x)^1*sqrt(f + g*x)*sqrt(a + c*x^2)), x, 4, (-2*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.Pi((2*e)/((sqrt(c)*d)/sqrt(-a) + e), asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (2*sqrt(-a)*g)/(sqrt(c)*f + sqrt(-a)*g)))/(((sqrt(c)*d)/sqrt(-a) + e)*sqrt(f + g*x)*sqrt(a + c*x^2))]
    @test_int [1/((d + e*x)^2*sqrt(f + g*x)*sqrt(a + c*x^2)), x, 14, -((e^2*sqrt(f + g*x)*sqrt(a + c*x^2))/((c*d^2 + a*e^2)*(e*f - d*g)*(d + e*x))) - (sqrt(-a)*sqrt(c)*e*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/((c*d^2 + a*e^2)*(e*f - d*g)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) + (sqrt(-a)*sqrt(c)*e*f*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/((c*d^2 + a*e^2)*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + c*x^2)) - (sqrt(-a)*sqrt(c)*d*g*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (-2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g)))/((c*d^2 + a*e^2)*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + c*x^2)) + ((a*e^2*g - c*d*(2*e*f - 3*d*g))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.Pi((2*e)/((sqrt(c)*d)/sqrt(-a) + e), asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (2*sqrt(-a)*g)/(sqrt(c)*f + sqrt(-a)*g)))/(((sqrt(c)*d)/sqrt(-a) + e)*(c*d^2 + a*e^2)*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + c*x^2))]
    @test_int [1/((d + e*x)^3*sqrt(f + g*x)*sqrt(a + c*x^2)), x, 23, -((e^2*sqrt(f + g*x)*sqrt(a + c*x^2))/(2*(c*d^2 + a*e^2)*(e*f - d*g)*(d + e*x)^2)) + (3*e^2*(a*e^2*g - c*d*(2*e*f - 3*d*g))*sqrt(f + g*x)*sqrt(a + c*x^2))/(4*(c*d^2 + a*e^2)^2*(e*f - d*g)^2*(d + e*x)) + (3*sqrt(-a)*sqrt(c)*e*(a*e^2*g - c*d*(2*e*f - 3*d*g))*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(4*(c*d^2 + a*e^2)^2*(e*f - d*g)^2*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) + (sqrt(-a)*sqrt(c)*g*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(2*(c*d^2 + a*e^2)*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + c*x^2)) - (3*sqrt(-a)*sqrt(c)*e*f*(a*e^2*g - c*d*(2*e*f - 3*d*g))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(4*(c*d^2 + a*e^2)^2*(e*f - d*g)^2*sqrt(f + g*x)*sqrt(a + c*x^2)) + (3*sqrt(-a)*sqrt(c)*d*g*(a*e^2*g - c*d*(2*e*f - 3*d*g))*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(4*(c*d^2 + a*e^2)^2*(e*f - d*g)^2*sqrt(f + g*x)*sqrt(a + c*x^2)) + (c*(e*f - 3*d*g)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.Pi((2*e)/((sqrt(c)*d)/sqrt(-a) + e), asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (2*sqrt(-a)*g)/(sqrt(c)*f + sqrt(-a)*g)))/(((sqrt(c)*d)/sqrt(-a) + e)*(c*d^2 + a*e^2)*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + c*x^2)) - (3*(a*e^2*g - c*d*(2*e*f - 3*d*g))^2*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.Pi((2*e)/((sqrt(c)*d)/sqrt(-a) + e), asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (2*sqrt(-a)*g)/(sqrt(c)*f + sqrt(-a)*g)))/(4*((sqrt(c)*d)/sqrt(-a) + e)*(c*d^2 + a*e^2)^2*(e*f - d*g)^2*sqrt(f + g*x)*sqrt(a + c*x^2))]

    @test_int [1/((f + g*x)^(3/2)*(d + e*x)*sqrt(a + c*x^2)), x, 10, (2*g^2*sqrt(a + c*x^2))/((e*f - d*g)*(c*f^2 + a*g^2)*sqrt(f + g*x)) + (2*sqrt(-a)*sqrt(c)*g*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/((e*f - d*g)*(c*f^2 + a*g^2)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) - (2*e*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.Pi((2*e)/((sqrt(c)*d)/sqrt(-a) + e), asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (2*sqrt(-a)*g)/(sqrt(c)*f + sqrt(-a)*g)))/(((sqrt(c)*d)/sqrt(-a) + e)*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + c*x^2))]

    @test_int [1/((f + g*x)^(5/2)*(d + e*x)*sqrt(a + c*x^2)), x, 17, (2*g^2*sqrt(a + c*x^2))/(3*(e*f - d*g)*(c*f^2 + a*g^2)*(f + g*x)^(3/2)) + (8*c*f*g^2*sqrt(a + c*x^2))/(3*(e*f - d*g)*(c*f^2 + a*g^2)^2*sqrt(f + g*x)) + (2*e*g^2*sqrt(a + c*x^2))/((e*f - d*g)^2*(c*f^2 + a*g^2)*sqrt(f + g*x)) + (8*sqrt(-a)*c^(3/2)*f*g*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(3*(e*f - d*g)*(c*f^2 + a*g^2)^2*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) + (2*sqrt(-a)*sqrt(c)*e*g*sqrt(f + g*x)*sqrt(1 + (c*x^2)/a)*Elliptic.E(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/((e*f - d*g)^2*(c*f^2 + a*g^2)*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(a + c*x^2)) - (2*sqrt(-a)*sqrt(c)*g*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.F(asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), -((2*a*g)/(sqrt(-a)*sqrt(c)*f - a*g))))/(3*(e*f - d*g)*(c*f^2 + a*g^2)*sqrt(f + g*x)*sqrt(a + c*x^2)) - (2*e^2*sqrt((sqrt(c)*(f + g*x))/(sqrt(c)*f + sqrt(-a)*g))*sqrt(1 + (c*x^2)/a)*Elliptic.Pi((2*e)/((sqrt(c)*d)/sqrt(-a) + e), asin(sqrt(1 - (sqrt(c)*x)/sqrt(-a))/sqrt(2)), (2*sqrt(-a)*g)/(sqrt(c)*f + sqrt(-a)*g)))/(((sqrt(c)*d)/sqrt(-a) + e)*(e*f - d*g)^2*sqrt(f + g*x)*sqrt(a + c*x^2))]

    @test_int [1/((d + e*x)*sqrt(f + g*x)*sqrt(1 + c*x^2)), x, 4, -((2*sqrt((sqrt(-c)*(f + g*x))/(sqrt(-c)*f + g))*Elliptic.Pi((2*e)/(sqrt(-c)*d + e), asin(sqrt(1 - sqrt(-c)*x)/sqrt(2)), (2*g)/(sqrt(-c)*f + g)))/((sqrt(-c)*d + e)*sqrt(f + g*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^(m/2)*(f+g*x)^(n/2)*(a+c*x^2)^(p/2)=#

    @test_int [1/(sqrt(d + e*x)*sqrt(f + g*x)*sqrt(a + c*x^2)), x, 2, -(((c*f^2 + a*g^2)^(1/4)*(d + e*x)*sqrt(((e*f - d*g)^2*(a + c*x^2))/((c*f^2 + a*g^2)*(d + e*x)^2))*(1 + (sqrt(c*d^2 + a*e^2)*(f + g*x))/(sqrt(c*f^2 + a*g^2)*(d + e*x)))*sqrt((1 - (2*(c*d*f + a*e*g)*(f + g*x))/((c*f^2 + a*g^2)*(d + e*x)) + ((c*d^2 + a*e^2)*(f + g*x)^2)/((c*f^2 + a*g^2)*(d + e*x)^2))/(1 + (sqrt(c*d^2 + a*e^2)*(f + g*x))/(sqrt(c*f^2 + a*g^2)*(d + e*x)))^2)*Elliptic.F(2*atan(((c*d^2 + a*e^2)^(1/4)*sqrt(f + g*x))/((c*f^2 + a*g^2)^(1/4)*sqrt(d + e*x))), (1/2)*(1 + (c*d*f + a*e*g)/(sqrt(c*d^2 + a*e^2)*sqrt(c*f^2 + a*g^2)))))/((c*d^2 + a*e^2)^(1/4)*(e*f - d*g)*sqrt(a + c*x^2)*sqrt(1 - (2*(c*d*f + a*e*g)*(f + g*x))/((c*f^2 + a*g^2)*(d + e*x)) + ((c*d^2 + a*e^2)*(f + g*x)^2)/((c*f^2 + a*g^2)*(d + e*x)^2))))]

    @test_int [1/(sqrt(-1 + x)*sqrt(1 + x)*sqrt(-1 + 2*x^2)), x, 4, (sqrt(1 - 2*x^2)*sqrt(1 - x^2)*Elliptic.F(asin(x), 2))/(sqrt(-1 + x)*sqrt(1 + x)*sqrt(-1 + 2*x^2))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(f+g*x)^n*(a+b*x+c*x^2)^p*when*c*d^2-b*d*e+a*e^2=0*and*m+p=0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^(m/2) / (a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(m/2)*(f+g*x)^n=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [sqrt(d + e*x)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)*(f + g*x)^3, x, 4, -((16*(c*d*f - a*e*g)^2*(2*a*e^2*g - c*d*(3*e*f - d*g))*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(35*c^4*d^4*e*sqrt(d + e*x))) + (16*g*(c*d*f - a*e*g)^2*sqrt(d + e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(35*c^3*d^3*e) + (12*(c*d*f - a*e*g)*(f + g*x)^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(35*c^2*d^2*sqrt(d + e*x)) + (2*(f + g*x)^3*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(7*c*d*sqrt(d + e*x))]
    @test_int [sqrt(d + e*x)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)*(f + g*x)^2, x, 3, -((8*(c*d*f - a*e*g)*(2*a*e^2*g - c*d*(3*e*f - d*g))*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(15*c^3*d^3*e*sqrt(d + e*x))) + (8*g*(c*d*f - a*e*g)*sqrt(d + e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(15*c^2*d^2*e) + (2*(f + g*x)^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(5*c*d*sqrt(d + e*x))]
    @test_int [sqrt(d + e*x)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)*(f + g*x)^1, x, 2, -((2*(2*a*e^2*g - c*d*(3*e*f - d*g))*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3*c^2*d^2*e*sqrt(d + e*x))) + (2*g*sqrt(d + e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3*c*d*e)]
    @test_int [sqrt(d + e*x)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)*(f + g*x)^0, x, 1, (2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(c*d*sqrt(d + e*x))]
    @test_int [sqrt(d + e*x)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(f + g*x)^1, x, 2, (2*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(sqrt(g)*sqrt(c*d*f - a*e*g))]
    @test_int [sqrt(d + e*x)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(f + g*x)^2, x, 3, sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/((c*d*f - a*e*g)*sqrt(d + e*x)*(f + g*x)) + (c*d*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(sqrt(g)*(c*d*f - a*e*g)^(3/2))]
    @test_int [sqrt(d + e*x)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(f + g*x)^3, x, 4, sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(2*(c*d*f - a*e*g)*sqrt(d + e*x)*(f + g*x)^2) + (3*c*d*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(4*(c*d*f - a*e*g)^2*sqrt(d + e*x)*(f + g*x)) + (3*c^2*d^2*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(4*sqrt(g)*(c*d*f - a*e*g)^(5/2))]
    @test_int [sqrt(d + e*x)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(f + g*x)^4, x, 5, sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(3*(c*d*f - a*e*g)*sqrt(d + e*x)*(f + g*x)^3) + (5*c*d*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(12*(c*d*f - a*e*g)^2*sqrt(d + e*x)*(f + g*x)^2) + (5*c^2*d^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(8*(c*d*f - a*e*g)^3*sqrt(d + e*x)*(f + g*x)) + (5*c^3*d^3*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(8*sqrt(g)*(c*d*f - a*e*g)^(7/2))]

    @test_int [(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)*(f + g*x)^3, x, 4, -((2*sqrt(d + e*x)*(f + g*x)^3)/(c*d*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))) - (16*g*(c*d*f - a*e*g)*(2*a*e^2*g - c*d*(3*e*f - d*g))*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(5*c^4*d^4*e*sqrt(d + e*x)) + (16*g^2*(c*d*f - a*e*g)*sqrt(d + e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(5*c^3*d^3*e) + (12*g*(f + g*x)^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(5*c^2*d^2*sqrt(d + e*x))]
    @test_int [(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)*(f + g*x)^2, x, 3, -((2*sqrt(d + e*x)*(f + g*x)^2)/(c*d*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))) - (8*g*(2*a*e^2*g - c*d*(3*e*f - d*g))*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3*c^3*d^3*e*sqrt(d + e*x)) + (8*g^2*sqrt(d + e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3*c^2*d^2*e)]
    @test_int [(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)*(f + g*x)^1, x, 2, -((2*(c*d*f - a*e*g)*(d + e*x)^(3/2))/(c*d*(c*d^2 - a*e^2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))) - (2*(2*a*e^2*g - c*d*(e*f + d*g))*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(c^2*d^2*(c*d^2 - a*e^2)*sqrt(d + e*x))]
    @test_int [(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)*(f + g*x)^0, x, 1, -((2*sqrt(d + e*x))/(c*d*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)))]
    @test_int [(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(f + g*x)^1, x, 3, -((2*sqrt(d + e*x))/((c*d*f - a*e*g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))) - (2*sqrt(g)*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(c*d*f - a*e*g)^(3/2)]
    @test_int [(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(f + g*x)^2, x, 4, -((2*sqrt(d + e*x))/((c*d*f - a*e*g)*(f + g*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))) - (3*g*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/((c*d*f - a*e*g)^2*sqrt(d + e*x)*(f + g*x)) - (3*c*d*sqrt(g)*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(c*d*f - a*e*g)^(5/2)]
    @test_int [(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(f + g*x)^3, x, 5, -((2*sqrt(d + e*x))/((c*d*f - a*e*g)*(f + g*x)^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))) - (5*g*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(2*(c*d*f - a*e*g)^2*sqrt(d + e*x)*(f + g*x)^2) - (15*c*d*g*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(4*(c*d*f - a*e*g)^3*sqrt(d + e*x)*(f + g*x)) - (15*c^2*d^2*sqrt(g)*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(4*(c*d*f - a*e*g)^(7/2))]

    @test_int [(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)*(f + g*x)^3, x, 4, -((2*(d + e*x)^(3/2)*(f + g*x)^3)/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))) - (4*g*sqrt(d + e*x)*(f + g*x)^2)/(c^2*d^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)) - (16*g^2*(2*a*e^2*g - c*d*(3*e*f - d*g))*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3*c^4*d^4*e*sqrt(d + e*x)) + (16*g^3*sqrt(d + e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3*c^3*d^3*e)]
    @test_int [(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)*(f + g*x)^2, x, 3, -((2*(d + e*x)^(3/2)*(f + g*x)^2)/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))) - (8*g*(c*d*f - a*e*g)*(d + e*x)^(3/2))/(3*c^2*d^2*(c*d^2 - a*e^2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)) - (8*g*(2*a*e^2*g - c*d*(e*f + d*g))*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3*c^3*d^3*(c*d^2 - a*e^2)*sqrt(d + e*x))]
    @test_int [(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)*(f + g*x)^1, x, 2, -((2*(c*d*f - a*e*g)*(d + e*x)^(5/2))/(3*c*d*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))) + (2*(2*a*e^2*g + c*d*(e*f - 3*d*g))*sqrt(d + e*x))/(3*c^2*d^2*(c*d^2 - a*e^2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)*(f + g*x)^0, x, 1, -((2*(d + e*x)^(3/2))/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)))]
    @test_int [(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(f + g*x)^1, x, 4, -((2*(d + e*x)^(3/2))/(3*(c*d*f - a*e*g)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))) + (2*g*sqrt(d + e*x))/((c*d*f - a*e*g)^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)) + (2*g^(3/2)*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(c*d*f - a*e*g)^(5/2)]
    @test_int [(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(f + g*x)^2, x, 5, -((2*(d + e*x)^(3/2))/(3*(c*d*f - a*e*g)*(f + g*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))) + (10*g*sqrt(d + e*x))/(3*(c*d*f - a*e*g)^2*(f + g*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)) + (5*g^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/((c*d*f - a*e*g)^3*sqrt(d + e*x)*(f + g*x)) + (5*c*d*g^(3/2)*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(c*d*f - a*e*g)^(7/2)]
    @test_int [(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(f + g*x)^3, x, 6, -((2*(d + e*x)^(3/2))/(3*(c*d*f - a*e*g)*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))) + (14*g*sqrt(d + e*x))/(3*(c*d*f - a*e*g)^2*(f + g*x)^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)) + (35*g^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(6*(c*d*f - a*e*g)^3*sqrt(d + e*x)*(f + g*x)^2) + (35*c*d*g^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(4*(c*d*f - a*e*g)^4*sqrt(d + e*x)*(f + g*x)) + (35*c^2*d^2*g^(3/2)*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(4*(c*d*f - a*e*g)^(9/2))]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/sqrt(d + e*x)*(f + g*x)^4, x, 5, -((128*(c*d*f - a*e*g)^3*(2*a*e^2*g - c*d*(5*e*f - 3*d*g))*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3465*c^5*d^5*e*(d + e*x)^(3/2))) + (128*g*(c*d*f - a*e*g)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(1155*c^4*d^4*e*sqrt(d + e*x)) + (32*(c*d*f - a*e*g)^2*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(231*c^3*d^3*(d + e*x)^(3/2)) + (16*(c*d*f - a*e*g)*(f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(99*c^2*d^2*(d + e*x)^(3/2)) + (2*(f + g*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(11*c*d*(d + e*x)^(3/2))]
    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/sqrt(d + e*x)*(f + g*x)^3, x, 4, -((16*(c*d*f - a*e*g)^2*(2*a*e^2*g - c*d*(5*e*f - 3*d*g))*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(315*c^4*d^4*e*(d + e*x)^(3/2))) + (16*g*(c*d*f - a*e*g)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(105*c^3*d^3*e*sqrt(d + e*x)) + (4*(c*d*f - a*e*g)*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(21*c^2*d^2*(d + e*x)^(3/2)) + (2*(f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(9*c*d*(d + e*x)^(3/2))]
    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/sqrt(d + e*x)*(f + g*x)^2, x, 3, -((8*(c*d*f - a*e*g)*(2*a*e^2*g - c*d*(5*e*f - 3*d*g))*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(105*c^3*d^3*e*(d + e*x)^(3/2))) + (8*g*(c*d*f - a*e*g)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(35*c^2*d^2*e*sqrt(d + e*x)) + (2*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(7*c*d*(d + e*x)^(3/2))]
    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/sqrt(d + e*x)*(f + g*x)^1, x, 2, -((2*(2*a*e^2*g - c*d*(5*e*f - 3*d*g))*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(15*c^2*d^2*e*(d + e*x)^(3/2))) + (2*g*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(5*c*d*e*sqrt(d + e*x))]
    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/sqrt(d + e*x)*(f + g*x)^0, x, 1, (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*c*d*(d + e*x)^(3/2))]
    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/sqrt(d + e*x)/(f + g*x)^1, x, 3, (2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(g*sqrt(d + e*x)) - (2*sqrt(c*d*f - a*e*g)*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/g^(3/2)]
    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/sqrt(d + e*x)/(f + g*x)^2, x, 3, -(sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(g*sqrt(d + e*x)*(f + g*x))) + (c*d*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(g^(3/2)*sqrt(c*d*f - a*e*g))]
    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/sqrt(d + e*x)/(f + g*x)^3, x, 4, -(sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(2*g*sqrt(d + e*x)*(f + g*x)^2)) + (c*d*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(4*g*(c*d*f - a*e*g)*sqrt(d + e*x)*(f + g*x)) + (c^2*d^2*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(4*g^(3/2)*(c*d*f - a*e*g)^(3/2))]
    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/sqrt(d + e*x)/(f + g*x)^4, x, 5, -(sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(3*g*sqrt(d + e*x)*(f + g*x)^3)) + (c*d*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(12*g*(c*d*f - a*e*g)*sqrt(d + e*x)*(f + g*x)^2) + (c^2*d^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(8*g*(c*d*f - a*e*g)^2*sqrt(d + e*x)*(f + g*x)) + (c^3*d^3*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(8*g^(3/2)*(c*d*f - a*e*g)^(5/2))]
    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/sqrt(d + e*x)/(f + g*x)^5, x, 6, -(sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(4*g*sqrt(d + e*x)*(f + g*x)^4)) + (c*d*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(24*g*(c*d*f - a*e*g)*sqrt(d + e*x)*(f + g*x)^3) + (5*c^2*d^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(96*g*(c*d*f - a*e*g)^2*sqrt(d + e*x)*(f + g*x)^2) + (5*c^3*d^3*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(64*g*(c*d*f - a*e*g)^3*sqrt(d + e*x)*(f + g*x)) + (5*c^4*d^4*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(64*g^(3/2)*(c*d*f - a*e*g)^(7/2))]

    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(3/2)*(f + g*x)^4, x, 5, -((128*(c*d*f - a*e*g)^3*(2*a*e^2*g - c*d*(7*e*f - 5*d*g))*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(15015*c^5*d^5*e*(d + e*x)^(5/2))) + (128*g*(c*d*f - a*e*g)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(3003*c^4*d^4*e*(d + e*x)^(3/2)) + (32*(c*d*f - a*e*g)^2*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(429*c^3*d^3*(d + e*x)^(5/2)) + (16*(c*d*f - a*e*g)*(f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(143*c^2*d^2*(d + e*x)^(5/2)) + (2*(f + g*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(13*c*d*(d + e*x)^(5/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(3/2)*(f + g*x)^3, x, 4, -((16*(c*d*f - a*e*g)^2*(2*a*e^2*g - c*d*(7*e*f - 5*d*g))*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(1155*c^4*d^4*e*(d + e*x)^(5/2))) + (16*g*(c*d*f - a*e*g)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(231*c^3*d^3*e*(d + e*x)^(3/2)) + (4*(c*d*f - a*e*g)*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(33*c^2*d^2*(d + e*x)^(5/2)) + (2*(f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(11*c*d*(d + e*x)^(5/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(3/2)*(f + g*x)^2, x, 3, -((8*(c*d*f - a*e*g)*(2*a*e^2*g - c*d*(7*e*f - 5*d*g))*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(315*c^3*d^3*e*(d + e*x)^(5/2))) + (8*g*(c*d*f - a*e*g)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(63*c^2*d^2*e*(d + e*x)^(3/2)) + (2*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(9*c*d*(d + e*x)^(5/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(3/2)*(f + g*x)^1, x, 2, -((2*(2*a*e^2*g - c*d*(7*e*f - 5*d*g))*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(35*c^2*d^2*e*(d + e*x)^(5/2))) + (2*g*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(7*c*d*e*(d + e*x)^(3/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(3/2)*(f + g*x)^0, x, 1, (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(5*c*d*(d + e*x)^(5/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(3/2)/(f + g*x)^1, x, 4, -((2*(c*d*f - a*e*g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(g^2*sqrt(d + e*x))) + (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*g*(d + e*x)^(3/2)) + (2*(c*d*f - a*e*g)^(3/2)*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/g^(5/2)]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(3/2)/(f + g*x)^2, x, 4, (3*c*d*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(g^2*sqrt(d + e*x)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(g*(d + e*x)^(3/2)*(f + g*x)) - (3*c*d*sqrt(c*d*f - a*e*g)*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/g^(5/2)]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(3/2)/(f + g*x)^3, x, 4, -((3*c*d*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(4*g^2*sqrt(d + e*x)*(f + g*x))) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(2*g*(d + e*x)^(3/2)*(f + g*x)^2) + (3*c^2*d^2*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(4*g^(5/2)*sqrt(c*d*f - a*e*g))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(3/2)/(f + g*x)^4, x, 5, -((c*d*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(4*g^2*sqrt(d + e*x)*(f + g*x)^2)) + (c^2*d^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(8*g^2*(c*d*f - a*e*g)*sqrt(d + e*x)*(f + g*x)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(3*g*(d + e*x)^(3/2)*(f + g*x)^3) + (c^3*d^3*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(8*g^(5/2)*(c*d*f - a*e*g)^(3/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(3/2)/(f + g*x)^5, x, 6, -((c*d*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(8*g^2*sqrt(d + e*x)*(f + g*x)^3)) + (c^2*d^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(32*g^2*(c*d*f - a*e*g)*sqrt(d + e*x)*(f + g*x)^2) + (3*c^3*d^3*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(64*g^2*(c*d*f - a*e*g)^2*sqrt(d + e*x)*(f + g*x)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(4*g*(d + e*x)^(3/2)*(f + g*x)^4) + (3*c^4*d^4*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(64*g^(5/2)*(c*d*f - a*e*g)^(5/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(3/2)/(f + g*x)^6, x, 7, -((3*c*d*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(40*g^2*sqrt(d + e*x)*(f + g*x)^4)) + (c^2*d^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(80*g^2*(c*d*f - a*e*g)*sqrt(d + e*x)*(f + g*x)^3) + (c^3*d^3*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(64*g^2*(c*d*f - a*e*g)^2*sqrt(d + e*x)*(f + g*x)^2) + (3*c^4*d^4*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(128*g^2*(c*d*f - a*e*g)^3*sqrt(d + e*x)*(f + g*x)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(5*g*(d + e*x)^(3/2)*(f + g*x)^5) + (3*c^5*d^5*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(128*g^(5/2)*(c*d*f - a*e*g)^(7/2))]

    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2)*(f + g*x)^4, x, 5, -((128*(c*d*f - a*e*g)^3*(2*a*e^2*g - c*d*(9*e*f - 7*d*g))*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(45045*c^5*d^5*e*(d + e*x)^(7/2))) + (128*g*(c*d*f - a*e*g)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(6435*c^4*d^4*e*(d + e*x)^(5/2)) + (32*(c*d*f - a*e*g)^2*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(715*c^3*d^3*(d + e*x)^(7/2)) + (16*(c*d*f - a*e*g)*(f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(195*c^2*d^2*(d + e*x)^(7/2)) + (2*(f + g*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(15*c*d*(d + e*x)^(7/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2)*(f + g*x)^3, x, 4, -((16*(c*d*f - a*e*g)^2*(2*a*e^2*g - c*d*(9*e*f - 7*d*g))*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(3003*c^4*d^4*e*(d + e*x)^(7/2))) + (16*g*(c*d*f - a*e*g)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(429*c^3*d^3*e*(d + e*x)^(5/2)) + (12*(c*d*f - a*e*g)*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(143*c^2*d^2*(d + e*x)^(7/2)) + (2*(f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(13*c*d*(d + e*x)^(7/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2)*(f + g*x)^2, x, 3, -((8*(c*d*f - a*e*g)*(2*a*e^2*g - c*d*(9*e*f - 7*d*g))*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(693*c^3*d^3*e*(d + e*x)^(7/2))) + (8*g*(c*d*f - a*e*g)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(99*c^2*d^2*e*(d + e*x)^(5/2)) + (2*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(11*c*d*(d + e*x)^(7/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2)*(f + g*x)^1, x, 2, -((2*(2*a*e^2*g - c*d*(9*e*f - 7*d*g))*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(63*c^2*d^2*e*(d + e*x)^(7/2))) + (2*g*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(9*c*d*e*(d + e*x)^(5/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2)*(f + g*x)^0, x, 1, (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(7*c*d*(d + e*x)^(7/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2)/(f + g*x)^1, x, 5, (2*(c*d*f - a*e*g)^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(g^3*sqrt(d + e*x)) - (2*(c*d*f - a*e*g)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*g^2*(d + e*x)^(3/2)) + (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(5*g*(d + e*x)^(5/2)) - (2*(c*d*f - a*e*g)^(5/2)*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/g^(7/2)]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2)/(f + g*x)^2, x, 5, -((5*c*d*(c*d*f - a*e*g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(g^3*sqrt(d + e*x))) + (5*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*g^2*(d + e*x)^(3/2)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(g*(d + e*x)^(5/2)*(f + g*x)) + (5*c*d*(c*d*f - a*e*g)^(3/2)*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/g^(7/2)]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2)/(f + g*x)^3, x, 5, (15*c^2*d^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(4*g^3*sqrt(d + e*x)) - (5*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(4*g^2*(d + e*x)^(3/2)*(f + g*x)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(2*g*(d + e*x)^(5/2)*(f + g*x)^2) - (15*c^2*d^2*sqrt(c*d*f - a*e*g)*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(4*g^(7/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2)/(f + g*x)^4, x, 5, -((5*c^2*d^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(8*g^3*sqrt(d + e*x)*(f + g*x))) - (5*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(12*g^2*(d + e*x)^(3/2)*(f + g*x)^2) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(3*g*(d + e*x)^(5/2)*(f + g*x)^3) + (5*c^3*d^3*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(8*g^(7/2)*sqrt(c*d*f - a*e*g))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2)/(f + g*x)^5, x, 6, -((5*c^2*d^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(32*g^3*sqrt(d + e*x)*(f + g*x)^2)) + (5*c^3*d^3*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(64*g^3*(c*d*f - a*e*g)*sqrt(d + e*x)*(f + g*x)) - (5*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(24*g^2*(d + e*x)^(3/2)*(f + g*x)^3) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(4*g*(d + e*x)^(5/2)*(f + g*x)^4) + (5*c^4*d^4*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(64*g^(7/2)*(c*d*f - a*e*g)^(3/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2)/(f + g*x)^6, x, 7, -((c^2*d^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(16*g^3*sqrt(d + e*x)*(f + g*x)^3)) + (c^3*d^3*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(64*g^3*(c*d*f - a*e*g)*sqrt(d + e*x)*(f + g*x)^2) + (3*c^4*d^4*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(128*g^3*(c*d*f - a*e*g)^2*sqrt(d + e*x)*(f + g*x)) - (c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(8*g^2*(d + e*x)^(3/2)*(f + g*x)^4) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(5*g*(d + e*x)^(5/2)*(f + g*x)^5) + (3*c^5*d^5*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(128*g^(7/2)*(c*d*f - a*e*g)^(5/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2)/(f + g*x)^7, x, 8, -((c^2*d^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(32*g^3*sqrt(d + e*x)*(f + g*x)^4)) + (c^3*d^3*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(192*g^3*(c*d*f - a*e*g)*sqrt(d + e*x)*(f + g*x)^3) + (5*c^4*d^4*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(768*g^3*(c*d*f - a*e*g)^2*sqrt(d + e*x)*(f + g*x)^2) + (5*c^5*d^5*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(512*g^3*(c*d*f - a*e*g)^3*sqrt(d + e*x)*(f + g*x)) - (c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(12*g^2*(d + e*x)^(3/2)*(f + g*x)^5) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(6*g*(d + e*x)^(5/2)*(f + g*x)^6) + (5*c^6*d^6*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(512*g^(7/2)*(c*d*f - a*e*g)^(7/2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^(m/2) / (a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(m/2)*(f+g*x)^(n/2)=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [sqrt(d + e*x)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)*(f + g*x)^(5/2), x, 7, (5*(c*d*f - a*e*g)^2*sqrt(f + g*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(8*c^3*d^3*sqrt(d + e*x)) + (5*(c*d*f - a*e*g)*(f + g*x)^(3/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(12*c^2*d^2*sqrt(d + e*x)) + ((f + g*x)^(5/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3*c*d*sqrt(d + e*x)) + (5*(c*d*f - a*e*g)^3*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(8*c^(7/2)*d^(7/2)*sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [sqrt(d + e*x)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)*(f + g*x)^(3/2), x, 6, (3*(c*d*f - a*e*g)*sqrt(f + g*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(4*c^2*d^2*sqrt(d + e*x)) + ((f + g*x)^(3/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(2*c*d*sqrt(d + e*x)) + (3*(c*d*f - a*e*g)^2*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(4*c^(5/2)*d^(5/2)*sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [sqrt(d + e*x)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)*(f + g*x)^(1/2), x, 5, (sqrt(f + g*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(c*d*sqrt(d + e*x)) + ((c*d*f - a*e*g)*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(c^(3/2)*d^(3/2)*sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [sqrt(d + e*x)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(f + g*x)^(1/2), x, 4, (2*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(sqrt(c)*sqrt(d)*sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [sqrt(d + e*x)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(f + g*x)^(3/2), x, 1, (2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/((c*d*f - a*e*g)*sqrt(d + e*x)*sqrt(f + g*x))]
    @test_int [sqrt(d + e*x)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(f + g*x)^(5/2), x, 2, (2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3*(c*d*f - a*e*g)*sqrt(d + e*x)*(f + g*x)^(3/2)) + (4*c*d*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3*(c*d*f - a*e*g)^2*sqrt(d + e*x)*sqrt(f + g*x))]
    @test_int [sqrt(d + e*x)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(f + g*x)^(7/2), x, 3, (2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(5*(c*d*f - a*e*g)*sqrt(d + e*x)*(f + g*x)^(5/2)) + (8*c*d*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(15*(c*d*f - a*e*g)^2*sqrt(d + e*x)*(f + g*x)^(3/2)) + (16*c^2*d^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(15*(c*d*f - a*e*g)^3*sqrt(d + e*x)*sqrt(f + g*x))]
    @test_int [sqrt(d + e*x)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(f + g*x)^(9/2), x, 4, (2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(7*(c*d*f - a*e*g)*sqrt(d + e*x)*(f + g*x)^(7/2)) + (12*c*d*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(35*(c*d*f - a*e*g)^2*sqrt(d + e*x)*(f + g*x)^(5/2)) + (16*c^2*d^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(35*(c*d*f - a*e*g)^3*sqrt(d + e*x)*(f + g*x)^(3/2)) + (32*c^3*d^3*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(35*(c*d*f - a*e*g)^4*sqrt(d + e*x)*sqrt(f + g*x))]

    @test_int [(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)*(f + g*x)^(5/2), x, 7, (15*g*(c*d*f - a*e*g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)*sqrt(f + g*x))/(4*c^3*d^3*sqrt(d + e*x)) + (5*g*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)*(f + g*x)^(3/2))/(2*c^2*d^2*sqrt(d + e*x)) - (2*sqrt(d + e*x)*(f + g*x)^(5/2))/(c*d*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)) + (15*sqrt(g)*(c*d*f - a*e*g)^2*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(4*c^(7/2)*d^(7/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)*(f + g*x)^(3/2), x, 6, -((2*sqrt(d + e*x)*(f + g*x)^(3/2))/(c*d*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))) + (3*g*sqrt(f + g*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(c^2*d^2*sqrt(d + e*x)) + (3*sqrt(g)*(c*d*f - a*e*g)*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(c^(5/2)*d^(5/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)*(f + g*x)^(1/2), x, 5, -((2*sqrt(d + e*x)*sqrt(f + g*x))/(c*d*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))) + (2*sqrt(g)*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(c^(3/2)*d^(3/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(f + g*x)^(1/2), x, 1, -((2*sqrt(d + e*x)*sqrt(f + g*x))/((c*d*f - a*e*g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)))]
    @test_int [(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(f + g*x)^(3/2), x, 2, -((2*sqrt(d + e*x))/((c*d*f - a*e*g)*sqrt(f + g*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))) - (4*g*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/((c*d*f - a*e*g)^2*sqrt(d + e*x)*sqrt(f + g*x))]
    @test_int [(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(f + g*x)^(5/2), x, 3, -((2*sqrt(d + e*x))/((c*d*f - a*e*g)*(f + g*x)^(3/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))) - (8*g*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3*(c*d*f - a*e*g)^2*sqrt(d + e*x)*(f + g*x)^(3/2)) - (16*c*d*g*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3*(c*d*f - a*e*g)^3*sqrt(d + e*x)*sqrt(f + g*x))]
    @test_int [(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(f + g*x)^(7/2), x, 4, -((2*sqrt(d + e*x))/((c*d*f - a*e*g)*(f + g*x)^(5/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))) - (12*g*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(5*(c*d*f - a*e*g)^2*sqrt(d + e*x)*(f + g*x)^(5/2)) - (16*c*d*g*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(5*(c*d*f - a*e*g)^3*sqrt(d + e*x)*(f + g*x)^(3/2)) - (32*c^2*d^2*g*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(5*(c*d*f - a*e*g)^4*sqrt(d + e*x)*sqrt(f + g*x))]

    @test_int [(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)*(f + g*x)^(5/2), x, 7, -((2*(d + e*x)^(3/2)*(f + g*x)^(5/2))/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))) - (10*g*sqrt(d + e*x)*(f + g*x)^(3/2))/(3*c^2*d^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)) + (5*g^2*sqrt(f + g*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(c^3*d^3*sqrt(d + e*x)) + (5*g^(3/2)*(c*d*f - a*e*g)*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(c^(7/2)*d^(7/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)*(f + g*x)^(3/2), x, 6, -((2*(d + e*x)^(3/2)*(f + g*x)^(3/2))/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))) - (2*g*sqrt(d + e*x)*sqrt(f + g*x))/(c^2*d^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)) + (2*g^(3/2)*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(c^(5/2)*d^(5/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)*(f + g*x)^(1/2), x, 1, -((2*(d + e*x)^(3/2)*(f + g*x)^(3/2))/(3*(c*d*f - a*e*g)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)))]
    @test_int [(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(f + g*x)^(1/2), x, 2, -((2*(d + e*x)^(3/2)*sqrt(f + g*x))/(3*(c*d*f - a*e*g)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))) + (4*g*sqrt(d + e*x)*sqrt(f + g*x))/(3*(c*d*f - a*e*g)^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(f + g*x)^(3/2), x, 3, -((2*(d + e*x)^(3/2))/(3*(c*d*f - a*e*g)*sqrt(f + g*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))) + (8*g*sqrt(d + e*x))/(3*(c*d*f - a*e*g)^2*sqrt(f + g*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)) + (16*g^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3*(c*d*f - a*e*g)^3*sqrt(d + e*x)*sqrt(f + g*x))]
    @test_int [(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(f + g*x)^(5/2), x, 4, -((2*(d + e*x)^(3/2))/(3*(c*d*f - a*e*g)*(f + g*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))) + (4*g*sqrt(d + e*x))/((c*d*f - a*e*g)^2*(f + g*x)^(3/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)) + (16*g^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3*(c*d*f - a*e*g)^3*sqrt(d + e*x)*(f + g*x)^(3/2)) + (32*c*d*g^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3*(c*d*f - a*e*g)^4*sqrt(d + e*x)*sqrt(f + g*x))]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/sqrt(d + e*x)*(f + g*x)^(5/2), x, 8, -((5*(c*d*f - a*e*g)^3*sqrt(f + g*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(64*c^3*d^3*g*sqrt(d + e*x))) - (5*(c*d*f - a*e*g)^2*(f + g*x)^(3/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(96*c^2*d^2*g*sqrt(d + e*x)) + (((a*e)/(c*d) - f/g)*(f + g*x)^(5/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(24*sqrt(d + e*x)) + ((f + g*x)^(7/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(4*g*sqrt(d + e*x)) - (5*(c*d*f - a*e*g)^4*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(64*c^(7/2)*d^(7/2)*g^(3/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/sqrt(d + e*x)*(f + g*x)^(3/2), x, 7, -(((c*d*f - a*e*g)^2*sqrt(f + g*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(8*c^2*d^2*g*sqrt(d + e*x))) + (((a*e)/(c*d) - f/g)*(f + g*x)^(3/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(12*sqrt(d + e*x)) + ((f + g*x)^(5/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3*g*sqrt(d + e*x)) - ((c*d*f - a*e*g)^3*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(8*c^(5/2)*d^(5/2)*g^(3/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/sqrt(d + e*x)*(f + g*x)^(1/2), x, 6, (((a*e)/(c*d) - f/g)*sqrt(f + g*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(4*sqrt(d + e*x)) + ((f + g*x)^(3/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(2*g*sqrt(d + e*x)) - ((c*d*f - a*e*g)^2*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(4*c^(3/2)*d^(3/2)*g^(3/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/sqrt(d + e*x)/(f + g*x)^(1/2), x, 5, (sqrt(f + g*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(g*sqrt(d + e*x)) - ((c*d*f - a*e*g)*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(sqrt(c)*sqrt(d)*g^(3/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/sqrt(d + e*x)/(f + g*x)^(3/2), x, 5, -((2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(g*sqrt(d + e*x)*sqrt(f + g*x))) + (2*sqrt(c)*sqrt(d)*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(g^(3/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/sqrt(d + e*x)/(f + g*x)^(5/2), x, 1, (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*(c*d*f - a*e*g)*(d + e*x)^(3/2)*(f + g*x)^(3/2))]
    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/sqrt(d + e*x)/(f + g*x)^(7/2), x, 2, (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(5*(c*d*f - a*e*g)*(d + e*x)^(3/2)*(f + g*x)^(5/2)) + (4*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(15*(c*d*f - a*e*g)^2*(d + e*x)^(3/2)*(f + g*x)^(3/2))]
    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/sqrt(d + e*x)/(f + g*x)^(9/2), x, 3, (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(7*(c*d*f - a*e*g)*(d + e*x)^(3/2)*(f + g*x)^(7/2)) + (8*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(35*(c*d*f - a*e*g)^2*(d + e*x)^(3/2)*(f + g*x)^(5/2)) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(105*(c*d*f - a*e*g)^3*(d + e*x)^(3/2)*(f + g*x)^(3/2))]
    @test_int [sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/sqrt(d + e*x)/(f + g*x)^(11/2), x, 4, (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(9*(c*d*f - a*e*g)*(d + e*x)^(3/2)*(f + g*x)^(9/2)) + (4*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(21*(c*d*f - a*e*g)^2*(d + e*x)^(3/2)*(f + g*x)^(7/2)) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(105*(c*d*f - a*e*g)^3*(d + e*x)^(3/2)*(f + g*x)^(5/2)) + (32*c^3*d^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(315*(c*d*f - a*e*g)^4*(d + e*x)^(3/2)*(f + g*x)^(3/2))]

    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(3/2)*(f + g*x)^(3/2), x, 8, (3*(c*d*f - a*e*g)^3*sqrt(f + g*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(64*c^2*d^2*g^2*sqrt(d + e*x)) + ((c*d*f - a*e*g)^2*(f + g*x)^(3/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(32*c*d*g^2*sqrt(d + e*x)) - ((c*d*f - a*e*g)*(f + g*x)^(5/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(8*g^2*sqrt(d + e*x)) + ((f + g*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(4*g*(d + e*x)^(3/2)) + (3*(c*d*f - a*e*g)^4*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(64*c^(5/2)*d^(5/2)*g^(5/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(3/2)*(f + g*x)^(1/2), x, 7, ((c*d*f - a*e*g)^2*sqrt(f + g*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(8*c*d*g^2*sqrt(d + e*x)) - ((c*d*f - a*e*g)*(f + g*x)^(3/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(4*g^2*sqrt(d + e*x)) + ((f + g*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*g*(d + e*x)^(3/2)) + ((c*d*f - a*e*g)^3*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(8*c^(3/2)*d^(3/2)*g^(5/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(3/2)/(f + g*x)^(1/2), x, 6, -((3*(c*d*f - a*e*g)*sqrt(f + g*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(4*g^2*sqrt(d + e*x))) + (sqrt(f + g*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(2*g*(d + e*x)^(3/2)) + (3*(c*d*f - a*e*g)^2*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(4*sqrt(c)*sqrt(d)*g^(5/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(3/2)/(f + g*x)^(3/2), x, 6, (3*c*d*sqrt(f + g*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(g^2*sqrt(d + e*x)) - (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(g*(d + e*x)^(3/2)*sqrt(f + g*x)) - (3*sqrt(c)*sqrt(d)*(c*d*f - a*e*g)*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(g^(5/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(3/2)/(f + g*x)^(5/2), x, 6, -((2*c*d*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(g^2*sqrt(d + e*x)*sqrt(f + g*x))) - (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*g*(d + e*x)^(3/2)*(f + g*x)^(3/2)) + (2*c^(3/2)*d^(3/2)*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(g^(5/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(3/2)/(f + g*x)^(7/2), x, 1, (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(5*(c*d*f - a*e*g)*(d + e*x)^(5/2)*(f + g*x)^(5/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(3/2)/(f + g*x)^(9/2), x, 2, (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(7*(c*d*f - a*e*g)*(d + e*x)^(5/2)*(f + g*x)^(7/2)) + (4*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(35*(c*d*f - a*e*g)^2*(d + e*x)^(5/2)*(f + g*x)^(5/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(3/2)/(f + g*x)^(11/2), x, 3, (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(9*(c*d*f - a*e*g)*(d + e*x)^(5/2)*(f + g*x)^(9/2)) + (8*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(63*(c*d*f - a*e*g)^2*(d + e*x)^(5/2)*(f + g*x)^(7/2)) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(315*(c*d*f - a*e*g)^3*(d + e*x)^(5/2)*(f + g*x)^(5/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(3/2)/(f + g*x)^(13/2), x, 4, (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(11*(c*d*f - a*e*g)*(d + e*x)^(5/2)*(f + g*x)^(11/2)) + (4*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(33*(c*d*f - a*e*g)^2*(d + e*x)^(5/2)*(f + g*x)^(9/2)) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(231*(c*d*f - a*e*g)^3*(d + e*x)^(5/2)*(f + g*x)^(7/2)) + (32*c^3*d^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(1155*(c*d*f - a*e*g)^4*(d + e*x)^(5/2)*(f + g*x)^(5/2))]

    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2)*(f + g*x)^(3/2), x, 9, -((3*(c*d*f - a*e*g)^4*sqrt(f + g*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(128*c^2*d^2*g^3*sqrt(d + e*x))) - ((c*d*f - a*e*g)^3*(f + g*x)^(3/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(64*c*d*g^3*sqrt(d + e*x)) + ((c*d*f - a*e*g)^2*(f + g*x)^(5/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(16*g^3*sqrt(d + e*x)) - ((c*d*f - a*e*g)*(f + g*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(8*g^2*(d + e*x)^(3/2)) + ((f + g*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(5*g*(d + e*x)^(5/2)) - (3*(c*d*f - a*e*g)^5*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(128*c^(5/2)*d^(5/2)*g^(7/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2)*(f + g*x)^(1/2), x, 8, -((5*(c*d*f - a*e*g)^3*sqrt(f + g*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(64*c*d*g^3*sqrt(d + e*x))) + (5*(c*d*f - a*e*g)^2*(f + g*x)^(3/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(32*g^3*sqrt(d + e*x)) - (5*(c*d*f - a*e*g)*(f + g*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(24*g^2*(d + e*x)^(3/2)) + ((f + g*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(4*g*(d + e*x)^(5/2)) - (5*(c*d*f - a*e*g)^4*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(64*c^(3/2)*d^(3/2)*g^(7/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2)/(f + g*x)^(1/2), x, 7, (5*(c*d*f - a*e*g)^2*sqrt(f + g*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(8*g^3*sqrt(d + e*x)) - (5*(c*d*f - a*e*g)*sqrt(f + g*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(12*g^2*(d + e*x)^(3/2)) + (sqrt(f + g*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(3*g*(d + e*x)^(5/2)) - (5*(c*d*f - a*e*g)^3*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(8*sqrt(c)*sqrt(d)*g^(7/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2)/(f + g*x)^(3/2), x, 7, -((15*c*d*(c*d*f - a*e*g)*sqrt(f + g*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(4*g^3*sqrt(d + e*x))) + (5*c*d*sqrt(f + g*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(2*g^2*(d + e*x)^(3/2)) - (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(g*(d + e*x)^(5/2)*sqrt(f + g*x)) + (15*sqrt(c)*sqrt(d)*(c*d*f - a*e*g)^2*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(4*g^(7/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2)/(f + g*x)^(5/2), x, 7, (5*c^2*d^2*sqrt(f + g*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(g^3*sqrt(d + e*x)) - (10*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*g^2*(d + e*x)^(3/2)*sqrt(f + g*x)) - (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(3*g*(d + e*x)^(5/2)*(f + g*x)^(3/2)) - (5*c^(3/2)*d^(3/2)*(c*d*f - a*e*g)*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(g^(7/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2)/(f + g*x)^(7/2), x, 7, -((2*c^2*d^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(g^3*sqrt(d + e*x)*sqrt(f + g*x))) - (2*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*g^2*(d + e*x)^(3/2)*(f + g*x)^(3/2)) - (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(5*g*(d + e*x)^(5/2)*(f + g*x)^(5/2)) + (2*c^(5/2)*d^(5/2)*sqrt(a*e + c*d*x)*sqrt(d + e*x)*atanh((sqrt(g)*sqrt(a*e + c*d*x))/(sqrt(c)*sqrt(d)*sqrt(f + g*x))))/(g^(7/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2)/(f + g*x)^(9/2), x, 1, (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(7*(c*d*f - a*e*g)*(d + e*x)^(7/2)*(f + g*x)^(7/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2)/(f + g*x)^(11/2), x, 2, (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(9*(c*d*f - a*e*g)*(d + e*x)^(7/2)*(f + g*x)^(9/2)) + (4*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(63*(c*d*f - a*e*g)^2*(d + e*x)^(7/2)*(f + g*x)^(7/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2)/(f + g*x)^(13/2), x, 3, (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(11*(c*d*f - a*e*g)*(d + e*x)^(7/2)*(f + g*x)^(11/2)) + (8*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(99*(c*d*f - a*e*g)^2*(d + e*x)^(7/2)*(f + g*x)^(9/2)) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(693*(c*d*f - a*e*g)^3*(d + e*x)^(7/2)*(f + g*x)^(7/2))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2)/(f + g*x)^(15/2), x, 4, (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(13*(c*d*f - a*e*g)*(d + e*x)^(7/2)*(f + g*x)^(13/2)) + (12*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(143*(c*d*f - a*e*g)^2*(d + e*x)^(7/2)*(f + g*x)^(11/2)) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(429*(c*d*f - a*e*g)^3*(d + e*x)^(7/2)*(f + g*x)^(9/2)) + (32*c^3*d^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(3003*(c*d*f - a*e*g)^4*(d + e*x)^(7/2)*(f + g*x)^(7/2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^(m/2) / (a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^(m/2)*(f+g*x)^n*when*n*symbolic=#

    @test_int [(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)*(f + g*x)^n, x, 3, -(((a*e + c*d*x)*(d + e*x)^(5/2)*(f + g*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, -(1/2) + n, 2 + n, (c*d*(f + g*x))/(c*d*f - a*e*g)))/((c*d*f - a*e*g)*(1 + n)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))), -((2*sqrt(d + e*x)*(f + g*x)^n*HypergeometricFunctions._₂F₁(-(3/2), -n, -(1/2), -((g*(a*e + c*d*x))/(c*d*f - a*e*g))))/(((c*d*(f + g*x))/(c*d*f - a*e*g))^n*(3*c*d*(a*e + c*d*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))]
    @test_int [(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)*(f + g*x)^n, x, 3, -(((a*e + c*d*x)*(d + e*x)^(3/2)*(f + g*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1/2 + n, 2 + n, (c*d*(f + g*x))/(c*d*f - a*e*g)))/((c*d*f - a*e*g)*(1 + n)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))), -((2*sqrt(d + e*x)*(f + g*x)^n*HypergeometricFunctions._₂F₁(-(1/2), -n, 1/2, -((g*(a*e + c*d*x))/(c*d*f - a*e*g))))/(((c*d*(f + g*x))/(c*d*f - a*e*g))^n*(c*d*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))))]
    @test_int [(d + e*x)^(1/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/2)*(f + g*x)^n, x, 3, -(((a*e + c*d*x)*sqrt(d + e*x)*(f + g*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 3/2 + n, 2 + n, (c*d*(f + g*x))/(c*d*f - a*e*g)))/((c*d*f - a*e*g)*(1 + n)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))), (2*(a*e + c*d*x)*sqrt(d + e*x)*(f + g*x)^n*HypergeometricFunctions._₂F₁(1/2, -n, 3/2, -((g*(a*e + c*d*x))/(c*d*f - a*e*g))))/(((c*d*(f + g*x))/(c*d*f - a*e*g))^n*(c*d*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/2)/(d + e*x)^(1/2)*(f + g*x)^n, x, 3, -(((a*e + c*d*x)*(f + g*x)^(1 + n)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)*HypergeometricFunctions._₂F₁(1, 5/2 + n, 2 + n, (c*d*(f + g*x))/(c*d*f - a*e*g)))/((c*d*f - a*e*g)*(1 + n)*sqrt(d + e*x))), (2*(a*e + c*d*x)*(f + g*x)^n*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)*HypergeometricFunctions._₂F₁(3/2, -n, 5/2, -((g*(a*e + c*d*x))/(c*d*f - a*e*g))))/(((c*d*(f + g*x))/(c*d*f - a*e*g))^n*(3*c*d*sqrt(d + e*x)))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(3/2)*(f + g*x)^n, x, 3, -(((a*e + c*d*x)*(f + g*x)^(1 + n)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)*HypergeometricFunctions._₂F₁(1, 7/2 + n, 2 + n, (c*d*(f + g*x))/(c*d*f - a*e*g)))/((c*d*f - a*e*g)*(1 + n)*(d + e*x)^(3/2))), (2*(a*e + c*d*x)^2*(f + g*x)^n*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)*HypergeometricFunctions._₂F₁(5/2, -n, 7/2, -((g*(a*e + c*d*x))/(c*d*f - a*e*g))))/(((c*d*(f + g*x))/(c*d*f - a*e*g))^n*(5*c*d*sqrt(d + e*x)))]
    @test_int [(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(5/2)*(f + g*x)^n, x, 3, -(((a*e + c*d*x)*(f + g*x)^(1 + n)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)*HypergeometricFunctions._₂F₁(1, 9/2 + n, 2 + n, (c*d*(f + g*x))/(c*d*f - a*e*g)))/((c*d*f - a*e*g)*(1 + n)*(d + e*x)^(5/2))), (2*(a*e + c*d*x)^3*(f + g*x)^n*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)*HypergeometricFunctions._₂F₁(7/2, -n, 9/2, -((g*(a*e + c*d*x))/(c*d*f - a*e*g))))/(((c*d*(f + g*x))/(c*d*f - a*e*g))^n*(7*c*d*sqrt(d + e*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m / (a*d*e+(c*d^2+a*e^2)*x+c*d*e*x^2)^m*(f+g*x)^n*when*m*symbolic=#

    @test_int [(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m*(f + g*x)^n, x, 3, -(((a*e + c*d*x)*(d + e*x)^m*(f + g*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 2 - m + n, 2 + n, (c*d*(f + g*x))/(c*d*f - a*e*g)))/((a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m*((c*d*f - a*e*g)*(1 + n)))), ((-((g*(a*e + c*d*x))/(c*d*f - a*e*g)))^m*(d + e*x)^m*(f + g*x)^(1 + n)*HypergeometricFunctions._₂F₁(m, 1 + n, 2 + n, (c*d*(f + g*x))/(c*d*f - a*e*g)))/((a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m*(g*(1 + n)))]

    @test_int [(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m*(f + g*x)^3, x, 4, If(13>=8, -((6*(c*d*f - a*e*g)^2*(a*e^2*g + c*d*(d*g*(1 - m) - e*f*(2 - m)))*(d + e*x)^(-1 + m)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 - m))/(c^4*d^4*e*(1 - m)*(2 - m)*(3 - m)*(4 - m))) + (6*g*(c*d*f - a*e*g)^2*(d + e*x)^m*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 - m))/(c^3*d^3*e*(2 - m)*(3 - m)*(4 - m)) + (3*(c*d*f - a*e*g)*(d + e*x)^(-1 + m)*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 - m))/(c^2*d^2*(3 - m)*(4 - m)) + ((d + e*x)^(-1 + m)*(f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 - m))/(c*d*(4 - m)), -((6*(c*d*f - a*e*g)^2*(a*e^2*g + c*d*(d*g*(1 - m) - e*f*(2 - m)))*(d + e*x)^(-1 + m)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 - m))/(c^4*d^4*e*(12 - 7*m + m^2)*(2 - 3*m + m^2))) + (6*g*(c*d*f - a*e*g)^2*(d + e*x)^m*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 - m))/(c^3*d^3*e*(2 - m)*(3 - m)*(4 - m)) + (3*(c*d*f - a*e*g)*(d + e*x)^(-1 + m)*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 - m))/(c^2*d^2*(3 - m)*(4 - m)) + ((d + e*x)^(-1 + m)*(f + g*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 - m))/(c*d*(4 - m)))]
    @test_int [(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m*(f + g*x)^2, x, 3, -((2*(c*d*f - a*e*g)*(a*e^2*g + c*d*(d*g*(1 - m) - e*f*(2 - m)))*(d + e*x)^(-1 + m)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 - m))/(c^3*d^3*e*(1 - m)*(2 - m)*(3 - m))) + (2*g*(c*d*f - a*e*g)*(d + e*x)^m*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 - m))/(c^2*d^2*e*(2 - m)*(3 - m)) + ((d + e*x)^(-1 + m)*(f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 - m))/(c*d*(3 - m))]
    @test_int [(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m*(f + g*x)^1, x, 2, -(((a*e^2*g + c*d*(d*g*(1 - m) - e*f*(2 - m)))*(d + e*x)^(-1 + m)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 - m))/(c^2*d^2*e*(1 - m)*(2 - m))) + (g*(d + e*x)^m*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 - m))/(c*d*e*(2 - m))]
    @test_int [(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m*(f + g*x)^0, x, 1, ((d + e*x)^(-1 + m)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 - m))/(c*d*(1 - m))]
    @test_int [(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m/(f + g*x)^1, x, 2, ((a*e + c*d*x)*(d + e*x)^m*HypergeometricFunctions._₂F₁(1, 1 - m, 2 - m, -((g*(a*e + c*d*x))/(c*d*f - a*e*g))))/((a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m*((c*d*f - a*e*g)*(1 - m)))]
    @test_int [(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m/(f + g*x)^2, x, 2, (c*d*(a*e + c*d*x)*(d + e*x)^m*HypergeometricFunctions._₂F₁(2, 1 - m, 2 - m, -((g*(a*e + c*d*x))/(c*d*f - a*e*g))))/((a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m*((c*d*f - a*e*g)^2*(1 - m)))]
    @test_int [(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m/(f + g*x)^3, x, 2, (c^2*d^2*(a*e + c*d*x)*(d + e*x)^m*HypergeometricFunctions._₂F₁(3, 1 - m, 2 - m, -((g*(a*e + c*d*x))/(c*d*f - a*e*g))))/((a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m*((c*d*f - a*e*g)^3*(1 - m)))]

    @test_int [(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m*(f + g*x)^(3/2), x, 3, (2*(-((g*(a*e + c*d*x))/(c*d*f - a*e*g)))^m*(d + e*x)^m*(f + g*x)^(5/2)*HypergeometricFunctions._₂F₁(5/2, m, 7/2, (c*d*(f + g*x))/(c*d*f - a*e*g)))/((a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m*(5*g))]
    @test_int [(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m*(f + g*x)^(1/2), x, 3, (2*(-((g*(a*e + c*d*x))/(c*d*f - a*e*g)))^m*(d + e*x)^m*(f + g*x)^(3/2)*HypergeometricFunctions._₂F₁(3/2, m, 5/2, (c*d*(f + g*x))/(c*d*f - a*e*g)))/((a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m*(3*g))]
    @test_int [(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m/(f + g*x)^(1/2), x, 3, (2*(-((g*(a*e + c*d*x))/(c*d*f - a*e*g)))^m*(d + e*x)^m*sqrt(f + g*x)*HypergeometricFunctions._₂F₁(1/2, m, 3/2, (c*d*(f + g*x))/(c*d*f - a*e*g)))/((a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m*g)]
    @test_int [(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m/(f + g*x)^(3/2), x, 3, -((2*(-((g*(a*e + c*d*x))/(c*d*f - a*e*g)))^m*(d + e*x)^m*HypergeometricFunctions._₂F₁(-(1/2), m, 1/2, (c*d*(f + g*x))/(c*d*f - a*e*g)))/((a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m*(g*sqrt(f + g*x))))]
    @test_int [(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m/(f + g*x)^(5/2), x, 3, -((2*(-((g*(a*e + c*d*x))/(c*d*f - a*e*g)))^m*(d + e*x)^m*HypergeometricFunctions._₂F₁(-(3/2), m, -(1/2), (c*d*(f + g*x))/(c*d*f - a*e*g)))/((a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m*(3*g*(f + g*x)^(3/2))))]

    @test_int [(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m*(a*e + c*d*x)^n, x, 1, ((a*e + c*d*x)^n*(d + e*x)^(-1 + m)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 - m))/(c*d*(1 - m + n))]

    @test_int [(d + e*x)^m/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m*(c*d^2*e*g - e*(c*d^2 + a*e^2)*g - c*d*e^2*g*x)^(m-1), x, 3, -(((d + e*x)^m*((-a)*e^3*g - c*d*e^2*g*x)^m*log(a*e + c*d*x))/((a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m*(c*d*e^2*g)))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(f+g*x)^n*(a+b*x+c*x^2)^p*when*c*d^2-b*d*e+a*e^2=0*and*m+p-1=0=#

    @test_int [(d + e*x)^(3/2)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)*(f + g*x)^n, x, 4, (2*e*(f + g*x)^(1 + n)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(c*d*g*(3 + 2*n)*sqrt(d + e*x)) + ((2*a*e^2*g*(1 + n) + c*d*(e*f - d*g*(3 + 2*n)))*(a*e + c*d*x)*sqrt(d + e*x)*(f + g*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 3/2 + n, 2 + n, (c*d*(f + g*x))/(c*d*f - a*e*g)))/(c*d*g*(c*d*f - a*e*g)*(1 + n)*(3 + 2*n)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)), (2*e*(f + g*x)^(1 + n)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(c*d*g*(3 + 2*n)*sqrt(d + e*x)) - (2*(2*a*e^2*g*(1 + n) + c*d*(e*f - d*g*(3 + 2*n)))*(a*e + c*d*x)*sqrt(d + e*x)*(f + g*x)^n*HypergeometricFunctions._₂F₁(1/2, -n, 3/2, -((g*(a*e + c*d*x))/(c*d*f - a*e*g))))/(((c*d*(f + g*x))/(c*d*f - a*e*g))^n*(c^2*d^2*g*(3 + 2*n)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)))]

    @test_int [(d + e*x)^(3/2)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)*(f + g*x)^4, x, 6, (128*(c*d*f - a*e*g)^3*(10*a*e^2*g + c*d*(e*f - 11*d*g))*(2*a*e^2*g - c*d*(3*e*f - d*g))*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3465*c^6*d^6*e*g*sqrt(d + e*x)) - (128*(c*d*f - a*e*g)^3*(10*a*e^2*g + c*d*(e*f - 11*d*g))*sqrt(d + e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3465*c^5*d^5*e) - (32*(c*d*f - a*e*g)^2*(10*a*e^2*g + c*d*(e*f - 11*d*g))*(f + g*x)^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(1155*c^4*d^4*g*sqrt(d + e*x)) - (16*(c*d*f - a*e*g)*(10*a*e^2*g + c*d*(e*f - 11*d*g))*(f + g*x)^3*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(693*c^3*d^3*g*sqrt(d + e*x)) - (2*(10*a*e^2*g + c*d*(e*f - 11*d*g))*(f + g*x)^4*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(99*c^2*d^2*g*sqrt(d + e*x)) + (2*e*(f + g*x)^5*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(11*c*d*g*sqrt(d + e*x))]
    @test_int [(d + e*x)^(3/2)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)*(f + g*x)^3, x, 5, (16*(c*d*f - a*e*g)^2*(8*a*e^2*g + c*d*(e*f - 9*d*g))*(2*a*e^2*g - c*d*(3*e*f - d*g))*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(315*c^5*d^5*e*g*sqrt(d + e*x)) - (16*(c*d*f - a*e*g)^2*(8*a*e^2*g + c*d*(e*f - 9*d*g))*sqrt(d + e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(315*c^4*d^4*e) - (4*(c*d*f - a*e*g)*(8*a*e^2*g + c*d*(e*f - 9*d*g))*(f + g*x)^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(105*c^3*d^3*g*sqrt(d + e*x)) - (2*(8*a*e^2*g + c*d*(e*f - 9*d*g))*(f + g*x)^3*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(63*c^2*d^2*g*sqrt(d + e*x)) + (2*e*(f + g*x)^4*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(9*c*d*g*sqrt(d + e*x))]
    @test_int [(d + e*x)^(3/2)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)*(f + g*x)^2, x, 4, (8*(c*d*f - a*e*g)*(6*a*e^2*g + c*d*(e*f - 7*d*g))*(2*a*e^2*g - c*d*(3*e*f - d*g))*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(105*c^4*d^4*e*g*sqrt(d + e*x)) - (8*(c*d*f - a*e*g)*(6*a*e^2*g + c*d*(e*f - 7*d*g))*sqrt(d + e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(105*c^3*d^3*e) - (2*(6*a*e^2*g + c*d*(e*f - 7*d*g))*(f + g*x)^2*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(35*c^2*d^2*g*sqrt(d + e*x)) + (2*e*(f + g*x)^3*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(7*c*d*g*sqrt(d + e*x))]
    @test_int [(d + e*x)^(3/2)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)*(f + g*x)^1, x, 3, -((4*(c*d^2 - a*e^2)*(4*a*e^2*g - c*d*(5*e*f - d*g))*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(15*c^3*d^3*e*sqrt(d + e*x))) - (2*(4*a*e^2*g - c*d*(5*e*f - d*g))*sqrt(d + e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(15*c^2*d^2*e) + (2*g*(d + e*x)^(3/2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(5*c*d*e)]
    @test_int [(d + e*x)^(3/2)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)*(f + g*x)^0, x, 2, (4*(c*d^2 - a*e^2)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3*c^2*d^2*sqrt(d + e*x)) + (2*sqrt(d + e*x)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3*c*d)]
    @test_int [(d + e*x)^(3/2)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(f + g*x)^1, x, 3, (2*e*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(c*d*g*sqrt(d + e*x)) - (2*(e*f - d*g)*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(g^(3/2)*sqrt(c*d*f - a*e*g))]
    @test_int [(d + e*x)^(3/2)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(f + g*x)^2, x, 3, -(((e*f - d*g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(g*(c*d*f - a*e*g)*sqrt(d + e*x)*(f + g*x))) - ((2*a*e^2*g - c*d*(e*f + d*g))*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(g^(3/2)*(c*d*f - a*e*g)^(3/2))]
    @test_int [(d + e*x)^(3/2)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(f + g*x)^3, x, 4, -(((e*f - d*g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(2*g*(c*d*f - a*e*g)*sqrt(d + e*x)*(f + g*x)^2)) - ((4*a*e^2*g - c*d*(e*f + 3*d*g))*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(4*g*(c*d*f - a*e*g)^2*sqrt(d + e*x)*(f + g*x)) - (c*d*(4*a*e^2*g - c*d*(e*f + 3*d*g))*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(4*g^(3/2)*(c*d*f - a*e*g)^(5/2))]
    @test_int [(d + e*x)^(3/2)/sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(f + g*x)^4, x, 5, -(((e*f - d*g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(3*g*(c*d*f - a*e*g)*sqrt(d + e*x)*(f + g*x)^3)) - ((6*a*e^2*g - c*d*(e*f + 5*d*g))*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(12*g*(c*d*f - a*e*g)^2*sqrt(d + e*x)*(f + g*x)^2) - (c*d*(6*a*e^2*g - c*d*(e*f + 5*d*g))*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(8*g*(c*d*f - a*e*g)^3*sqrt(d + e*x)*(f + g*x)) - (c^2*d^2*(6*a*e^2*g - c*d*(e*f + 5*d*g))*atan((sqrt(g)*sqrt(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/(sqrt(c*d*f - a*e*g)*sqrt(d + e*x))))/(8*g^(3/2)*(c*d*f - a*e*g)^(7/2))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(f+g*x)^m*(a+b*x+c*x^2)^p*when*e*f+d*g=0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(1+d*x)^(m/2)*(1-d*x)^(m/2)*(a+b*x+c*x^2)^p=#

    #= ::Subsubsection:: =#
    #=m>0=#

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [(a + b*x + c*x^2)^3/(sqrt(1 + d*x)*sqrt(1 - d*x)), x, 8, -((b*(24*c^2 + 10*b^2*d^2 + 60*a*c*d^2 + 45*a^2*d^4)*sqrt(1 - d^2*x^2))/(15*d^6)) - ((5*c^3 + 18*b^2*c*d^2 + 18*a*c^2*d^2 + 24*a*b^2*d^4 + 24*a^2*c*d^4)*x*sqrt(1 - d^2*x^2))/(16*d^6) - (b*(12*c^2 + 5*b^2*d^2 + 30*a*c*d^2)*x^2*sqrt(1 - d^2*x^2))/(15*d^4) - (c*(5*c^2 + 18*b^2*d^2 + 18*a*c*d^2)*x^3*sqrt(1 - d^2*x^2))/(24*d^4) - (3*b*c^2*x^4*sqrt(1 - d^2*x^2))/(5*d^2) - (c^3*x^5*sqrt(1 - d^2*x^2))/(6*d^2) + ((5*c^3 + 18*b^2*c*d^2 + 18*a*c^2*d^2 + 24*a*b^2*d^4 + 24*a^2*c*d^4 + 16*a^3*d^6)*asin(d*x))/(16*d^7)]
    @test_int [(a + b*x + c*x^2)^2/(sqrt(1 + d*x)*sqrt(1 - d*x)), x, 6, -((2*b*(2*c + 3*a*d^2)*sqrt(1 - d^2*x^2))/(3*d^4)) - ((4*b^2 + c*(8*a + (3*c)/d^2))*x*sqrt(1 - d^2*x^2))/(8*d^2) - (2*b*c*x^2*sqrt(1 - d^2*x^2))/(3*d^2) - (c^2*x^3*sqrt(1 - d^2*x^2))/(4*d^2) + ((3*c^2 + 4*b^2*d^2 + 8*a*c*d^2 + 8*a^2*d^4)*asin(d*x))/(8*d^5)]
    @test_int [(a + b*x + c*x^2)^1/(sqrt(1 + d*x)*sqrt(1 - d*x)), x, 4, -((b*sqrt(1 - d^2*x^2))/d^2) - (c*x*sqrt(1 - d^2*x^2))/(2*d^2) + ((c + 2*a*d^2)*asin(d*x))/(2*d^3)]
    @test_int [1/((a + b*x + c*x^2)^1*sqrt(1 + d*x)*sqrt(1 - d*x)), x, 6, -((sqrt(2)*c*atanh((2*c + (b - sqrt(b^2 - 4*a*c))*d^2*x)/(sqrt(2)*sqrt(2*c^2 + 2*a*c*d^2 - b*(b - sqrt(b^2 - 4*a*c))*d^2)*sqrt(1 - d^2*x^2))))/(sqrt(b^2 - 4*a*c)*sqrt(2*c^2 + 2*a*c*d^2 - b*(b - sqrt(b^2 - 4*a*c))*d^2))) + (sqrt(2)*c*atanh((2*c + (b + sqrt(b^2 - 4*a*c))*d^2*x)/(sqrt(2)*sqrt(2*c^2 + 2*a*c*d^2 - b*(b + sqrt(b^2 - 4*a*c))*d^2)*sqrt(1 - d^2*x^2))))/(sqrt(b^2 - 4*a*c)*sqrt(2*c^2 + 2*a*c*d^2 - b*(b + sqrt(b^2 - 4*a*c))*d^2))]
    @test_int [1/((a + b*x + c*x^2)^2*sqrt(1 + d*x)*sqrt(1 - d*x)), x, 7, -(((b*(b^2*d^2 - c*(c + 3*a*d^2)) - c*(2*c^2 - b^2*d^2 + 2*a*c*d^2)*x)*sqrt(1 - d^2*x^2))/((b^2 - 4*a*c)*(b^2*d^2 - (c + a*d^2)^2)*(a + b*x + c*x^2))) - (c*(4*c^3 + 12*a*c^2*d^2 - a*b*(b + sqrt(b^2 - 4*a*c))*d^4 - c*d^2*(5*b^2 - b*sqrt(b^2 - 4*a*c) - 8*a^2*d^2))*atanh((2*c + (b - sqrt(b^2 - 4*a*c))*d^2*x)/(sqrt(2)*sqrt(2*c^2 + 2*a*c*d^2 - b*(b - sqrt(b^2 - 4*a*c))*d^2)*sqrt(1 - d^2*x^2))))/(sqrt(2)*(b^2 - 4*a*c)^(3/2)*sqrt(2*c^2 + 2*a*c*d^2 - b*(b - sqrt(b^2 - 4*a*c))*d^2)*(b^2*d^2 - (c + a*d^2)^2)) + (c*(4*c^3 + 12*a*c^2*d^2 - 2*a*b^2*d^4 - b*(b + sqrt(b^2 - 4*a*c))*d^2*(c - a*d^2) - 4*c*d^2*(b^2 - 2*a^2*d^2))*atanh((2*c + (b + sqrt(b^2 - 4*a*c))*d^2*x)/(sqrt(2)*sqrt(2*c^2 + 2*a*c*d^2 - b*(b + sqrt(b^2 - 4*a*c))*d^2)*sqrt(1 - d^2*x^2))))/(sqrt(2)*(b^2 - 4*a*c)^(3/2)*sqrt(2*c^2 + 2*a*c*d^2 - b*(b + sqrt(b^2 - 4*a*c))*d^2)*(b^2*d^2 - (c + a*d^2)^2))]

    @test_int [(a + b*x + c*x^2)^3/((1 + d*x)^(3/2)*(1 - d*x)^(3/2)), x, 7, (b*(3*a^2 + (3*c^2)/d^4 + b^2/d^2 + (6*a*c)/d^2)*d^4 + (c + a*d^2)*(c^2 + 3*b^2*d^2 + 2*a*c*d^2 + a^2*d^4)*x)/(d^6*sqrt(1 - d^2*x^2)) + (b*(5*c^2 + b^2*d^2 + 6*a*c*d^2)*sqrt(1 - d^2*x^2))/d^6 + (c*(7*c^2 + 12*b^2*d^2 + 12*a*c*d^2)*x*sqrt(1 - d^2*x^2))/(8*d^6) + (b*c^2*x^2*sqrt(1 - d^2*x^2))/d^4 + (c^3*x^3*sqrt(1 - d^2*x^2))/(4*d^4) - (3*(5*c^3 + 12*b^2*c*d^2 + 12*a*c^2*d^2 + 8*a*b^2*d^4 + 8*a^2*c*d^4)*asin(d*x))/(8*d^7)]
    @test_int [(a + b*x + c*x^2)^2/((1 + d*x)^(3/2)*(1 - d*x)^(3/2)), x, 5, (2*b*(a + c/d^2)*d^2 + (c^2 + b^2*d^2 + 2*a*c*d^2 + a^2*d^4)*x)/(d^4*sqrt(1 - d^2*x^2)) + (2*b*c*sqrt(1 - d^2*x^2))/d^4 + (c^2*x*sqrt(1 - d^2*x^2))/(2*d^4) - ((2*b^2 + c*(4*a + (3*c)/d^2))*asin(d*x))/(2*d^3)]
    @test_int [(a + b*x + c*x^2)^1/((1 + d*x)^(3/2)*(1 - d*x)^(3/2)), x, 4, (b + (c + a*d^2)*x)/(d^2*sqrt(1 - d^2*x^2)) - (c*asin(d*x))/d^3]
    @test_int [1/((a + b*x + c*x^2)^1*(1 + d*x)^(3/2)*(1 - d*x)^(3/2)), x, 7, (d^2*(b - (c + a*d^2)*x))/((b^2*d^2 - (c + a*d^2)^2)*sqrt(1 - d^2*x^2)) + (c*(2*c^2 + 2*a*c*d^2 - b*(b + sqrt(b^2 - 4*a*c))*d^2)*atanh((2*c + (b - sqrt(b^2 - 4*a*c))*d^2*x)/(sqrt(2)*sqrt(2*c^2 + 2*a*c*d^2 - b*(b - sqrt(b^2 - 4*a*c))*d^2)*sqrt(1 - d^2*x^2))))/(sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(2*c^2 + 2*a*c*d^2 - b*(b - sqrt(b^2 - 4*a*c))*d^2)*(b^2*d^2 - (c + a*d^2)^2)) - (c*(2*c^2 + 2*a*c*d^2 - b*(b - sqrt(b^2 - 4*a*c))*d^2)*atanh((2*c + (b + sqrt(b^2 - 4*a*c))*d^2*x)/(sqrt(2)*sqrt(2*c^2 + 2*a*c*d^2 - b*(b + sqrt(b^2 - 4*a*c))*d^2)*sqrt(1 - d^2*x^2))))/(sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(2*c^2 + 2*a*c*d^2 - b*(b + sqrt(b^2 - 4*a*c))*d^2)*(b^2*d^2 - (c + a*d^2)^2))]
    @test_int [1/((a + b*x + c*x^2)^2*(1 + d*x)^(3/2)*(1 - d*x)^(3/2)), x, 8, -((d^2*(b*(c^3 + 2*b^2*c*d^2 - 10*a*c^2*d^2 + 3*a*b^2*d^4 - 11*a^2*c*d^4) - (2*c^4 + b^2*d^4*(2*b^2 + a^2*d^2) - c^2*d^2*(b^2 + 6*a^2*d^2) - c*(6*a*b^2*d^4 + 4*a^3*d^6))*x))/((b^2 - 4*a*c)*(c - b*d + a*d^2)^2*(c + b*d + a*d^2)^2*sqrt(1 - d^2*x^2))) - (b*(b^2*d^2 - c*(c + 3*a*d^2)) - c*(2*c^2 - b^2*d^2 + 2*a*c*d^2)*x)/((b^2 - 4*a*c)*(b^2*d^2 - (c + a*d^2)^2)*(a + b*x + c*x^2)*sqrt(1 - d^2*x^2)) + (c*(4*c^5 + 24*a*c^4*d^2 + 3*a*b^3*(b + sqrt(b^2 - 4*a*c))*d^6 - c^3*d^2*(9*b^2 - b*sqrt(b^2 - 4*a*c) - 36*a^2*d^2) - 2*a*c^2*d^4*(7*b^2 + 5*b*sqrt(b^2 - 4*a*c) - 8*a^2*d^2) + b*c*d^4*(2*b^3 + 2*b^2*sqrt(b^2 - 4*a*c) - 17*a^2*b*d^2 - 11*a^2*sqrt(b^2 - 4*a*c)*d^2))*atanh((2*c + (b - sqrt(b^2 - 4*a*c))*d^2*x)/(sqrt(2)*sqrt(2*c^2 + 2*a*c*d^2 - b*(b - sqrt(b^2 - 4*a*c))*d^2)*sqrt(1 - d^2*x^2))))/(sqrt(2)*(b^2 - 4*a*c)^(3/2)*sqrt(2*c^2 + 2*a*c*d^2 - b*(b - sqrt(b^2 - 4*a*c))*d^2)*(c^2 - b^2*d^2 + 2*a*c*d^2 + a^2*d^4)^2) + (c*(b*(b + sqrt(b^2 - 4*a*c))*d^4*(c^3 + 2*b^2*c*d^2 - 10*a*c^2*d^2 + 3*a*b^2*d^4 - 11*a^2*c*d^4) - 2*(2*c^5*d^2 + 12*a*c^4*d^4 + 3*a*b^4*d^8 + 2*b^2*c*d^6*(b^2 - 7*a^2*d^2) - c^3*(4*b^2*d^4 - 18*a^2*d^6) - 4*c^2*(3*a*b^2*d^6 - 2*a^3*d^8)))*atanh((2*c + (b + sqrt(b^2 - 4*a*c))*d^2*x)/(sqrt(2)*sqrt(2*c^2 + 2*a*c*d^2 - b*(b + sqrt(b^2 - 4*a*c))*d^2)*sqrt(1 - d^2*x^2))))/(sqrt(2)*(b^2 - 4*a*c)^(3/2)*d^2*sqrt(2*c^2 + 2*a*c*d^2 - b*(b + sqrt(b^2 - 4*a*c))*d^2)*(c^2 - b^2*d^2 + 2*a*c*d^2 + a^2*d^4)^2)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(1+d*x)^m*(1-d*x)^m*(a+b*x+c*x^2)^p*when*m*symbolic=#

    @test_int [(1 + e*x)^m*(1 - e*x)^m*(a + c*x^2)^p, x, 3, (x*(a + c*x^2)^p*AppellF1(1/2, -p, -m, 3/2, -((c*x^2)/a), e^2*x^2))/(1 + (c*x^2)/a)^p]
    @test_int [(d + e*x)^m*(d - e*x)^m*(a + c*x^2)^p, x, 4, (x*(d - e*x)^m*(d + e*x)^m*(a + c*x^2)^p*AppellF1(1/2, -p, -m, 3/2, -((c*x^2)/a), (e^2*x^2)/d^2))/((1 + (c*x^2)/a)^p*(1 - (e^2*x^2)/d^2)^m)]
    @test_int [(d + e*x)^m*(d*f - e*f*x)^m*(a + c*x^2)^p, x, 4, (x*(d + e*x)^m*(d*f - e*f*x)^m*(a + c*x^2)^p*AppellF1(1/2, -p, -m, 3/2, -((c*x^2)/a), (e^2*x^2)/d^2))/((1 + (c*x^2)/a)^p*(1 - (e^2*x^2)/d^2)^m)]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(f+g*x)^n*(a+b*x+c*x^2)^p*when*2*c*d-b*e=0=#

    @test_int [(d + e*x)^3*(f + g*x)^n*(a + 2*c*d*x + c*e*x^2), x, 2, -(((e*f - d*g)^3*(a*g^2 + c*f*(e*f - 2*d*g))*(f + g*x)^(1 + n))/(g^6*(1 + n))) + ((e*f - d*g)^2*(3*a*e*g^2 + c*(5*e^2*f^2 - 10*d*e*f*g + 2*d^2*g^2))*(f + g*x)^(2 + n))/(g^6*(2 + n)) - (e*(e*f - d*g)*(3*a*e*g^2 + c*(10*e^2*f^2 - 20*d*e*f*g + 7*d^2*g^2))*(f + g*x)^(3 + n))/(g^6*(3 + n)) + (e^2*(a*e*g^2 + c*(10*e^2*f^2 - 20*d*e*f*g + 9*d^2*g^2))*(f + g*x)^(4 + n))/(g^6*(4 + n)) - (5*c*e^3*(e*f - d*g)*(f + g*x)^(5 + n))/(g^6*(5 + n)) + (c*e^4*(f + g*x)^(6 + n))/(g^6*(6 + n))]
    @test_int [(d + e*x)^2*(f + g*x)^n*(a + 2*c*d*x + c*e*x^2), x, 2, ((e*f - d*g)^2*(a*g^2 + c*f*(e*f - 2*d*g))*(f + g*x)^(1 + n))/(g^5*(1 + n)) - (2*(e*f - d*g)*(a*e*g^2 + c*(2*e^2*f^2 - 4*d*e*f*g + d^2*g^2))*(f + g*x)^(2 + n))/(g^5*(2 + n)) + (e*(a*e*g^2 + c*(6*e^2*f^2 - 12*d*e*f*g + 5*d^2*g^2))*(f + g*x)^(3 + n))/(g^5*(3 + n)) - (4*c*e^2*(e*f - d*g)*(f + g*x)^(4 + n))/(g^5*(4 + n)) + (c*e^3*(f + g*x)^(5 + n))/(g^5*(5 + n))]
    @test_int [(d + e*x)^1*(f + g*x)^n*(a + 2*c*d*x + c*e*x^2), x, 2, -(((e*f - d*g)*(a*g^2 + c*f*(e*f - 2*d*g))*(f + g*x)^(1 + n))/(g^4*(1 + n))) + ((a*e*g^2 + c*(3*e^2*f^2 - 6*d*e*f*g + 2*d^2*g^2))*(f + g*x)^(2 + n))/(g^4*(2 + n)) - (3*c*e*(e*f - d*g)*(f + g*x)^(3 + n))/(g^4*(3 + n)) + (c*e^2*(f + g*x)^(4 + n))/(g^4*(4 + n))]
    @test_int [(d + e*x)^0*(f + g*x)^n*(a + 2*c*d*x + c*e*x^2), x, 2, ((a*g^2 + c*f*(e*f - 2*d*g))*(f + g*x)^(1 + n))/(g^3*(1 + n)) - (2*c*(e*f - d*g)*(f + g*x)^(2 + n))/(g^3*(2 + n)) + (c*e*(f + g*x)^(3 + n))/(g^3*(3 + n))]
    @test_int [(f + g*x)^n*(a + 2*c*d*x + c*e*x^2)/(d + e*x)^1, x, 3, -((c*(e*f - d*g)*(f + g*x)^(1 + n))/(e*g^2*(1 + n))) + (c*(f + g*x)^(2 + n))/(g^2*(2 + n)) + ((c*d^2 - a*e)*(f + g*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (e*(f + g*x))/(e*f - d*g)))/(e*(e*f - d*g)*(1 + n))]
    @test_int [(f + g*x)^n*(a + 2*c*d*x + c*e*x^2)/(d + e*x)^2, x, 3, (c*(f + g*x)^(1 + n))/(e*g*(1 + n)) - ((c*d^2 - a*e)*g*(f + g*x)^(1 + n)*HypergeometricFunctions._₂F₁(2, 1 + n, 2 + n, (e*(f + g*x))/(e*f - d*g)))/(e*(e*f - d*g)^2*(1 + n))]
    @test_int [(f + g*x)^n*(a + 2*c*d*x + c*e*x^2)/(d + e*x)^3, x, 3, -(((a - (c*d^2)/e)*(f + g*x)^(1 + n))/(2*(e*f - d*g)*(d + e*x)^2)) - ((c*d^2 - a*e)*g*(1 - n)*(f + g*x)^(1 + n))/(2*e*(e*f - d*g)^2*(d + e*x)) + ((a*e*g^2*(1 - n)*n - c*(2*e^2*f^2 - 4*d*e*f*g + d^2*g^2*(2 + n - n^2)))*(f + g*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (e*(f + g*x))/(e*f - d*g)))/(2*e*(e*f - d*g)^3*(1 + n))]
    @test_int [(f + g*x)^n*(a + 2*c*d*x + c*e*x^2)/(d + e*x)^4, x, 3, -(((a - (c*d^2)/e)*(f + g*x)^(1 + n))/(3*(e*f - d*g)*(d + e*x)^3)) - ((c*d^2 - a*e)*g*(2 - n)*(f + g*x)^(1 + n))/(6*e*(e*f - d*g)^2*(d + e*x)^2) + (g*(a*e*g^2*(2 - 3*n + n^2) + c*(6*e^2*f^2 - 12*d*e*f*g + d^2*g^2*(4 + 3*n - n^2)))*(f + g*x)^(1 + n)*HypergeometricFunctions._₂F₁(2, 1 + n, 2 + n, (e*(f + g*x))/(e*f - d*g)))/(6*e*(e*f - d*g)^4*(1 + n))]

    @test_int [(d + e*x)^m*(f + g*x)^n*(a + 2*c*d*x + c*e*x^2), x, 4, -((c*(e*f - d*g)*(2 + m)*(d + e*x)^(1 + m)*(f + g*x)^(1 + n))/(e*g^2*(2 + m + n)*(3 + m + n))) + (c*(d + e*x)^(2 + m)*(f + g*x)^(1 + n))/(e*g*(3 + m + n)) + (1/(e^2*g^2*(1 + m)*(2 + m + n)*(3 + m + n)))*(((c*(e*f - d*g)*(2 + m)*(e*f*(1 + m) + d*g*(1 + n)) + g*(2 + m + n)*(a*e*g*(3 + m + n) - c*d*(e*f*(2 + m) + d*g*(1 + n))))*(d + e*x)^(1 + m)*(f + g*x)^n*HypergeometricFunctions._₂F₁(1 + m, -n, 2 + m, -((g*(d + e*x))/(e*f - d*g))))/((e*(f + g*x))/(e*f - d*g))^n)]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(f+g*x)^n*(a+b*x+c*x^2)^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(f+g*x)^n*(a+b*x+c*x^2)^p=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [(a + b*x + c*x^2)/((d + e*x)*(f + g*x)), x, 2, (c*x)/(e*g) + ((c*d^2 - b*d*e + a*e^2)*log(d + e*x))/(e^2*(e*f - d*g)) - ((c*f^2 - b*f*g + a*g^2)*log(f + g*x))/(g^2*(e*f - d*g))]

    @test_int [(a + b*x + c*x^2)^2/((d + e*x)*(f + g*x)), x, 2, ((b^2*e^2*g^2 - 2*c*e*g*(b*e*f + b*d*g - a*e*g) + c^2*(e^2*f^2 + d*e*f*g + d^2*g^2))*x)/(e^3*g^3) - (c*(c*e*f + c*d*g - 2*b*e*g)*x^2)/(2*e^2*g^2) + (c^2*x^3)/(3*e*g) + ((c*d^2 - b*d*e + a*e^2)^2*log(d + e*x))/(e^4*(e*f - d*g)) - ((c*f^2 - b*f*g + a*g^2)^2*log(f + g*x))/(g^4*(e*f - d*g))]

    @test_int [(a + b*x + c*x^2)^3/((d + e*x)*(f + g*x)), x, 2, -((1/(e^5*g^5))*((b^2*e^3*g^3*(b*e*f + b*d*g - 3*a*e*g) - c^3*(e^4*f^4 + d*e^3*f^3*g + d^2*e^2*f^2*g^2 + d^3*e*f*g^3 + d^4*g^4) - 3*c*e^2*g^2*(a^2*e^2*g^2 - 2*a*b*e*g*(e*f + d*g) + b^2*(e^2*f^2 + d*e*f*g + d^2*g^2)) - 3*c^2*e*g*(a*e*g*(e^2*f^2 + d*e*f*g + d^2*g^2) - b*(e^3*f^3 + d*e^2*f^2*g + d^2*e*f*g^2 + d^3*g^3)))*x)) + ((b^3*e^3*g^3 - 3*b*c*e^2*g^2*(b*e*f + b*d*g - 2*a*e*g) - c^3*(e^3*f^3 + d*e^2*f^2*g + d^2*e*f*g^2 + d^3*g^3) - 3*c^2*e*g*(a*e*g*(e*f + d*g) - b*(e^2*f^2 + d*e*f*g + d^2*g^2)))*x^2)/(2*e^4*g^4) + (c*(3*b^2*e^2*g^2 - 3*c*e*g*(b*e*f + b*d*g - a*e*g) + c^2*(e^2*f^2 + d*e*f*g + d^2*g^2))*x^3)/(3*e^3*g^3) - (c^2*(c*e*f + c*d*g - 3*b*e*g)*x^4)/(4*e^2*g^2) + (c^3*x^5)/(5*e*g) + ((c*d^2 - b*d*e + a*e^2)^3*log(d + e*x))/(e^6*(e*f - d*g)) - ((c*f^2 - b*f*g + a*g^2)^3*log(f + g*x))/(g^6*(e*f - d*g))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [1/((a + b*x + c*x^2)*(d + e*x)*(f + g*x)), x, 6, -(((2*c^2*d*f + b^2*e*g - c*(b*e*f + b*d*g + 2*a*e*g))*atanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(sqrt(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(c*f^2 - g*(b*f - a*g)))) + (e^2*log(d + e*x))/((c*d^2 - b*d*e + a*e^2)*(e*f - d*g)) - (g^2*log(f + g*x))/((e*f - d*g)*(c*f^2 - b*f*g + a*g^2)) - ((c*e*f + c*d*g - b*e*g)*log(a + b*x + c*x^2))/(2*(c*d^2 - b*d*e + a*e^2)*(c*f^2 - g*(b*f - a*g)))]

    @test_int [1/((a + b*x + c*x^2)^2*(d + e*x)*(f + g*x)), x, 9, -((b^3*e*g - b^2*c*(e*f + d*g) + 2*a*c^2*(e*f + d*g) + b*c*(c*d*f - 3*a*e*g) + c*(2*c^2*d*f + b^2*e*g - c*(b*e*f + b*d*g + 2*a*e*g))*x)/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(c*f^2 - g*(b*f - a*g))*(a + b*x + c*x^2))) + (2*c*(2*c^2*d*f + b^2*e*g - c*(b*e*f + b*d*g + 2*a*e*g))*atanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/((b^2 - 4*a*c)^(3/2)*(c*d^2 - b*d*e + a*e^2)*(c*f^2 - g*(b*f - a*g))) + ((b^2*e^2*g^2*(b*e*f + b*d*g - 2*a*e*g) - 2*c^3*d*f*(e^2*f^2 + d*e*f*g + d^2*g^2) + 2*c*e*g*(a^2*e^2*g^2 + a*b*e*g*(e*f + d*g) - b^2*(e*f + d*g)^2) - c^2*(4*a*d*e^2*f*g^2 - b*(e^3*f^3 + 5*d*e^2*f^2*g + 5*d^2*e*f*g^2 + d^3*g^3)))*atanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(sqrt(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(c*f^2 - g*(b*f - a*g))^2) + (e^4*log(d + e*x))/((c*d^2 - b*d*e + a*e^2)^2*(e*f - d*g)) - (g^4*log(f + g*x))/((e*f - d*g)*(c*f^2 - b*f*g + a*g^2)^2) - ((c*e*f + c*d*g - b*e*g)*(c*(e^2*f^2 + d^2*g^2) + e*g*(2*a*e*g - b*(e*f + d*g)))*log(a + b*x + c*x^2))/(2*(c*d^2 - b*d*e + a*e^2)^2*(c*f^2 - g*(b*f - a*g))^2)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(f+g*x)^(n/2)*(a+b*x+c*x^2)^p=#

    #= ::Subsubsection:: =#
    #=p>0 & n>0=#

    #= ::Subsubsection::Closed:: =#
    #=p>0 & n<0=#

    @test_int [(d + e*x)^3*(a + b*x + c*x^2)/sqrt(f + g*x), x, 3, -((2*(e*f - d*g)^3*(c*f^2 - b*f*g + a*g^2)*sqrt(f + g*x))/g^6) + (2*(e*f - d*g)^2*(c*f*(5*e*f - 2*d*g) - g*(4*b*e*f - b*d*g - 3*a*e*g))*(f + g*x)^(3/2))/(3*g^6) + (2*(e*f - d*g)*(3*e*g*(2*b*e*f - b*d*g - a*e*g) - c*(10*e^2*f^2 - 8*d*e*f*g + d^2*g^2))*(f + g*x)^(5/2))/(5*g^6) - (2*e*(e*g*(4*b*e*f - 3*b*d*g - a*e*g) - c*(10*e^2*f^2 - 12*d*e*f*g + 3*d^2*g^2))*(f + g*x)^(7/2))/(7*g^6) - (2*e^2*(5*c*e*f - 3*c*d*g - b*e*g)*(f + g*x)^(9/2))/(9*g^6) + (2*c*e^3*(f + g*x)^(11/2))/(11*g^6)]
    @test_int [(d + e*x)^2*(a + b*x + c*x^2)/sqrt(f + g*x), x, 3, (2*(e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)*sqrt(f + g*x))/g^5 - (2*(e*f - d*g)*(2*c*f*(2*e*f - d*g) - g*(3*b*e*f - b*d*g - 2*a*e*g))*(f + g*x)^(3/2))/(3*g^5) - (2*(e*g*(3*b*e*f - 2*b*d*g - a*e*g) - c*(6*e^2*f^2 - 6*d*e*f*g + d^2*g^2))*(f + g*x)^(5/2))/(5*g^5) - (2*e*(4*c*e*f - 2*c*d*g - b*e*g)*(f + g*x)^(7/2))/(7*g^5) + (2*c*e^2*(f + g*x)^(9/2))/(9*g^5)]
    @test_int [(d + e*x)^1*(a + b*x + c*x^2)/sqrt(f + g*x), x, 2, -((2*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)*sqrt(f + g*x))/g^4) + (2*(c*f*(3*e*f - 2*d*g) - g*(2*b*e*f - b*d*g - a*e*g))*(f + g*x)^(3/2))/(3*g^4) - (2*(3*c*e*f - c*d*g - b*e*g)*(f + g*x)^(5/2))/(5*g^4) + (2*c*e*(f + g*x)^(7/2))/(7*g^4)]
    @test_int [(d + e*x)^0*(a + b*x + c*x^2)/sqrt(f + g*x), x, 2, (2*(c*f^2 - b*f*g + a*g^2)*sqrt(f + g*x))/g^3 - (2*(2*c*f - b*g)*(f + g*x)^(3/2))/(3*g^3) + (2*c*(f + g*x)^(5/2))/(5*g^3)]
    @test_int [(a + b*x + c*x^2)/((d + e*x)^1*sqrt(f + g*x)), x, 4, (2*(b*e*g - c*(e*f + d*g))*sqrt(f + g*x))/(e^2*g^2) + (2*c*(f + g*x)^(3/2))/(3*e*g^2) - (2*(c*d^2 - b*d*e + a*e^2)*atanh((sqrt(e)*sqrt(f + g*x))/sqrt(e*f - d*g)))/(e^(5/2)*sqrt(e*f - d*g))]
    @test_int [(a + b*x + c*x^2)/((d + e*x)^2*sqrt(f + g*x)), x, 4, (2*c*sqrt(f + g*x))/(e^2*g) - ((a + (d*(c*d - b*e))/e^2)*sqrt(f + g*x))/((e*f - d*g)*(d + e*x)) + ((c*d*(4*e*f - 3*d*g) - e*(2*b*e*f - b*d*g - a*e*g))*atanh((sqrt(e)*sqrt(f + g*x))/sqrt(e*f - d*g)))/(e^(5/2)*(e*f - d*g)^(3/2))]
    @test_int [(a + b*x + c*x^2)/((d + e*x)^3*sqrt(f + g*x)), x, 4, -(((a + (d*(c*d - b*e))/e^2)*sqrt(f + g*x))/(2*(e*f - d*g)*(d + e*x)^2)) + ((c*d*(8*e*f - 5*d*g) - e*(4*b*e*f - b*d*g - 3*a*e*g))*sqrt(f + g*x))/(4*e^2*(e*f - d*g)^2*(d + e*x)) + ((e*g*(4*b*e*f - b*d*g - 3*a*e*g) - c*(8*e^2*f^2 - 8*d*e*f*g + 3*d^2*g^2))*atanh((sqrt(e)*sqrt(f + g*x))/sqrt(e*f - d*g)))/(4*e^(5/2)*(e*f - d*g)^(5/2))]

    @test_int [(d + e*x)^3*(a + b*x + c*x^2)/(f + g*x)^(3/2), x, 3, (2*(e*f - d*g)^3*(c*f^2 - b*f*g + a*g^2))/(g^6*sqrt(f + g*x)) + (2*(e*f - d*g)^2*(c*f*(5*e*f - 2*d*g) - g*(4*b*e*f - b*d*g - 3*a*e*g))*sqrt(f + g*x))/g^6 + (2*(e*f - d*g)*(3*e*g*(2*b*e*f - b*d*g - a*e*g) - c*(10*e^2*f^2 - 8*d*e*f*g + d^2*g^2))*(f + g*x)^(3/2))/(3*g^6) - (2*e*(e*g*(4*b*e*f - 3*b*d*g - a*e*g) - c*(10*e^2*f^2 - 12*d*e*f*g + 3*d^2*g^2))*(f + g*x)^(5/2))/(5*g^6) - (2*e^2*(5*c*e*f - 3*c*d*g - b*e*g)*(f + g*x)^(7/2))/(7*g^6) + (2*c*e^3*(f + g*x)^(9/2))/(9*g^6)]
    @test_int [(d + e*x)^2*(a + b*x + c*x^2)/(f + g*x)^(3/2), x, 3, -((2*(e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2))/(g^5*sqrt(f + g*x))) - (2*(e*f - d*g)*(2*c*f*(2*e*f - d*g) - g*(3*b*e*f - b*d*g - 2*a*e*g))*sqrt(f + g*x))/g^5 - (2*(e*g*(3*b*e*f - 2*b*d*g - a*e*g) - c*(6*e^2*f^2 - 6*d*e*f*g + d^2*g^2))*(f + g*x)^(3/2))/(3*g^5) - (2*e*(4*c*e*f - 2*c*d*g - b*e*g)*(f + g*x)^(5/2))/(5*g^5) + (2*c*e^2*(f + g*x)^(7/2))/(7*g^5)]
    @test_int [(d + e*x)^1*(a + b*x + c*x^2)/(f + g*x)^(3/2), x, 2, (2*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2))/(g^4*sqrt(f + g*x)) + (2*(c*f*(3*e*f - 2*d*g) - g*(2*b*e*f - b*d*g - a*e*g))*sqrt(f + g*x))/g^4 - (2*(3*c*e*f - c*d*g - b*e*g)*(f + g*x)^(3/2))/(3*g^4) + (2*c*e*(f + g*x)^(5/2))/(5*g^4)]
    @test_int [(d + e*x)^0*(a + b*x + c*x^2)/(f + g*x)^(3/2), x, 2, -((2*(c*f^2 - b*f*g + a*g^2))/(g^3*sqrt(f + g*x))) - (2*(2*c*f - b*g)*sqrt(f + g*x))/g^3 + (2*c*(f + g*x)^(3/2))/(3*g^3)]
    @test_int [(a + b*x + c*x^2)/((d + e*x)^1*(f + g*x)^(3/2)), x, 4, (2*(c*f^2 - b*f*g + a*g^2))/(g^2*(e*f - d*g)*sqrt(f + g*x)) + (2*c*sqrt(f + g*x))/(e*g^2) - (2*(c*d^2 - b*d*e + a*e^2)*atanh((sqrt(e)*sqrt(f + g*x))/sqrt(e*f - d*g)))/(e^(3/2)*(e*f - d*g)^(3/2))]
    @test_int [(a + b*x + c*x^2)/((d + e*x)^2*(f + g*x)^(3/2)), x, 4, -((2*(c*f^2 - b*f*g + a*g^2))/(g*(e*f - d*g)^2*sqrt(f + g*x))) - ((c*d^2 - b*d*e + a*e^2)*sqrt(f + g*x))/(e*(e*f - d*g)^2*(d + e*x)) + ((c*d*(4*e*f - d*g) - e*(2*b*e*f + b*d*g - 3*a*e*g))*atanh((sqrt(e)*sqrt(f + g*x))/sqrt(e*f - d*g)))/(e^(3/2)*(e*f - d*g)^(5/2))]
    @test_int [(a + b*x + c*x^2)/((d + e*x)^3*(f + g*x)^(3/2)), x, 5, (2*(c*f^2 - b*f*g + a*g^2))/((e*f - d*g)^3*sqrt(f + g*x)) - ((c*d^2 - b*d*e + a*e^2)*sqrt(f + g*x))/(2*e*(e*f - d*g)^2*(d + e*x)^2) + ((c*d*(8*e*f - d*g) - e*(4*b*e*f + 3*b*d*g - 7*a*e*g))*sqrt(f + g*x))/(4*e*(e*f - d*g)^3*(d + e*x)) - ((c*(8*e^2*f^2 + 8*d*e*f*g - d^2*g^2) + 3*e*g*(5*a*e*g - b*(4*e*f + d*g)))*atanh((sqrt(e)*sqrt(f + g*x))/sqrt(e*f - d*g)))/(4*e^(3/2)*(e*f - d*g)^(7/2))]

    #= ::Subsubsection:: =#
    #=p<0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^(m/2)*(f+g*x)^(n/2)*(a+b*x+c*x^2)^p=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [sqrt(-1 + x)*sqrt(1 + x)/(1 + x - x^2), x, -9, -acosh(x) + sqrt((2/5)*(-1 + sqrt(5)))*atan(sqrt(1 + x)/(sqrt(-2 + sqrt(5))*sqrt(-1 + x))) + sqrt((2/5)*(1 + sqrt(5)))*atanh(sqrt(1 + x)/(sqrt(2 + sqrt(5))*sqrt(-1 + x)))]

    @test_int [(a + b*x + c*x^2)/(sqrt(d + e*x)*sqrt(f + g*x)), x, 5, -(((3*c*e*f + 5*c*d*g - 4*b*e*g)*sqrt(d + e*x)*sqrt(f + g*x))/(4*e^2*g^2)) + (c*(d + e*x)^(3/2)*sqrt(f + g*x))/(2*e^2*g) + ((c*(3*e^2*f^2 + 2*d*e*f*g + 3*d^2*g^2) + 4*e*g*(2*a*e*g - b*(e*f + d*g)))*atanh((sqrt(g)*sqrt(d + e*x))/(sqrt(e)*sqrt(f + g*x))))/(4*e^(5/2)*g^(5/2))]

    @test_int [(a + b*x + c*x^2)*(d + e*x)^(3/2)/sqrt(f + g*x), x, 7, -(((e*f - d*g)*(c*(35*e^2*f^2 + 10*d*e*f*g + 3*d^2*g^2) + 8*e*g*(6*a*e*g - b*(5*e*f + d*g)))*sqrt(d + e*x)*sqrt(f + g*x))/(64*e^2*g^4)) + ((c*(35*e^2*f^2 + 10*d*e*f*g + 3*d^2*g^2) + 8*e*g*(6*a*e*g - b*(5*e*f + d*g)))*(d + e*x)^(3/2)*sqrt(f + g*x))/(96*e^2*g^3) - ((7*c*e*f + 9*c*d*g - 8*b*e*g)*(d + e*x)^(5/2)*sqrt(f + g*x))/(24*e^2*g^2) + (c*(d + e*x)^(7/2)*sqrt(f + g*x))/(4*e^2*g) + ((e*f - d*g)^2*(c*(35*e^2*f^2 + 10*d*e*f*g + 3*d^2*g^2) + 8*e*g*(6*a*e*g - b*(5*e*f + d*g)))*atanh((sqrt(g)*sqrt(d + e*x))/(sqrt(e)*sqrt(f + g*x))))/(64*e^(5/2)*g^(9/2))]
    @test_int [(a + b*x + c*x^2)*(d + e*x)^(1/2)/sqrt(f + g*x), x, 6, ((c*(5*e^2*f^2 + 2*d*e*f*g + d^2*g^2) + 2*e*g*(4*a*e*g - b*(3*e*f + d*g)))*sqrt(d + e*x)*sqrt(f + g*x))/(8*e^2*g^3) - ((5*c*e*f + 7*c*d*g - 6*b*e*g)*(d + e*x)^(3/2)*sqrt(f + g*x))/(12*e^2*g^2) + (c*(d + e*x)^(5/2)*sqrt(f + g*x))/(3*e^2*g) - ((e*f - d*g)*(c*(5*e^2*f^2 + 2*d*e*f*g + d^2*g^2) + 2*e*g*(4*a*e*g - b*(3*e*f + d*g)))*atanh((sqrt(g)*sqrt(d + e*x))/(sqrt(e)*sqrt(f + g*x))))/(8*e^(5/2)*g^(7/2))]
    @test_int [(a + b*x + c*x^2)/((d + e*x)^(1/2)*sqrt(f + g*x)), x, 5, -(((3*c*e*f + 5*c*d*g - 4*b*e*g)*sqrt(d + e*x)*sqrt(f + g*x))/(4*e^2*g^2)) + (c*(d + e*x)^(3/2)*sqrt(f + g*x))/(2*e^2*g) + ((c*(3*e^2*f^2 + 2*d*e*f*g + 3*d^2*g^2) + 4*e*g*(2*a*e*g - b*(e*f + d*g)))*atanh((sqrt(g)*sqrt(d + e*x))/(sqrt(e)*sqrt(f + g*x))))/(4*e^(5/2)*g^(5/2))]
    @test_int [(a + b*x + c*x^2)/((d + e*x)^(3/2)*sqrt(f + g*x)), x, 5, -((2*(a + (d*(c*d - b*e))/e^2)*sqrt(f + g*x))/((e*f - d*g)*sqrt(d + e*x))) + (c*sqrt(d + e*x)*sqrt(f + g*x))/(e^2*g) - ((c*e*f + 3*c*d*g - 2*b*e*g)*atanh((sqrt(g)*sqrt(d + e*x))/(sqrt(e)*sqrt(f + g*x))))/(e^(5/2)*g^(3/2))]
    @test_int [(a + b*x + c*x^2)/((d + e*x)^(5/2)*sqrt(f + g*x)), x, 5, -((2*(a + (d*(c*d - b*e))/e^2)*sqrt(f + g*x))/(3*(e*f - d*g)*(d + e*x)^(3/2))) + (2*(c*(6*d*e*f - 4*d^2*g) - e*(3*b*e*f - b*d*g - 2*a*e*g))*sqrt(f + g*x))/(3*e^2*(e*f - d*g)^2*sqrt(d + e*x)) + (2*c*atanh((sqrt(g)*sqrt(d + e*x))/(sqrt(e)*sqrt(f + g*x))))/(e^(5/2)*sqrt(g))]
    @test_int [(a + b*x + c*x^2)/((d + e*x)^(7/2)*sqrt(f + g*x)), x, 3, -((2*(a + (d*(c*d - b*e))/e^2)*sqrt(f + g*x))/(5*(e*f - d*g)*(d + e*x)^(5/2))) + (2*(2*c*d*(5*e*f - 3*d*g) - e*(5*b*e*f - b*d*g - 4*a*e*g))*sqrt(f + g*x))/(15*e^2*(e*f - d*g)^2*(d + e*x)^(3/2)) + (2*(2*e*g*(5*b*e*f - b*d*g - 4*a*e*g) - c*(15*e^2*f^2 - 10*d*e*f*g + 3*d^2*g^2))*sqrt(f + g*x))/(15*e^2*(e*f - d*g)^3*sqrt(d + e*x))]
    @test_int [(a + b*x + c*x^2)/((d + e*x)^(9/2)*sqrt(f + g*x)), x, 4, -((2*(a + (d*(c*d - b*e))/e^2)*sqrt(f + g*x))/(7*(e*f - d*g)*(d + e*x)^(7/2))) + (2*(2*c*d*(7*e*f - 4*d*g) - e*(7*b*e*f - b*d*g - 6*a*e*g))*sqrt(f + g*x))/(35*e^2*(e*f - d*g)^2*(d + e*x)^(5/2)) + (2*(4*e*g*(7*b*e*f - b*d*g - 6*a*e*g) - c*(35*e^2*f^2 - 14*d*e*f*g + 3*d^2*g^2))*sqrt(f + g*x))/(105*e^2*(e*f - d*g)^3*(d + e*x)^(3/2)) - (4*g*(4*e*g*(7*b*e*f - b*d*g - 6*a*e*g) - c*(35*e^2*f^2 - 14*d*e*f*g + 3*d^2*g^2))*sqrt(f + g*x))/(105*e^2*(e*f - d*g)^4*sqrt(d + e*x))]

    @test_int [(d + e*x)^(1/2)*(a + b*x + c*x^2)/(e + f*x)^(3/2), x, 6, (2*(a + (e*(c*e - b*f))/f^2)*(d + e*x)^(3/2))/((e^2 - d*f)*sqrt(e + f*x)) + ((4*e*f*(3*b*e^2 - b*d*f - 2*a*e*f) - c*(15*e^4 - 6*d*e^2*f - d^2*f^2))*sqrt(d + e*x)*sqrt(e + f*x))/(4*e*f^3*(e^2 - d*f)) + (c*(d + e*x)^(3/2)*sqrt(e + f*x))/(2*e*f^2) - ((4*e*f*(3*b*e^2 - b*d*f - 2*a*e*f) - c*(15*e^4 - 6*d*e^2*f - d^2*f^2))*atanh((sqrt(f)*sqrt(d + e*x))/(sqrt(e)*sqrt(e + f*x))))/(4*e^(3/2)*f^(7/2))]

    @test_int [(15*d^2 + 20*d*e*x + 8*e^2*x^2)*(d + e*x)^(3/2)/sqrt(a + b*x), x, 7, ((b*d - a*e)*(73*b^2*d^2 - 90*a*b*d*e + 35*a^2*e^2)*sqrt(a + b*x)*sqrt(d + e*x))/(8*b^4) + ((73*b^2*d^2 - 90*a*b*d*e + 35*a^2*e^2)*sqrt(a + b*x)*(d + e*x)^(3/2))/(12*b^3) + ((17*b*d - 13*a*e)*sqrt(a + b*x)*(d + e*x)^(5/2))/(3*b^2) + (2*e*(a + b*x)^(3/2)*(d + e*x)^(5/2))/b^2 + ((b*d - a*e)^2*(73*b^2*d^2 - 90*a*b*d*e + 35*a^2*e^2)*atanh((sqrt(e)*sqrt(a + b*x))/(sqrt(b)*sqrt(d + e*x))))/(8*b^(9/2)*sqrt(e))]
    @test_int [(15*d^2 + 20*d*e*x + 8*e^2*x^2)*(d + e*x)^(1/2)/sqrt(a + b*x), x, 6, ((11*b^2*d^2 - 13*a*b*d*e + 5*a^2*e^2)*sqrt(a + b*x)*sqrt(d + e*x))/b^3 + (2*(4*b*d - 3*a*e)*sqrt(a + b*x)*(d + e*x)^(3/2))/b^2 + (8*e*(a + b*x)^(3/2)*(d + e*x)^(3/2))/(3*b^2) + ((b*d - a*e)*(11*b^2*d^2 - 13*a*b*d*e + 5*a^2*e^2)*atanh((sqrt(e)*sqrt(a + b*x))/(sqrt(b)*sqrt(d + e*x))))/(b^(7/2)*sqrt(e))]
    @test_int [(15*d^2 + 20*d*e*x + 8*e^2*x^2)/((d + e*x)^(1/2)*sqrt(a + b*x)), x, 5, (2*(7*b*d - 5*a*e)*sqrt(a + b*x)*sqrt(d + e*x))/b^2 + (4*e*(a + b*x)^(3/2)*sqrt(d + e*x))/b^2 + (2*(8*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2)*atanh((sqrt(e)*sqrt(a + b*x))/(sqrt(b)*sqrt(d + e*x))))/(b^(5/2)*sqrt(e))]
    @test_int [(15*d^2 + 20*d*e*x + 8*e^2*x^2)/((d + e*x)^(3/2)*sqrt(a + b*x)), x, 5, (6*d^2*sqrt(a + b*x))/((b*d - a*e)*sqrt(d + e*x)) + (8*sqrt(a + b*x)*sqrt(d + e*x))/b + (8*(2*b*d - a*e)*atanh((sqrt(e)*sqrt(a + b*x))/(sqrt(b)*sqrt(d + e*x))))/(b^(3/2)*sqrt(e))]
    @test_int [(15*d^2 + 20*d*e*x + 8*e^2*x^2)/((d + e*x)^(5/2)*sqrt(a + b*x)), x, 5, (2*d^2*sqrt(a + b*x))/((b*d - a*e)*(d + e*x)^(3/2)) + (4*d*(3*b*d - 2*a*e)*sqrt(a + b*x))/((b*d - a*e)^2*sqrt(d + e*x)) + (16*atanh((sqrt(e)*sqrt(a + b*x))/(sqrt(b)*sqrt(d + e*x))))/(sqrt(b)*sqrt(e))]
    @test_int [(15*d^2 + 20*d*e*x + 8*e^2*x^2)/((d + e*x)^(7/2)*sqrt(a + b*x)), x, 3, (6*d^2*sqrt(a + b*x))/(5*(b*d - a*e)*(d + e*x)^(5/2)) + (8*d*(8*b*d - 5*a*e)*sqrt(a + b*x))/(15*(b*d - a*e)^2*(d + e*x)^(3/2)) + (16*(23*b^2*d^2 - 35*a*b*d*e + 15*a^2*e^2)*sqrt(a + b*x))/(15*(b*d - a*e)^3*sqrt(d + e*x))]
    @test_int [(15*d^2 + 20*d*e*x + 8*e^2*x^2)/((d + e*x)^(9/2)*sqrt(a + b*x)), x, 4, (6*d^2*sqrt(a + b*x))/(7*(b*d - a*e)*(d + e*x)^(7/2)) + (4*d*(23*b*d - 14*a*e)*sqrt(a + b*x))/(35*(b*d - a*e)^2*(d + e*x)^(5/2)) + (16*(58*b^2*d^2 - 84*a*b*d*e + 35*a^2*e^2)*sqrt(a + b*x))/(105*(b*d - a*e)^3*(d + e*x)^(3/2)) + (32*b*(58*b^2*d^2 - 84*a*b*d*e + 35*a^2*e^2)*sqrt(a + b*x))/(105*(b*d - a*e)^4*sqrt(d + e*x))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(d + e*x)^(3/2)/(sqrt(f + g*x)*(a + b*x + c*x^2)), x, 11, (2*e^(3/2)*atanh((sqrt(g)*sqrt(d + e*x))/(sqrt(e)*sqrt(f + g*x))))/(c*sqrt(g)) - (2*(e*(2*c*d - b*e) + (2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))/sqrt(b^2 - 4*a*c))*atanh((sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*sqrt(d + e*x))/(sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*sqrt(f + g*x))))/(c*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)) - (2*(e*(2*c*d - b*e) - (2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))/sqrt(b^2 - 4*a*c))*atanh((sqrt(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)*sqrt(d + e*x))/(sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*sqrt(f + g*x))))/(c*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*sqrt(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))]
    @test_int [(d + e*x)^(1/2)/(sqrt(f + g*x)*(a + b*x + c*x^2)), x, 6, -((2*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*atanh((sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*sqrt(d + e*x))/(sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*sqrt(f + g*x))))/(sqrt(b^2 - 4*a*c)*sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))) + (2*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*atanh((sqrt(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)*sqrt(d + e*x))/(sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*sqrt(f + g*x))))/(sqrt(b^2 - 4*a*c)*sqrt(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))]
    @test_int [1/((d + e*x)^(1/2)*sqrt(f + g*x)*(a + b*x + c*x^2)), x, 6, -((4*c*atanh((sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*sqrt(d + e*x))/(sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*sqrt(f + g*x))))/(sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))) + (4*c*atanh((sqrt(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)*sqrt(d + e*x))/(sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*sqrt(f + g*x))))/(sqrt(b^2 - 4*a*c)*sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*sqrt(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))]
    @test_int [1/((d + e*x)^(3/2)*sqrt(f + g*x)*(a + b*x + c*x^2)), x, 8, (4*c*e*sqrt(f + g*x))/(sqrt(b^2 - 4*a*c)*(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*(e*f - d*g)*sqrt(d + e*x)) - (4*c*e*sqrt(f + g*x))/(sqrt(b^2 - 4*a*c)*(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*(e*f - d*g)*sqrt(d + e*x)) - (8*c^2*atanh((sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*sqrt(d + e*x))/(sqrt(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)*sqrt(f + g*x))))/(sqrt(b^2 - 4*a*c)*(2*c*d - (b - sqrt(b^2 - 4*a*c))*e)^(3/2)*sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)) + (8*c^2*atanh((sqrt(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)*sqrt(d + e*x))/(sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*sqrt(f + g*x))))/(sqrt(b^2 - 4*a*c)*(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)^(3/2)*sqrt(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(f+g*x)^n*(a+b*x+c*x^2)^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [(f + g*x)^3*sqrt(a + b*x + c*x^2)/(d + e*x), x, 8, ((5*b^3*e^3*g^3 + 64*c^3*(e*f - d*g)^3 - 4*b*c*e^2*g^2*(6*b*e*f - 2*b*d*g + a*e*g) + 16*b*c^2*e*g*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2) + 2*c*e*g*(5*b^2*e^2*g^2 - 4*c*e*g*(6*b*e*f - 2*b*d*g + a*e*g) + 16*c^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2))*x)*sqrt(a + b*x + c*x^2))/(64*c^3*e^4) + (g^2*(24*c*e*f - 14*c*d*g - 5*b*e*g)*(a + b*x + c*x^2)^(3/2))/(24*c^2*e^2) + (g^3*(d + e*x)*(a + b*x + c*x^2)^(3/2))/(4*c*e^2) - (1/(128*c^(7/2)*e^5))*((4*c*e*(2*c*d - b*e)*(16*c^2*e^2*f^3 + 5*b^2*d*e*g^3 - 4*c*d*g^2*(6*b*e*f - 2*b*d*g + a*e*g)) - 2*(4*c^2*d^2 - (b^2*e^2)/2 - 2*c*e*(b*d - a*e))*g*(5*b^2*e^2*g^2 - 4*c*e*g*(6*b*e*f - 2*b*d*g + a*e*g) + 16*c^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2)))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2)))) + (sqrt(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)^3*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/e^5]
    @test_int [(f + g*x)^2*sqrt(a + b*x + c*x^2)/(d + e*x), x, 7, -(((b^2*e^2*g^2 - 8*c^2*(e*f - d*g)^2 - 2*b*c*e*g*(2*e*f - d*g) - 2*c*e*g*(4*c*e*f - 2*c*d*g - b*e*g)*x)*sqrt(a + b*x + c*x^2))/(8*c^2*e^3)) + (g^2*(a + b*x + c*x^2)^(3/2))/(3*c*e) + (((8*c^2*d^2 - b^2*e^2 - 4*c*e*(b*d - a*e))*g*(4*c*e*f - 2*c*d*g - b*e*g) - 4*c*e*(2*c*d - b*e)*(2*c*e*f^2 - b*d*g^2))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(16*c^(5/2)*e^4) + (sqrt(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)^2*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/e^4]
    @test_int [(f + g*x)^1*sqrt(a + b*x + c*x^2)/(d + e*x), x, 6, ((4*c*e*f - 4*c*d*g + b*e*g + 2*c*e*g*x)*sqrt(a + b*x + c*x^2))/(4*c*e^2) - ((b^2*e^2*g + 8*c^2*d*(e*f - d*g) - 4*c*e*(b*e*f - b*d*g + a*e*g))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(8*c^(3/2)*e^3) + (sqrt(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/e^3]
    @test_int [(f + g*x)^0*sqrt(a + b*x + c*x^2)/(d + e*x), x, 6, sqrt(a + b*x + c*x^2)/e - ((2*c*d - b*e)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*e^2) + (sqrt(c*d^2 - b*d*e + a*e^2)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/e^2]
    @test_int [sqrt(a + b*x + c*x^2)/((d + e*x)*(f + g*x)^1), x, 8, (sqrt(c)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(e*g) + (sqrt(c*d^2 - b*d*e + a*e^2)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/(e*(e*f - d*g)) - (sqrt(c*f^2 - b*f*g + a*g^2)*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/(g*(e*f - d*g))]
    @test_int [sqrt(a + b*x + c*x^2)/((d + e*x)*(f + g*x)^2), x, 20, sqrt(a + b*x + c*x^2)/((e*f - d*g)*(f + g*x)) - ((2*c*d - b*e)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*(e*f - d*g)^2) + (e*(2*c*f - b*g)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*g*(e*f - d*g)^2) - (sqrt(c)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(g*(e*f - d*g)) + (sqrt(c*d^2 - b*d*e + a*e^2)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/(e*f - d*g)^2 + ((2*c*f - b*g)*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/(2*g*(e*f - d*g)*sqrt(c*f^2 - b*f*g + a*g^2)) - (e*sqrt(c*f^2 - b*f*g + a*g^2)*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/(g*(e*f - d*g)^2)]
    @test_int [sqrt(a + b*x + c*x^2)/((d + e*x)*(f + g*x)^3), x, 23, (e*sqrt(a + b*x + c*x^2))/((e*f - d*g)^2*(f + g*x)) - (g*(b*f - 2*a*g + (2*c*f - b*g)*x)*sqrt(a + b*x + c*x^2))/(4*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)*(f + g*x)^2) - (e*(2*c*d - b*e)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*(e*f - d*g)^3) + (e^2*(2*c*f - b*g)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*g*(e*f - d*g)^3) - (sqrt(c)*e*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(g*(e*f - d*g)^2) + (e*sqrt(c*d^2 - b*d*e + a*e^2)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/(e*f - d*g)^3 + ((b^2 - 4*a*c)*g*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/(8*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)^(3/2)) + (e*(2*c*f - b*g)*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/(2*g*(e*f - d*g)^2*sqrt(c*f^2 - b*f*g + a*g^2)) - (e^2*sqrt(c*f^2 - b*f*g + a*g^2)*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/(g*(e*f - d*g)^3)]
    @test_int [sqrt(a + b*x + c*x^2)/((d + e*x)*(f + g*x)^4), x, 27, (e^2*sqrt(a + b*x + c*x^2))/((e*f - d*g)^3*(f + g*x)) - (g*(2*c*f - b*g)*(b*f - 2*a*g + (2*c*f - b*g)*x)*sqrt(a + b*x + c*x^2))/(8*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)^2*(f + g*x)^2) - (e*g*(b*f - 2*a*g + (2*c*f - b*g)*x)*sqrt(a + b*x + c*x^2))/(4*(e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)*(f + g*x)^2) + (g^2*(a + b*x + c*x^2)^(3/2))/(3*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)*(f + g*x)^3) - (e^2*(2*c*d - b*e)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*(e*f - d*g)^4) + (e^3*(2*c*f - b*g)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*g*(e*f - d*g)^4) - (sqrt(c)*e^2*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(g*(e*f - d*g)^3) + (e^2*sqrt(c*d^2 - b*d*e + a*e^2)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/(e*f - d*g)^4 + ((b^2 - 4*a*c)*g*(2*c*f - b*g)*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/(16*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)^(5/2)) + ((b^2 - 4*a*c)*e*g*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/(8*(e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)^(3/2)) + (e^2*(2*c*f - b*g)*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/(2*g*(e*f - d*g)^3*sqrt(c*f^2 - b*f*g + a*g^2)) - (e^3*sqrt(c*f^2 - b*f*g + a*g^2)*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/(g*(e*f - d*g)^4)]

    @test_int [(f + g*x)^3*(a + b*x + c*x^2)^(3/2)/(d + e*x), x, 9, -((1/(1536*c^4*e^6))*((3*(7*b^5*e^5*g^3 - 512*c^5*d^2*(e*f - d*g)^3 + 128*c^4*e*(5*b*d - 4*a*e)*(e*f - d*g)^3 - 4*b^3*c*e^4*g^2*(9*b*e*f - 3*b*d*g + 8*a*e*g) + 8*b*c^2*e^3*g*(2*a^2*e^2*g^2 + 6*a*b*e*g*(3*e*f - d*g) + 3*b^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2)) - 32*b*c^3*e^2*(2*b*(e*f - d*g)^3 + 3*a*e*g*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2))) + 2*c*e*(8*c*e*(2*c*d - b*e)*(24*c^2*e^2*f^3 + 7*b^2*d*e*g^3 - 4*c*d*g^2*(9*b*e*f - 3*b*d*g + a*e*g)) - 2*(8*c^2*d^2 - 4*b*c*d*e - (3*b^2*e^2)/2 + 6*a*c*e^2)*g*(7*b^2*e^2*g^2 - 4*c*e*g*(9*b*e*f - 3*b*d*g + a*e*g) + 24*c^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2)))*x)*sqrt(a + b*x + c*x^2))) + (1/(192*c^3*e^4))*((7*b^3*e^3*g^3 + 64*c^3*(e*f - d*g)^3 - 4*b*c*e^2*g^2*(9*b*e*f - 3*b*d*g + a*e*g) + 24*b*c^2*e*g*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2) + 2*c*e*g*(7*b^2*e^2*g^2 - 4*c*e*g*(9*b*e*f - 3*b*d*g + a*e*g) + 24*c^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2))*x)*(a + b*x + c*x^2)^(3/2)) + (g^2*(36*c*e*f - 22*c*d*g - 7*b*e*g)*(a + b*x + c*x^2)^(5/2))/(60*c^2*e^2) + (g^3*(d + e*x)*(a + b*x + c*x^2)^(5/2))/(6*c*e^2) + (1/(3072*c^(9/2)*e^7))*((4*c*e*(2*c*d - b*e)*(8*c*e*(b*d - 2*a*e)*(24*c^2*e^2*f^3 + 7*b^2*d*e*g^3 - 4*c*d*g^2*(9*b*e*f - 3*b*d*g + a*e*g)) - d*(8*b*c*d - 3*b^2*e - 4*a*c*e)*g*(7*b^2*e^2*g^2 - 4*c*e*g*(9*b*e*f - 3*b*d*g + a*e*g) + 24*c^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2))) - 2*(4*c^2*d^2 - (b^2*e^2)/2 - 2*c*e*(b*d - a*e))*(8*c*e*(2*c*d - b*e)*(24*c^2*e^2*f^3 + 7*b^2*d*e*g^3 - 4*c*d*g^2*(9*b*e*f - 3*b*d*g + a*e*g)) - 2*(8*c^2*d^2 - 4*b*c*d*e - (3*b^2*e^2)/2 + 6*a*c*e^2)*g*(7*b^2*e^2*g^2 - 4*c*e*g*(9*b*e*f - 3*b*d*g + a*e*g) + 24*c^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2))))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2)))) + ((c*d^2 - b*d*e + a*e^2)^(3/2)*(e*f - d*g)^3*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/e^7]
    @test_int [(f + g*x)^2*(a + b*x + c*x^2)^(3/2)/(d + e*x), x, 8, (1/(128*c^3*e^5))*((3*b^4*e^4*g^2 + 128*c^4*d^2*(e*f - d*g)^2 - 32*c^3*e*(5*b*d - 4*a*e)*(e*f - d*g)^2 - 6*b^2*c*e^3*g*(2*b*e*f - b*d*g + 2*a*e*g) + 8*b*c^2*e^2*(2*b*(e*f - d*g)^2 + 3*a*e*g*(2*e*f - d*g)) + 2*c*e*((16*c^2*d^2 - 3*b^2*e^2 - 4*c*e*(2*b*d - 3*a*e))*g*(4*c*e*f - 2*c*d*g - b*e*g) - 8*c*e*(2*c*d - b*e)*(2*c*e*f^2 - b*d*g^2))*x)*sqrt(a + b*x + c*x^2)) - ((3*b^2*e^2*g^2 - 16*c^2*(e*f - d*g)^2 - 6*b*c*e*g*(2*e*f - d*g) - 6*c*e*g*(4*c*e*f - 2*c*d*g - b*e*g)*x)*(a + b*x + c*x^2)^(3/2))/(48*c^2*e^3) + (g^2*(a + b*x + c*x^2)^(5/2))/(5*c*e) - (1/(256*c^(7/2)*e^6))*((3*b^5*e^5*g^2 + 256*c^5*d^3*(e*f - d*g)^2 - 384*c^4*d*e*(b*d - a*e)*(e*f - d*g)^2 - 6*b^3*c*e^4*g*(2*b*e*f - b*d*g + 4*a*e*g) + 16*b*c^2*e^3*(3*a^2*e^2*g^2 + b^2*(e*f - d*g)^2 + 3*a*b*e*g*(2*e*f - d*g)) + 96*c^3*e^2*(b^2*d*(e*f - d*g)^2 - 2*a*b*e*(e*f - d*g)^2 - a^2*e^2*g*(2*e*f - d*g)))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2)))) + ((c*d^2 - b*d*e + a*e^2)^(3/2)*(e*f - d*g)^2*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/e^6]
    @test_int [(f + g*x)^1*(a + b*x + c*x^2)^(3/2)/(d + e*x), x, 7, -((1/(64*c^2*e^4))*((3*b^3*e^3*g - 64*c^3*d^2*(e*f - d*g) + 16*c^2*e*(5*b*d - 4*a*e)*(e*f - d*g) - 4*b*c*e^2*(2*b*e*f - 2*b*d*g + 3*a*e*g) + 2*c*e*(3*b^2*e^2*g + 16*c^2*d*(e*f - d*g) - 4*c*e*(2*b*e*f - 2*b*d*g + 3*a*e*g))*x)*sqrt(a + b*x + c*x^2))) + ((8*c*e*f - 8*c*d*g + 3*b*e*g + 6*c*e*g*x)*(a + b*x + c*x^2)^(3/2))/(24*c*e^2) + (1/(128*c^(5/2)*e^5))*((3*b^4*e^4*g - 128*c^4*d^3*(e*f - d*g) + 192*c^3*d*e*(b*d - a*e)*(e*f - d*g) - 8*b^2*c*e^3*(b*e*f - b*d*g + 3*a*e*g) + 48*c^2*e^2*(a^2*e^2*g - b^2*d*(e*f - d*g) + 2*a*b*e*(e*f - d*g)))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2)))) + ((c*d^2 - b*d*e + a*e^2)^(3/2)*(e*f - d*g)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/e^5]
    @test_int [(f + g*x)^0*(a + b*x + c*x^2)^(3/2)/(d + e*x), x, 7, ((8*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 4*a*e) - 2*c*e*(2*c*d - b*e)*x)*sqrt(a + b*x + c*x^2))/(8*c*e^3) + (a + b*x + c*x^2)^(3/2)/(3*e) - ((2*c*d - b*e)*(8*c^2*d^2 - b^2*e^2 - 4*c*e*(2*b*d - 3*a*e))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(16*c^(3/2)*e^4) + ((c*d^2 - b*d*e + a*e^2)^(3/2)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/e^4]
    @test_int [(a + b*x + c*x^2)^(3/2)/((d + e*x)*(f + g*x)^1), x, 13, ((c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))/(e^2*(e*f - d*g)) - ((4*c*e*f^2 - g*(5*b*e*f - b*d*g - 4*a*e*g) - 2*c*g*(e*f - d*g)*x)*sqrt(a + b*x + c*x^2))/(4*e*g^2*(e*f - d*g)) - ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*e^3*(e*f - d*g)) + ((8*c^2*e*f^3 + b*g^2*(3*b*e*f + b*d*g - 4*a*e*g) - 4*c*g*(3*b*e*f^2 - a*g*(3*e*f - d*g)))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(8*sqrt(c)*e*g^3*(e*f - d*g)) + ((c*d^2 - b*d*e + a*e^2)^(3/2)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/(e^3*(e*f - d*g)) - ((c*f^2 - b*f*g + a*g^2)^(3/2)*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/(g^3*(e*f - d*g))]
    @test_int [(a + b*x + c*x^2)^(3/2)/((d + e*x)*(f + g*x)^2), x, 23, ((8*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 4*a*e) - 2*c*e*(2*c*d - b*e)*x)*sqrt(a + b*x + c*x^2))/(8*c*e*(e*f - d*g)^2) + (3*(4*c*f - 3*b*g - 2*c*g*x)*sqrt(a + b*x + c*x^2))/(4*g^2*(e*f - d*g)) - (e*(8*c^2*f^2 + b^2*g^2 - 2*c*g*(5*b*f - 4*a*g) - 2*c*g*(2*c*f - b*g)*x)*sqrt(a + b*x + c*x^2))/(8*c*g^2*(e*f - d*g)^2) + (a + b*x + c*x^2)^(3/2)/((e*f - d*g)*(f + g*x)) - ((2*c*d - b*e)*(8*c^2*d^2 - b^2*e^2 - 4*c*e*(2*b*d - 3*a*e))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(16*c^(3/2)*e^2*(e*f - d*g)^2) + (e*(2*c*f - b*g)*(8*c^2*f^2 - b^2*g^2 - 4*c*g*(2*b*f - 3*a*g))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(16*c^(3/2)*g^3*(e*f - d*g)^2) - (3*(8*c^2*f^2 + b^2*g^2 - 4*c*g*(2*b*f - a*g))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(8*sqrt(c)*g^3*(e*f - d*g)) + ((c*d^2 - b*d*e + a*e^2)^(3/2)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/(e^2*(e*f - d*g)^2) + (3*(2*c*f - b*g)*sqrt(c*f^2 - b*f*g + a*g^2)*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/(2*g^3*(e*f - d*g)) - (e*(c*f^2 - b*f*g + a*g^2)^(3/2)*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/(g^3*(e*f - d*g)^2)]
    @test_int [(a + b*x + c*x^2)^(3/2)/((d + e*x)*(f + g*x)^3), x, 30, ((8*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 4*a*e) - 2*c*e*(2*c*d - b*e)*x)*sqrt(a + b*x + c*x^2))/(8*c*(e*f - d*g)^3) + (3*e*(4*c*f - 3*b*g - 2*c*g*x)*sqrt(a + b*x + c*x^2))/(4*g^2*(e*f - d*g)^2) - (3*(4*c*f - b*g + 2*c*g*x)*sqrt(a + b*x + c*x^2))/(4*g^2*(e*f - d*g)*(f + g*x)) - (e^2*(8*c^2*f^2 + b^2*g^2 - 2*c*g*(5*b*f - 4*a*g) - 2*c*g*(2*c*f - b*g)*x)*sqrt(a + b*x + c*x^2))/(8*c*g^2*(e*f - d*g)^3) + (a + b*x + c*x^2)^(3/2)/(2*(e*f - d*g)*(f + g*x)^2) + (e*(a + b*x + c*x^2)^(3/2))/((e*f - d*g)^2*(f + g*x)) - ((2*c*d - b*e)*(8*c^2*d^2 - b^2*e^2 - 4*c*e*(2*b*d - 3*a*e))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(16*c^(3/2)*e*(e*f - d*g)^3) + (3*sqrt(c)*(2*c*f - b*g)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*g^3*(e*f - d*g)) + (e^2*(2*c*f - b*g)*(8*c^2*f^2 - b^2*g^2 - 4*c*g*(2*b*f - 3*a*g))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(16*c^(3/2)*g^3*(e*f - d*g)^3) - (3*e*(8*c^2*f^2 + b^2*g^2 - 4*c*g*(2*b*f - a*g))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(8*sqrt(c)*g^3*(e*f - d*g)^2) + ((c*d^2 - b*d*e + a*e^2)^(3/2)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/(e*(e*f - d*g)^3) + (3*e*(2*c*f - b*g)*sqrt(c*f^2 - b*f*g + a*g^2)*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/(2*g^3*(e*f - d*g)^2) - (e^2*(c*f^2 - b*f*g + a*g^2)^(3/2)*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/(g^3*(e*f - d*g)^3) - (3*(8*c^2*f^2 + b^2*g^2 - 4*c*g*(2*b*f - a*g))*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/(8*g^3*(e*f - d*g)*sqrt(c*f^2 - b*f*g + a*g^2))]

    @test_int [(a + b*x + c*x^2)^(5/2)/((d + e*x)*(f + g*x)), x, 15, ((c*d^2 - b*d*e + a*e^2)*(8*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 4*a*e) - 2*c*e*(2*c*d - b*e)*x)*sqrt(a + b*x + c*x^2))/(8*c*e^4*(e*f - d*g)) - (1/(64*c*e*g^4*(e*f - d*g)))*((64*c^3*e*f^4 - 16*c^2*e*f^2*g*(9*b*f - 8*a*g) - b^2*g^3*(5*b*e*f + 3*b*d*g - 8*a*e*g) + 4*c*g^2*(22*b^2*e*f^2 + 16*a^2*e*g^2 - 3*a*b*g*(13*e*f - d*g)) - 2*c*g*(16*c^2*e*f^3 + b*g^2*(5*b*e*f + 3*b*d*g - 8*a*e*g) - 4*c*g*(6*b*e*f^2 - a*g*(7*e*f - 3*d*g)))*x)*sqrt(a + b*x + c*x^2)) + ((c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^(3/2))/(3*e^2*(e*f - d*g)) - ((8*c*e*f^2 - g*(11*b*e*f - 3*b*d*g - 8*a*e*g) - 6*c*g*(e*f - d*g)*x)*(a + b*x + c*x^2)^(3/2))/(24*e*g^2*(e*f - d*g)) - ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(8*c^2*d^2 - b^2*e^2 - 4*c*e*(2*b*d - 3*a*e))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(16*c^(3/2)*e^5*(e*f - d*g)) + ((128*c^4*e*f^5 - 320*c^3*e*f^3*g*(b*f - a*g) - b^3*g^4*(5*b*e*f + 3*b*d*g - 8*a*e*g) + 48*c^2*g^2*(5*b^2*e*f^3 - 10*a*b*e*f^2*g + a^2*g^2*(5*e*f - d*g)) - 8*b*c*g^3*(5*b^2*e*f^2 + 12*a^2*e*g^2 - 3*a*b*g*(5*e*f + d*g)))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(128*c^(3/2)*e*g^5*(e*f - d*g)) + ((c*d^2 - b*d*e + a*e^2)^(5/2)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/(e^5*(e*f - d*g)) - ((c*f^2 - b*f*g + a*g^2)^(5/2)*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/(g^5*(e*f - d*g))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(f + g*x)^4/((d + e*x)*sqrt(a + b*x + c*x^2)), x, 8, (g^2*(15*b^2*e^2*g^2 - 4*c*e*g*(18*b*e*f - 7*b*d*g + 4*a*e*g) + 4*c^2*(36*e^2*f^2 - 36*d*e*f*g + 11*d^2*g^2))*sqrt(a + b*x + c*x^2))/(24*c^3*e^3) + (g^3*(24*c*e*f - 14*c*d*g - 5*b*e*g)*(d + e*x)*sqrt(a + b*x + c*x^2))/(12*c^2*e^3) + (g^4*(d + e*x)^2*sqrt(a + b*x + c*x^2))/(3*c*e^3) - (1/(16*c^(7/2)*e^4))*(g*(5*b^3*e^3*g^3 - 6*b*c*e^2*g^2*(4*b*e*f - b*d*g + 2*a*e*g) - 16*c^3*(4*e^3*f^3 - 6*d*e^2*f^2*g + 4*d^2*e*f*g^2 - d^3*g^3) + 8*c^2*e*g*(a*e*g*(4*e*f - d*g) + b*(6*e^2*f^2 - 4*d*e*f*g + d^2*g^2)))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2)))) + ((e*f - d*g)^4*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/(e^4*sqrt(c*d^2 - b*d*e + a*e^2))]
    @test_int [(f + g*x)^3/((d + e*x)*sqrt(a + b*x + c*x^2)), x, 7, (3*g^2*(4*c*e*f - 2*c*d*g - b*e*g)*sqrt(a + b*x + c*x^2))/(4*c^2*e^2) + (g^3*(d + e*x)*sqrt(a + b*x + c*x^2))/(2*c*e^2) + (g*(3*b^2*e^2*g^2 - 4*c*e*g*(3*b*e*f - b*d*g + a*e*g) + 8*c^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(8*c^(5/2)*e^3) + ((e*f - d*g)^3*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/(e^3*sqrt(c*d^2 - b*d*e + a*e^2))]
    @test_int [(f + g*x)^2/((d + e*x)*sqrt(a + b*x + c*x^2)), x, 6, (g^2*sqrt(a + b*x + c*x^2))/(c*e) + (g*(4*c*e*f - 2*c*d*g - b*e*g)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*c^(3/2)*e^2) + ((e*f - d*g)^2*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/(e^2*sqrt(c*d^2 - b*d*e + a*e^2))]
    @test_int [(f + g*x)^1/((d + e*x)*sqrt(a + b*x + c*x^2)), x, 5, (g*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(sqrt(c)*e) + ((e*f - d*g)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/(e*sqrt(c*d^2 - b*d*e + a*e^2))]
    @test_int [(f + g*x)^0/((d + e*x)*sqrt(a + b*x + c*x^2)), x, 2, atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2)))/sqrt(c*d^2 - b*d*e + a*e^2)]
    @test_int [1/((d + e*x)*(f + g*x)^1*sqrt(a + b*x + c*x^2)), x, 6, (e*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/(sqrt(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)) - (g*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/((e*f - d*g)*sqrt(c*f^2 - b*f*g + a*g^2))]
    @test_int [1/((d + e*x)*(f + g*x)^2*sqrt(a + b*x + c*x^2)), x, 9, (g^2*sqrt(a + b*x + c*x^2))/((e*f - d*g)*(c*f^2 - b*f*g + a*g^2)*(f + g*x)) + (e^2*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/(sqrt(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)^2) - (g*(2*c*f - b*g)*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/(2*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)^(3/2)) - (e*g*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/((e*f - d*g)^2*sqrt(c*f^2 - b*f*g + a*g^2))]
    @test_int [1/((d + e*x)*(f + g*x)^3*sqrt(a + b*x + c*x^2)), x, 13, (g^2*sqrt(a + b*x + c*x^2))/(2*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)*(f + g*x)^2) + (3*g^2*(2*c*f - b*g)*sqrt(a + b*x + c*x^2))/(4*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)^2*(f + g*x)) + (e*g^2*sqrt(a + b*x + c*x^2))/((e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)*(f + g*x)) + (e^3*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/(sqrt(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)^3) - (e*g*(2*c*f - b*g)*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/(2*(e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)^(3/2)) - (e^2*g*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/((e*f - d*g)^3*sqrt(c*f^2 - b*f*g + a*g^2)) - (g*(8*c^2*f^2 + 3*b^2*g^2 - 4*c*g*(2*b*f + a*g))*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/(8*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)^(5/2))]

    @test_int [(f + g*x)^4/((d + e*x)*(a + b*x + c*x^2)^(3/2)), x, 7, -((2*(a*b^3*d*g^4 - b^2*(c^2*e*f^4 + 4*a*c*d*f*g^3 + a^2*e*g^4) + 2*a*c*(a^2*e*g^4 + c^2*f^3*(e*f - 4*d*g) - 2*a*c*f*g^2*(3*e*f - 2*d*g)) + b*c*(c^2*d*f^4 + a^2*g^3*(4*e*f - 3*d*g) + 2*a*c*f^2*g*(2*e*f + 3*d*g)) + (2*c^4*d*f^4 + b^3*(b*d - a*e)*g^4 - b*c*g^3*(4*b^2*d*f - 3*a^2*e*g - 4*a*b*(e*f - d*g)) + 2*c^2*g^2*(3*b^2*d*f^2 - 3*a*b*f*(e*f - 2*d*g) - a^2*g*(4*e*f - d*g)) + c^3*f^2*(4*a*g*(2*e*f - 3*d*g) - b*f*(e*f + 4*d*g)))*x))/(c^2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))) + (g^4*sqrt(a + b*x + c*x^2))/(c^2*e) + (g^3*(8*c*e*f - 2*c*d*g - 3*b*e*g)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*c^(5/2)*e^2) + ((e*f - d*g)^4*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/(e^2*(c*d^2 - b*d*e + a*e^2)^(3/2))]
    @test_int [(f + g*x)^3/((d + e*x)*(a + b*x + c*x^2)^(3/2)), x, 6, (2*(b^2*(c*e*f^3 + a*d*g^3) - 2*a*c*(c*f^2*(e*f - 3*d*g) - a*g^2*(3*e*f - d*g)) - b*(c^2*d*f^3 + a^2*e*g^3 + 3*a*c*f*g*(e*f + d*g)) - (2*c^3*d*f^3 - b^2*(b*d - a*e)*g^3 + c*g^2*(3*b^2*d*f - 3*a*b*e*f + 3*a*b*d*g - 2*a^2*e*g) + c^2*f*(6*a*g*(e*f - d*g) - b*f*(e*f + 3*d*g)))*x))/(c*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2)) + (g^3*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(c^(3/2)*e) + ((e*f - d*g)^3*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/(e*(c*d^2 - b*d*e + a*e^2)^(3/2))]
    @test_int [(f + g*x)^2/((d + e*x)*(a + b*x + c*x^2)^(3/2)), x, 4, (2*(b^2*e*f^2 + 2*a*(a*e*g^2 - c*f*(e*f - 2*d*g)) - b*(c*d*f^2 + a*g*(2*e*f + d*g)) - (2*c^2*d*f^2 + b*(b*d - a*e)*g^2 + c*(2*a*g*(2*e*f - d*g) - b*f*(e*f + 2*d*g)))*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2)) + ((e*f - d*g)^2*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/(c*d^2 - b*d*e + a*e^2)^(3/2)]
    @test_int [(f + g*x)^1/((d + e*x)*(a + b*x + c*x^2)^(3/2)), x, 4, -((2*(b*c*d*f - b^2*e*f + 2*a*c*e*f - 2*a*c*d*g + a*b*e*g + c*(2*c*d*f + 2*a*e*g - b*(e*f + d*g))*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))) + (e*(e*f - d*g)*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/(c*d^2 - b*d*e + a*e^2)^(3/2)]
    @test_int [(f + g*x)^0/((d + e*x)*(a + b*x + c*x^2)^(3/2)), x, 4, -((2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))) + (e^2*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/(c*d^2 - b*d*e + a*e^2)^(3/2)]
    @test_int [1/((d + e*x)*(f + g*x)^1*(a + b*x + c*x^2)^(3/2)), x, 10, -((2*e*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*sqrt(a + b*x + c*x^2))) + (2*g*(b*c*f - b^2*g + 2*a*c*g + c*(2*c*f - b*g)*x))/((b^2 - 4*a*c)*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2)) + (e^3*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/((c*d^2 - b*d*e + a*e^2)^(3/2)*(e*f - d*g)) - (g^3*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/((e*f - d*g)*(c*f^2 - b*f*g + a*g^2)^(3/2))]
    @test_int [1/((d + e*x)*(f + g*x)^2*(a + b*x + c*x^2)^(3/2)), x, 14, -((2*e^2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)^2*sqrt(a + b*x + c*x^2))) + (2*e*g*(b*c*f - b^2*g + 2*a*c*g + c*(2*c*f - b*g)*x))/((b^2 - 4*a*c)*(e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2)) + (2*g*(b*c*f - b^2*g + 2*a*c*g + c*(2*c*f - b*g)*x))/((b^2 - 4*a*c)*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)*(f + g*x)*sqrt(a + b*x + c*x^2)) + (g^2*(4*c^2*f^2 + 3*b^2*g^2 - 4*c*g*(b*f + 2*a*g))*sqrt(a + b*x + c*x^2))/((b^2 - 4*a*c)*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)^2*(f + g*x)) + (e^4*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/((c*d^2 - b*d*e + a*e^2)^(3/2)*(e*f - d*g)^2) - (3*g^3*(2*c*f - b*g)*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/(2*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)^(5/2)) - (e*g^3*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/((e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)^(3/2))]
    @test_int [1/((d + e*x)*(f + g*x)^3*(a + b*x + c*x^2)^(3/2)), x, 19, -((2*e^3*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)^3*sqrt(a + b*x + c*x^2))) + (2*e^2*g*(b*c*f - b^2*g + 2*a*c*g + c*(2*c*f - b*g)*x))/((b^2 - 4*a*c)*(e*f - d*g)^3*(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2)) + (2*g*(b*c*f - b^2*g + 2*a*c*g + c*(2*c*f - b*g)*x))/((b^2 - 4*a*c)*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)*(f + g*x)^2*sqrt(a + b*x + c*x^2)) + (2*e*g*(b*c*f - b^2*g + 2*a*c*g + c*(2*c*f - b*g)*x))/((b^2 - 4*a*c)*(e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)*(f + g*x)*sqrt(a + b*x + c*x^2)) + (g^2*(8*c^2*f^2 + 5*b^2*g^2 - 4*c*g*(2*b*f + 3*a*g))*sqrt(a + b*x + c*x^2))/(2*(b^2 - 4*a*c)*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)^2*(f + g*x)^2) + (e*g^2*(4*c^2*f^2 + 3*b^2*g^2 - 4*c*g*(b*f + 2*a*g))*sqrt(a + b*x + c*x^2))/((b^2 - 4*a*c)*(e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)^2*(f + g*x)) + (g^2*(2*c*f - b*g)*(8*c^2*f^2 + 15*b^2*g^2 - 4*c*g*(2*b*f + 13*a*g))*sqrt(a + b*x + c*x^2))/(4*(b^2 - 4*a*c)*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)^3*(f + g*x)) + (e^5*atanh((b*d - 2*a*e + (2*c*d - b*e)*x)/(2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(a + b*x + c*x^2))))/((c*d^2 - b*d*e + a*e^2)^(3/2)*(e*f - d*g)^3) - (3*e*g^3*(2*c*f - b*g)*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/(2*(e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)^(5/2)) - (e^2*g^3*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/((e*f - d*g)^3*(c*f^2 - b*f*g + a*g^2)^(3/2)) - (3*g^3*(16*c^2*f^2 + 5*b^2*g^2 - 4*c*g*(4*b*f + a*g))*atanh((b*f - 2*a*g + (2*c*f - b*g)*x)/(2*sqrt(c*f^2 - b*f*g + a*g^2)*sqrt(a + b*x + c*x^2))))/(8*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)^(7/2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(f+g*x)^(n/2)*(a+b*x+c*x^2)^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0 & n>0=#

    @test_int [(d + e*x)^3*sqrt(f + g*x)*sqrt(a + b*x + c*x^2), x, 10, (-2*(64*b^4*e^4*g^4 + 4*b^2*c*e^3*g^3*(7*b*e*f - 66*b*d*g - 69*a*e*g) + c^4*(187*e^4*f^4 - 732*d*e^3*f^3*g + 1098*d^2*e^2*f^2*g^2 - 798*d^3*e*f*g^3 + 315*d^4*g^4) + 3*c^2*e^2*g^2*(50*a^2*e^2*g^2 - a*b*e*g*(29*e*f - 297*d*g) + 3*b^2*(e^2*f^2 - 11*d*e*f*g + 44*d^2*g^2)) - c^3*e*g*(6*a*e*g*(2*e^2*f^2 - 33*d*e*f*g + 165*d^2*g^2) + b*(8*e^3*f^3 - 99*d^2*e*f*g^2 + 231*d^3*g^3)))*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(3465*c^4*e*g^4) + (2*(d + e*x)^4*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(11*e) + (2*(48*b^3*e^3*g^3 + b*c*e^2*g^2*(67*b*e*f - 198*b*d*g - 157*a*e*g) + c^3*(233*e^3*f^3 - 843*d*e^2*f^2*g + 1107*d^2*e*f*g^2 - 567*d^3*g^3) - c^2*e*g*(2*a*e*g*(74*e*f - 231*d*g) - 3*b*(24*e^2*f^2 - 88*d*e*f*g + 99*d^2*g^2)))*(f + g*x)^(3/2)*sqrt(a + b*x + c*x^2))/(3465*c^3*g^4) - (2*e*(8*b^2*e^2*g^2 + c*e*g*(19*b*e*f - 33*b*d*g - 18*a*e*g) + c^2*(29*e^2*f^2 - 96*d*e*f*g + 81*d^2*g^2))*(f + g*x)^(5/2)*sqrt(a + b*x + c*x^2))/(693*c^2*g^4) + (2*e^2*(c*e*f - 3*c*d*g + b*e*g)*(f + g*x)^(7/2)*sqrt(a + b*x + c*x^2))/(99*c*g^4) + (sqrt(2)*sqrt(b^2 - 4*a*c)*(128*b^5*e^3*g^5 - 8*b^3*c*e^2*g^4*(7*b*e*f + 66*b*d*g + 87*a*e*g) + 2*c^5*f^2*(64*e^3*f^3 - 264*d*e^2*f^2*g + 396*d^2*e*f*g^2 - 231*d^3*g^3) + b*c^2*e*g^3*(771*a^2*e^2*g^2 + 6*a*b*e*g*(43*e*f + 396*d*g) - b^2*(37*e^2*f^2 - 264*d*e*f*g - 792*d^2*g^2)) - c^4*g*(b*f*(56*e^3*f^3 - 264*d*e^2*f^2*g + 495*d^2*e*f*g^2 - 462*d^3*g^3) - 18*a*g*(6*e^3*f^3 - 33*d*e^2*f^2*g + 88*d^2*e*f*g^2 + 77*d^3*g^3)) - c^3*g^2*(6*a^2*e^2*g^2*(26*e*f + 231*d*g) - 9*a*b*e*g*(15*e^2*f^2 - 110*d*e*f*g - 319*d^2*g^2) + b^2*(37*e^3*f^3 - 198*d*e^2*f^2*g + 495*d^2*e*f*g^2 + 462*d^3*g^3)))*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(3465*c^5*g^5*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) + (2*sqrt(2)*sqrt(b^2 - 4*a*c)*(c*f^2 - b*f*g + a*g^2)*(64*b^4*e^3*g^4 + 4*b^2*c*e^2*g^3*(7*b*e*f - 66*b*d*g - 69*a*e*g) - 2*c^4*f*(64*e^3*f^3 - 264*d*e^2*f^2*g + 396*d^2*e*f*g^2 - 231*d^3*g^3) + 3*c^2*e*g^2*(50*a^2*e^2*g^2 - a*b*e*g*(29*e*f - 297*d*g) + 3*b^2*(e^2*f^2 - 11*d*e*f*g + 44*d^2*g^2)) - c^3*g*(6*a*e*g*(2*e^2*f^2 - 33*d*e*f*g + 165*d^2*g^2) + b*(8*e^3*f^3 - 99*d^2*e*f*g^2 + 231*d^3*g^3)))*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(3465*c^5*g^5*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))]
    @test_int [(d + e*x)^2*sqrt(f + g*x)*sqrt(a + b*x + c*x^2), x, 9, (2*(8*b^3*e^3*g^3 + 3*b*c*e^2*g^2*(b*e*f - 8*b*d*g - 9*a*e*g) + c^3*(19*e^3*f^3 - 57*d*e^2*f^2*g + 63*d^2*e*f*g^2 - 35*d^3*g^3) - 3*c^2*e*g^2*(2*a*e*(e*f - 10*d*g) + b*d*(2*e*f - 7*d*g)))*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(315*c^3*e*g^3) + (2*(d + e*x)^3*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(9*e) - (4*(3*b^2*e^2*g^2 + c*e*g*(4*b*e*f - 9*b*d*g - 7*a*e*g) + c^2*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2))*(f + g*x)^(3/2)*sqrt(a + b*x + c*x^2))/(315*c^2*g^3) + (2*e*(c*e*f - 3*c*d*g + b*e*g)*(f + g*x)^(5/2)*sqrt(a + b*x + c*x^2))/(63*c*g^3) - (2*sqrt(2)*sqrt(b^2 - 4*a*c)*(8*b^4*e^2*g^4 - 4*b^2*c*e*g^3*(b*e*f + 6*b*d*g + 9*a*e*g) + c^4*f^2*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2) + 3*c^2*g^2*(7*a^2*e^2*g^2 + a*b*e*g*(5*e*f + 29*d*g) - b^2*(e^2*f^2 - 5*d*e*f*g - 7*d^2*g^2)) + c^3*g*(3*a*g*(3*e^2*f^2 - 16*d*e*f*g - 21*d^2*g^2) - b*f*(4*e^2*f^2 - 15*d*e*f*g + 21*d^2*g^2)))*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(315*c^4*g^4*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) - (2*sqrt(2)*sqrt(b^2 - 4*a*c)*(c*f^2 - b*f*g + a*g^2)*(8*b^3*e^2*g^3 + 3*b*c*e*g^2*(b*e*f - 8*b*d*g - 9*a*e*g) - 2*c^3*f*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2) - 3*c^2*g^2*(2*a*e*(e*f - 10*d*g) + b*d*(2*e*f - 7*d*g)))*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(315*c^4*g^4*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))]
    @test_int [(d + e*x)^1*sqrt(f + g*x)*sqrt(a + b*x + c*x^2), x, 7, (-2*sqrt(f + g*x)*(4*b^2*e*g^2 + c^2*f*(4*e*f - 7*d*g) - c*g*(2*b*e*f + 7*b*d*g - 5*a*e*g) - 3*c*g*(c*e*f + 7*c*d*g - 4*b*e*g)*x)*sqrt(a + b*x + c*x^2))/(105*c^2*g^2) + (2*e*sqrt(f + g*x)*(a + b*x + c*x^2)^(3/2))/(7*c) + (sqrt(2)*sqrt(b^2 - 4*a*c)*((c*e*f + 7*c*d*g - 4*b*e*g)*(8*c^2*f^2 - 2*b^2*g^2 - 3*c*g*(b*f - 2*a*g)) - 5*c*g*(2*c*f - b*g)*(7*c*d*f - e*(3*b*f + a*g)))*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(105*c^3*g^3*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) + (2*sqrt(2)*sqrt(b^2 - 4*a*c)*(c*f^2 - b*f*g + a*g^2)*(4*b^2*e*g^2 - 2*c^2*f*(4*e*f - 7*d*g) + c*g*(b*e*f - 7*b*d*g - 10*a*e*g))*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(105*c^3*g^3*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))]
    @test_int [(d + e*x)^0*sqrt(f + g*x)*sqrt(a + b*x + c*x^2), x, 7, (-2*(2*c*f - b*g)*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(15*c*g) + (2*(f + g*x)^(3/2)*sqrt(a + b*x + c*x^2))/(5*g) - (2*sqrt(2)*sqrt(b^2 - 4*a*c)*(c^2*f^2 + b^2*g^2 - c*g*(b*f + 3*a*g))*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(15*c^2*g^2*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) + (2*sqrt(2)*sqrt(b^2 - 4*a*c)*(2*c*f - b*g)*(c*f^2 - b*f*g + a*g^2)*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(15*c^2*g^2*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))]
    @test_int [sqrt(f + g*x)*sqrt(a + b*x + c*x^2)/(d + e*x)^1, x, 15, (2*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(3*e) + (sqrt(2)*sqrt(b^2 - 4*a*c)*(c*e*f - 3*c*d*g + b*e*g)*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(3*c*e^2*g*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) + (1/(3*c*e^3*g*sqrt(f + g*x)*sqrt(a + x*(b + c*x))))*2*sqrt(2)*sqrt(b^2 - 4*a*c)*(e*g*(b*e*f - 3*b*d*g + 2*a*e*g) + c*((-e^2)*f^2 + 3*d^2*g^2))*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (2*sqrt(b^2 - 4*a*c)*g)/(-2*c*f + (b + sqrt(b^2 - 4*a*c))*g)) - (1/(sqrt(c)*e^3*sqrt(a + b*x + c*x^2)))*sqrt(2)*(c*d^2 - b*d*e + a*e^2)*sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((e*(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g))/(2*c*(e*f - d*g)), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)), (b - sqrt(b^2 - 4*a*c) - (2*c*f)/g)/(b + sqrt(b^2 - 4*a*c) - (2*c*f)/g)), (2*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(3*e) + (sqrt(2)*sqrt(b^2 - 4*a*c)*(c*e*f - 3*c*d*g + b*e*g)*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(3*c*e^2*g*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) - (2*sqrt(2)*sqrt(b^2 - 4*a*c)*f*(c*e*f - 3*c*d*g + b*e*g)*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(3*c*e^2*g*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) - (2*sqrt(2)*sqrt(b^2 - 4*a*c)*(3*c*d*(e*f - d*g) - e*(2*b*e*f - 3*b*d*g + 2*a*e*g))*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(3*c*e^3*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) - (1/(sqrt(c)*e^3*sqrt(a + b*x + c*x^2)))*(sqrt(2)*(c*d^2 - b*d*e + a*e^2)*sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((e*(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g))/(2*c*(e*f - d*g)), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)), (b - sqrt(b^2 - 4*a*c) - (2*c*f)/g)/(b + sqrt(b^2 - 4*a*c) - (2*c*f)/g)))]
    @test_int [sqrt(f + g*x)*sqrt(a + b*x + c*x^2)/(d + e*x)^2, x, 15, -((sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(e*(d + e*x))) + (3*sqrt(b^2 - 4*a*c)*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(sqrt(2)*e^2*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) + (sqrt(2)*sqrt(b^2 - 4*a*c)*(2*b*e*g - c*(e*f + 3*d*g))*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (2*sqrt(b^2 - 4*a*c)*g)/(-2*c*f + (b + sqrt(b^2 - 4*a*c))*g)))/(c*e^3*sqrt(f + g*x)*sqrt(a + x*(b + c*x))) + (1/(sqrt(2)*sqrt(c)*e^3*(e*f - d*g)*sqrt(a + b*x + c*x^2)))*(sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*(c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((e*(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g))/(2*c*(e*f - d*g)), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)), (b - sqrt(b^2 - 4*a*c) - (2*c*f)/g)/(b + sqrt(b^2 - 4*a*c) - (2*c*f)/g))), -((sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(e*(d + e*x))) + (3*sqrt(b^2 - 4*a*c)*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(sqrt(2)*e^2*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) - (3*sqrt(2)*sqrt(b^2 - 4*a*c)*f*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(e^2*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) + (sqrt(2)*sqrt(b^2 - 4*a*c)*(2*c*e*f - 3*c*d*g + 2*b*e*g)*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(c*e^3*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) + (1/(sqrt(2)*sqrt(c)*e^3*(e*f - d*g)*sqrt(a + b*x + c*x^2)))*(sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*(c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((e*(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g))/(2*c*(e*f - d*g)), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)), (b - sqrt(b^2 - 4*a*c) - (2*c*f)/g)/(b + sqrt(b^2 - 4*a*c) - (2*c*f)/g)))]
    @test_int [sqrt(f + g*x)*sqrt(a + b*x + c*x^2)/(d + e*x)^3, x, 25, -((sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(2*e*(d + e*x)^2)) + ((c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(4*e*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*(d + e*x)) - (sqrt(b^2 - 4*a*c)*(c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(4*sqrt(2)*e^2*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) - (sqrt(b^2 - 4*a*c)*((-c)*d*(2*e*f + 3*d*g) + e*(b*e*f + 4*b*d*g - 5*a*e*g))*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (2*sqrt(b^2 - 4*a*c)*g)/(-2*c*f + (b + sqrt(b^2 - 4*a*c))*g)))/(2*sqrt(2)*e^3*(c*d^2 + e*((-b)*d + a*e))*sqrt(f + g*x)*sqrt(a + x*(b + c*x))) + (sqrt(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g)*(b^2*e^4*f^2 + a^2*e^4*g^2 + c^2*d^3*g*(4*e*f - 3*d*g) - 2*a*c*e^2*(2*e^2*f^2 - 6*d*e*f*g + 3*d^2*g^2) - 2*b*e*g*(a*e^3*f + c*d^2*(3*e*f - 2*d*g)))*sqrt((g*(-b + sqrt(b^2 - 4*a*c) - 2*c*x))/(2*c*f + (-b + sqrt(b^2 - 4*a*c))*g))*sqrt((g*(b + sqrt(b^2 - 4*a*c) + 2*c*x))/(-2*c*f + (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((2*c*e*f - b*e*g + sqrt(b^2 - 4*a*c)*e*g)/(2*c*e*f - 2*c*d*g), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g)), (2*c*f + (-b + sqrt(b^2 - 4*a*c))*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(4*sqrt(2)*sqrt(c)*e^3*(c*d^2 + e*((-b)*d + a*e))*(e*f - d*g)^2*sqrt(a + x*(b + c*x))), -((sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(2*e*(d + e*x)^2)) + ((c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(4*e*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*(d + e*x)) - (sqrt(b^2 - 4*a*c)*(c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(4*sqrt(2)*e^2*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) + (3*sqrt(b^2 - 4*a*c)*g*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(sqrt(2)*e^3*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) + (sqrt(b^2 - 4*a*c)*f*(c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(2*sqrt(2)*e^2*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) - (sqrt(b^2 - 4*a*c)*d*g*(c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(2*sqrt(2)*e^3*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) - (1/(sqrt(2)*sqrt(c)*e^3*(e*f - d*g)*sqrt(a + b*x + c*x^2)))*(sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*(c*e*f - 3*c*d*g + b*e*g)*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((e*(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g))/(2*c*(e*f - d*g)), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)), (b - sqrt(b^2 - 4*a*c) - (2*c*f)/g)/(b + sqrt(b^2 - 4*a*c) - (2*c*f)/g))) + (sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*(c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))^2*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((e*(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g))/(2*c*(e*f - d*g)), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)), (b - sqrt(b^2 - 4*a*c) - (2*c*f)/g)/(b + sqrt(b^2 - 4*a*c) - (2*c*f)/g)))/(4*sqrt(2)*sqrt(c)*e^3*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)^2*sqrt(a + b*x + c*x^2))]

    #= ::Subsubsection::Closed:: =#
    #=p>0 & n<0=#

    @test_int [(d + e*x)^3*sqrt(a + b*x + c*x^2)/sqrt(f + g*x), x, 9, (2*(8*b^3*e^3*g^3 + 3*b*c*e^2*g^2*(5*b*e*f - 12*b*d*g - 9*a*e*g) - c^3*(152*e^3*f^3 - 408*d*e^2*f^2*g + 336*d^2*e*f*g^2 - 70*d^3*g^3) - 3*c^2*e*g*(6*a*e*g*(2*e*f - 5*d*g) - b*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2)))*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(315*c^3*g^4) + (2*(d + e*x)^3*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(9*g) - (2*e*(6*b^2*e^2*g^2 + c*e*g*(17*b*e*f - 27*b*d*g - 14*a*e*g) - 2*c^2*(64*e^2*f^2 - 111*d*e*f*g + 42*d^2*g^2))*(f + g*x)^(3/2)*sqrt(a + b*x + c*x^2))/(315*c^2*g^4) - (2*e^2*(8*c*e*f - 6*c*d*g - b*e*g)*(f + g*x)^(5/2)*sqrt(a + b*x + c*x^2))/(63*c*g^4) - (sqrt(2)*sqrt(b^2 - 4*a*c)*(16*b^4*e^3*g^4 + 8*b^2*c*e^2*g^3*(2*b*e*f - 9*b*d*g - 9*a*e*g) - 2*c^4*f*(64*e^3*f^3 - 216*d*e^2*f^2*g + 252*d^2*e*f*g^2 - 105*d^3*g^3) + 3*c^2*e*g^2*(14*a^2*e^2*g^2 - a*b*e*g*(19*e*f - 87*d*g) + b^2*(7*e^2*f^2 - 27*d*e*f*g + 42*d^2*g^2)) - c^3*g*(6*a*e*g*(10*e^2*f^2 - 39*d*e*f*g + 63*d^2*g^2) - b*(40*e^3*f^3 - 144*d*e^2*f^2*g + 189*d^2*e*f*g^2 - 105*d^3*g^3)))*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(315*c^4*g^5*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) - (2*sqrt(2)*sqrt(b^2 - 4*a*c)*(c*f^2 - b*f*g + a*g^2)*(8*b^3*e^3*g^3 + 3*b*c*e^2*g^2*(5*b*e*f - 12*b*d*g - 9*a*e*g) + 2*c^3*(64*e^3*f^3 - 216*d*e^2*f^2*g + 252*d^2*e*f*g^2 - 105*d^3*g^3) - 3*c^2*e*g*(6*a*e*g*(2*e*f - 5*d*g) - b*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2)))*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(315*c^4*g^5*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))]
    @test_int [(d + e*x)^2*sqrt(a + b*x + c*x^2)/sqrt(f + g*x), x, 8, (-4*(2*b^2*e^2*g^2 + c*e*g*(4*b*e*f - 7*b*d*g - 5*a*e*g) - c^2*(21*e^2*f^2 - 34*d*e*f*g + 10*d^2*g^2))*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(105*c^2*g^3) + (2*(d + e*x)^2*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(7*g) - (2*e*(6*c*e*f - 4*c*d*g - b*e*g)*(f + g*x)^(3/2)*sqrt(a + b*x + c*x^2))/(35*c*g^3) + (sqrt(2)*sqrt(b^2 - 4*a*c)*(8*b^3*e^2*g^3 + b*c*e*g^2*(9*b*e*f - 28*b*d*g - 29*a*e*g) - 2*c^3*f*(24*e^2*f^2 - 56*d*e*f*g + 35*d^2*g^2) - c^2*g*(2*a*e*g*(13*e*f - 42*d*g) - b*(16*e^2*f^2 - 42*d*e*f*g + 35*d^2*g^2)))*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(105*c^3*g^4*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) + (4*sqrt(2)*sqrt(b^2 - 4*a*c)*(c*f^2 - b*f*g + a*g^2)*(2*b^2*e^2*g^2 + c*e*g*(4*b*e*f - 7*b*d*g - 5*a*e*g) + c^2*(24*e^2*f^2 - 56*d*e*f*g + 35*d^2*g^2))*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(105*c^3*g^4*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))]
    @test_int [(d + e*x)^1*sqrt(a + b*x + c*x^2)/sqrt(f + g*x), x, 6, (-2*sqrt(f + g*x)*(4*c*e*f - 5*c*d*g - b*e*g - 3*c*e*g*x)*sqrt(a + b*x + c*x^2))/(15*c*g^2) - (sqrt(2)*sqrt(b^2 - 4*a*c)*(2*b^2*e*g^2 - 2*c^2*f*(4*e*f - 5*d*g) + c*g*(3*b*e*f - 5*b*d*g - 6*a*e*g))*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(15*c^2*g^3*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) - (2*sqrt(2)*sqrt(b^2 - 4*a*c)*(8*c*e*f - 10*c*d*g + b*e*g)*(c*f^2 - b*f*g + a*g^2)*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(15*c^2*g^3*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))]
    @test_int [(d + e*x)^0*sqrt(a + b*x + c*x^2)/sqrt(f + g*x), x, 6, (2*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(3*g) - (sqrt(2)*sqrt(b^2 - 4*a*c)*(2*c*f - b*g)*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(3*c*g^2*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) + (4*sqrt(2)*sqrt(b^2 - 4*a*c)*(c*f^2 - b*f*g + a*g^2)*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(3*c*g^2*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))]
    @test_int [sqrt(a + b*x + c*x^2)/((d + e*x)^1*sqrt(f + g*x)), x, 11, (sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(e*g*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) - (2*sqrt(2)*sqrt(b^2 - 4*a*c)*(c*e*f + c*d*g - b*e*g)*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(c*e^2*g*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) - (sqrt(2)*(c*d^2 - b*d*e + a*e^2)*sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((e*(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g))/(2*c*(e*f - d*g)), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)), (b - sqrt(b^2 - 4*a*c) - (2*c*f)/g)/(b + sqrt(b^2 - 4*a*c) - (2*c*f)/g)))/(sqrt(c)*e^2*(e*f - d*g)*sqrt(a + b*x + c*x^2))]
    @test_int [sqrt(a + b*x + c*x^2)/((d + e*x)^2*sqrt(f + g*x)), x, 15, -((sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/((e*f - d*g)*(d + e*x))) + (sqrt(b^2 - 4*a*c)*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(sqrt(2)*e*(e*f - d*g)*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) + (sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (2*sqrt(b^2 - 4*a*c)*g)/(-2*c*f + (b + sqrt(b^2 - 4*a*c))*g)))/(e^2*sqrt(f + g*x)*sqrt(a + x*(b + c*x))) - (sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*(e^2*(b*f - a*g) - c*d*(2*e*f - d*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((e*(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g))/(2*c*(e*f - d*g)), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)), (b - sqrt(b^2 - 4*a*c) - (2*c*f)/g)/(b + sqrt(b^2 - 4*a*c) - (2*c*f)/g)))/(sqrt(2)*sqrt(c)*e^2*(e*f - d*g)^2*sqrt(a + b*x + c*x^2)), -((sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/((e*f - d*g)*(d + e*x))) + (sqrt(b^2 - 4*a*c)*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(sqrt(2)*e*(e*f - d*g)*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) - (sqrt(2)*sqrt(b^2 - 4*a*c)*f*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(e*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) + (sqrt(2)*sqrt(b^2 - 4*a*c)*(2*e*f - d*g)*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(e^2*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) - (sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*(e^2*(b*f - a*g) - c*d*(2*e*f - d*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((e*(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g))/(2*c*(e*f - d*g)), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)), (b - sqrt(b^2 - 4*a*c) - (2*c*f)/g)/(b + sqrt(b^2 - 4*a*c) - (2*c*f)/g)))/(sqrt(2)*sqrt(c)*e^2*(e*f - d*g)^2*sqrt(a + b*x + c*x^2))]
    @test_int [sqrt(a + b*x + c*x^2)/((d + e*x)^3*sqrt(f + g*x)), x, 25, -((sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(2*(e*f - d*g)*(d + e*x)^2)) + ((c*d*(2*e*f + d*g) - e*(b*e*f + 2*b*d*g - 3*a*e*g))*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(4*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)^2*(d + e*x)) - (sqrt(b^2 - 4*a*c)*(c*d*(2*e*f + d*g) - e*(b*e*f + 2*b*d*g - 3*a*e*g))*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(4*sqrt(2)*e*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)^2*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) - (sqrt(b^2 - 4*a*c)*(e^2*(b*f - a*g) + c*d*(-2*e*f + d*g))*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (2*sqrt(b^2 - 4*a*c)*g)/(-2*c*f + (b + sqrt(b^2 - 4*a*c))*g)))/(2*sqrt(2)*e^2*(c*d^2 + e*((-b)*d + a*e))*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + x*(b + c*x))) - (sqrt(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g)*(3*a^2*e^4*g^2 + c^2*d^3*g*(4*e*f - d*g) + b^2*e^3*f*((-e)*f + 4*d*g) + 2*a*c*e^2*(2*e^2*f^2 - 2*d*e*f*g + 3*d^2*g^2) - 2*b*e^2*g*(3*c*d^2*f + a*e*(e*f + 2*d*g)))*sqrt((g*(-b + sqrt(b^2 - 4*a*c) - 2*c*x))/(2*c*f + (-b + sqrt(b^2 - 4*a*c))*g))*sqrt((g*(b + sqrt(b^2 - 4*a*c) + 2*c*x))/(-2*c*f + (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((2*c*e*f - b*e*g + sqrt(b^2 - 4*a*c)*e*g)/(2*c*e*f - 2*c*d*g), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g)), (2*c*f + (-b + sqrt(b^2 - 4*a*c))*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(4*sqrt(2)*sqrt(c)*e^2*(c*d^2 + e*((-b)*d + a*e))*(e*f - d*g)^3*sqrt(a + x*(b + c*x))), -((sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(2*(e*f - d*g)*(d + e*x)^2)) + ((c*d*(2*e*f + d*g) - e*(b*e*f + 2*b*d*g - 3*a*e*g))*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(4*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)^2*(d + e*x)) - (sqrt(b^2 - 4*a*c)*(c*d*(2*e*f + d*g) - e*(b*e*f + 2*b*d*g - 3*a*e*g))*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(4*sqrt(2)*e*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)^2*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) - (sqrt(b^2 - 4*a*c)*g*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(sqrt(2)*e^2*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) + (sqrt(b^2 - 4*a*c)*f*(c*d*(2*e*f + d*g) - e*(b*e*f + 2*b*d*g - 3*a*e*g))*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(2*sqrt(2)*e*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)^2*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) - (sqrt(b^2 - 4*a*c)*d*g*(c*d*(2*e*f + d*g) - e*(b*e*f + 2*b*d*g - 3*a*e*g))*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(2*sqrt(2)*e^2*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)^2*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) - (1/(sqrt(2)*sqrt(c)*e^2*(e*f - d*g)^2*sqrt(a + b*x + c*x^2)))*(sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*(c*e*f + c*d*g - b*e*g)*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((e*(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g))/(2*c*(e*f - d*g)), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)), (b - sqrt(b^2 - 4*a*c) - (2*c*f)/g)/(b + sqrt(b^2 - 4*a*c) - (2*c*f)/g))) + (sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*(c*d*(2*e*f + d*g) - e*(b*e*f + 2*b*d*g - 3*a*e*g))*(c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((e*(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g))/(2*c*(e*f - d*g)), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)), (b - sqrt(b^2 - 4*a*c) - (2*c*f)/g)/(b + sqrt(b^2 - 4*a*c) - (2*c*f)/g)))/(4*sqrt(2)*sqrt(c)*e^2*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)^3*sqrt(a + b*x + c*x^2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0 & n>0=#

    @test_int [(d + e*x)^3*sqrt(f + g*x)/sqrt(a + b*x + c*x^2), x, 8, (2*e*(24*b^2*e^2*g^2 + c*e*g*(13*b*e*f - 84*b*d*g - 25*a*e*g) - c^2*(7*e^2*f^2 + 12*d*e*f*g - 90*d^2*g^2))*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(105*c^3*g^2) + (2*e*(d + e*x)^2*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(7*c) + (2*e^2*(c*e*f + 11*c*d*g - 6*b*e*g)*(f + g*x)^(3/2)*sqrt(a + b*x + c*x^2))/(35*c^2*g^2) - (sqrt(2)*sqrt(b^2 - 4*a*c)*(48*b^3*e^3*g^3 - 8*b*c*e^2*g^2*(2*b*e*f + 21*b*d*g + 13*a*e*g) - c^3*(8*e^3*f^3 - 42*d*e^2*f^2*g + 105*d^2*e*f*g^2 + 105*d^3*g^3) + c^2*e*g*(a*e*g*(19*e*f + 189*d*g) - b*(9*e^2*f^2 - 63*d*e*f*g - 210*d^2*g^2)))*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(105*c^4*g^3*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) - (2*sqrt(2)*sqrt(b^2 - 4*a*c)*e*(c*f^2 - b*f*g + a*g^2)*(24*b^2*e^2*g^2 + c*e*g*(13*b*e*f - 84*b*d*g - 25*a*e*g) + c^2*(8*e^2*f^2 - 42*d*e*f*g + 105*d^2*g^2))*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(105*c^4*g^3*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))]
    @test_int [(d + e*x)^2*sqrt(f + g*x)/sqrt(a + b*x + c*x^2), x, 7, (2*e*(c*e*f + 7*c*d*g - 4*b*e*g)*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(15*c^2*g) + (2*e*(d + e*x)*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(5*c) + (sqrt(2)*sqrt(b^2 - 4*a*c)*(8*b^2*e^2*g^2 - c*e*g*(3*b*e*f + 20*b*d*g + 9*a*e*g) - c^2*(2*e^2*f^2 - 10*d*e*f*g - 15*d^2*g^2))*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(15*c^3*g^2*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) + (4*sqrt(2)*sqrt(b^2 - 4*a*c)*e*(c*e*f - 5*c*d*g + 2*b*e*g)*(c*f^2 - b*f*g + a*g^2)*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(15*c^3*g^2*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))]
    @test_int [(d + e*x)^1*sqrt(f + g*x)/sqrt(a + b*x + c*x^2), x, 6, (2*e*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(3*c) + (sqrt(2)*sqrt(b^2 - 4*a*c)*(c*e*f + 3*c*d*g - 2*b*e*g)*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(3*c^2*g*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) - (2*sqrt(2)*sqrt(b^2 - 4*a*c)*e*(c*f^2 - b*f*g + a*g^2)*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(3*c^2*g*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))]
    @test_int [(d + e*x)^0*sqrt(f + g*x)/sqrt(a + b*x + c*x^2), x, 2, (sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(c*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2))]
    @test_int [sqrt(f + g*x)/((d + e*x)^1*sqrt(a + b*x + c*x^2)), x, 8, (2*sqrt(2)*sqrt(b^2 - 4*a*c)*g*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(c*e*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) - (sqrt(2)*sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((e*(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g))/(2*c*(e*f - d*g)), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)), (b - sqrt(b^2 - 4*a*c) - (2*c*f)/g)/(b + sqrt(b^2 - 4*a*c) - (2*c*f)/g)))/(sqrt(c)*e*sqrt(a + b*x + c*x^2))]
    @test_int [sqrt(f + g*x)/((d + e*x)^2*sqrt(a + b*x + c*x^2)), x, 15, -((e*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/((c*d^2 - b*d*e + a*e^2)*(d + e*x))) + (sqrt(b^2 - 4*a*c)*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(sqrt(2)*(c*d^2 - b*d*e + a*e^2)*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) - (sqrt(2)*sqrt(b^2 - 4*a*c)*f*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/((c*d^2 - b*d*e + a*e^2)*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) + (sqrt(2)*sqrt(b^2 - 4*a*c)*d*g*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(e*(c*d^2 - b*d*e + a*e^2)*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) + (sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*(e^2*(b*f - a*g) - c*d*(2*e*f - d*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((e*(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g))/(2*c*(e*f - d*g)), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)), (b - sqrt(b^2 - 4*a*c) - (2*c*f)/g)/(b + sqrt(b^2 - 4*a*c) - (2*c*f)/g)))/(sqrt(2)*sqrt(c)*e*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*sqrt(a + b*x + c*x^2))]
    @test_int [sqrt(f + g*x)/((d + e*x)^3*sqrt(a + b*x + c*x^2)), x, 25, -(e*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) - (e*(c*d*(6*e*f - 5*d*g) - e*(3*b*e*f - 2*b*d*g - a*e*g))*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(4*(c*d^2 - b*d*e + a*e^2)^2*(e*f - d*g)*(d + e*x)) + (sqrt(b^2 - 4*a*c)*(c*d*(6*e*f - 5*d*g) - e*(3*b*e*f - 2*b*d*g - a*e*g))*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(4*sqrt(2)*(c*d^2 - b*d*e + a*e^2)^2*(e*f - d*g)*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) - (sqrt(b^2 - 4*a*c)*g*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(sqrt(2)*e*(c*d^2 - b*d*e + a*e^2)*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) - (sqrt(b^2 - 4*a*c)*f*(c*d*(6*e*f - 5*d*g) - e*(3*b*e*f - 2*b*d*g - a*e*g))*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(2*sqrt(2)*(c*d^2 - b*d*e + a*e^2)^2*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) + (sqrt(b^2 - 4*a*c)*d*g*(c*d*(6*e*f - 5*d*g) - e*(3*b*e*f - 2*b*d*g - a*e*g))*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(2*sqrt(2)*e*(c*d^2 - b*d*e + a*e^2)^2*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) + (sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*(c*e*f - 3*c*d*g + b*e*g)*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((e*(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g))/(2*c*(e*f - d*g)), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)), (b - sqrt(b^2 - 4*a*c) - (2*c*f)/g)/(b + sqrt(b^2 - 4*a*c) - (2*c*f)/g)))/(sqrt(2)*sqrt(c)*e*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*sqrt(a + b*x + c*x^2)) - (sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*(c*d*(6*e*f - 5*d*g) - e*(3*b*e*f - 2*b*d*g - a*e*g))*(c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((e*(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g))/(2*c*(e*f - d*g)), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)), (b - sqrt(b^2 - 4*a*c) - (2*c*f)/g)/(b + sqrt(b^2 - 4*a*c) - (2*c*f)/g)))/(4*sqrt(2)*sqrt(c)*e*(c*d^2 - b*d*e + a*e^2)^2*(e*f - d*g)^2*sqrt(a + b*x + c*x^2))]

    @test_int [(f + g*x)^(3/2)/((d + e*x)*sqrt(a + b*x + c*x^2)), x, 11, (sqrt(2)*sqrt(b^2 - 4*a*c)*g*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(c*e*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) + (2*sqrt(2)*sqrt(b^2 - 4*a*c)*g*(e*f - d*g)*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(c*e^2*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) - (sqrt(2)*sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*(e*f - d*g)*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((e*(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g))/(2*c*(e*f - d*g)), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)), (b - sqrt(b^2 - 4*a*c) - (2*c*f)/g)/(b + sqrt(b^2 - 4*a*c) - (2*c*f)/g)))/(sqrt(c)*e^2*sqrt(a + b*x + c*x^2))]

    @test_int [(f + g*x)^(5/2)/((d + e*x)*sqrt(a + b*x + c*x^2)), x, 17, (2*g^2*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(3*c*e) + (2*sqrt(2)*sqrt(b^2 - 4*a*c)*g*(2*c*f - b*g)*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(3*c^2*e*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) + (sqrt(2)*sqrt(b^2 - 4*a*c)*g*(e*f - d*g)*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(c*e^2*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) + (2*sqrt(2)*sqrt(b^2 - 4*a*c)*g*(e*f - d*g)^2*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(c*e^3*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) - (2*sqrt(2)*sqrt(b^2 - 4*a*c)*g*(c*f^2 - b*f*g + a*g^2)*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(3*c^2*e*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) - (sqrt(2)*sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*(e*f - d*g)^2*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((e*(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g))/(2*c*(e*f - d*g)), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)), (b - sqrt(b^2 - 4*a*c) - (2*c*f)/g)/(b + sqrt(b^2 - 4*a*c) - (2*c*f)/g)))/(sqrt(c)*e^3*sqrt(a + b*x + c*x^2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0 & n<0=#

    @test_int [(d + e*x)^3/(sqrt(f + g*x)*sqrt(a + b*x + c*x^2)), x, 7, (-8*e^2*(c*e*f - 3*c*d*g + b*e*g)*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(15*c^2*g^2) + (2*e^2*(d + e*x)*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(5*c*g) + (sqrt(2)*sqrt(b^2 - 4*a*c)*e*(8*b^2*e^2*g^2 + c*e*g*(7*b*e*f - 30*b*d*g - 9*a*e*g) + c^2*(8*e^2*f^2 - 30*d*e*f*g + 45*d^2*g^2))*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(15*c^3*g^3*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) - (2*sqrt(2)*sqrt(b^2 - 4*a*c)*(4*b*e^3*g^2*(b*f - a*g) + c^2*(8*e^3*f^3 - 30*d*e^2*f^2*g + 45*d^2*e*f*g^2 - 15*d^3*g^3) - c*e^2*g*(a*g*(7*e*f - 15*d*g) - 3*b*f*(e*f - 5*d*g)))*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(15*c^3*g^3*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))]
    @test_int [(d + e*x)^2/(sqrt(f + g*x)*sqrt(a + b*x + c*x^2)), x, 7, (2*e^2*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(3*c*g) - (2*sqrt(2)*sqrt(b^2 - 4*a*c)*e*(c*e*f - 3*c*d*g + b*e*g)*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(3*c^2*g^2*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) + (2*sqrt(2)*sqrt(b^2 - 4*a*c)*(e^2*g*(b*f - a*g) + c*(2*e^2*f^2 - 6*d*e*f*g + 3*d^2*g^2))*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(3*c^2*g^2*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))]
    @test_int [(d + e*x)^1/(sqrt(f + g*x)*sqrt(a + b*x + c*x^2)), x, 5, (sqrt(2)*sqrt(b^2 - 4*a*c)*e*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(c*g*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) - (2*sqrt(2)*sqrt(b^2 - 4*a*c)*(e*f - d*g)*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(c*g*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))]
    @test_int [(d + e*x)^0/(sqrt(f + g*x)*sqrt(a + b*x + c*x^2)), x, 2, (2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(c*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))]
    @test_int [1/((d + e*x)^1*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)), x, 5, -((sqrt(2)*sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((e*(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g))/(2*c*(e*f - d*g)), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)), (b - sqrt(b^2 - 4*a*c) - (2*c*f)/g)/(b + sqrt(b^2 - 4*a*c) - (2*c*f)/g)))/(sqrt(c)*(e*f - d*g)*sqrt(a + b*x + c*x^2)))]
    @test_int [1/((d + e*x)^2*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)), x, 15, -((e^2*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/((c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*(d + e*x))) + (sqrt(b^2 - 4*a*c)*e*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(sqrt(2)*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) - (sqrt(2)*sqrt(b^2 - 4*a*c)*e*f*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/((c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) + (sqrt(2)*sqrt(b^2 - 4*a*c)*d*g*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (-2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/((c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) - (sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*(c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((e*(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g))/(2*c*(e*f - d*g)), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)), (b - sqrt(b^2 - 4*a*c) - (2*c*f)/g)/(b + sqrt(b^2 - 4*a*c) - (2*c*f)/g)))/(sqrt(2)*sqrt(c)*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)^2*sqrt(a + b*x + c*x^2))]
    @test_int [1/((d + e*x)^3*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)), x, 25, -((e^2*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(2*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*(d + e*x)^2)) - (3*e^2*(c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(4*(c*d^2 - b*d*e + a*e^2)^2*(e*f - d*g)^2*(d + e*x)) + (3*sqrt(b^2 - 4*a*c)*e*(c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(4*sqrt(2)*(c*d^2 - b*d*e + a*e^2)^2*(e*f - d*g)^2*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) + (sqrt(b^2 - 4*a*c)*(c*d*(-6*e*f + 7*d*g) + e*(3*b*e*f - 4*b*d*g + a*e*g))*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), (2*sqrt(b^2 - 4*a*c)*g)/(-2*c*f + (b + sqrt(b^2 - 4*a*c))*g)))/(2*sqrt(2)*(c*d^2 + e*((-b)*d + a*e))^2*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + x*(b + c*x))) + (sqrt(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g)*(c^2*d^2*(8*e^2*f^2 - 20*d*e*f*g + 15*d^2*g^2) + 2*c*e*(b*d*(-4*e^2*f^2 + 11*d*e*f*g - 10*d^2*g^2) + a*e*(-2*e^2*f^2 + 2*d*e*f*g + 3*d^2*g^2)) + e^2*(3*a^2*e^2*g^2 + 2*a*b*e*g*(e*f - 4*d*g) + b^2*(3*e^2*f^2 - 8*d*e*f*g + 8*d^2*g^2)))*sqrt((g*(-b + sqrt(b^2 - 4*a*c) - 2*c*x))/(2*c*f + (-b + sqrt(b^2 - 4*a*c))*g))*sqrt((g*(b + sqrt(b^2 - 4*a*c) + 2*c*x))/(-2*c*f + (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((2*c*e*f - b*e*g + sqrt(b^2 - 4*a*c)*e*g)/(2*c*e*f - 2*c*d*g), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g)), (2*c*f + (-b + sqrt(b^2 - 4*a*c))*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)))/(4*sqrt(2)*sqrt(c)*(c*d^2 + e*((-b)*d + a*e))^2*((-e)*f + d*g)^3*sqrt(a + x*(b + c*x))), -((e^2*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(2*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*(d + e*x)^2)) - (3*e^2*(c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*sqrt(f + g*x)*sqrt(a + b*x + c*x^2))/(4*(c*d^2 - b*d*e + a*e^2)^2*(e*f - d*g)^2*(d + e*x)) + (3*sqrt(b^2 - 4*a*c)*e*(c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(4*sqrt(2)*(c*d^2 - b*d*e + a*e^2)^2*(e*f - d*g)^2*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) - (sqrt(b^2 - 4*a*c)*g*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(sqrt(2)*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) - (3*sqrt(b^2 - 4*a*c)*e*f*(c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(2*sqrt(2)*(c*d^2 - b*d*e + a*e^2)^2*(e*f - d*g)^2*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) + (3*sqrt(b^2 - 4*a*c)*d*g*(c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(2*sqrt(2)*(c*d^2 - b*d*e + a*e^2)^2*(e*f - d*g)^2*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) + (sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*(c*e*f - 3*c*d*g + b*e*g)*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((e*(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g))/(2*c*(e*f - d*g)), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)), (b - sqrt(b^2 - 4*a*c) - (2*c*f)/g)/(b + sqrt(b^2 - 4*a*c) - (2*c*f)/g)))/(sqrt(2)*sqrt(c)*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)^2*sqrt(a + b*x + c*x^2)) - (3*sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*(c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))^2*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((e*(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g))/(2*c*(e*f - d*g)), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)), (b - sqrt(b^2 - 4*a*c) - (2*c*f)/g)/(b + sqrt(b^2 - 4*a*c) - (2*c*f)/g)))/(4*sqrt(2)*sqrt(c)*(c*d^2 - b*d*e + a*e^2)^2*(e*f - d*g)^3*sqrt(a + b*x + c*x^2))]

    @test_int [1/((f + g*x)^(3/2)*(d + e*x)*sqrt(a + b*x + c*x^2)), x, 11, (2*g^2*sqrt(a + b*x + c*x^2))/((e*f - d*g)*(c*f^2 - b*f*g + a*g^2)*sqrt(f + g*x)) - (sqrt(2)*sqrt(b^2 - 4*a*c)*g*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/((e*f - d*g)*(c*f^2 - b*f*g + a*g^2)*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) - (sqrt(2)*e*sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((e*(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g))/(2*c*(e*f - d*g)), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)), (b - sqrt(b^2 - 4*a*c) - (2*c*f)/g)/(b + sqrt(b^2 - 4*a*c) - (2*c*f)/g)))/(sqrt(c)*(e*f - d*g)^2*sqrt(a + b*x + c*x^2))]

    @test_int [1/((f + g*x)^(5/2)*(d + e*x)*sqrt(a + b*x + c*x^2)), x, 18, (2*g^2*sqrt(a + b*x + c*x^2))/(3*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)*(f + g*x)^(3/2)) + (4*g^2*(2*c*f - b*g)*sqrt(a + b*x + c*x^2))/(3*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)^2*sqrt(f + g*x)) + (2*e*g^2*sqrt(a + b*x + c*x^2))/((e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)*sqrt(f + g*x)) - (2*sqrt(2)*sqrt(b^2 - 4*a*c)*g*(2*c*f - b*g)*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(3*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)^2*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) - (sqrt(2)*sqrt(b^2 - 4*a*c)*e*g*sqrt(f + g*x)*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.E(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/((e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2)*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(a + b*x + c*x^2)) + (2*sqrt(2)*sqrt(b^2 - 4*a*c)*g*sqrt((c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*sqrt(-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c)))*Elliptic.F(asin(sqrt((b + sqrt(b^2 - 4*a*c) + 2*c*x)/sqrt(b^2 - 4*a*c))/sqrt(2)), -((2*sqrt(b^2 - 4*a*c)*g)/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))))/(3*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)) - (sqrt(2)*e^2*sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b - sqrt(b^2 - 4*a*c))*g))*sqrt(1 - (2*c*(f + g*x))/(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))*Elliptic.Pi((e*(2*c*f - b*g + sqrt(b^2 - 4*a*c)*g))/(2*c*(e*f - d*g)), asin((sqrt(2)*sqrt(c)*sqrt(f + g*x))/sqrt(2*c*f - (b - sqrt(b^2 - 4*a*c))*g)), (b - sqrt(b^2 - 4*a*c) - (2*c*f)/g)/(b + sqrt(b^2 - 4*a*c) - (2*c*f)/g)))/(sqrt(c)*(e*f - d*g)^3*sqrt(a + b*x + c*x^2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^(m/2)*(f+g*x)^(n/2)*(a+b*x+c*x^2)^(p/2)=#

    #= ::Subsubsection:: =#
    #=p>0 & n>0=#

    #= ::Subsubsection:: =#
    #=p>0 & n<0=#

    #= ::Subsubsection:: =#
    #=p<0 & n>0=#

    #= ::Subsubsection::Closed:: =#
    #=p<0 & n<0=#

    #= [(d + e*x)^(5/2)/(sqrt(f + g*x)*sqrt(a + b*x + c*x^2)), x, 0, 0]
    @test_int [(d + e*x)^(3/2)/(sqrt(f + g*x)*sqrt(a + b*x + c*x^2)), x, 0, 0] =#
    @test_int [(d + e*x)^(1/2)/(sqrt(f + g*x)*sqrt(a + b*x + c*x^2)), x, 1, (sqrt(2)*sqrt(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)*sqrt(b - sqrt(b^2 - 4*a*c) + 2*c*x)*sqrt(((e*f - d*g)*(b + sqrt(b^2 - 4*a*c) + 2*c*x))/((2*c*f - (b + sqrt(b^2 - 4*a*c))*g)*(d + e*x)))*sqrt(((e*f - d*g)*(2*a + (b + sqrt(b^2 - 4*a*c))*x))/((b*f + sqrt(b^2 - 4*a*c)*f - 2*a*g)*(d + e*x)))*(d + e*x)*Elliptic.Pi((e*(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))/((2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*g), asin((sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*sqrt(f + g*x))/(sqrt(2*c*f - (b + sqrt(b^2 - 4*a*c))*g)*sqrt(d + e*x))), ((b*d + sqrt(b^2 - 4*a*c)*d - 2*a*e)*(2*c*f - (b + sqrt(b^2 - 4*a*c))*g))/((2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*(b*f + sqrt(b^2 - 4*a*c)*f - 2*a*g))))/(sqrt(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)*g*sqrt((2*a*c)/(b + sqrt(b^2 - 4*a*c)) + c*x)*sqrt(a + b*x + c*x^2))]
    @test_int [1/((d + e*x)^(1/2)*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)), x, 2, -(((c*f^2 - g*(b*f - a*g))^(1/4)*(d + e*x)*sqrt(((e*f - d*g)^2*(a + b*x + c*x^2))/((c*f^2 - b*f*g + a*g^2)*(d + e*x)^2))*(1 + (sqrt(c*d^2 - b*d*e + a*e^2)*(f + g*x))/(sqrt(c*f^2 - g*(b*f - a*g))*(d + e*x)))*sqrt((1 - ((2*c*d*f + 2*a*e*g - b*(e*f + d*g))*(f + g*x))/((c*f^2 - b*f*g + a*g^2)*(d + e*x)) + ((c*d^2 - b*d*e + a*e^2)*(f + g*x)^2)/((c*f^2 - g*(b*f - a*g))*(d + e*x)^2))/(1 + (sqrt(c*d^2 - b*d*e + a*e^2)*(f + g*x))/(sqrt(c*f^2 - g*(b*f - a*g))*(d + e*x)))^2)*Elliptic.F(2*atan(((c*d^2 - b*d*e + a*e^2)^(1/4)*sqrt(f + g*x))/((c*f^2 - b*f*g + a*g^2)^(1/4)*sqrt(d + e*x))), (1/4)*(2 + (2*c*d*f + 2*a*e*g - b*(e*f + d*g))/(sqrt(c*d^2 - e*(b*d - a*e))*sqrt(c*f^2 - g*(b*f - a*g))))))/((c*d^2 - b*d*e + a*e^2)^(1/4)*(e*f - d*g)*sqrt(a + b*x + c*x^2)*sqrt(1 - ((2*c*d*f + 2*a*e*g - b*(e*f + d*g))*(f + g*x))/((c*f^2 - b*f*g + a*g^2)*(d + e*x)) + ((c*d^2 - b*d*e + a*e^2)*(f + g*x)^2)/((c*f^2 - g*(b*f - a*g))*(d + e*x)^2))))]
    #= [1/((d + e*x)^(3/2)*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)), x, 0, 0]
    @test_int [1/((d + e*x)^(5/2)*sqrt(f + g*x)*sqrt(a + b*x + c*x^2)), x, 0, 0] =#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(f+g*x)^n*(a+b*x+c*x^2)^p*when*m*symbolic=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(f+g*x)^n*(a+b*x+c*x^2)^p*when*m*symbolic=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [(d + e*x)^m*(a + b*x + c*x^2)*(f + g*x)^2, x, 2, ((c*d^2 - b*d*e + a*e^2)*(e*f - d*g)^2*(d + e*x)^(1 + m))/(e^5*(1 + m)) - ((e*f - d*g)*(2*c*d*(e*f - 2*d*g) - e*(b*e*f - 3*b*d*g + 2*a*e*g))*(d + e*x)^(2 + m))/(e^5*(2 + m)) + ((e*g*(2*b*e*f - 3*b*d*g + a*e*g) + c*(e^2*f^2 - 6*d*e*f*g + 6*d^2*g^2))*(d + e*x)^(3 + m))/(e^5*(3 + m)) + (g*(2*c*e*f - 4*c*d*g + b*e*g)*(d + e*x)^(4 + m))/(e^5*(4 + m)) + (c*g^2*(d + e*x)^(5 + m))/(e^5*(5 + m))]
    @test_int [(d + e*x)^m*(a + b*x + c*x^2)*(f + g*x)^1, x, 2, ((c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*(d + e*x)^(1 + m))/(e^4*(1 + m)) - ((c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*(d + e*x)^(2 + m))/(e^4*(2 + m)) + ((c*e*f - 3*c*d*g + b*e*g)*(d + e*x)^(3 + m))/(e^4*(3 + m)) + (c*g*(d + e*x)^(4 + m))/(e^4*(4 + m))]
    @test_int [(d + e*x)^m*(a + b*x + c*x^2)/(f + g*x)^1, x, 3, -(((c*e*f + c*d*g - b*e*g)*(d + e*x)^(1 + m))/(e^2*g^2*(1 + m))) + (c*(d + e*x)^(2 + m))/(e^2*g*(2 + m)) + ((c*f^2 - b*f*g + a*g^2)*(d + e*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, -((g*(d + e*x))/(e*f - d*g))))/(g^2*(e*f - d*g)*(1 + m))]
    @test_int [(d + e*x)^m*(a + b*x + c*x^2)/(f + g*x)^2, x, 3, (c*(d + e*x)^(1 + m))/(e*g^2*(1 + m)) + ((a + (f*(c*f - b*g))/g^2)*(d + e*x)^(1 + m))/((e*f - d*g)*(f + g*x)) + ((c*f*(2*d*g - e*f*(2 + m)) - g*(a*e*g*m + b*(d*g - e*f*(1 + m))))*(d + e*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, -((g*(d + e*x))/(e*f - d*g))))/(g^2*(e*f - d*g)^2*(1 + m))]
    @test_int [(d + e*x)^m*(a + b*x + c*x^2)/(f + g*x)^3, x, 3, ((a + (f*(c*f - b*g))/g^2)*(d + e*x)^(1 + m))/(2*(e*f - d*g)*(f + g*x)^2) + ((c*f*(4*d*g - e*f*(3 + m)) + g*(a*e*g*(1 - m) - b*(2*d*g - e*f*(1 + m))))*(d + e*x)^(1 + m))/(2*g^2*(e*f - d*g)^2*(f + g*x)) + ((c*(2*d^2*g^2 - 4*d*e*f*g*(1 + m) + e^2*f^2*(2 + 3*m + m^2)) - e*g*m*(a*e*g*(1 - m) - b*(2*d*g - e*f*(1 + m))))*(d + e*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, -((g*(d + e*x))/(e*f - d*g))))/(2*g^2*(e*f - d*g)^3*(1 + m))]

    @test_int [(d + e*x)^m*(a + b*x + c*x^2)^2*(f + g*x)^2, x, 2, ((c*d^2 - b*d*e + a*e^2)^2*(e*f - d*g)^2*(d + e*x)^(1 + m))/(e^7*(1 + m)) - (2*(c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*(c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*(d + e*x)^(2 + m))/(e^7*(2 + m)) + ((c^2*d^2*(6*e^2*f^2 - 20*d*e*f*g + 15*d^2*g^2) + e^2*(a^2*e^2*g^2 + 2*a*b*e*g*(2*e*f - 3*d*g) + b^2*(e^2*f^2 - 6*d*e*f*g + 6*d^2*g^2)) + 2*c*e*(a*e*(e^2*f^2 - 6*d*e*f*g + 6*d^2*g^2) - b*d*(3*e^2*f^2 - 12*d*e*f*g + 10*d^2*g^2)))*(d + e*x)^(3 + m))/(e^7*(3 + m)) + (2*(b*e^2*g*(b*e*f - 2*b*d*g + a*e*g) - 2*c^2*d*(e^2*f^2 - 5*d*e*f*g + 5*d^2*g^2) + c*e*(2*a*e*g*(e*f - 2*d*g) + b*(e^2*f^2 - 8*d*e*f*g + 10*d^2*g^2)))*(d + e*x)^(4 + m))/(e^7*(4 + m)) + ((b^2*e^2*g^2 + 2*c*e*g*(2*b*e*f - 5*b*d*g + a*e*g) + c^2*(e^2*f^2 - 10*d*e*f*g + 15*d^2*g^2))*(d + e*x)^(5 + m))/(e^7*(5 + m)) + (2*c*g*(c*e*f - 3*c*d*g + b*e*g)*(d + e*x)^(6 + m))/(e^7*(6 + m)) + (c^2*g^2*(d + e*x)^(7 + m))/(e^7*(7 + m))]
    @test_int [(d + e*x)^m*(a + b*x + c*x^2)^2*(f + g*x)^1, x, 2, ((c*d^2 - b*d*e + a*e^2)^2*(e*f - d*g)*(d + e*x)^(1 + m))/(e^6*(1 + m)) - ((c*d^2 - b*d*e + a*e^2)*(c*d*(4*e*f - 5*d*g) - e*(2*b*e*f - 3*b*d*g + a*e*g))*(d + e*x)^(2 + m))/(e^6*(2 + m)) + ((2*c^2*d^2*(3*e*f - 5*d*g) + b*e^2*(b*e*f - 3*b*d*g + 2*a*e*g) + 2*c*e*(a*e*(e*f - 3*d*g) - 3*b*d*(e*f - 2*d*g)))*(d + e*x)^(3 + m))/(e^6*(3 + m)) + ((b^2*e^2*g - 2*c^2*d*(2*e*f - 5*d*g) + 2*c*e*(b*e*f - 4*b*d*g + a*e*g))*(d + e*x)^(4 + m))/(e^6*(4 + m)) + (c*(c*e*f - 5*c*d*g + 2*b*e*g)*(d + e*x)^(5 + m))/(e^6*(5 + m)) + (c^2*g*(d + e*x)^(6 + m))/(e^6*(6 + m))]
    @test_int [(d + e*x)^m*(a + b*x + c*x^2)^2/(f + g*x)^1, x, 4, ((b*e*g - c*(e*f + d*g))*(c*(e^2*f^2 + d^2*g^2) + e*g*(2*a*e*g - b*(e*f + d*g)))*(d + e*x)^(1 + m))/(e^4*g^4*(1 + m)) + ((b^2*e^2*g^2 + c^2*(e^2*f^2 + 2*d*e*f*g + 3*d^2*g^2) + 2*c*e*g*(a*e*g - b*(e*f + 2*d*g)))*(d + e*x)^(2 + m))/(e^4*g^3*(2 + m)) - (c*(c*e*f + 3*c*d*g - 2*b*e*g)*(d + e*x)^(3 + m))/(e^4*g^2*(3 + m)) + (c^2*(d + e*x)^(4 + m))/(e^4*g*(4 + m)) + ((c*f^2 - b*f*g + a*g^2)^2*(d + e*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, -((g*(d + e*x))/(e*f - d*g))))/(g^4*(e*f - d*g)*(1 + m))]
    @test_int [(d + e*x)^m*(a + b*x + c*x^2)^2/(f + g*x)^2, x, 4, ((b^2*e^2*g^2 + c^2*(3*e^2*f^2 + 2*d*e*f*g + d^2*g^2) + 2*c*e*g*(a*e*g - b*(2*e*f + d*g)))*(d + e*x)^(1 + m))/(e^3*g^4*(1 + m)) - (2*c*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(2 + m))/(e^3*g^3*(2 + m)) + (c^2*(d + e*x)^(3 + m))/(e^3*g^2*(3 + m)) + ((c*f^2 - b*f*g + a*g^2)^2*(d + e*x)^(1 + m))/(g^4*(e*f - d*g)*(f + g*x)) + ((c*f^2 - b*f*g + a*g^2)*(c*f*(4*d*g - e*f*(4 + m)) - g*(a*e*g*m + b*(2*d*g - e*f*(2 + m))))*(d + e*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, -((g*(d + e*x))/(e*f - d*g))))/(g^4*(e*f - d*g)^2*(1 + m))]
    @test_int [(d + e*x)^m*(a + b*x + c*x^2)^2/(f + g*x)^3, x, 5, -((c*(3*c*e*f + c*d*g - 2*b*e*g)*(d + e*x)^(1 + m))/(e^2*g^4*(1 + m))) + (c^2*(d + e*x)^(2 + m))/(e^2*g^3*(2 + m)) + ((c*f^2 - b*f*g + a*g^2)^2*(d + e*x)^(1 + m))/(2*g^4*(e*f - d*g)*(f + g*x)^2) + ((c*f^2 - b*f*g + a*g^2)*(c*f*(8*d*g - e*f*(7 + m)) + g*(a*e*g*(1 - m) - b*(4*d*g - e*f*(3 + m))))*(d + e*x)^(1 + m))/(2*g^4*(e*f - d*g)^2*(f + g*x)) + (1/(2*g^4*(e*f - d*g)^3*(1 + m)))*((c^2*f^2*(12*d^2*g^2 - 8*d*e*f*g*(3 + m) + e^2*f^2*(12 + 7*m + m^2)) - g^2*(a^2*e^2*g^2*(1 - m)*m - 2*a*b*e*g*m*(2*d*g - e*f*(1 + m)) - b^2*(2*d^2*g^2 - 4*d*e*f*g*(1 + m) + e^2*f^2*(2 + 3*m + m^2))) + 2*c*g*(a*g*(2*d^2*g^2 - 4*d*e*f*g*(1 + m) + e^2*f^2*(2 + 3*m + m^2)) - b*f*(6*d^2*g^2 - 6*d*e*f*g*(2 + m) + e^2*f^2*(6 + 5*m + m^2))))*(d + e*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, -((g*(d + e*x))/(e*f - d*g))))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(1 + 4*x)^m*(2 + 3*x)^4/(1 - 5*x + 3*x^2), x, 4, (3687*(1 + 4*x)^(1 + m))/(64*(1 + m)) + (207*(1 + 4*x)^(2 + m))/(32*(2 + m)) + (27*(1 + 4*x)^(3 + m))/(64*(3 + m)) - (3*(5499 - 1631*sqrt(13))*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*sqrt(13))))/(26*(13 - 2*sqrt(13))*(1 + m)) - (3*(5499 + 1631*sqrt(13))*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*sqrt(13))))/(26*(13 + 2*sqrt(13))*(1 + m))]
    @test_int [(1 + 4*x)^m*(2 + 3*x)^3/(1 - 5*x + 3*x^2), x, 4, (123*(1 + 4*x)^(1 + m))/(16*(1 + m)) + (9*(1 + 4*x)^(2 + m))/(16*(2 + m)) - (3*(416 - 135*sqrt(13))*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*sqrt(13))))/(13*(13 - 2*sqrt(13))*(1 + m)) - (3*(416 + 135*sqrt(13))*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*sqrt(13))))/(13*(13 + 2*sqrt(13))*(1 + m))]
    @test_int [(1 + 4*x)^m*(2 + 3*x)^2/(1 - 5*x + 3*x^2), x, 4, (3*(1 + 4*x)^(1 + m))/(4*(1 + m)) - (3*(117 - 47*sqrt(13))*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*sqrt(13))))/(26*(13 - 2*sqrt(13))*(1 + m)) - (3*(117 + 47*sqrt(13))*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*sqrt(13))))/(26*(13 + 2*sqrt(13))*(1 + m))]
    @test_int [(1 + 4*x)^m*(2 + 3*x)^1/(1 - 5*x + 3*x^2), x, 4, -((3*(13 - 9*sqrt(13))*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*sqrt(13))))/(26*(13 - 2*sqrt(13))*(1 + m))) - (3*(13 + 9*sqrt(13))*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*sqrt(13))))/(26*(13 + 2*sqrt(13))*(1 + m))]
    @test_int [(1 + 4*x)^m*(2 + 3*x)^0/(1 - 5*x + 3*x^2), x, 4, (3*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*sqrt(13))))/(sqrt(13)*(13 - 2*sqrt(13))*(1 + m)) - (3*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*sqrt(13))))/(sqrt(13)*(13 + 2*sqrt(13))*(1 + m))]
    @test_int [(1 + 4*x)^m/((2 + 3*x)^1*(1 - 5*x + 3*x^2)), x, 7, (3*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (-(3/5))*(1 + 4*x)))/(85*(1 + m)) + (3*(13 + 9*sqrt(13))*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*sqrt(13))))/(442*(13 - 2*sqrt(13))*(1 + m)) + (3*(13 - 9*sqrt(13))*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*sqrt(13))))/(442*(13 + 2*sqrt(13))*(1 + m))]
    @test_int [(1 + 4*x)^m/((2 + 3*x)^2*(1 - 5*x + 3*x^2)), x, 8, (27*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (-(3/5))*(1 + 4*x)))/(1445*(1 + m)) + (3*(117 + 47*sqrt(13))*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*sqrt(13))))/(7514*(13 - 2*sqrt(13))*(1 + m)) + (3*(117 - 47*sqrt(13))*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*sqrt(13))))/(7514*(13 + 2*sqrt(13))*(1 + m)) + (12*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(2, 1 + m, 2 + m, (-(3/5))*(1 + 4*x)))/(425*(1 + m))]

    @test_int [(1 + 4*x)^m*(2 + 3*x)^4/(1 - 5*x + 3*x^2)^2, x, 5, (9*(1 + 4*x)^(1 + m))/(4*(1 + m)) + ((844 - 2355*x)*(1 + 4*x)^(1 + m))/(39*(1 - 5*x + 3*x^2)) - ((13689 - sqrt(13)*(297 + 4474*m - 1570*sqrt(13)*m))*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*sqrt(13))))/(169*(13 - 2*sqrt(13))*(1 + m)) - ((13689 + sqrt(13)*(297 + 4474*m + 1570*sqrt(13)*m))*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*sqrt(13))))/(169*(13 + 2*sqrt(13))*(1 + m))]
    @test_int [(1 + 4*x)^m*(2 + 3*x)^3/(1 - 5*x + 3*x^2)^2, x, 5, ((209 - 426*x)*(1 + 4*x)^(1 + m))/(39*(1 - 5*x + 3*x^2)) - ((1521 + sqrt(13)*(1701 - 1168*m + 568*sqrt(13)*m))*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*sqrt(13))))/(338*(13 - 2*sqrt(13))*(1 + m)) + ((sqrt(13)*(1701 - 1168*m) - 13*(117 + 568*m))*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*sqrt(13))))/(338*(13 + 2*sqrt(13))*(1 + m))]
    @test_int [(1 + 4*x)^m*(2 + 3*x)^2/(1 - 5*x + 3*x^2)^2, x, 5, ((61 - 87*x)*(1 + 4*x)^(1 + m))/(39*(1 - 5*x + 3*x^2)) - (2*(153 - (23 - 29*sqrt(13))*m)*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*sqrt(13))))/(13*sqrt(13)*(13 - 2*sqrt(13))*(1 + m)) + (2*(153 - (23 + 29*sqrt(13))*m)*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*sqrt(13))))/(13*sqrt(13)*(13 + 2*sqrt(13))*(1 + m))]
    @test_int [(1 + 4*x)^m*(2 + 3*x)^1/(1 - 5*x + 3*x^2)^2, x, 5, ((20 - 21*x)*(1 + 4*x)^(1 + m))/(39*(1 - 5*x + 3*x^2)) - ((81 + 2*(5 + 7*sqrt(13))*m)*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*sqrt(13))))/(13*sqrt(13)*(13 - 2*sqrt(13))*(1 + m)) + ((81 + 2*(5 - 7*sqrt(13))*m)*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*sqrt(13))))/(13*sqrt(13)*(13 + 2*sqrt(13))*(1 + m))]
    @test_int [(1 + 4*x)^m*(2 + 3*x)^0/(1 - 5*x + 3*x^2)^2, x, 5, ((7 - 6*x)*(1 + 4*x)^(1 + m))/(39*(1 - 5*x + 3*x^2)) - (2*(9 + 2*(2 + sqrt(13))*m)*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*sqrt(13))))/(13*sqrt(13)*(13 - 2*sqrt(13))*(1 + m)) + (2*(9 + 2*(2 - sqrt(13))*m)*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*sqrt(13))))/(13*sqrt(13)*(13 + 2*sqrt(13))*(1 + m))]
    @test_int [(1 + 4*x)^m/((2 + 3*x)^1*(1 - 5*x + 3*x^2)^2), x, 12, ((43 - 33*x)*(1 + 4*x)^(1 + m))/(663*(1 - 5*x + 3*x^2)) + (9*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (-(3/5))*(1 + 4*x)))/(1445*(1 + m)) + (9*(13 + 9*sqrt(13))*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*sqrt(13))))/(7514*(13 - 2*sqrt(13))*(1 + m)) - ((81 + (62 + 22*sqrt(13))*m)*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*sqrt(13))))/(221*sqrt(13)*(13 - 2*sqrt(13))*(1 + m)) + (9*(13 - 9*sqrt(13))*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*sqrt(13))))/(7514*(13 + 2*sqrt(13))*(1 + m)) + ((81 + (62 - 22*sqrt(13))*m)*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*sqrt(13))))/(221*sqrt(13)*(13 + 2*sqrt(13))*(1 + m))]
    @test_int [(1 + 4*x)^m/((2 + 3*x)^2*(1 - 5*x + 3*x^2)^2), x, 13, ((268 - 195*x)*(1 + 4*x)^(1 + m))/(11271*(1 - 5*x + 3*x^2)) + (162*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (-(3/5))*(1 + 4*x)))/(24565*(1 + m)) + (9*(117 + 64*sqrt(13))*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*sqrt(13))))/(63869*(13 - 2*sqrt(13))*(1 + m)) - ((423 + 2*(211 + 65*sqrt(13))*m)*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*sqrt(13))))/(3757*sqrt(13)*(13 - 2*sqrt(13))*(1 + m)) + (9*(117 - 64*sqrt(13))*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*sqrt(13))))/(63869*(13 + 2*sqrt(13))*(1 + m)) + ((423 + (422 - 130*sqrt(13))*m)*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*sqrt(13))))/(3757*sqrt(13)*(13 + 2*sqrt(13))*(1 + m)) + (36*(1 + 4*x)^(1 + m)*HypergeometricFunctions._₂F₁(2, 1 + m, 2 + m, (-(3/5))*(1 + 4*x)))/(7225*(1 + m))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(f+g*x)^(n/2)*(a+b*x+c*x^2)^p*when*m*symbolic=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [(d + e*x)^m*(a + b*x + c*x^2)/(e + f*x)^(3/2), x, 4, (2*(a + (e*(c*e - b*f))/f^2)*(d + e*x)^(1 + m))/((e^2 - d*f)*sqrt(e + f*x)) + (2*c*(d + e*x)^(1 + m)*sqrt(e + f*x))/(e*f^2*(3 + 2*m)) + (2*(c*(d^2*f^2 + 4*d*e^2*f*(1 + m) - 4*e^4*(2 + 3*m + m^2)) - e*f*(3 + 2*m)*(a*e*f*(1 + 2*m) + b*(d*f - 2*e^2*(1 + m))))*(d + e*x)^m*sqrt(e + f*x)*HypergeometricFunctions._₂F₁(1/2, -m, 3/2, (e*(e + f*x))/(e^2 - d*f)))/((-((f*(d + e*x))/(e^2 - d*f)))^m*(e*f^3*(e^2 - d*f)*(3 + 2*m)))]

    #= ::Subsubsection:: =#
    #=p<0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(f+g*x)^n*(a+b*x+c*x^2)^(p/2)*when*m*symbolic=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [(d + e*x)^m*sqrt(a + b*x + c*x^2)*(f + g*x)^2, x, 6, (g^2*(d + e*x)^(1 + m)*(a + b*x + c*x^2)^(3/2))/(c*e*(4 + m)) + ((e*(b*d - a*e)*g^2*(1 + m) + c*(3*d^2*g^2 + e^2*f^2*(4 + m) - 2*d*e*f*g*(4 + m)))*(d + e*x)^(1 + m)*sqrt(a + b*x + c*x^2)*AppellF1(1 + m, -(1/2), -(1/2), 2 + m, (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e), (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(c*e^3*(1 + m)*(4 + m)*sqrt(1 - (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e))*sqrt(1 - (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))) - (g*(b*e*g*(5 + 2*m) + 2*c*(3*d*g - 2*e*f*(4 + m)))*(d + e*x)^(2 + m)*sqrt(a + b*x + c*x^2)*AppellF1(2 + m, -(1/2), -(1/2), 3 + m, (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e), (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(2*c*e^3*(2 + m)*(4 + m)*sqrt(1 - (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e))*sqrt(1 - (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))]
    @test_int [(d + e*x)^m*sqrt(a + b*x + c*x^2)*(f + g*x)^1, x, 5, ((e*f - d*g)*(d + e*x)^(1 + m)*sqrt(a + b*x + c*x^2)*AppellF1(1 + m, -(1/2), -(1/2), 2 + m, (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e), (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(e^2*(1 + m)*sqrt(1 - (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e))*sqrt(1 - (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))) + (g*(d + e*x)^(2 + m)*sqrt(a + b*x + c*x^2)*AppellF1(2 + m, -(1/2), -(1/2), 3 + m, (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e), (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(e^2*(2 + m)*sqrt(1 - (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e))*sqrt(1 - (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))]
    @test_int [(d + e*x)^m*sqrt(a + b*x + c*x^2)*(f + g*x)^0, x, 2, ((d + e*x)^(1 + m)*sqrt(a + b*x + c*x^2)*AppellF1(1 + m, -(1/2), -(1/2), 2 + m, (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e), (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(e*(1 + m)*sqrt(1 - (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e))*sqrt(1 - (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))]
    @test_int [(d + e*x)^m*sqrt(a + b*x + c*x^2)/(f + g*x)^1, x, 0, Unintegrable(((d + e*x)^m*sqrt(a + b*x + c*x^2))/(f + g*x), x)]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(d + e*x)^m*(f + g*x)^2/sqrt(a + b*x + c*x^2), x, 6, (g^2*(d + e*x)^(1 + m)*sqrt(a + b*x + c*x^2))/(c*e*(2 + m)) + (1/(c*e^3*(1 + m)*(2 + m)*sqrt(a + b*x + c*x^2)))*((e*(b*d - a*e)*g^2*(1 + m) + c*(d^2*g^2 + e^2*f^2*(2 + m) - 2*d*e*f*g*(2 + m)))*(d + e*x)^(1 + m)*sqrt(1 - (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e))*sqrt(1 - (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))*AppellF1(1 + m, 1/2, 1/2, 2 + m, (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e), (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))) - (1/(2*c*e^3*(2 + m)^2*sqrt(a + b*x + c*x^2)))*(g*(b*e*g*(3 + 2*m) + c*(2*d*g - 4*e*f*(2 + m)))*(d + e*x)^(2 + m)*sqrt(1 - (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e))*sqrt(1 - (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))*AppellF1(2 + m, 1/2, 1/2, 3 + m, (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e), (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))]
    @test_int [(d + e*x)^m*(f + g*x)^1/sqrt(a + b*x + c*x^2), x, 5, ((e*f - d*g)*(d + e*x)^(1 + m)*sqrt(1 - (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e))*sqrt(1 - (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))*AppellF1(1 + m, 1/2, 1/2, 2 + m, (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e), (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(e^2*(1 + m)*sqrt(a + b*x + c*x^2)) + (g*(d + e*x)^(2 + m)*sqrt(1 - (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e))*sqrt(1 - (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))*AppellF1(2 + m, 1/2, 1/2, 3 + m, (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e), (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(e^2*(2 + m)*sqrt(a + b*x + c*x^2))]
    @test_int [(d + e*x)^m*(f + g*x)^0/sqrt(a + b*x + c*x^2), x, 2, ((d + e*x)^(1 + m)*sqrt(1 - (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e))*sqrt(1 - (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))*AppellF1(1 + m, 1/2, 1/2, 2 + m, (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e), (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/(e*(1 + m)*sqrt(a + b*x + c*x^2))]
    @test_int [(d + e*x)^m/((f + g*x)^1*sqrt(a + b*x + c*x^2)), x, 0, Unintegrable((d + e*x)^m/((f + g*x)*sqrt(a + b*x + c*x^2)), x)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(f+g*x)^n*(a+b*x+c*x^2)^p*when*m*and*n*symbolic=#

    @test_int [(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^1, x, 4, ((b*e*g*(3 + m + n) - c*(e*f*(2 + m) + d*g*(4 + m + 2*n)))*(d + e*x)^(1 + m)*(f + g*x)^(1 + n))/(e^2*g^2*(2 + m + n)*(3 + m + n)) + (c*(d + e*x)^(2 + m)*(f + g*x)^(1 + n))/(e^2*g*(3 + m + n)) + (1/(e^3*g^2*(1 + m)*(2 + m + n)*(3 + m + n)))*(((g*(2 + m + n)*(a*e^2*g*(3 + m + n) - c*d*(e*f*(2 + m) + d*g*(1 + n))) - (e*f*(1 + m) + d*g*(1 + n))*(b*e*g*(3 + m + n) - c*(e*f*(2 + m) + d*g*(4 + m + 2*n))))*(d + e*x)^(1 + m)*(f + g*x)^n*HypergeometricFunctions._₂F₁(1 + m, -n, 2 + m, -((g*(d + e*x))/(e*f - d*g))))/((e*(f + g*x))/(e*f - d*g))^n)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^m*(f+g*x)^n*(a+b*x+c*x^2)^p*when*m*and*p*symbolic=#

    @test_int [(d + e*x)^m*(a + b*x + c*x^2)^p*(f + g*x)^2, x, 6, (g^2*(d + e*x)^(1 + m)*(a + b*x + c*x^2)^(1 + p))/(c*e*(3 + m + 2*p)) + (1/(c*e^3*(1 + m)*(3 + m + 2*p)))*(((e*(b*d - a*e)*g^2*(1 + m) + c*(2*d^2*g^2*(1 + p) + e^2*f^2*(3 + m + 2*p) - 2*d*e*f*g*(3 + m + 2*p)))*(d + e*x)^(1 + m)*(a + b*x + c*x^2)^p*AppellF1(1 + m, -p, -p, 2 + m, (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e), (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/((1 - (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e))^p*(1 - (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))^p)) - (1/(c*e^3*(2 + m)*(3 + m + 2*p)))*((g*(b*e*g*(2 + m + p) + 2*c*(d*g*(1 + p) - e*f*(3 + m + 2*p)))*(d + e*x)^(2 + m)*(a + b*x + c*x^2)^p*AppellF1(2 + m, -p, -p, 3 + m, (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e), (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/((1 - (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e))^p*(1 - (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))^p))]
    @test_int [(d + e*x)^m*(a + b*x + c*x^2)^p*(f + g*x)^1, x, 5, (1/(e^2*(1 + m)))*(((e*f - d*g)*(d + e*x)^(1 + m)*(a + b*x + c*x^2)^p*AppellF1(1 + m, -p, -p, 2 + m, (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e), (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/((1 - (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e))^p*(1 - (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))^p)) + (1/(e^2*(2 + m)))*((g*(d + e*x)^(2 + m)*(a + b*x + c*x^2)^p*AppellF1(2 + m, -p, -p, 3 + m, (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e), (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/((1 - (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e))^p*(1 - (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))^p))]
    @test_int [(d + e*x)^m*(a + b*x + c*x^2)^p*(f + g*x)^0, x, 2, (1/(e*(1 + m)))*(((d + e*x)^(1 + m)*(a + b*x + c*x^2)^p*AppellF1(1 + m, -p, -p, 2 + m, (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e), (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e)))/((1 - (2*c*(d + e*x))/(2*c*d - (b - sqrt(b^2 - 4*a*c))*e))^p*(1 - (2*c*(d + e*x))/(2*c*d - (b + sqrt(b^2 - 4*a*c))*e))^p))]
    @test_int [(d + e*x)^m*(a + b*x + c*x^2)^p/(f + g*x)^1, x, 0, Unintegrable(((d + e*x)^m*(a + b*x + c*x^2)^p)/(f + g*x), x)]

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(g*x)^m*(d+e*x)^n*(a+b/x+c/x^2)^p=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(g*x)^m*(d+e*x)^n*(a+b/x+c/x^2)^p*with*b=0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(d+e*x)^(n/2)*(a+c/x^2)^p=#

    @test_int [1/(x^2*sqrt(1 - 1/(c^2*x^2))*sqrt(d + e*x)), x, 5, -((2*sqrt((c*(d + e*x))/(c*d + e))*sqrt(1 - c^2*x^2)*Elliptic.Pi(2, asin(sqrt(1 - c*x)/sqrt(2)), (2*e)/(c*d + e)))/(sqrt(1 - 1/(c^2*x^2))*x*sqrt(d + e*x)))]
    end

max_stars_repo_path: test/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.4 (d+e x)^m (f+g x)^n (a+b x+c x^2)^p.jl