text:
@testset "1.1.2.2 (c x)^m (a+b x^2)^p" begin
    (a, b, c, d, m, p, x, ) = @variables a b c d m p x

    #= ::Package:: =#

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

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

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

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

    @test_int [x^4*(a + b*x^2), x, 2, (a*x^5)/5 + (b*x^7)/7]
    @test_int [x^3*(a + b*x^2), x, 2, (a*x^4)/4 + (b*x^6)/6]
    @test_int [x^2*(a + b*x^2), x, 2, (a*x^3)/3 + (b*x^5)/5]
    @test_int [x^1*(a + b*x^2), x, 2, (a*x^2)/2 + (b*x^4)/4]
    @test_int [x^0*(a + b*x^2), x, 1, a*x + (b*x^3)/3]
    @test_int [(a + b*x^2)/x^1, x, 2, (b*x^2)/2 + a*log(x)]
    @test_int [(a + b*x^2)/x^2, x, 2, -(a/x) + b*x]
    @test_int [(a + b*x^2)/x^3, x, 2, -(a/(2*x^2)) + b*log(x)]
    @test_int [(a + b*x^2)/x^4, x, 2, -(a/(3*x^3)) - b/x]
    @test_int [(a + b*x^2)/x^5, x, 2, -(a/(4*x^4)) - b/(2*x^2)]
    @test_int [(a + b*x^2)/x^6, x, 2, -(a/(5*x^5)) - b/(3*x^3)]
    @test_int [(a + b*x^2)/x^7, x, 2, -(a/(6*x^6)) - b/(4*x^4)]

    @test_int [x^5*(a + b*x^2)^2, x, 3, (a^2*x^6)/6 + (1/4)*a*b*x^8 + (b^2*x^10)/10]
    @test_int [x^4*(a + b*x^2)^2, x, 2, (a^2*x^5)/5 + (2/7)*a*b*x^7 + (b^2*x^9)/9]
    @test_int [x^3*(a + b*x^2)^2, x, 3, (a^2*x^4)/4 + (1/3)*a*b*x^6 + (b^2*x^8)/8]
    @test_int [x^2*(a + b*x^2)^2, x, 2, (a^2*x^3)/3 + (2*a*b*x^5)/5 + (b^2*x^7)/7]
    @test_int [x^1*(a + b*x^2)^2, x, 1, (a + b*x^2)^3/(6*b)]
    @test_int [x^0*(a + b*x^2)^2, x, 2, a^2*x + (2*a*b*x^3)/3 + (b^2*x^5)/5]
    @test_int [(a + b*x^2)^2/x^1, x, 3, a*b*x^2 + (b^2*x^4)/4 + a^2*log(x)]
    @test_int [(a + b*x^2)^2/x^2, x, 2, -(a^2/x) + 2*a*b*x + (b^2*x^3)/3]
    @test_int [(a + b*x^2)^2/x^3, x, 3, -(a^2/(2*x^2)) + (b^2*x^2)/2 + 2*a*b*log(x)]
    @test_int [(a + b*x^2)^2/x^4, x, 2, -(a^2/(3*x^3)) - (2*a*b)/x + b^2*x]
    @test_int [(a + b*x^2)^2/x^5, x, 3, -(a^2/(4*x^4)) - (a*b)/x^2 + b^2*log(x)]
    @test_int [(a + b*x^2)^2/x^6, x, 2, -(a^2/(5*x^5)) - (2*a*b)/(3*x^3) - b^2/x]
    @test_int [(a + b*x^2)^2/x^7, x, 1, -((a + b*x^2)^3/(6*a*x^6))]
    @test_int [(a + b*x^2)^2/x^8, x, 2, -(a^2/(7*x^7)) - (2*a*b)/(5*x^5) - b^2/(3*x^3)]
    @test_int [(a + b*x^2)^2/x^9, x, 3, -(a^2/(8*x^8)) - (a*b)/(3*x^6) - b^2/(4*x^4)]
    @test_int [(a + b*x^2)^2/x^10, x, 2, -(a^2/(9*x^9)) - (2*a*b)/(7*x^7) - b^2/(5*x^5)]

    @test_int [x^9*(a + b*x^2)^3, x, 3, (a^3*x^10)/10 + (1/4)*a^2*b*x^12 + (3/14)*a*b^2*x^14 + (b^3*x^16)/16]
    @test_int [x^7*(a + b*x^2)^3, x, 3, (a^3*x^8)/8 + (3/10)*a^2*b*x^10 + (1/4)*a*b^2*x^12 + (b^3*x^14)/14]
    @test_int [x^5*(a + b*x^2)^3, x, 3, (a^3*x^6)/6 + (3/8)*a^2*b*x^8 + (3/10)*a*b^2*x^10 + (b^3*x^12)/12]
    @test_int [x^3*(a + b*x^2)^3, x, 3, -((a*(a + b*x^2)^4)/(8*b^2)) + (a + b*x^2)^5/(10*b^2)]
    @test_int [x^1*(a + b*x^2)^3, x, 1, (a + b*x^2)^4/(8*b)]
    @test_int [(a + b*x^2)^3/x^1, x, 3, (3/2)*a^2*b*x^2 + (3/4)*a*b^2*x^4 + (b^3*x^6)/6 + a^3*log(x)]
    @test_int [(a + b*x^2)^3/x^3, x, 3, -(a^3/(2*x^2)) + (3/2)*a*b^2*x^2 + (b^3*x^4)/4 + 3*a^2*b*log(x)]
    @test_int [(a + b*x^2)^3/x^5, x, 3, -(a^3/(4*x^4)) - (3*a^2*b)/(2*x^2) + (b^3*x^2)/2 + 3*a*b^2*log(x)]
    @test_int [(a + b*x^2)^3/x^7, x, 3, -(a^3/(6*x^6)) - (3*a^2*b)/(4*x^4) - (3*a*b^2)/(2*x^2) + b^3*log(x)]
    @test_int [(a + b*x^2)^3/x^9, x, 1, -((a + b*x^2)^4/(8*a*x^8))]
    @test_int [(a + b*x^2)^3/x^11, x, 3, -((a + b*x^2)^4/(10*a*x^10)) + (b*(a + b*x^2)^4)/(40*a^2*x^8)]
    @test_int [(a + b*x^2)^3/x^13, x, 3, -(a^3/(12*x^12)) - (3*a^2*b)/(10*x^10) - (3*a*b^2)/(8*x^8) - b^3/(6*x^6)]
    @test_int [(a + b*x^2)^3/x^15, x, 3, -(a^3/(14*x^14)) - (a^2*b)/(4*x^12) - (3*a*b^2)/(10*x^10) - b^3/(8*x^8)]

    @test_int [x^6*(a + b*x^2)^3, x, 2, (a^3*x^7)/7 + (1/3)*a^2*b*x^9 + (3/11)*a*b^2*x^11 + (b^3*x^13)/13]
    @test_int [x^4*(a + b*x^2)^3, x, 2, (a^3*x^5)/5 + (3/7)*a^2*b*x^7 + (1/3)*a*b^2*x^9 + (b^3*x^11)/11]
    @test_int [x^2*(a + b*x^2)^3, x, 2, (a^3*x^3)/3 + (3*a^2*b*x^5)/5 + (3*a*b^2*x^7)/7 + (b^3*x^9)/9]
    @test_int [x^0*(a + b*x^2)^3, x, 2, a^3*x + a^2*b*x^3 + (3*a*b^2*x^5)/5 + (b^3*x^7)/7]
    @test_int [(a + b*x^2)^3/x^2, x, 2, -(a^3/x) + 3*a^2*b*x + a*b^2*x^3 + (b^3*x^5)/5]
    @test_int [(a + b*x^2)^3/x^4, x, 2, -(a^3/(3*x^3)) - (3*a^2*b)/x + 3*a*b^2*x + (b^3*x^3)/3]
    @test_int [(a + b*x^2)^3/x^6, x, 2, -(a^3/(5*x^5)) - (a^2*b)/x^3 - (3*a*b^2)/x + b^3*x]
    @test_int [(a + b*x^2)^3/x^8, x, 2, -(a^3/(7*x^7)) - (3*a^2*b)/(5*x^5) - (a*b^2)/x^3 - b^3/x]
    @test_int [(a + b*x^2)^3/x^10, x, 2, -(a^3/(9*x^9)) - (3*a^2*b)/(7*x^7) - (3*a*b^2)/(5*x^5) - b^3/(3*x^3)]
    @test_int [(a + b*x^2)^3/x^12, x, 2, -(a^3/(11*x^11)) - (a^2*b)/(3*x^9) - (3*a*b^2)/(7*x^7) - b^3/(5*x^5)]

    @test_int [x^13*(a + b*x^2)^5, x, 3, (a^5*x^14)/14 + (5/16)*a^4*b*x^16 + (5/9)*a^3*b^2*x^18 + (1/2)*a^2*b^3*x^20 + (5/22)*a*b^4*x^22 + (b^5*x^24)/24]
    @test_int [x^11*(a + b*x^2)^5, x, 3, (a^5*x^12)/12 + (5/14)*a^4*b*x^14 + (5/8)*a^3*b^2*x^16 + (5/9)*a^2*b^3*x^18 + (1/4)*a*b^4*x^20 + (b^5*x^22)/22]
    @test_int [x^9*(a + b*x^2)^5, x, 3, (a^5*x^10)/10 + (5/12)*a^4*b*x^12 + (5/7)*a^3*b^2*x^14 + (5/8)*a^2*b^3*x^16 + (5/18)*a*b^4*x^18 + (b^5*x^20)/20]
    @test_int [x^7*(a + b*x^2)^5, x, 3, -((a^3*(a + b*x^2)^6)/(12*b^4)) + (3*a^2*(a + b*x^2)^7)/(14*b^4) - (3*a*(a + b*x^2)^8)/(16*b^4) + (a + b*x^2)^9/(18*b^4)]
    @test_int [x^5*(a + b*x^2)^5, x, 3, (a^2*(a + b*x^2)^6)/(12*b^3) - (a*(a + b*x^2)^7)/(7*b^3) + (a + b*x^2)^8/(16*b^3)]
    @test_int [x^3*(a + b*x^2)^5, x, 3, -((a*(a + b*x^2)^6)/(12*b^2)) + (a + b*x^2)^7/(14*b^2)]
    @test_int [x^1*(a + b*x^2)^5, x, 1, (a + b*x^2)^6/(12*b)]
    @test_int [(a + b*x^2)^5/x^1, x, 3, (5*a^4*b*x^2)/2 + (5*a^3*b^2*x^4)/2 + (5*a^2*b^3*x^6)/3 + (5*a*b^4*x^8)/8 + (b^5*x^10)/10 + a^5*log(x)]
    @test_int [(a + b*x^2)^5/x^3, x, 3, -a^5/(2*x^2) + 5*a^3*b^2*x^2 + (5*a^2*b^3*x^4)/2 + (5*a*b^4*x^6)/6 + (b^5*x^8)/8 + 5*a^4*b*log(x)]
    @test_int [(a + b*x^2)^5/x^5, x, 3, -a^5/(4*x^4) - (5*a^4*b)/(2*x^2) + 5*a^2*b^3*x^2 + (5*a*b^4*x^4)/4 + (b^5*x^6)/6 + 10*a^3*b^2*log(x)]
    @test_int [(a + b*x^2)^5/x^7, x, 3, -(a^5/(6*x^6)) - (5*a^4*b)/(4*x^4) - (5*a^3*b^2)/x^2 + (5/2)*a*b^4*x^2 + (b^5*x^4)/4 + 10*a^2*b^3*log(x)]
    @test_int [(a + b*x^2)^5/x^9, x, 3, -(a^5/(8*x^8)) - (5*a^4*b)/(6*x^6) - (5*a^3*b^2)/(2*x^4) - (5*a^2*b^3)/x^2 + (b^5*x^2)/2 + 5*a*b^4*log(x)]
    @test_int [(a + b*x^2)^5/x^11, x, 3, -(a^5/(10*x^10)) - (5*a^4*b)/(8*x^8) - (5*a^3*b^2)/(3*x^6) - (5*a^2*b^3)/(2*x^4) - (5*a*b^4)/(2*x^2) + b^5*log(x)]
    @test_int [(a + b*x^2)^5/x^13, x, 1, -(a + b*x^2)^6/(12*a*x^12)]
    @test_int [(a + b*x^2)^5/x^15, x, 3, -((a + b*x^2)^6/(14*a*x^14)) + (b*(a + b*x^2)^6)/(84*a^2*x^12)]
    @test_int [(a + b*x^2)^5/x^17, x, 4, -((a + b*x^2)^6/(16*a*x^16)) + (b*(a + b*x^2)^6)/(56*a^2*x^14) - (b^2*(a + b*x^2)^6)/(336*a^3*x^12)]
    @test_int [(a + b*x^2)^5/x^19, x, 3, -(a^5/(18*x^18)) - (5*a^4*b)/(16*x^16) - (5*a^3*b^2)/(7*x^14) - (5*a^2*b^3)/(6*x^12) - (a*b^4)/(2*x^10) - b^5/(8*x^8)]
    @test_int [(a + b*x^2)^5/x^21, x, 3, -(a^5/(20*x^20)) - (5*a^4*b)/(18*x^18) - (5*a^3*b^2)/(8*x^16) - (5*a^2*b^3)/(7*x^14) - (5*a*b^4)/(12*x^12) - b^5/(10*x^10)]

    @test_int [x^8*(a + b*x^2)^5, x, 2, (a^5*x^9)/9 + (5*a^4*b*x^11)/11 + (10*a^3*b^2*x^13)/13 + (2*a^2*b^3*x^15)/3 + (5*a*b^4*x^17)/17 + (b^5*x^19)/19]
    @test_int [x^6*(a + b*x^2)^5, x, 2, (a^5*x^7)/7 + (5*a^4*b*x^9)/9 + (10*a^3*b^2*x^11)/11 + (10*a^2*b^3*x^13)/13 + (a*b^4*x^15)/3 + (b^5*x^17)/17]
    @test_int [x^4*(a + b*x^2)^5, x, 2, (a^5*x^5)/5 + (5*a^4*b*x^7)/7 + (10*a^3*b^2*x^9)/9 + (10*a^2*b^3*x^11)/11 + (5*a*b^4*x^13)/13 + (b^5*x^15)/15]
    @test_int [x^2*(a + b*x^2)^5, x, 2, (a^5*x^3)/3 + a^4*b*x^5 + (10*a^3*b^2*x^7)/7 + (10*a^2*b^3*x^9)/9 + (5*a*b^4*x^11)/11 + (b^5*x^13)/13]
    @test_int [x^0*(a + b*x^2)^5, x, 2, a^5*x + (5*a^4*b*x^3)/3 + 2*a^3*b^2*x^5 + (10*a^2*b^3*x^7)/7 + (5*a*b^4*x^9)/9 + (b^5*x^11)/11]
    @test_int [(a + b*x^2)^5/x^2, x, 2, -(a^5/x) + 5*a^4*b*x + (10*a^3*b^2*x^3)/3 + 2*a^2*b^3*x^5 + (5*a*b^4*x^7)/7 + (b^5*x^9)/9]
    @test_int [(a + b*x^2)^5/x^4, x, 2, -a^5/(3*x^3) - (5*a^4*b)/x + 10*a^3*b^2*x + (10*a^2*b^3*x^3)/3 + a*b^4*x^5 + (b^5*x^7)/7]
    @test_int [(a + b*x^2)^5/x^6, x, 2, -a^5/(5*x^5) - (5*a^4*b)/(3*x^3) - (10*a^3*b^2)/x + 10*a^2*b^3*x + (5*a*b^4*x^3)/3 + (b^5*x^5)/5]
    @test_int [(a + b*x^2)^5/x^8, x, 2, -a^5/(7*x^7) - (a^4*b)/x^5 - (10*a^3*b^2)/(3*x^3) - (10*a^2*b^3)/x + 5*a*b^4*x + (b^5*x^3)/3]
    @test_int [(a + b*x^2)^5/x^10, x, 2, -a^5/(9*x^9) - (5*a^4*b)/(7*x^7) - (2*a^3*b^2)/x^5 - (10*a^2*b^3)/(3*x^3) - (5*a*b^4)/x + b^5*x]
    @test_int [(a + b*x^2)^5/x^12, x, 2, -(a^5/(11*x^11)) - (5*a^4*b)/(9*x^9) - (10*a^3*b^2)/(7*x^7) - (2*a^2*b^3)/x^5 - (5*a*b^4)/(3*x^3) - b^5/x]
    @test_int [(a + b*x^2)^5/x^14, x, 2, -a^5/(13*x^13) - (5*a^4*b)/(11*x^11) - (10*a^3*b^2)/(9*x^9) - (10*a^2*b^3)/(7*x^7) - (a*b^4)/x^5 - b^5/(3*x^3)]
    @test_int [(a + b*x^2)^5/x^16, x, 2, -a^5/(15*x^15) - (5*a^4*b)/(13*x^13) - (10*a^3*b^2)/(11*x^11) - (10*a^2*b^3)/(9*x^9) - (5*a*b^4)/(7*x^7) - b^5/(5*x^5)]
    @test_int [(a + b*x^2)^5/x^18, x, 2, -a^5/(17*x^17) - (a^4*b)/(3*x^15) - (10*a^3*b^2)/(13*x^13) - (10*a^2*b^3)/(11*x^11) - (5*a*b^4)/(9*x^9) - b^5/(7*x^7)]
    @test_int [(a + b*x^2)^5/x^20, x, 2, -a^5/(19*x^19) - (5*a^4*b)/(17*x^17) - (2*a^3*b^2)/(3*x^15) - (10*a^2*b^3)/(13*x^13) - (5*a*b^4)/(11*x^11) - b^5/(9*x^9)]

    @test_int [x^13*(a + b*x^2)^8, x, 3, (a^6*(a + b*x^2)^9)/(18*b^7) - (3*a^5*(a + b*x^2)^10)/(10*b^7) + (15*a^4*(a + b*x^2)^11)/(22*b^7) - (5*a^3*(a + b*x^2)^12)/(6*b^7) + (15*a^2*(a + b*x^2)^13)/(26*b^7) - (3*a*(a + b*x^2)^14)/(14*b^7) + (a + b*x^2)^15/(30*b^7)]
    @test_int [x^11*(a + b*x^2)^8, x, 3, -((a^5*(a + b*x^2)^9)/(18*b^6)) + (a^4*(a + b*x^2)^10)/(4*b^6) - (5*a^3*(a + b*x^2)^11)/(11*b^6) + (5*a^2*(a + b*x^2)^12)/(12*b^6) - (5*a*(a + b*x^2)^13)/(26*b^6) + (a + b*x^2)^14/(28*b^6)]
    @test_int [x^9*(a + b*x^2)^8, x, 3, (a^4*(a + b*x^2)^9)/(18*b^5) - (a^3*(a + b*x^2)^10)/(5*b^5) + (3*a^2*(a + b*x^2)^11)/(11*b^5) - (a*(a + b*x^2)^12)/(6*b^5) + (a + b*x^2)^13/(26*b^5)]
    @test_int [x^7*(a + b*x^2)^8, x, 3, -((a^3*(a + b*x^2)^9)/(18*b^4)) + (3*a^2*(a + b*x^2)^10)/(20*b^4) - (3*a*(a + b*x^2)^11)/(22*b^4) + (a + b*x^2)^12/(24*b^4)]
    @test_int [x^5*(a + b*x^2)^8, x, 3, (a^2*(a + b*x^2)^9)/(18*b^3) - (a*(a + b*x^2)^10)/(10*b^3) + (a + b*x^2)^11/(22*b^3)]
    @test_int [x^3*(a + b*x^2)^8, x, 3, -((a*(a + b*x^2)^9)/(18*b^2)) + (a + b*x^2)^10/(20*b^2)]
    @test_int [x^1*(a + b*x^2)^8, x, 1, (a + b*x^2)^9/(18*b)]
    @test_int [(a + b*x^2)^8/x^1, x, 3, 4*a^7*b*x^2 + 7*a^6*b^2*x^4 + (28*a^5*b^3*x^6)/3 + (35*a^4*b^4*x^8)/4 + (28*a^3*b^5*x^10)/5 + (7*a^2*b^6*x^12)/3 + (4*a*b^7*x^14)/7 + (b^8*x^16)/16 + a^8*log(x)]
    @test_int [(a + b*x^2)^8/x^3, x, 3, -a^8/(2*x^2) + 14*a^6*b^2*x^2 + 14*a^5*b^3*x^4 + (35*a^4*b^4*x^6)/3 + 7*a^3*b^5*x^8 + (14*a^2*b^6*x^10)/5 + (2*a*b^7*x^12)/3 + (b^8*x^14)/14 + 8*a^7*b*log(x)]
    @test_int [(a + b*x^2)^8/x^5, x, 3, -a^8/(4*x^4) - (4*a^7*b)/x^2 + 28*a^5*b^3*x^2 + (35*a^4*b^4*x^4)/2 + (28*a^3*b^5*x^6)/3 + (7*a^2*b^6*x^8)/2 + (4*a*b^7*x^10)/5 + (b^8*x^12)/12 + 28*a^6*b^2*log(x)]
    @test_int [(a + b*x^2)^8/x^7, x, 3, -a^8/(6*x^6) - (2*a^7*b)/x^4 - (14*a^6*b^2)/x^2 + 35*a^4*b^4*x^2 + 14*a^3*b^5*x^4 + (14*a^2*b^6*x^6)/3 + a*b^7*x^8 + (b^8*x^10)/10 + 56*a^5*b^3*log(x)]
    @test_int [(a + b*x^2)^8/x^9, x, 3, -a^8/(8*x^8) - (4*a^7*b)/(3*x^6) - (7*a^6*b^2)/x^4 - (28*a^5*b^3)/x^2 + 28*a^3*b^5*x^2 + 7*a^2*b^6*x^4 + (4*a*b^7*x^6)/3 + (b^8*x^8)/8 + 70*a^4*b^4*log(x)]
    @test_int [(a + b*x^2)^8/x^11, x, 3, -a^8/(10*x^10) - (a^7*b)/x^8 - (14*a^6*b^2)/(3*x^6) - (14*a^5*b^3)/x^4 - (35*a^4*b^4)/x^2 + 14*a^2*b^6*x^2 + 2*a*b^7*x^4 + (b^8*x^6)/6 + 56*a^3*b^5*log(x)]
    @test_int [(a + b*x^2)^8/x^13, x, 3, -a^8/(12*x^12) - (4*a^7*b)/(5*x^10) - (7*a^6*b^2)/(2*x^8) - (28*a^5*b^3)/(3*x^6) - (35*a^4*b^4)/(2*x^4) - (28*a^3*b^5)/x^2 + 4*a*b^7*x^2 + (b^8*x^4)/4 + 28*a^2*b^6*log(x)]
    @test_int [(a + b*x^2)^8/x^15, x, 3, -a^8/(14*x^14) - (2*a^7*b)/(3*x^12) - (14*a^6*b^2)/(5*x^10) - (7*a^5*b^3)/x^8 - (35*a^4*b^4)/(3*x^6) - (14*a^3*b^5)/x^4 - (14*a^2*b^6)/x^2 + (b^8*x^2)/2 + 8*a*b^7*log(x)]
    @test_int [(a + b*x^2)^8/x^17, x, 3, -a^8/(16*x^16) - (4*a^7*b)/(7*x^14) - (7*a^6*b^2)/(3*x^12) - (28*a^5*b^3)/(5*x^10) - (35*a^4*b^4)/(4*x^8) - (28*a^3*b^5)/(3*x^6) - (7*a^2*b^6)/x^4 - (4*a*b^7)/x^2 + b^8*log(x)]
    @test_int [(a + b*x^2)^8/x^19, x, 1, -(a + b*x^2)^9/(18*a*x^18)]
    @test_int [(a + b*x^2)^8/x^21, x, 3, -(a + b*x^2)^9/(20*a*x^20) + (b*(a + b*x^2)^9)/(180*a^2*x^18)]
    @test_int [(a + b*x^2)^8/x^23, x, 4, -(a + b*x^2)^9/(22*a*x^22) + (b*(a + b*x^2)^9)/(110*a^2*x^20) - (b^2*(a + b*x^2)^9)/(990*a^3*x^18)]
    @test_int [(a + b*x^2)^8/x^25, x, 5, -(a + b*x^2)^9/(24*a*x^24) + (b*(a + b*x^2)^9)/(88*a^2*x^22) - (b^2*(a + b*x^2)^9)/(440*a^3*x^20) + (b^3*(a + b*x^2)^9)/(3960*a^4*x^18)]
    @test_int [(a + b*x^2)^8/x^27, x, 6, -((a + b*x^2)^9/(26*a*x^26)) + (b*(a + b*x^2)^9)/(78*a^2*x^24) - (b^2*(a + b*x^2)^9)/(286*a^3*x^22) + (b^3*(a + b*x^2)^9)/(1430*a^4*x^20) - (b^4*(a + b*x^2)^9)/(12870*a^5*x^18)]
    @test_int [(a + b*x^2)^8/x^29, x, 3, -a^8/(28*x^28) - (4*a^7*b)/(13*x^26) - (7*a^6*b^2)/(6*x^24) - (28*a^5*b^3)/(11*x^22) - (7*a^4*b^4)/(2*x^20) - (28*a^3*b^5)/(9*x^18) - (7*a^2*b^6)/(4*x^16) - (4*a*b^7)/(7*x^14) - b^8/(12*x^12)]
    @test_int [(a + b*x^2)^8/x^31, x, 3, -a^8/(30*x^30) - (2*a^7*b)/(7*x^28) - (14*a^6*b^2)/(13*x^26) - (7*a^5*b^3)/(3*x^24) - (35*a^4*b^4)/(11*x^22) - (14*a^3*b^5)/(5*x^20) - (14*a^2*b^6)/(9*x^18) - (a*b^7)/(2*x^16) - b^8/(14*x^14)]
    @test_int [(a + b*x^2)^8/x^33, x, 3, -(a^8/(32*x^32)) - (4*a^7*b)/(15*x^30) - (a^6*b^2)/x^28 - (28*a^5*b^3)/(13*x^26) - (35*a^4*b^4)/(12*x^24) - (28*a^3*b^5)/(11*x^22) - (7*a^2*b^6)/(5*x^20) - (4*a*b^7)/(9*x^18) - b^8/(16*x^16)]

    @test_int [x^8*(a + b*x^2)^8, x, 2, (a^8*x^9)/9 + (8*a^7*b*x^11)/11 + (28*a^6*b^2*x^13)/13 + (56*a^5*b^3*x^15)/15 + (70*a^4*b^4*x^17)/17 + (56*a^3*b^5*x^19)/19 + (4*a^2*b^6*x^21)/3 + (8*a*b^7*x^23)/23 + (b^8*x^25)/25]
    @test_int [x^6*(a + b*x^2)^8, x, 2, (a^8*x^7)/7 + (8*a^7*b*x^9)/9 + (28*a^6*b^2*x^11)/11 + (56*a^5*b^3*x^13)/13 + (14*a^4*b^4*x^15)/3 + (56*a^3*b^5*x^17)/17 + (28*a^2*b^6*x^19)/19 + (8*a*b^7*x^21)/21 + (b^8*x^23)/23]
    @test_int [x^4*(a + b*x^2)^8, x, 2, (a^8*x^5)/5 + (8*a^7*b*x^7)/7 + (28*a^6*b^2*x^9)/9 + (56*a^5*b^3*x^11)/11 + (70*a^4*b^4*x^13)/13 + (56*a^3*b^5*x^15)/15 + (28*a^2*b^6*x^17)/17 + (8*a*b^7*x^19)/19 + (b^8*x^21)/21]
    @test_int [x^2*(a + b*x^2)^8, x, 2, (a^8*x^3)/3 + (8*a^7*b*x^5)/5 + 4*a^6*b^2*x^7 + (56*a^5*b^3*x^9)/9 + (70*a^4*b^4*x^11)/11 + (56*a^3*b^5*x^13)/13 + (28*a^2*b^6*x^15)/15 + (8*a*b^7*x^17)/17 + (b^8*x^19)/19]
    @test_int [x^0*(a + b*x^2)^8, x, 2, a^8*x + (8*a^7*b*x^3)/3 + (28*a^6*b^2*x^5)/5 + 8*a^5*b^3*x^7 + (70*a^4*b^4*x^9)/9 + (56*a^3*b^5*x^11)/11 + (28*a^2*b^6*x^13)/13 + (8*a*b^7*x^15)/15 + (b^8*x^17)/17]
    @test_int [(a + b*x^2)^8/x^2, x, 2, -(a^8/x) + 8*a^7*b*x + (28*a^6*b^2*x^3)/3 + (56*a^5*b^3*x^5)/5 + 10*a^4*b^4*x^7 + (56*a^3*b^5*x^9)/9 + (28*a^2*b^6*x^11)/11 + (8*a*b^7*x^13)/13 + (b^8*x^15)/15]
    @test_int [(a + b*x^2)^8/x^4, x, 2, -a^8/(3*x^3) - (8*a^7*b)/x + 28*a^6*b^2*x + (56*a^5*b^3*x^3)/3 + 14*a^4*b^4*x^5 + 8*a^3*b^5*x^7 + (28*a^2*b^6*x^9)/9 + (8*a*b^7*x^11)/11 + (b^8*x^13)/13]
    @test_int [(a + b*x^2)^8/x^6, x, 2, -a^8/(5*x^5) - (8*a^7*b)/(3*x^3) - (28*a^6*b^2)/x + 56*a^5*b^3*x + (70*a^4*b^4*x^3)/3 + (56*a^3*b^5*x^5)/5 + 4*a^2*b^6*x^7 + (8*a*b^7*x^9)/9 + (b^8*x^11)/11]
    @test_int [(a + b*x^2)^8/x^8, x, 2, -a^8/(7*x^7) - (8*a^7*b)/(5*x^5) - (28*a^6*b^2)/(3*x^3) - (56*a^5*b^3)/x + 70*a^4*b^4*x + (56*a^3*b^5*x^3)/3 + (28*a^2*b^6*x^5)/5 + (8*a*b^7*x^7)/7 + (b^8*x^9)/9]
    @test_int [(a + b*x^2)^8/x^10, x, 2, -a^8/(9*x^9) - (8*a^7*b)/(7*x^7) - (28*a^6*b^2)/(5*x^5) - (56*a^5*b^3)/(3*x^3) - (70*a^4*b^4)/x + 56*a^3*b^5*x + (28*a^2*b^6*x^3)/3 + (8*a*b^7*x^5)/5 + (b^8*x^7)/7]
    @test_int [(a + b*x^2)^8/x^12, x, 2, -a^8/(11*x^11) - (8*a^7*b)/(9*x^9) - (4*a^6*b^2)/x^7 - (56*a^5*b^3)/(5*x^5) - (70*a^4*b^4)/(3*x^3) - (56*a^3*b^5)/x + 28*a^2*b^6*x + (8*a*b^7*x^3)/3 + (b^8*x^5)/5]
    @test_int [(a + b*x^2)^8/x^14, x, 2, -a^8/(13*x^13) - (8*a^7*b)/(11*x^11) - (28*a^6*b^2)/(9*x^9) - (8*a^5*b^3)/x^7 - (14*a^4*b^4)/x^5 - (56*a^3*b^5)/(3*x^3) - (28*a^2*b^6)/x + 8*a*b^7*x + (b^8*x^3)/3]
    @test_int [(a + b*x^2)^8/x^16, x, 2, -a^8/(15*x^15) - (8*a^7*b)/(13*x^13) - (28*a^6*b^2)/(11*x^11) - (56*a^5*b^3)/(9*x^9) - (10*a^4*b^4)/x^7 - (56*a^3*b^5)/(5*x^5) - (28*a^2*b^6)/(3*x^3) - (8*a*b^7)/x + b^8*x]
    @test_int [(a + b*x^2)^8/x^18, x, 2, -a^8/(17*x^17) - (8*a^7*b)/(15*x^15) - (28*a^6*b^2)/(13*x^13) - (56*a^5*b^3)/(11*x^11) - (70*a^4*b^4)/(9*x^9) - (8*a^3*b^5)/x^7 - (28*a^2*b^6)/(5*x^5) - (8*a*b^7)/(3*x^3) - b^8/x]
    @test_int [(a + b*x^2)^8/x^20, x, 2, -a^8/(19*x^19) - (8*a^7*b)/(17*x^17) - (28*a^6*b^2)/(15*x^15) - (56*a^5*b^3)/(13*x^13) - (70*a^4*b^4)/(11*x^11) - (56*a^3*b^5)/(9*x^9) - (4*a^2*b^6)/x^7 - (8*a*b^7)/(5*x^5) - b^8/(3*x^3)]

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

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

    @test_int [x^13/(a + b*x^2)^2, x, 3, (5*a^4*x^2)/(2*b^6) - (a^3*x^4)/b^5 + (a^2*x^6)/(2*b^4) - (a*x^8)/(4*b^3) + x^10/(10*b^2) - a^6/(2*b^7*(a + b*x^2)) - (3*a^5*log(a + b*x^2))/b^7]
    @test_int [x^12/(a + b*x^2)^2, x, 4, (11*a^4*x)/(2*b^6) - (11*a^3*x^3)/(6*b^5) + (11*a^2*x^5)/(10*b^4) - (11*a*x^7)/(14*b^3) + (11*x^9)/(18*b^2) - x^11/(2*b*(a + b*x^2)) - (11*a^(9/2)*atan((sqrt(b)*x)/sqrt(a)))/(2*b^(13/2))]
    @test_int [x^11/(a + b*x^2)^2, x, 3, -((2*a^3*x^2)/b^5) + (3*a^2*x^4)/(4*b^4) - (a*x^6)/(3*b^3) + x^8/(8*b^2) + a^5/(2*b^6*(a + b*x^2)) + (5*a^4*log(a + b*x^2))/(2*b^6)]
    @test_int [x^10/(a + b*x^2)^2, x, 4, -((9*a^3*x)/(2*b^5)) + (3*a^2*x^3)/(2*b^4) - (9*a*x^5)/(10*b^3) + (9*x^7)/(14*b^2) - x^9/(2*b*(a + b*x^2)) + (9*a^(7/2)*atan((sqrt(b)*x)/sqrt(a)))/(2*b^(11/2))]
    @test_int [x^9/(a + b*x^2)^2, x, 3, (3*a^2*x^2)/(2*b^4) - (a*x^4)/(2*b^3) + x^6/(6*b^2) - a^4/(2*b^5*(a + b*x^2)) - (2*a^3*log(a + b*x^2))/b^5]
    @test_int [x^8/(a + b*x^2)^2, x, 4, (7*a^2*x)/(2*b^4) - (7*a*x^3)/(6*b^3) + (7*x^5)/(10*b^2) - x^7/(2*b*(a + b*x^2)) - (7*a^(5/2)*atan((sqrt(b)*x)/sqrt(a)))/(2*b^(9/2))]
    @test_int [x^7/(a + b*x^2)^2, x, 3, -((a*x^2)/b^3) + x^4/(4*b^2) + a^3/(2*b^4*(a + b*x^2)) + (3*a^2*log(a + b*x^2))/(2*b^4)]
    @test_int [x^6/(a + b*x^2)^2, x, 4, -((5*a*x)/(2*b^3)) + (5*x^3)/(6*b^2) - x^5/(2*b*(a + b*x^2)) + (5*a^(3/2)*atan((sqrt(b)*x)/sqrt(a)))/(2*b^(7/2))]
    @test_int [x^5/(a + b*x^2)^2, x, 3, x^2/(2*b^2) - a^2/(2*b^3*(a + b*x^2)) - (a*log(a + b*x^2))/b^3]
    @test_int [x^4/(a + b*x^2)^2, x, 3, (3*x)/(2*b^2) - x^3/(2*b*(a + b*x^2)) - (3*sqrt(a)*atan((sqrt(b)*x)/sqrt(a)))/(2*b^(5/2))]
    @test_int [x^3/(a + b*x^2)^2, x, 3, a/(2*b^2*(a + b*x^2)) + log(a + b*x^2)/(2*b^2)]
    @test_int [x^2/(a + b*x^2)^2, x, 2, -x/(2*b*(a + b*x^2)) + atan((sqrt(b)*x)/sqrt(a))/(2*sqrt(a)*b^(3/2))]
    @test_int [x^1/(a + b*x^2)^2, x, 1, -1/(2*b*(a + b*x^2))]
    @test_int [x^0/(a + b*x^2)^2, x, 2, x/(2*a*(a + b*x^2)) + atan((sqrt(b)*x)/sqrt(a))/(2*a^(3/2)*sqrt(b))]
    @test_int [1/(x^1*(a + b*x^2)^2), x, 3, 1/(2*a*(a + b*x^2)) + log(x)/a^2 - log(a + b*x^2)/(2*a^2)]
    @test_int [1/(x^2*(a + b*x^2)^2), x, 3, -(3/(2*a^2*x)) + 1/(2*a*x*(a + b*x^2)) - (3*sqrt(b)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(5/2))]
    @test_int [1/(x^3*(a + b*x^2)^2), x, 3, -(1/(2*a^2*x^2)) - b/(2*a^2*(a + b*x^2)) - (2*b*log(x))/a^3 + (b*log(a + b*x^2))/a^3]
    @test_int [1/(x^4*(a + b*x^2)^2), x, 4, -(5/(6*a^2*x^3)) + (5*b)/(2*a^3*x) + 1/(2*a*x^3*(a + b*x^2)) + (5*b^(3/2)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(7/2))]
    @test_int [1/(x^5*(a + b*x^2)^2), x, 3, -(1/(4*a^2*x^4)) + b/(a^3*x^2) + b^2/(2*a^3*(a + b*x^2)) + (3*b^2*log(x))/a^4 - (3*b^2*log(a + b*x^2))/(2*a^4)]
    @test_int [1/(x^6*(a + b*x^2)^2), x, 5, -(7/(10*a^2*x^5)) + (7*b)/(6*a^3*x^3) - (7*b^2)/(2*a^4*x) + 1/(2*a*x^5*(a + b*x^2)) - (7*b^(5/2)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(9/2))]
    @test_int [1/(x^7*(a + b*x^2)^2), x, 3, -(1/(6*a^2*x^6)) + b/(2*a^3*x^4) - (3*b^2)/(2*a^4*x^2) - b^3/(2*a^4*(a + b*x^2)) - (4*b^3*log(x))/a^5 + (2*b^3*log(a + b*x^2))/a^5]
    @test_int [1/(x^8*(a + b*x^2)^2), x, 6, -(9/(14*a^2*x^7)) + (9*b)/(10*a^3*x^5) - (3*b^2)/(2*a^4*x^3) + (9*b^3)/(2*a^5*x) + 1/(2*a*x^7*(a + b*x^2)) + (9*b^(7/2)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(11/2))]
    @test_int [1/(x^9*(a + b*x^2)^2), x, 3, -(1/(8*a^2*x^8)) + b/(3*a^3*x^6) - (3*b^2)/(4*a^4*x^4) + (2*b^3)/(a^5*x^2) + b^4/(2*a^5*(a + b*x^2)) + (5*b^4*log(x))/a^6 - (5*b^4*log(a + b*x^2))/(2*a^6)]

    @test_int [x^15/(a + b*x^2)^3, x, 3, (15*a^4*x^2)/(2*b^7) - (5*a^3*x^4)/(2*b^6) + (a^2*x^6)/b^5 - (3*a*x^8)/(8*b^4) + x^10/(10*b^3) + a^7/(4*b^8*(a + b*x^2)^2) - (7*a^6)/(2*b^8*(a + b*x^2)) - (21*a^5*log(a + b*x^2))/(2*b^8)]
    @test_int [x^13/(a + b*x^2)^3, x, 3, -((5*a^3*x^2)/b^6) + (3*a^2*x^4)/(2*b^5) - (a*x^6)/(2*b^4) + x^8/(8*b^3) - a^6/(4*b^7*(a + b*x^2)^2) + (3*a^5)/(b^7*(a + b*x^2)) + (15*a^4*log(a + b*x^2))/(2*b^7)]
    @test_int [x^11/(a + b*x^2)^3, x, 3, (3*a^2*x^2)/b^5 - (3*a*x^4)/(4*b^4) + x^6/(6*b^3) + a^5/(4*b^6*(a + b*x^2)^2) - (5*a^4)/(2*b^6*(a + b*x^2)) - (5*a^3*log(a + b*x^2))/b^6]
    @test_int [x^9/(a + b*x^2)^3, x, 3, -((3*a*x^2)/(2*b^4)) + x^4/(4*b^3) - a^4/(4*b^5*(a + b*x^2)^2) + (2*a^3)/(b^5*(a + b*x^2)) + (3*a^2*log(a + b*x^2))/b^5]
    @test_int [x^7/(a + b*x^2)^3, x, 3, x^2/(2*b^3) + a^3/(4*b^4*(a + b*x^2)^2) - (3*a^2)/(2*b^4*(a + b*x^2)) - (3*a*log(a + b*x^2))/(2*b^4)]
    @test_int [x^5/(a + b*x^2)^3, x, 3, -(a^2/(4*b^3*(a + b*x^2)^2)) + a/(b^3*(a + b*x^2)) + log(a + b*x^2)/(2*b^3)]
    @test_int [x^3/(a + b*x^2)^3, x, 1, x^4/(4*a*(a + b*x^2)^2)]
    @test_int [x^1/(a + b*x^2)^3, x, 1, -(1/(4*b*(a + b*x^2)^2))]
    @test_int [1/(x^1*(a + b*x^2)^3), x, 3, 1/(4*a*(a + b*x^2)^2) + 1/(2*a^2*(a + b*x^2)) + log(x)/a^3 - log(a + b*x^2)/(2*a^3)]
    @test_int [1/(x^3*(a + b*x^2)^3), x, 3, -(1/(2*a^3*x^2)) - b/(4*a^2*(a + b*x^2)^2) - b/(a^3*(a + b*x^2)) - (3*b*log(x))/a^4 + (3*b*log(a + b*x^2))/(2*a^4)]
    @test_int [1/(x^5*(a + b*x^2)^3), x, 3, -(1/(4*a^3*x^4)) + (3*b)/(2*a^4*x^2) + b^2/(4*a^3*(a + b*x^2)^2) + (3*b^2)/(2*a^4*(a + b*x^2)) + (6*b^2*log(x))/a^5 - (3*b^2*log(a + b*x^2))/a^5]
    @test_int [1/(x^7*(a + b*x^2)^3), x, 3, -(1/(6*a^3*x^6)) + (3*b)/(4*a^4*x^4) - (3*b^2)/(a^5*x^2) - b^3/(4*a^4*(a + b*x^2)^2) - (2*b^3)/(a^5*(a + b*x^2)) - (10*b^3*log(x))/a^6 + (5*b^3*log(a + b*x^2))/a^6]
    @test_int [1/(x^9*(a + b*x^2)^3), x, 3, -(1/(8*a^3*x^8)) + b/(2*a^4*x^6) - (3*b^2)/(2*a^5*x^4) + (5*b^3)/(a^6*x^2) + b^4/(4*a^5*(a + b*x^2)^2) + (5*b^4)/(2*a^6*(a + b*x^2)) + (15*b^4*log(x))/a^7 - (15*b^4*log(a + b*x^2))/(2*a^7)]

    @test_int [x^12/(a + b*x^2)^3, x, 5, -((99*a^3*x)/(8*b^6)) + (33*a^2*x^3)/(8*b^5) - (99*a*x^5)/(40*b^4) + (99*x^7)/(56*b^3) - x^11/(4*b*(a + b*x^2)^2) - (11*x^9)/(8*b^2*(a + b*x^2)) + (99*a^(7/2)*atan((sqrt(b)*x)/sqrt(a)))/(8*b^(13/2))]
    @test_int [x^10/(a + b*x^2)^3, x, 5, (63*a^2*x)/(8*b^5) - (21*a*x^3)/(8*b^4) + (63*x^5)/(40*b^3) - x^9/(4*b*(a + b*x^2)^2) - (9*x^7)/(8*b^2*(a + b*x^2)) - (63*a^(5/2)*atan((sqrt(b)*x)/sqrt(a)))/(8*b^(11/2))]
    @test_int [x^8/(a + b*x^2)^3, x, 5, -((35*a*x)/(8*b^4)) + (35*x^3)/(24*b^3) - x^7/(4*b*(a + b*x^2)^2) - (7*x^5)/(8*b^2*(a + b*x^2)) + (35*a^(3/2)*atan((sqrt(b)*x)/sqrt(a)))/(8*b^(9/2))]
    @test_int [x^6/(a + b*x^2)^3, x, 4, (15*x)/(8*b^3) - x^5/(4*b*(a + b*x^2)^2) - (5*x^3)/(8*b^2*(a + b*x^2)) - (15*sqrt(a)*atan((sqrt(b)*x)/sqrt(a)))/(8*b^(7/2))]
    @test_int [x^4/(a + b*x^2)^3, x, 3, -(x^3/(4*b*(a + b*x^2)^2)) - (3*x)/(8*b^2*(a + b*x^2)) + (3*atan((sqrt(b)*x)/sqrt(a)))/(8*sqrt(a)*b^(5/2))]
    @test_int [x^2/(a + b*x^2)^3, x, 3, -(x/(4*b*(a + b*x^2)^2)) + x/(8*a*b*(a + b*x^2)) + atan((sqrt(b)*x)/sqrt(a))/(8*a^(3/2)*b^(3/2))]
    @test_int [x^0/(a + b*x^2)^3, x, 3, x/(4*a*(a + b*x^2)^2) + (3*x)/(8*a^2*(a + b*x^2)) + (3*atan((sqrt(b)*x)/sqrt(a)))/(8*a^(5/2)*sqrt(b))]
    @test_int [1/(x^2*(a + b*x^2)^3), x, 4, -(15/(8*a^3*x)) + 1/(4*a*x*(a + b*x^2)^2) + 5/(8*a^2*x*(a + b*x^2)) - (15*sqrt(b)*atan((sqrt(b)*x)/sqrt(a)))/(8*a^(7/2))]
    @test_int [1/(x^4*(a + b*x^2)^3), x, 5, -(35/(24*a^3*x^3)) + (35*b)/(8*a^4*x) + 1/(4*a*x^3*(a + b*x^2)^2) + 7/(8*a^2*x^3*(a + b*x^2)) + (35*b^(3/2)*atan((sqrt(b)*x)/sqrt(a)))/(8*a^(9/2))]
    @test_int [1/(x^6*(a + b*x^2)^3), x, 6, -(63/(40*a^3*x^5)) + (21*b)/(8*a^4*x^3) - (63*b^2)/(8*a^5*x) + 1/(4*a*x^5*(a + b*x^2)^2) + 9/(8*a^2*x^5*(a + b*x^2)) - (63*b^(5/2)*atan((sqrt(b)*x)/sqrt(a)))/(8*a^(11/2))]
    @test_int [1/(x^8*(a + b*x^2)^3), x, 7, -(99/(56*a^3*x^7)) + (99*b)/(40*a^4*x^5) - (33*b^2)/(8*a^5*x^3) + (99*b^3)/(8*a^6*x) + 1/(4*a*x^7*(a + b*x^2)^2) + 11/(8*a^2*x^7*(a + b*x^2)) + (99*b^(7/2)*atan((sqrt(b)*x)/sqrt(a)))/(8*a^(13/2))]

    @test_int [x^25/(a + b*x^2)^10, x, 3, (55*a^2*x^2)/(2*b^12) - (5*a*x^4)/(2*b^11) + x^6/(6*b^10) - a^12/(18*b^13*(a + b*x^2)^9) + (3*a^11)/(4*b^13*(a + b*x^2)^8) - (33*a^10)/(7*b^13*(a + b*x^2)^7) + (55*a^9)/(3*b^13*(a + b*x^2)^6) - (99*a^8)/(2*b^13*(a + b*x^2)^5) + (99*a^7)/(b^13*(a + b*x^2)^4) - (154*a^6)/(b^13*(a + b*x^2)^3) + (198*a^5)/(b^13*(a + b*x^2)^2) - (495*a^4)/(2*b^13*(a + b*x^2)) - (110*a^3*log(a + b*x^2))/b^13]
    @test_int [x^23/(a + b*x^2)^10, x, 3, -((5*a*x^2)/b^11) + x^4/(4*b^10) + a^11/(18*b^12*(a + b*x^2)^9) - (11*a^10)/(16*b^12*(a + b*x^2)^8) + (55*a^9)/(14*b^12*(a + b*x^2)^7) - (55*a^8)/(4*b^12*(a + b*x^2)^6) + (33*a^7)/(b^12*(a + b*x^2)^5) - (231*a^6)/(4*b^12*(a + b*x^2)^4) + (77*a^5)/(b^12*(a + b*x^2)^3) - (165*a^4)/(2*b^12*(a + b*x^2)^2) + (165*a^3)/(2*b^12*(a + b*x^2)) + (55*a^2*log(a + b*x^2))/(2*b^12)]
    @test_int [x^21/(a + b*x^2)^10, x, 3, x^2/(2*b^10) - a^10/(18*b^11*(a + b*x^2)^9) + (5*a^9)/(8*b^11*(a + b*x^2)^8) - (45*a^8)/(14*b^11*(a + b*x^2)^7) + (10*a^7)/(b^11*(a + b*x^2)^6) - (21*a^6)/(b^11*(a + b*x^2)^5) + (63*a^5)/(2*b^11*(a + b*x^2)^4) - (35*a^4)/(b^11*(a + b*x^2)^3) + (30*a^3)/(b^11*(a + b*x^2)^2) - (45*a^2)/(2*b^11*(a + b*x^2)) - (5*a*log(a + b*x^2))/b^11]
    @test_int [x^19/(a + b*x^2)^10, x, 3, a^9/(18*b^10*(a + b*x^2)^9) - (9*a^8)/(16*b^10*(a + b*x^2)^8) + (18*a^7)/(7*b^10*(a + b*x^2)^7) - (7*a^6)/(b^10*(a + b*x^2)^6) + (63*a^5)/(5*b^10*(a + b*x^2)^5) - (63*a^4)/(4*b^10*(a + b*x^2)^4) + (14*a^3)/(b^10*(a + b*x^2)^3) - (9*a^2)/(b^10*(a + b*x^2)^2) + (9*a)/(2*b^10*(a + b*x^2)) + log(a + b*x^2)/(2*b^10)]
    @test_int [x^17/(a + b*x^2)^10, x, 1, x^18/(18*a*(a + b*x^2)^9)]
    @test_int [x^15/(a + b*x^2)^10, x, 3, x^16/(18*a*(a + b*x^2)^9) + x^16/(144*a^2*(a + b*x^2)^8)]
    @test_int [x^13/(a + b*x^2)^10, x, 4, x^14/(18*a*(a + b*x^2)^9) + x^14/(72*a^2*(a + b*x^2)^8) + x^14/(504*a^3*(a + b*x^2)^7)]
    @test_int [x^11/(a + b*x^2)^10, x, 5, x^12/(18*a*(a + b*x^2)^9) + x^12/(48*a^2*(a + b*x^2)^8) + x^12/(168*a^3*(a + b*x^2)^7) + x^12/(1008*a^4*(a + b*x^2)^6)]
    @test_int [x^9/(a + b*x^2)^10, x, 3, -(a^4/(18*b^5*(a + b*x^2)^9)) + a^3/(4*b^5*(a + b*x^2)^8) - (3*a^2)/(7*b^5*(a + b*x^2)^7) + a/(3*b^5*(a + b*x^2)^6) - 1/(10*b^5*(a + b*x^2)^5)]
    @test_int [x^7/(a + b*x^2)^10, x, 3, a^3/(18*b^4*(a + b*x^2)^9) - (3*a^2)/(16*b^4*(a + b*x^2)^8) + (3*a)/(14*b^4*(a + b*x^2)^7) - 1/(12*b^4*(a + b*x^2)^6)]
    @test_int [x^5/(a + b*x^2)^10, x, 3, -(a^2/(18*b^3*(a + b*x^2)^9)) + a/(8*b^3*(a + b*x^2)^8) - 1/(14*b^3*(a + b*x^2)^7)]
    @test_int [x^3/(a + b*x^2)^10, x, 3, a/(18*b^2*(a + b*x^2)^9) - 1/(16*b^2*(a + b*x^2)^8)]
    @test_int [x^1/(a + b*x^2)^10, x, 1, -(1/(18*b*(a + b*x^2)^9))]
    @test_int [1/(x^1*(a + b*x^2)^10), x, 3, 1/(18*a*(a + b*x^2)^9) + 1/(16*a^2*(a + b*x^2)^8) + 1/(14*a^3*(a + b*x^2)^7) + 1/(12*a^4*(a + b*x^2)^6) + 1/(10*a^5*(a + b*x^2)^5) + 1/(8*a^6*(a + b*x^2)^4) + 1/(6*a^7*(a + b*x^2)^3) + 1/(4*a^8*(a + b*x^2)^2) + 1/(2*a^9*(a + b*x^2)) + log(x)/a^10 - log(a + b*x^2)/(2*a^10)]
    @test_int [1/(x^3*(a + b*x^2)^10), x, 3, -(1/(2*a^10*x^2)) - b/(18*a^2*(a + b*x^2)^9) - b/(8*a^3*(a + b*x^2)^8) - (3*b)/(14*a^4*(a + b*x^2)^7) - b/(3*a^5*(a + b*x^2)^6) - b/(2*a^6*(a + b*x^2)^5) - (3*b)/(4*a^7*(a + b*x^2)^4) - (7*b)/(6*a^8*(a + b*x^2)^3) - (2*b)/(a^9*(a + b*x^2)^2) - (9*b)/(2*a^10*(a + b*x^2)) - (10*b*log(x))/a^11 + (5*b*log(a + b*x^2))/a^11]
    @test_int [1/(x^5*(a + b*x^2)^10), x, 3, -(1/(4*a^10*x^4)) + (5*b)/(a^11*x^2) + b^2/(18*a^3*(a + b*x^2)^9) + (3*b^2)/(16*a^4*(a + b*x^2)^8) + (3*b^2)/(7*a^5*(a + b*x^2)^7) + (5*b^2)/(6*a^6*(a + b*x^2)^6) + (3*b^2)/(2*a^7*(a + b*x^2)^5) + (21*b^2)/(8*a^8*(a + b*x^2)^4) + (14*b^2)/(3*a^9*(a + b*x^2)^3) + (9*b^2)/(a^10*(a + b*x^2)^2) + (45*b^2)/(2*a^11*(a + b*x^2)) + (55*b^2*log(x))/a^12 - (55*b^2*log(a + b*x^2))/(2*a^12)]
    @test_int [1/(x^7*(a + b*x^2)^10), x, 3, -(1/(6*a^10*x^6)) + (5*b)/(2*a^11*x^4) - (55*b^2)/(2*a^12*x^2) - b^3/(18*a^4*(a + b*x^2)^9) - b^3/(4*a^5*(a + b*x^2)^8) - (5*b^3)/(7*a^6*(a + b*x^2)^7) - (5*b^3)/(3*a^7*(a + b*x^2)^6) - (7*b^3)/(2*a^8*(a + b*x^2)^5) - (7*b^3)/(a^9*(a + b*x^2)^4) - (14*b^3)/(a^10*(a + b*x^2)^3) - (30*b^3)/(a^11*(a + b*x^2)^2) - (165*b^3)/(2*a^12*(a + b*x^2)) - (220*b^3*log(x))/a^13 + (110*b^3*log(a + b*x^2))/a^13]

    @test_int [x^24/(a + b*x^2)^10, x, 12, (7436429*a^2*x)/(65536*b^12) - (7436429*a*x^3)/(196608*b^11) + (7436429*x^5)/(327680*b^10) - x^23/(18*b*(a + b*x^2)^9) - (23*x^21)/(288*b^2*(a + b*x^2)^8) - (23*x^19)/(192*b^3*(a + b*x^2)^7) - (437*x^17)/(2304*b^4*(a + b*x^2)^6) - (7429*x^15)/(23040*b^5*(a + b*x^2)^5) - (7429*x^13)/(12288*b^6*(a + b*x^2)^4) - (96577*x^11)/(73728*b^7*(a + b*x^2)^3) - (1062347*x^9)/(294912*b^8*(a + b*x^2)^2) - (1062347*x^7)/(65536*b^9*(a + b*x^2)) - (7436429*a^(5/2)*atan((sqrt(b)*x)/sqrt(a)))/(65536*b^(25/2))]
    @test_int [x^22/(a + b*x^2)^10, x, 12, -((1616615*a*x)/(65536*b^11)) + (1616615*x^3)/(196608*b^10) - x^21/(18*b*(a + b*x^2)^9) - (7*x^19)/(96*b^2*(a + b*x^2)^8) - (19*x^17)/(192*b^3*(a + b*x^2)^7) - (323*x^15)/(2304*b^4*(a + b*x^2)^6) - (323*x^13)/(1536*b^5*(a + b*x^2)^5) - (4199*x^11)/(12288*b^6*(a + b*x^2)^4) - (46189*x^9)/(73728*b^7*(a + b*x^2)^3) - (46189*x^7)/(32768*b^8*(a + b*x^2)^2) - (323323*x^5)/(65536*b^9*(a + b*x^2)) + (1616615*a^(3/2)*atan((sqrt(b)*x)/sqrt(a)))/(65536*b^(23/2))]
    @test_int [x^20/(a + b*x^2)^10, x, 11, (230945*x)/(65536*b^10) - x^19/(18*b*(a + b*x^2)^9) - (19*x^17)/(288*b^2*(a + b*x^2)^8) - (323*x^15)/(4032*b^3*(a + b*x^2)^7) - (1615*x^13)/(16128*b^4*(a + b*x^2)^6) - (4199*x^11)/(32256*b^5*(a + b*x^2)^5) - (46189*x^9)/(258048*b^6*(a + b*x^2)^4) - (46189*x^7)/(172032*b^7*(a + b*x^2)^3) - (46189*x^5)/(98304*b^8*(a + b*x^2)^2) - (230945*x^3)/(196608*b^9*(a + b*x^2)) - (230945*sqrt(a)*atan((sqrt(b)*x)/sqrt(a)))/(65536*b^(21/2))]
    @test_int [x^18/(a + b*x^2)^10, x, 10, -(x^17/(18*b*(a + b*x^2)^9)) - (17*x^15)/(288*b^2*(a + b*x^2)^8) - (85*x^13)/(1344*b^3*(a + b*x^2)^7) - (1105*x^11)/(16128*b^4*(a + b*x^2)^6) - (2431*x^9)/(32256*b^5*(a + b*x^2)^5) - (2431*x^7)/(28672*b^6*(a + b*x^2)^4) - (2431*x^5)/(24576*b^7*(a + b*x^2)^3) - (12155*x^3)/(98304*b^8*(a + b*x^2)^2) - (12155*x)/(65536*b^9*(a + b*x^2)) + (12155*atan((sqrt(b)*x)/sqrt(a)))/(65536*sqrt(a)*b^(19/2))]
    @test_int [x^16/(a + b*x^2)^10, x, 10, -(x^15/(18*b*(a + b*x^2)^9)) - (5*x^13)/(96*b^2*(a + b*x^2)^8) - (65*x^11)/(1344*b^3*(a + b*x^2)^7) - (715*x^9)/(16128*b^4*(a + b*x^2)^6) - (143*x^7)/(3584*b^5*(a + b*x^2)^5) - (143*x^5)/(4096*b^6*(a + b*x^2)^4) - (715*x^3)/(24576*b^7*(a + b*x^2)^3) - (715*x)/(32768*b^8*(a + b*x^2)^2) + (715*x)/(65536*a*b^8*(a + b*x^2)) + (715*atan((sqrt(b)*x)/sqrt(a)))/(65536*a^(3/2)*b^(17/2))]
    @test_int [x^14/(a + b*x^2)^10, x, 10, -(x^13/(18*b*(a + b*x^2)^9)) - (13*x^11)/(288*b^2*(a + b*x^2)^8) - (143*x^9)/(4032*b^3*(a + b*x^2)^7) - (143*x^7)/(5376*b^4*(a + b*x^2)^6) - (143*x^5)/(7680*b^5*(a + b*x^2)^5) - (143*x^3)/(12288*b^6*(a + b*x^2)^4) - (143*x)/(24576*b^7*(a + b*x^2)^3) + (143*x)/(98304*a*b^7*(a + b*x^2)^2) + (143*x)/(65536*a^2*b^7*(a + b*x^2)) + (143*atan((sqrt(b)*x)/sqrt(a)))/(65536*a^(5/2)*b^(15/2))]
    @test_int [x^12/(a + b*x^2)^10, x, 10, -(x^11/(18*b*(a + b*x^2)^9)) - (11*x^9)/(288*b^2*(a + b*x^2)^8) - (11*x^7)/(448*b^3*(a + b*x^2)^7) - (11*x^5)/(768*b^4*(a + b*x^2)^6) - (11*x^3)/(1536*b^5*(a + b*x^2)^5) - (11*x)/(4096*b^6*(a + b*x^2)^4) + (11*x)/(24576*a*b^6*(a + b*x^2)^3) + (55*x)/(98304*a^2*b^6*(a + b*x^2)^2) + (55*x)/(65536*a^3*b^6*(a + b*x^2)) + (55*atan((sqrt(b)*x)/sqrt(a)))/(65536*a^(7/2)*b^(13/2))]
    @test_int [x^10/(a + b*x^2)^10, x, 10, -(x^9/(18*b*(a + b*x^2)^9)) - x^7/(32*b^2*(a + b*x^2)^8) - x^5/(64*b^3*(a + b*x^2)^7) - (5*x^3)/(768*b^4*(a + b*x^2)^6) - x/(512*b^5*(a + b*x^2)^5) + x/(4096*a*b^5*(a + b*x^2)^4) + (7*x)/(24576*a^2*b^5*(a + b*x^2)^3) + (35*x)/(98304*a^3*b^5*(a + b*x^2)^2) + (35*x)/(65536*a^4*b^5*(a + b*x^2)) + (35*atan((sqrt(b)*x)/sqrt(a)))/(65536*a^(9/2)*b^(11/2))]
    @test_int [x^8/(a + b*x^2)^10, x, 10, -(x^7/(18*b*(a + b*x^2)^9)) - (7*x^5)/(288*b^2*(a + b*x^2)^8) - (5*x^3)/(576*b^3*(a + b*x^2)^7) - (5*x)/(2304*b^4*(a + b*x^2)^6) + x/(4608*a*b^4*(a + b*x^2)^5) + x/(4096*a^2*b^4*(a + b*x^2)^4) + (7*x)/(24576*a^3*b^4*(a + b*x^2)^3) + (35*x)/(98304*a^4*b^4*(a + b*x^2)^2) + (35*x)/(65536*a^5*b^4*(a + b*x^2)) + (35*atan((sqrt(b)*x)/sqrt(a)))/(65536*a^(11/2)*b^(9/2))]
    @test_int [x^6/(a + b*x^2)^10, x, 10, -(x^5/(18*b*(a + b*x^2)^9)) - (5*x^3)/(288*b^2*(a + b*x^2)^8) - (5*x)/(1344*b^3*(a + b*x^2)^7) + (5*x)/(16128*a*b^3*(a + b*x^2)^6) + (11*x)/(32256*a^2*b^3*(a + b*x^2)^5) + (11*x)/(28672*a^3*b^3*(a + b*x^2)^4) + (11*x)/(24576*a^4*b^3*(a + b*x^2)^3) + (55*x)/(98304*a^5*b^3*(a + b*x^2)^2) + (55*x)/(65536*a^6*b^3*(a + b*x^2)) + (55*atan((sqrt(b)*x)/sqrt(a)))/(65536*a^(13/2)*b^(7/2))]
    @test_int [x^4/(a + b*x^2)^10, x, 10, -(x^3/(18*b*(a + b*x^2)^9)) - x/(96*b^2*(a + b*x^2)^8) + x/(1344*a*b^2*(a + b*x^2)^7) + (13*x)/(16128*a^2*b^2*(a + b*x^2)^6) + (143*x)/(161280*a^3*b^2*(a + b*x^2)^5) + (143*x)/(143360*a^4*b^2*(a + b*x^2)^4) + (143*x)/(122880*a^5*b^2*(a + b*x^2)^3) + (143*x)/(98304*a^6*b^2*(a + b*x^2)^2) + (143*x)/(65536*a^7*b^2*(a + b*x^2)) + (143*atan((sqrt(b)*x)/sqrt(a)))/(65536*a^(15/2)*b^(5/2))]
    @test_int [x^2/(a + b*x^2)^10, x, 10, -(x/(18*b*(a + b*x^2)^9)) + x/(288*a*b*(a + b*x^2)^8) + (5*x)/(1344*a^2*b*(a + b*x^2)^7) + (65*x)/(16128*a^3*b*(a + b*x^2)^6) + (143*x)/(32256*a^4*b*(a + b*x^2)^5) + (143*x)/(28672*a^5*b*(a + b*x^2)^4) + (143*x)/(24576*a^6*b*(a + b*x^2)^3) + (715*x)/(98304*a^7*b*(a + b*x^2)^2) + (715*x)/(65536*a^8*b*(a + b*x^2)) + (715*atan((sqrt(b)*x)/sqrt(a)))/(65536*a^(17/2)*b^(3/2))]
    @test_int [x^0/(a + b*x^2)^10, x, 10, x/(18*a*(a + b*x^2)^9) + (17*x)/(288*a^2*(a + b*x^2)^8) + (85*x)/(1344*a^3*(a + b*x^2)^7) + (1105*x)/(16128*a^4*(a + b*x^2)^6) + (2431*x)/(32256*a^5*(a + b*x^2)^5) + (2431*x)/(28672*a^6*(a + b*x^2)^4) + (2431*x)/(24576*a^7*(a + b*x^2)^3) + (12155*x)/(98304*a^8*(a + b*x^2)^2) + (12155*x)/(65536*a^9*(a + b*x^2)) + (12155*atan((sqrt(b)*x)/sqrt(a)))/(65536*a^(19/2)*sqrt(b))]
    @test_int [1/(x^2*(a + b*x^2)^10), x, 11, -(230945/(65536*a^10*x)) + 1/(18*a*x*(a + b*x^2)^9) + 19/(288*a^2*x*(a + b*x^2)^8) + 323/(4032*a^3*x*(a + b*x^2)^7) + 1615/(16128*a^4*x*(a + b*x^2)^6) + 4199/(32256*a^5*x*(a + b*x^2)^5) + 46189/(258048*a^6*x*(a + b*x^2)^4) + 46189/(172032*a^7*x*(a + b*x^2)^3) + 46189/(98304*a^8*x*(a + b*x^2)^2) + 230945/(196608*a^9*x*(a + b*x^2)) - (230945*sqrt(b)*atan((sqrt(b)*x)/sqrt(a)))/(65536*a^(21/2))]
    @test_int [1/(x^4*(a + b*x^2)^10), x, 12, -(1616615/(196608*a^10*x^3)) + (1616615*b)/(65536*a^11*x) + 1/(18*a*x^3*(a + b*x^2)^9) + 7/(96*a^2*x^3*(a + b*x^2)^8) + 19/(192*a^3*x^3*(a + b*x^2)^7) + 323/(2304*a^4*x^3*(a + b*x^2)^6) + 323/(1536*a^5*x^3*(a + b*x^2)^5) + 4199/(12288*a^6*x^3*(a + b*x^2)^4) + 46189/(73728*a^7*x^3*(a + b*x^2)^3) + 46189/(32768*a^8*x^3*(a + b*x^2)^2) + 323323/(65536*a^9*x^3*(a + b*x^2)) + (1616615*b^(3/2)*atan((sqrt(b)*x)/sqrt(a)))/(65536*a^(23/2))]
    @test_int [1/(x^6*(a + b*x^2)^10), x, 13, -(7436429/(327680*a^10*x^5)) + (7436429*b)/(196608*a^11*x^3) - (7436429*b^2)/(65536*a^12*x) + 1/(18*a*x^5*(a + b*x^2)^9) + 23/(288*a^2*x^5*(a + b*x^2)^8) + 23/(192*a^3*x^5*(a + b*x^2)^7) + 437/(2304*a^4*x^5*(a + b*x^2)^6) + 7429/(23040*a^5*x^5*(a + b*x^2)^5) + 7429/(12288*a^6*x^5*(a + b*x^2)^4) + 96577/(73728*a^7*x^5*(a + b*x^2)^3) + 1062347/(294912*a^8*x^5*(a + b*x^2)^2) + 1062347/(65536*a^9*x^5*(a + b*x^2)) - (7436429*b^(5/2)*atan((sqrt(b)*x)/sqrt(a)))/(65536*a^(25/2))]

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

    @test_int [x^3/(a - b*x^2)^2, x, 3, a/(2*b^2*(a - b*x^2)) + log(a - b*x^2)/(2*b^2)]
    @test_int [x^2/(a - b*x^2)^2, x, 2, x/(2*b*(a - b*x^2)) - atanh((sqrt(b)*x)/sqrt(a))/(2*sqrt(a)*b^(3/2))]
    @test_int [x/(a - b*x^2)^2, x, 1, 1/(2*b*(a - b*x^2))]
    @test_int [1/(a - b*x^2)^2, x, 2, x/(2*a*(a - b*x^2)) + atanh((sqrt(b)*x)/sqrt(a))/(2*a^(3/2)*sqrt(b))]
    @test_int [1/(x*(a - b*x^2)^2), x, 3, 1/(2*a*(a - b*x^2)) + log(x)/a^2 - log(a - b*x^2)/(2*a^2)]
    @test_int [1/(x^2*(a - b*x^2)^2), x, 3, -(3/(2*a^2*x)) + 1/(2*a*x*(a - b*x^2)) + (3*sqrt(b)*atanh((sqrt(b)*x)/sqrt(a)))/(2*a^(5/2))]
    @test_int [1/(x^3*(a - b*x^2)^2), x, 3, -(1/(2*a^2*x^2)) + b/(2*a^2*(a - b*x^2)) + (2*b*log(x))/a^3 - (b*log(a - b*x^2))/a^3]

    @test_int [x^3/(a - b*x^2)^3, x, 1, x^4/(4*a*(a - b*x^2)^2)]
    @test_int [x^2/(a - b*x^2)^3, x, 3, x/(4*b*(a - b*x^2)^2) - x/(8*a*b*(a - b*x^2)) - atanh((sqrt(b)*x)/sqrt(a))/(8*a^(3/2)*b^(3/2))]
    @test_int [x/(a - b*x^2)^3, x, 1, 1/(4*b*(a - b*x^2)^2)]
    @test_int [1/(a - b*x^2)^3, x, 3, x/(4*a*(a - b*x^2)^2) + (3*x)/(8*a^2*(a - b*x^2)) + (3*atanh((sqrt(b)*x)/sqrt(a)))/(8*a^(5/2)*sqrt(b))]
    @test_int [1/(x*(a - b*x^2)^3), x, 3, 1/(4*a*(a - b*x^2)^2) + 1/(2*a^2*(a - b*x^2)) + log(x)/a^3 - log(a - b*x^2)/(2*a^3)]
    @test_int [1/(x^2*(a - b*x^2)^3), x, 4, -(15/(8*a^3*x)) + 1/(4*a*x*(a - b*x^2)^2) + 5/(8*a^2*x*(a - b*x^2)) + (15*sqrt(b)*atanh((sqrt(b)*x)/sqrt(a)))/(8*a^(7/2))]
    @test_int [1/(x^3*(a - b*x^2)^3), x, 3, -(1/(2*a^3*x^2)) + b/(4*a^2*(a - b*x^2)^2) + b/(a^3*(a - b*x^2)) + (3*b*log(x))/a^4 - (3*b*log(a - b*x^2))/(2*a^4)]

    @test_int [x^3/(a - b*x^2)^5, x, 3, a/(8*b^2*(a - b*x^2)^4) - 1/(6*b^2*(a - b*x^2)^3)]
    @test_int [x^2/(a - b*x^2)^5, x, 5, x/(8*b*(a - b*x^2)^4) - x/(48*a*b*(a - b*x^2)^3) - (5*x)/(192*a^2*b*(a - b*x^2)^2) - (5*x)/(128*a^3*b*(a - b*x^2)) - (5*atanh((sqrt(b)*x)/sqrt(a)))/(128*a^(7/2)*b^(3/2))]
    @test_int [x/(a - b*x^2)^5, x, 1, 1/(8*b*(a - b*x^2)^4)]
    @test_int [1/(a - b*x^2)^5, x, 5, x/(8*a*(a - b*x^2)^4) + (7*x)/(48*a^2*(a - b*x^2)^3) + (35*x)/(192*a^3*(a - b*x^2)^2) + (35*x)/(128*a^4*(a - b*x^2)) + (35*atanh((sqrt(b)*x)/sqrt(a)))/(128*a^(9/2)*sqrt(b))]
    @test_int [1/(x*(a - b*x^2)^5), x, 3, 1/(8*a*(a - b*x^2)^4) + 1/(6*a^2*(a - b*x^2)^3) + 1/(4*a^3*(a - b*x^2)^2) + 1/(2*a^4*(a - b*x^2)) + log(x)/a^5 - log(a - b*x^2)/(2*a^5)]
    @test_int [1/(x^2*(a - b*x^2)^5), x, 6, -(315/(128*a^5*x)) + 1/(8*a*x*(a - b*x^2)^4) + 3/(16*a^2*x*(a - b*x^2)^3) + 21/(64*a^3*x*(a - b*x^2)^2) + 105/(128*a^4*x*(a - b*x^2)) + (315*sqrt(b)*atanh((sqrt(b)*x)/sqrt(a)))/(128*a^(11/2))]
    @test_int [1/(x^3*(a - b*x^2)^5), x, 3, -(1/(2*a^5*x^2)) + b/(8*a^2*(a - b*x^2)^4) + b/(3*a^3*(a - b*x^2)^3) + (3*b)/(4*a^4*(a - b*x^2)^2) + (2*b)/(a^5*(a - b*x^2)) + (5*b*log(x))/a^6 - (5*b*log(a - b*x^2))/(2*a^6)]

    @test_int [1/(x*(1 + b*x^2)), x, 4, log(x) - (1/2)*log(1 + b*x^2)]
    @test_int [1/(x*(-1 + b*x^2)), x, 4, -log(x) + (1/2)*log(1 - b*x^2)]
    @test_int [1/(x^3*(1 + b*x^2)), x, 3, -(1/(2*x^2)) - b*log(x) + (1/2)*b*log(1 + b*x^2)]
    @test_int [1/(x^3*(-1 + b*x^2)), x, 3, 1/(2*x^2) - b*log(x) + (1/2)*b*log(1 - b*x^2)]

    #= Formerly*failed*because*both*PosQ((1-a)/a)*and*PosQ(-(1-a)/a)*returned*False. =#
    @test_int [1/(-1 + a + a*x^2), x, 1, -(atanh((sqrt(a)*x)/sqrt(1 - a))/sqrt((1 - a)*a))]

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

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

    #= Checks*to*ensure*PosQ(a/(b-a*c))*returns*True. =#
    @test_int [1/(a + (b - a*c)*x^2), x, 1, atan((sqrt(b - a*c)*x)/sqrt(a))/(sqrt(a)*sqrt(b - a*c))]
    @test_int [1/(a - (b - a*c)*x^2), x, 1, atanh((sqrt(b - a*c)*x)/sqrt(a))/(sqrt(a)*sqrt(b - a*c))]

    #= Checks*to*ensure*Rt(-c*(a-d)/(-b+c),2)*returns*sqrt(c)*sqrt(a-d)/sqrt(b-c). =#
    @test_int [1/(c*(a - d) - (b - c)*x^2), x, 1, atanh((sqrt(b - c)*x)/(sqrt(c)*sqrt(a - d)))/(sqrt(c)*sqrt(b - c)*sqrt(a - d))]

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

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

    @test_int [x^(7/2)*(a + b*x^2), x, 2, (2*a*x^(9/2))/9 + (2*b*x^(13/2))/13]
    @test_int [x^(5/2)*(a + b*x^2), x, 2, (2*a*x^(7/2))/7 + (2*b*x^(11/2))/11]
    @test_int [x^(3/2)*(a + b*x^2), x, 2, (2*a*x^(5/2))/5 + (2*b*x^(9/2))/9]
    @test_int [sqrt(x)*(a + b*x^2), x, 2, (2*a*x^(3/2))/3 + (2*b*x^(7/2))/7]
    @test_int [(a + b*x^2)/sqrt(x), x, 2, 2*a*sqrt(x) + (2*b*x^(5/2))/5]
    @test_int [(a + b*x^2)/x^(3/2), x, 2, (-2*a)/sqrt(x) + (2*b*x^(3/2))/3]
    @test_int [(a + b*x^2)/x^(5/2), x, 2, (-2*a)/(3*x^(3/2)) + 2*b*sqrt(x)]
    @test_int [(a + b*x^2)/x^(7/2), x, 2, (-2*a)/(5*x^(5/2)) - (2*b)/sqrt(x)]

    @test_int [x^(7/2)*(a + b*x^2)^2, x, 2, (2*a^2*x^(9/2))/9 + (4*a*b*x^(13/2))/13 + (2*b^2*x^(17/2))/17]
    @test_int [x^(5/2)*(a + b*x^2)^2, x, 2, (2*a^2*x^(7/2))/7 + (4*a*b*x^(11/2))/11 + (2*b^2*x^(15/2))/15]
    @test_int [x^(3/2)*(a + b*x^2)^2, x, 2, (2*a^2*x^(5/2))/5 + (4*a*b*x^(9/2))/9 + (2*b^2*x^(13/2))/13]
    @test_int [sqrt(x)*(a + b*x^2)^2, x, 2, (2*a^2*x^(3/2))/3 + (4*a*b*x^(7/2))/7 + (2*b^2*x^(11/2))/11]
    @test_int [(a + b*x^2)^2/sqrt(x), x, 2, 2*a^2*sqrt(x) + (4*a*b*x^(5/2))/5 + (2*b^2*x^(9/2))/9]
    @test_int [(a + b*x^2)^2/x^(3/2), x, 2, (-2*a^2)/sqrt(x) + (4*a*b*x^(3/2))/3 + (2*b^2*x^(7/2))/7]
    @test_int [(a + b*x^2)^2/x^(5/2), x, 2, (-2*a^2)/(3*x^(3/2)) + 4*a*b*sqrt(x) + (2*b^2*x^(5/2))/5]
    @test_int [(a + b*x^2)^2/x^(7/2), x, 2, (-2*a^2)/(5*x^(5/2)) - (4*a*b)/sqrt(x) + (2*b^2*x^(3/2))/3]

    @test_int [x^(7/2)*(a + b*x^2)^3, x, 2, (2*a^3*x^(9/2))/9 + (6*a^2*b*x^(13/2))/13 + (6*a*b^2*x^(17/2))/17 + (2*b^3*x^(21/2))/21]
    @test_int [x^(5/2)*(a + b*x^2)^3, x, 2, (2*a^3*x^(7/2))/7 + (6*a^2*b*x^(11/2))/11 + (2*a*b^2*x^(15/2))/5 + (2*b^3*x^(19/2))/19]
    @test_int [x^(3/2)*(a + b*x^2)^3, x, 2, (2*a^3*x^(5/2))/5 + (2*a^2*b*x^(9/2))/3 + (6*a*b^2*x^(13/2))/13 + (2*b^3*x^(17/2))/17]
    @test_int [sqrt(x)*(a + b*x^2)^3, x, 2, (2*a^3*x^(3/2))/3 + (6*a^2*b*x^(7/2))/7 + (6*a*b^2*x^(11/2))/11 + (2*b^3*x^(15/2))/15]
    @test_int [(a + b*x^2)^3/sqrt(x), x, 2, 2*a^3*sqrt(x) + (6*a^2*b*x^(5/2))/5 + (2*a*b^2*x^(9/2))/3 + (2*b^3*x^(13/2))/13]
    @test_int [(a + b*x^2)^3/x^(3/2), x, 2, (-2*a^3)/sqrt(x) + 2*a^2*b*x^(3/2) + (6*a*b^2*x^(7/2))/7 + (2*b^3*x^(11/2))/11]
    @test_int [(a + b*x^2)^3/x^(5/2), x, 2, (-2*a^3)/(3*x^(3/2)) + 6*a^2*b*sqrt(x) + (6*a*b^2*x^(5/2))/5 + (2*b^3*x^(9/2))/9]
    @test_int [(a + b*x^2)^3/x^(7/2), x, 2, (-2*a^3)/(5*x^(5/2)) - (6*a^2*b)/sqrt(x) + 2*a*b^2*x^(3/2) + (2*b^3*x^(7/2))/7]

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

    @test_int [x^(7/2)/(a + b*x^2), x, 12, -((2*a*sqrt(x))/b^2) + (2*x^(5/2))/(5*b) - (a^(5/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(9/4)) + (a^(5/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(9/4)) - (a^(5/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(9/4)) + (a^(5/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(9/4))]
    @test_int [x^(5/2)/(a + b*x^2), x, 11, (2*x^(3/2))/(3*b) + (a^(3/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(7/4)) - (a^(3/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(7/4)) - (a^(3/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(7/4)) + (a^(3/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(7/4))]
    @test_int [x^(3/2)/(a + b*x^2), x, 11, (2*sqrt(x))/b + (a^(1/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(5/4)) - (a^(1/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*b^(5/4)) + (a^(1/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(5/4)) - (a^(1/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*b^(5/4))]
    @test_int [sqrt(x)/(a + b*x^2), x, 10, -(atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4))/(sqrt(2)*a^(1/4)*b^(3/4))) + atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4))/(sqrt(2)*a^(1/4)*b^(3/4)) + log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x)/(2*sqrt(2)*a^(1/4)*b^(3/4)) - log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x)/(2*sqrt(2)*a^(1/4)*b^(3/4))]
    @test_int [1/(sqrt(x)*(a + b*x^2)), x, 10, -(atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4))/(sqrt(2)*a^(3/4)*b^(1/4))) + atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4))/(sqrt(2)*a^(3/4)*b^(1/4)) - log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x)/(2*sqrt(2)*a^(3/4)*b^(1/4)) + log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x)/(2*sqrt(2)*a^(3/4)*b^(1/4))]
    @test_int [1/(x^(3/2)*(a + b*x^2)), x, 11, -(2/(a*sqrt(x))) + (b^(1/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(5/4)) - (b^(1/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(5/4)) - (b^(1/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(5/4)) + (b^(1/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(5/4))]
    @test_int [1/(x^(5/2)*(a + b*x^2)), x, 11, -(2/(3*a*x^(3/2))) + (b^(3/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(7/4)) - (b^(3/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(7/4)) + (b^(3/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(7/4)) - (b^(3/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(7/4))]
    @test_int [1/(x^(7/2)*(a + b*x^2)), x, 12, -(2/(5*a*x^(5/2))) + (2*b)/(a^2*sqrt(x)) - (b^(5/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(9/4)) + (b^(5/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(sqrt(2)*a^(9/4)) + (b^(5/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(9/4)) - (b^(5/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(2*sqrt(2)*a^(9/4))]

    @test_int [x^(7/2)/(a + b*x^2)^2, x, 12, (5*sqrt(x))/(2*b^2) - x^(5/2)/(2*b*(a + b*x^2)) + (5*a^(1/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*b^(9/4)) - (5*a^(1/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*b^(9/4)) + (5*a^(1/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*b^(9/4)) - (5*a^(1/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*b^(9/4))]
    @test_int [x^(5/2)/(a + b*x^2)^2, x, 11, -(x^(3/2)/(2*b*(a + b*x^2))) - (3*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(1/4)*b^(7/4)) + (3*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(1/4)*b^(7/4)) + (3*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(1/4)*b^(7/4)) - (3*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(1/4)*b^(7/4))]
    @test_int [x^(3/2)/(a + b*x^2)^2, x, 11, -(sqrt(x)/(2*b*(a + b*x^2))) - atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4))/(4*sqrt(2)*a^(3/4)*b^(5/4)) + atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4))/(4*sqrt(2)*a^(3/4)*b^(5/4)) - log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x)/(8*sqrt(2)*a^(3/4)*b^(5/4)) + log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x)/(8*sqrt(2)*a^(3/4)*b^(5/4))]
    @test_int [sqrt(x)/(a + b*x^2)^2, x, 11, x^(3/2)/(2*a*(a + b*x^2)) - atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4))/(4*sqrt(2)*a^(5/4)*b^(3/4)) + atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4))/(4*sqrt(2)*a^(5/4)*b^(3/4)) + log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x)/(8*sqrt(2)*a^(5/4)*b^(3/4)) - log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x)/(8*sqrt(2)*a^(5/4)*b^(3/4))]
    @test_int [1/(sqrt(x)*(a + b*x^2)^2), x, 11, sqrt(x)/(2*a*(a + b*x^2)) - (3*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(7/4)*b^(1/4)) + (3*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(7/4)*b^(1/4)) - (3*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(7/4)*b^(1/4)) + (3*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(7/4)*b^(1/4))]
    @test_int [1/(x^(3/2)*(a + b*x^2)^2), x, 12, -(5/(2*a^2*sqrt(x))) + 1/(2*a*sqrt(x)*(a + b*x^2)) + (5*b^(1/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(9/4)) - (5*b^(1/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(9/4)) - (5*b^(1/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(9/4)) + (5*b^(1/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(9/4))]
    @test_int [1/(x^(5/2)*(a + b*x^2)^2), x, 12, -(7/(6*a^2*x^(3/2))) + 1/(2*a*x^(3/2)*(a + b*x^2)) + (7*b^(3/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(11/4)) - (7*b^(3/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(11/4)) + (7*b^(3/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(11/4)) - (7*b^(3/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(11/4))]
    @test_int [1/(x^(7/2)*(a + b*x^2)^2), x, 13, -(9/(10*a^2*x^(5/2))) + (9*b)/(2*a^3*sqrt(x)) + 1/(2*a*x^(5/2)*(a + b*x^2)) - (9*b^(5/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(13/4)) + (9*b^(5/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(4*sqrt(2)*a^(13/4)) + (9*b^(5/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(13/4)) - (9*b^(5/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(8*sqrt(2)*a^(13/4))]

    @test_int [x^(7/2)/(a + b*x^2)^3, x, 12, -(x^(5/2)/(4*b*(a + b*x^2)^2)) - (5*sqrt(x))/(16*b^2*(a + b*x^2)) - (5*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(3/4)*b^(9/4)) + (5*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(3/4)*b^(9/4)) - (5*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(3/4)*b^(9/4)) + (5*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(3/4)*b^(9/4))]
    @test_int [x^(5/2)/(a + b*x^2)^3, x, 12, -(x^(3/2)/(4*b*(a + b*x^2)^2)) + (3*x^(3/2))/(16*a*b*(a + b*x^2)) - (3*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(5/4)*b^(7/4)) + (3*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(5/4)*b^(7/4)) + (3*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(5/4)*b^(7/4)) - (3*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(5/4)*b^(7/4))]
    @test_int [x^(3/2)/(a + b*x^2)^3, x, 12, -(sqrt(x)/(4*b*(a + b*x^2)^2)) + sqrt(x)/(16*a*b*(a + b*x^2)) - (3*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(7/4)*b^(5/4)) + (3*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(7/4)*b^(5/4)) - (3*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(7/4)*b^(5/4)) + (3*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(7/4)*b^(5/4))]
    @test_int [sqrt(x)/(a + b*x^2)^3, x, 12, x^(3/2)/(4*a*(a + b*x^2)^2) + (5*x^(3/2))/(16*a^2*(a + b*x^2)) - (5*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(9/4)*b^(3/4)) + (5*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(9/4)*b^(3/4)) + (5*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(9/4)*b^(3/4)) - (5*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(9/4)*b^(3/4))]
    @test_int [1/(sqrt(x)*(a + b*x^2)^3), x, 12, sqrt(x)/(4*a*(a + b*x^2)^2) + (7*sqrt(x))/(16*a^2*(a + b*x^2)) - (21*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(11/4)*b^(1/4)) + (21*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(11/4)*b^(1/4)) - (21*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(11/4)*b^(1/4)) + (21*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(11/4)*b^(1/4))]
    @test_int [1/(x^(3/2)*(a + b*x^2)^3), x, 13, -(45/(16*a^3*sqrt(x))) + 1/(4*a*sqrt(x)*(a + b*x^2)^2) + 9/(16*a^2*sqrt(x)*(a + b*x^2)) + (45*b^(1/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(13/4)) - (45*b^(1/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(13/4)) - (45*b^(1/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(13/4)) + (45*b^(1/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(13/4))]
    @test_int [1/(x^(5/2)*(a + b*x^2)^3), x, 13, -(77/(48*a^3*x^(3/2))) + 1/(4*a*x^(3/2)*(a + b*x^2)^2) + 11/(16*a^2*x^(3/2)*(a + b*x^2)) + (77*b^(3/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(15/4)) - (77*b^(3/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(15/4)) + (77*b^(3/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(15/4)) - (77*b^(3/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(15/4))]
    @test_int [1/(x^(7/2)*(a + b*x^2)^3), x, 14, -(117/(80*a^3*x^(5/2))) + (117*b)/(16*a^4*sqrt(x)) + 1/(4*a*x^(5/2)*(a + b*x^2)^2) + 13/(16*a^2*x^(5/2)*(a + b*x^2)) - (117*b^(5/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(17/4)) + (117*b^(5/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(x))/a^(1/4)))/(32*sqrt(2)*a^(17/4)) + (117*b^(5/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(17/4)) - (117*b^(5/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x))/(64*sqrt(2)*a^(17/4))]

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

    @test_int [x^(7/2)/(1 + x^2), x, 12, -2*sqrt(x) + (2*x^(5/2))/5 - atan(1 - sqrt(2)*sqrt(x))/sqrt(2) + atan(1 + sqrt(2)*sqrt(x))/sqrt(2) - log(1 - sqrt(2)*sqrt(x) + x)/(2*sqrt(2)) + log(1 + sqrt(2)*sqrt(x) + x)/(2*sqrt(2))]
    @test_int [x^(5/2)/(1 + x^2), x, 11, (2*x^(3/2))/3 + atan(1 - sqrt(2)*sqrt(x))/sqrt(2) - atan(1 + sqrt(2)*sqrt(x))/sqrt(2) - log(1 - sqrt(2)*sqrt(x) + x)/(2*sqrt(2)) + log(1 + sqrt(2)*sqrt(x) + x)/(2*sqrt(2))]
    @test_int [x^(3/2)/(1 + x^2), x, 11, 2*sqrt(x) + atan(1 - sqrt(2)*sqrt(x))/sqrt(2) - atan(1 + sqrt(2)*sqrt(x))/sqrt(2) + log(1 - sqrt(2)*sqrt(x) + x)/(2*sqrt(2)) - log(1 + sqrt(2)*sqrt(x) + x)/(2*sqrt(2))]
    @test_int [sqrt(x)/(1 + x^2), x, 10, -(atan(1 - sqrt(2)*sqrt(x))/sqrt(2)) + atan(1 + sqrt(2)*sqrt(x))/sqrt(2) + log(1 - sqrt(2)*sqrt(x) + x)/(2*sqrt(2)) - log(1 + sqrt(2)*sqrt(x) + x)/(2*sqrt(2))]
    @test_int [1/(sqrt(x)*(1 + x^2)), x, 10, -(atan(1 - sqrt(2)*sqrt(x))/sqrt(2)) + atan(1 + sqrt(2)*sqrt(x))/sqrt(2) - log(1 - sqrt(2)*sqrt(x) + x)/(2*sqrt(2)) + log(1 + sqrt(2)*sqrt(x) + x)/(2*sqrt(2))]
    @test_int [1/(x^(3/2)*(1 + x^2)), x, 11, -2/sqrt(x) + atan(1 - sqrt(2)*sqrt(x))/sqrt(2) - atan(1 + sqrt(2)*sqrt(x))/sqrt(2) - log(1 - sqrt(2)*sqrt(x) + x)/(2*sqrt(2)) + log(1 + sqrt(2)*sqrt(x) + x)/(2*sqrt(2))]
    @test_int [1/(x^(5/2)*(1 + x^2)), x, 11, -2/(3*x^(3/2)) + atan(1 - sqrt(2)*sqrt(x))/sqrt(2) - atan(1 + sqrt(2)*sqrt(x))/sqrt(2) + log(1 - sqrt(2)*sqrt(x) + x)/(2*sqrt(2)) - log(1 + sqrt(2)*sqrt(x) + x)/(2*sqrt(2))]
    @test_int [1/(x^(7/2)*(1 + x^2)), x, 12, -2/(5*x^(5/2)) + 2/sqrt(x) - atan(1 - sqrt(2)*sqrt(x))/sqrt(2) + atan(1 + sqrt(2)*sqrt(x))/sqrt(2) + log(1 - sqrt(2)*sqrt(x) + x)/(2*sqrt(2)) - log(1 + sqrt(2)*sqrt(x) + x)/(2*sqrt(2))]

    @test_int [x^(7/2)/(1 + x^2)^2, x, 12, (5*sqrt(x))/2 - x^(5/2)/(2*(1 + x^2)) + (5*atan(1 - sqrt(2)*sqrt(x)))/(4*sqrt(2)) - (5*atan(1 + sqrt(2)*sqrt(x)))/(4*sqrt(2)) + (5*log(1 - sqrt(2)*sqrt(x) + x))/(8*sqrt(2)) - (5*log(1 + sqrt(2)*sqrt(x) + x))/(8*sqrt(2))]
    @test_int [x^(5/2)/(1 + x^2)^2, x, 11, -x^(3/2)/(2*(1 + x^2)) - (3*atan(1 - sqrt(2)*sqrt(x)))/(4*sqrt(2)) + (3*atan(1 + sqrt(2)*sqrt(x)))/(4*sqrt(2)) + (3*log(1 - sqrt(2)*sqrt(x) + x))/(8*sqrt(2)) - (3*log(1 + sqrt(2)*sqrt(x) + x))/(8*sqrt(2))]
    @test_int [x^(3/2)/(1 + x^2)^2, x, 11, -sqrt(x)/(2*(1 + x^2)) - atan(1 - sqrt(2)*sqrt(x))/(4*sqrt(2)) + atan(1 + sqrt(2)*sqrt(x))/(4*sqrt(2)) - log(1 - sqrt(2)*sqrt(x) + x)/(8*sqrt(2)) + log(1 + sqrt(2)*sqrt(x) + x)/(8*sqrt(2))]
    @test_int [sqrt(x)/(1 + x^2)^2, x, 11, x^(3/2)/(2*(1 + x^2)) - atan(1 - sqrt(2)*sqrt(x))/(4*sqrt(2)) + atan(1 + sqrt(2)*sqrt(x))/(4*sqrt(2)) + log(1 - sqrt(2)*sqrt(x) + x)/(8*sqrt(2)) - log(1 + sqrt(2)*sqrt(x) + x)/(8*sqrt(2))]
    @test_int [1/(sqrt(x)*(1 + x^2)^2), x, 11, sqrt(x)/(2*(1 + x^2)) - (3*atan(1 - sqrt(2)*sqrt(x)))/(4*sqrt(2)) + (3*atan(1 + sqrt(2)*sqrt(x)))/(4*sqrt(2)) - (3*log(1 - sqrt(2)*sqrt(x) + x))/(8*sqrt(2)) + (3*log(1 + sqrt(2)*sqrt(x) + x))/(8*sqrt(2))]
    @test_int [1/(x^(3/2)*(1 + x^2)^2), x, 12, -5/(2*sqrt(x)) + 1/(2*sqrt(x)*(1 + x^2)) + (5*atan(1 - sqrt(2)*sqrt(x)))/(4*sqrt(2)) - (5*atan(1 + sqrt(2)*sqrt(x)))/(4*sqrt(2)) - (5*log(1 - sqrt(2)*sqrt(x) + x))/(8*sqrt(2)) + (5*log(1 + sqrt(2)*sqrt(x) + x))/(8*sqrt(2))]
    @test_int [1/(x^(5/2)*(1 + x^2)^2), x, 12, -7/(6*x^(3/2)) + 1/(2*x^(3/2)*(1 + x^2)) + (7*atan(1 - sqrt(2)*sqrt(x)))/(4*sqrt(2)) - (7*atan(1 + sqrt(2)*sqrt(x)))/(4*sqrt(2)) + (7*log(1 - sqrt(2)*sqrt(x) + x))/(8*sqrt(2)) - (7*log(1 + sqrt(2)*sqrt(x) + x))/(8*sqrt(2))]
    @test_int [1/(x^(7/2)*(1 + x^2)^2), x, 13, -9/(10*x^(5/2)) + 9/(2*sqrt(x)) + 1/(2*x^(5/2)*(1 + x^2)) - (9*atan(1 - sqrt(2)*sqrt(x)))/(4*sqrt(2)) + (9*atan(1 + sqrt(2)*sqrt(x)))/(4*sqrt(2)) + (9*log(1 - sqrt(2)*sqrt(x) + x))/(8*sqrt(2)) - (9*log(1 + sqrt(2)*sqrt(x) + x))/(8*sqrt(2))]

    @test_int [x^(7/2)/(1 + x^2)^3, x, 12, -x^(5/2)/(4*(1 + x^2)^2) - (5*sqrt(x))/(16*(1 + x^2)) - (5*atan(1 - sqrt(2)*sqrt(x)))/(32*sqrt(2)) + (5*atan(1 + sqrt(2)*sqrt(x)))/(32*sqrt(2)) - (5*log(1 - sqrt(2)*sqrt(x) + x))/(64*sqrt(2)) + (5*log(1 + sqrt(2)*sqrt(x) + x))/(64*sqrt(2))]
    @test_int [x^(5/2)/(1 + x^2)^3, x, 12, -x^(3/2)/(4*(1 + x^2)^2) + (3*x^(3/2))/(16*(1 + x^2)) - (3*atan(1 - sqrt(2)*sqrt(x)))/(32*sqrt(2)) + (3*atan(1 + sqrt(2)*sqrt(x)))/(32*sqrt(2)) + (3*log(1 - sqrt(2)*sqrt(x) + x))/(64*sqrt(2)) - (3*log(1 + sqrt(2)*sqrt(x) + x))/(64*sqrt(2))]
    @test_int [x^(3/2)/(1 + x^2)^3, x, 12, -sqrt(x)/(4*(1 + x^2)^2) + sqrt(x)/(16*(1 + x^2)) - (3*atan(1 - sqrt(2)*sqrt(x)))/(32*sqrt(2)) + (3*atan(1 + sqrt(2)*sqrt(x)))/(32*sqrt(2)) - (3*log(1 - sqrt(2)*sqrt(x) + x))/(64*sqrt(2)) + (3*log(1 + sqrt(2)*sqrt(x) + x))/(64*sqrt(2))]
    @test_int [sqrt(x)/(1 + x^2)^3, x, 12, x^(3/2)/(4*(1 + x^2)^2) + (5*x^(3/2))/(16*(1 + x^2)) - (5*atan(1 - sqrt(2)*sqrt(x)))/(32*sqrt(2)) + (5*atan(1 + sqrt(2)*sqrt(x)))/(32*sqrt(2)) + (5*log(1 - sqrt(2)*sqrt(x) + x))/(64*sqrt(2)) - (5*log(1 + sqrt(2)*sqrt(x) + x))/(64*sqrt(2))]
    @test_int [1/(sqrt(x)*(1 + x^2)^3), x, 12, sqrt(x)/(4*(1 + x^2)^2) + (7*sqrt(x))/(16*(1 + x^2)) - (21*atan(1 - sqrt(2)*sqrt(x)))/(32*sqrt(2)) + (21*atan(1 + sqrt(2)*sqrt(x)))/(32*sqrt(2)) - (21*log(1 - sqrt(2)*sqrt(x) + x))/(64*sqrt(2)) + (21*log(1 + sqrt(2)*sqrt(x) + x))/(64*sqrt(2))]
    @test_int [1/(x^(3/2)*(1 + x^2)^3), x, 13, -45/(16*sqrt(x)) + 1/(4*sqrt(x)*(1 + x^2)^2) + 9/(16*sqrt(x)*(1 + x^2)) + (45*atan(1 - sqrt(2)*sqrt(x)))/(32*sqrt(2)) - (45*atan(1 + sqrt(2)*sqrt(x)))/(32*sqrt(2)) - (45*log(1 - sqrt(2)*sqrt(x) + x))/(64*sqrt(2)) + (45*log(1 + sqrt(2)*sqrt(x) + x))/(64*sqrt(2))]
    @test_int [1/(x^(5/2)*(1 + x^2)^3), x, 13, -77/(48*x^(3/2)) + 1/(4*x^(3/2)*(1 + x^2)^2) + 11/(16*x^(3/2)*(1 + x^2)) + (77*atan(1 - sqrt(2)*sqrt(x)))/(32*sqrt(2)) - (77*atan(1 + sqrt(2)*sqrt(x)))/(32*sqrt(2)) + (77*log(1 - sqrt(2)*sqrt(x) + x))/(64*sqrt(2)) - (77*log(1 + sqrt(2)*sqrt(x) + x))/(64*sqrt(2))]
    @test_int [1/(x^(7/2)*(1 + x^2)^3), x, 14, -117/(80*x^(5/2)) + 117/(16*sqrt(x)) + 1/(4*x^(5/2)*(1 + x^2)^2) + 13/(16*x^(5/2)*(1 + x^2)) - (117*atan(1 - sqrt(2)*sqrt(x)))/(32*sqrt(2)) + (117*atan(1 + sqrt(2)*sqrt(x)))/(32*sqrt(2)) + (117*log(1 - sqrt(2)*sqrt(x) + x))/(64*sqrt(2)) - (117*log(1 + sqrt(2)*sqrt(x) + x))/(64*sqrt(2))]

    @test_int [sqrt(x)/(1 - x^2), x, 4, -atan(sqrt(x)) + atanh(sqrt(x))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(c*x)^(m/3)*(a+b*x^2)^p=#

    @test_int [x^(2/3)/(1 + x^2), x, 11, (-(1/2))*atan(sqrt(3) - 2*x^(1/3)) + (1/2)*atan(sqrt(3) + 2*x^(1/3)) + atan(x^(1/3)) - (1/2)*sqrt(3)*atanh((sqrt(3)*x^(1/3))/(1 + x^(2/3))), (-(1/2))*atan(sqrt(3) - 2*x^(1/3)) + (1/2)*atan(sqrt(3) + 2*x^(1/3)) + atan(x^(1/3)) + (1/4)*sqrt(3)*log(1 - sqrt(3)*x^(1/3) + x^(2/3)) - (1/4)*sqrt(3)*log(1 + sqrt(3)*x^(1/3) + x^(2/3))]

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

    @test_int [x^m*(a + b*x^2)^5, x, 2, (a^5*x^(1 + m))/(1 + m) + (5*a^4*b*x^(3 + m))/(3 + m) + (10*a^3*b^2*x^(5 + m))/(5 + m) + (10*a^2*b^3*x^(7 + m))/(7 + m) + (5*a*b^4*x^(9 + m))/(9 + m) + (b^5*x^(11 + m))/(11 + m)]
    @test_int [x^m*(a + b*x^2)^4, x, 2, (a^4*x^(1 + m))/(1 + m) + (4*a^3*b*x^(3 + m))/(3 + m) + (6*a^2*b^2*x^(5 + m))/(5 + m) + (4*a*b^3*x^(7 + m))/(7 + m) + (b^4*x^(9 + m))/(9 + m)]
    @test_int [x^m*(a + b*x^2)^3, x, 2, (a^3*x^(1 + m))/(1 + m) + (3*a^2*b*x^(3 + m))/(3 + m) + (3*a*b^2*x^(5 + m))/(5 + m) + (b^3*x^(7 + m))/(7 + m)]
    @test_int [x^m*(a + b*x^2)^2, x, 2, (a^2*x^(1 + m))/(1 + m) + (2*a*b*x^(3 + m))/(3 + m) + (b^2*x^(5 + m))/(5 + m)]
    @test_int [x^m*(a + b*x^2)^1, x, 2, (a*x^(1 + m))/(1 + m) + (b*x^(3 + m))/(3 + m)]
    @test_int [x^m/(a + b*x^2)^1, x, 1, (x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(a*(1 + m))]
    @test_int [x^m/(a + b*x^2)^2, x, 1, (x^(1 + m)*HypergeometricFunctions._₂F₁(2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(a^2*(1 + m))]
    @test_int [x^m/(a + b*x^2)^3, x, 1, (x^(1 + m)*HypergeometricFunctions._₂F₁(3, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)))/(a^3*(1 + m))]

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

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

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

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

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

    @test_int [x^7*sqrt(a + b*x^2), x, 3, -((a^3*(a + b*x^2)^(3/2))/(3*b^4)) + (3*a^2*(a + b*x^2)^(5/2))/(5*b^4) - (3*a*(a + b*x^2)^(7/2))/(7*b^4) + (a + b*x^2)^(9/2)/(9*b^4)]
    @test_int [x^5*sqrt(a + b*x^2), x, 3, (a^2*(a + b*x^2)^(3/2))/(3*b^3) - (2*a*(a + b*x^2)^(5/2))/(5*b^3) + (a + b*x^2)^(7/2)/(7*b^3)]
    @test_int [x^3*sqrt(a + b*x^2), x, 3, -((a*(a + b*x^2)^(3/2))/(3*b^2)) + (a + b*x^2)^(5/2)/(5*b^2)]
    @test_int [x^1*sqrt(a + b*x^2), x, 1, (a + b*x^2)^(3/2)/(3*b)]
    @test_int [sqrt(a + b*x^2)/x^1, x, 4, sqrt(a + b*x^2) - sqrt(a)*atanh(sqrt(a + b*x^2)/sqrt(a))]
    @test_int [sqrt(a + b*x^2)/x^3, x, 4, -sqrt(a + b*x^2)/(2*x^2) - (b*atanh(sqrt(a + b*x^2)/sqrt(a)))/(2*sqrt(a))]
    @test_int [sqrt(a + b*x^2)/x^5, x, 5, -sqrt(a + b*x^2)/(4*x^4) - (b*sqrt(a + b*x^2))/(8*a*x^2) + (b^2*atanh(sqrt(a + b*x^2)/sqrt(a)))/(8*a^(3/2))]
    @test_int [sqrt(a + b*x^2)/x^7, x, 6, -(sqrt(a + b*x^2)/(6*x^6)) - (b*sqrt(a + b*x^2))/(24*a*x^4) + (b^2*sqrt(a + b*x^2))/(16*a^2*x^2) - (b^3*atanh(sqrt(a + b*x^2)/sqrt(a)))/(16*a^(5/2))]

    @test_int [x^4*sqrt(a + b*x^2), x, 5, -(a^2*x*sqrt(a + b*x^2))/(16*b^2) + (a*x^3*sqrt(a + b*x^2))/(24*b) + (x^5*sqrt(a + b*x^2))/6 + (a^3*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(16*b^(5/2))]
    @test_int [x^2*sqrt(a + b*x^2), x, 4, (a*x*sqrt(a + b*x^2))/(8*b) + (x^3*sqrt(a + b*x^2))/4 - (a^2*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(8*b^(3/2))]
    @test_int [x^0*sqrt(a + b*x^2), x, 3, (x*sqrt(a + b*x^2))/2 + (a*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(2*sqrt(b))]
    @test_int [sqrt(a + b*x^2)/x^2, x, 3, -(sqrt(a + b*x^2)/x) + sqrt(b)*atanh((sqrt(b)*x)/sqrt(a + b*x^2))]
    @test_int [sqrt(a + b*x^2)/x^4, x, 1, -(a + b*x^2)^(3/2)/(3*a*x^3)]
    @test_int [sqrt(a + b*x^2)/x^6, x, 2, -((a + b*x^2)^(3/2)/(5*a*x^5)) + (2*b*(a + b*x^2)^(3/2))/(15*a^2*x^3)]
    @test_int [sqrt(a + b*x^2)/x^8, x, 3, -((a + b*x^2)^(3/2)/(7*a*x^7)) + (4*b*(a + b*x^2)^(3/2))/(35*a^2*x^5) - (8*b^2*(a + b*x^2)^(3/2))/(105*a^3*x^3)]
    @test_int [sqrt(a + b*x^2)/x^10, x, 4, -((a + b*x^2)^(3/2)/(9*a*x^9)) + (2*b*(a + b*x^2)^(3/2))/(21*a^2*x^7) - (8*b^2*(a + b*x^2)^(3/2))/(105*a^3*x^5) + (16*b^3*(a + b*x^2)^(3/2))/(315*a^4*x^3)]

    @test_int [x^7*(a + b*x^2)^(3/2), x, 3, -((a^3*(a + b*x^2)^(5/2))/(5*b^4)) + (3*a^2*(a + b*x^2)^(7/2))/(7*b^4) - (a*(a + b*x^2)^(9/2))/(3*b^4) + (a + b*x^2)^(11/2)/(11*b^4)]
    @test_int [x^5*(a + b*x^2)^(3/2), x, 3, (a^2*(a + b*x^2)^(5/2))/(5*b^3) - (2*a*(a + b*x^2)^(7/2))/(7*b^3) + (a + b*x^2)^(9/2)/(9*b^3)]
    @test_int [x^3*(a + b*x^2)^(3/2), x, 3, -((a*(a + b*x^2)^(5/2))/(5*b^2)) + (a + b*x^2)^(7/2)/(7*b^2)]
    @test_int [x^1*(a + b*x^2)^(3/2), x, 1, (a + b*x^2)^(5/2)/(5*b)]
    @test_int [(a + b*x^2)^(3/2)/x^1, x, 5, a*sqrt(a + b*x^2) + (a + b*x^2)^(3/2)/3 - a^(3/2)*atanh(sqrt(a + b*x^2)/sqrt(a))]
    @test_int [(a + b*x^2)^(3/2)/x^3, x, 5, (3*b*sqrt(a + b*x^2))/2 - (a + b*x^2)^(3/2)/(2*x^2) - (3*sqrt(a)*b*atanh(sqrt(a + b*x^2)/sqrt(a)))/2]
    @test_int [(a + b*x^2)^(3/2)/x^5, x, 5, (-3*b*sqrt(a + b*x^2))/(8*x^2) - (a + b*x^2)^(3/2)/(4*x^4) - (3*b^2*atanh(sqrt(a + b*x^2)/sqrt(a)))/(8*sqrt(a))]
    @test_int [(a + b*x^2)^(3/2)/x^7, x, 6, -((b*sqrt(a + b*x^2))/(8*x^4)) - (b^2*sqrt(a + b*x^2))/(16*a*x^2) - (a + b*x^2)^(3/2)/(6*x^6) + (b^3*atanh(sqrt(a + b*x^2)/sqrt(a)))/(16*a^(3/2))]
    @test_int [(a + b*x^2)^(3/2)/x^9, x, 7, -((b*sqrt(a + b*x^2))/(16*x^6)) - (b^2*sqrt(a + b*x^2))/(64*a*x^4) + (3*b^3*sqrt(a + b*x^2))/(128*a^2*x^2) - (a + b*x^2)^(3/2)/(8*x^8) - (3*b^4*atanh(sqrt(a + b*x^2)/sqrt(a)))/(128*a^(5/2))]

    @test_int [x^4*(a + b*x^2)^(3/2), x, 6, (-3*a^3*x*sqrt(a + b*x^2))/(128*b^2) + (a^2*x^3*sqrt(a + b*x^2))/(64*b) + (a*x^5*sqrt(a + b*x^2))/16 + (x^5*(a + b*x^2)^(3/2))/8 + (3*a^4*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(128*b^(5/2))]
    @test_int [x^2*(a + b*x^2)^(3/2), x, 5, (a^2*x*sqrt(a + b*x^2))/(16*b) + (a*x^3*sqrt(a + b*x^2))/8 + (x^3*(a + b*x^2)^(3/2))/6 - (a^3*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(16*b^(3/2))]
    @test_int [x^0*(a + b*x^2)^(3/2), x, 4, (3*a*x*sqrt(a + b*x^2))/8 + (x*(a + b*x^2)^(3/2))/4 + (3*a^2*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(8*sqrt(b))]
    @test_int [(a + b*x^2)^(3/2)/x^2, x, 4, (3*b*x*sqrt(a + b*x^2))/2 - (a + b*x^2)^(3/2)/x + (3*a*sqrt(b)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/2]
    @test_int [(a + b*x^2)^(3/2)/x^4, x, 4, -((b*sqrt(a + b*x^2))/x) - (a + b*x^2)^(3/2)/(3*x^3) + b^(3/2)*atanh((sqrt(b)*x)/sqrt(a + b*x^2))]
    @test_int [(a + b*x^2)^(3/2)/x^6, x, 1, -((a + b*x^2)^(5/2)/(5*a*x^5))]
    @test_int [(a + b*x^2)^(3/2)/x^8, x, 2, -((a + b*x^2)^(5/2)/(7*a*x^7)) + (2*b*(a + b*x^2)^(5/2))/(35*a^2*x^5)]
    @test_int [(a + b*x^2)^(3/2)/x^10, x, 3, -((a + b*x^2)^(5/2)/(9*a*x^9)) + (4*b*(a + b*x^2)^(5/2))/(63*a^2*x^7) - (8*b^2*(a + b*x^2)^(5/2))/(315*a^3*x^5)]
    @test_int [(a + b*x^2)^(3/2)/x^12, x, 4, -((a + b*x^2)^(5/2)/(11*a*x^11)) + (2*b*(a + b*x^2)^(5/2))/(33*a^2*x^9) - (8*b^2*(a + b*x^2)^(5/2))/(231*a^3*x^7) + (16*b^3*(a + b*x^2)^(5/2))/(1155*a^4*x^5)]

    @test_int [x^7*(a + b*x^2)^(5/2), x, 3, -((a^3*(a + b*x^2)^(7/2))/(7*b^4)) + (a^2*(a + b*x^2)^(9/2))/(3*b^4) - (3*a*(a + b*x^2)^(11/2))/(11*b^4) + (a + b*x^2)^(13/2)/(13*b^4)]
    @test_int [x^5*(a + b*x^2)^(5/2), x, 3, (a^2*(a + b*x^2)^(7/2))/(7*b^3) - (2*a*(a + b*x^2)^(9/2))/(9*b^3) + (a + b*x^2)^(11/2)/(11*b^3)]
    @test_int [x^3*(a + b*x^2)^(5/2), x, 3, -((a*(a + b*x^2)^(7/2))/(7*b^2)) + (a + b*x^2)^(9/2)/(9*b^2)]
    @test_int [x^1*(a + b*x^2)^(5/2), x, 1, (a + b*x^2)^(7/2)/(7*b)]
    @test_int [(a + b*x^2)^(5/2)/x^1, x, 6, a^2*sqrt(a + b*x^2) + (a*(a + b*x^2)^(3/2))/3 + (a + b*x^2)^(5/2)/5 - a^(5/2)*atanh(sqrt(a + b*x^2)/sqrt(a))]
    @test_int [(a + b*x^2)^(5/2)/x^3, x, 6, (5*a*b*sqrt(a + b*x^2))/2 + (5*b*(a + b*x^2)^(3/2))/6 - (a + b*x^2)^(5/2)/(2*x^2) - (5*a^(3/2)*b*atanh(sqrt(a + b*x^2)/sqrt(a)))/2]
    @test_int [(a + b*x^2)^(5/2)/x^5, x, 6, (15*b^2*sqrt(a + b*x^2))/8 - (5*b*(a + b*x^2)^(3/2))/(8*x^2) - (a + b*x^2)^(5/2)/(4*x^4) - (15*sqrt(a)*b^2*atanh(sqrt(a + b*x^2)/sqrt(a)))/8]
    @test_int [(a + b*x^2)^(5/2)/x^7, x, 6, -((5*b^2*sqrt(a + b*x^2))/(16*x^2)) - (5*b*(a + b*x^2)^(3/2))/(24*x^4) - (a + b*x^2)^(5/2)/(6*x^6) - (5*b^3*atanh(sqrt(a + b*x^2)/sqrt(a)))/(16*sqrt(a))]
    @test_int [(a + b*x^2)^(5/2)/x^9, x, 7, -((5*b^2*sqrt(a + b*x^2))/(64*x^4)) - (5*b^3*sqrt(a + b*x^2))/(128*a*x^2) - (5*b*(a + b*x^2)^(3/2))/(48*x^6) - (a + b*x^2)^(5/2)/(8*x^8) + (5*b^4*atanh(sqrt(a + b*x^2)/sqrt(a)))/(128*a^(3/2))]
    @test_int [(a + b*x^2)^(5/2)/x^11, x, 8, -((b^2*sqrt(a + b*x^2))/(32*x^6)) - (b^3*sqrt(a + b*x^2))/(128*a*x^4) + (3*b^4*sqrt(a + b*x^2))/(256*a^2*x^2) - (b*(a + b*x^2)^(3/2))/(16*x^8) - (a + b*x^2)^(5/2)/(10*x^10) - (3*b^5*atanh(sqrt(a + b*x^2)/sqrt(a)))/(256*a^(5/2))]

    @test_int [x^4*(a + b*x^2)^(5/2), x, 7, -((3*a^4*x*sqrt(a + b*x^2))/(256*b^2)) + (a^3*x^3*sqrt(a + b*x^2))/(128*b) + (1/32)*a^2*x^5*sqrt(a + b*x^2) + (1/16)*a*x^5*(a + b*x^2)^(3/2) + (1/10)*x^5*(a + b*x^2)^(5/2) + (3*a^5*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(256*b^(5/2))]
    @test_int [x^2*(a + b*x^2)^(5/2), x, 6, (5*a^3*x*sqrt(a + b*x^2))/(128*b) + (5*a^2*x^3*sqrt(a + b*x^2))/64 + (5*a*x^3*(a + b*x^2)^(3/2))/48 + (x^3*(a + b*x^2)^(5/2))/8 - (5*a^4*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(128*b^(3/2))]
    @test_int [x^0*(a + b*x^2)^(5/2), x, 5, (5*a^2*x*sqrt(a + b*x^2))/16 + (5*a*x*(a + b*x^2)^(3/2))/24 + (x*(a + b*x^2)^(5/2))/6 + (5*a^3*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(16*sqrt(b))]
    @test_int [(a + b*x^2)^(5/2)/x^2, x, 5, (15*a*b*x*sqrt(a + b*x^2))/8 + (5*b*x*(a + b*x^2)^(3/2))/4 - (a + b*x^2)^(5/2)/x + (15*a^2*sqrt(b)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/8]
    @test_int [(a + b*x^2)^(5/2)/x^4, x, 5, (5*b^2*x*sqrt(a + b*x^2))/2 - (5*b*(a + b*x^2)^(3/2))/(3*x) - (a + b*x^2)^(5/2)/(3*x^3) + (5*a*b^(3/2)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/2]
    @test_int [(a + b*x^2)^(5/2)/x^6, x, 5, -((b^2*sqrt(a + b*x^2))/x) - (b*(a + b*x^2)^(3/2))/(3*x^3) - (a + b*x^2)^(5/2)/(5*x^5) + b^(5/2)*atanh((sqrt(b)*x)/sqrt(a + b*x^2))]
    @test_int [(a + b*x^2)^(5/2)/x^8, x, 1, -((a + b*x^2)^(7/2)/(7*a*x^7))]
    @test_int [(a + b*x^2)^(5/2)/x^10, x, 2, -((a + b*x^2)^(7/2)/(9*a*x^9)) + (2*b*(a + b*x^2)^(7/2))/(63*a^2*x^7)]
    @test_int [(a + b*x^2)^(5/2)/x^12, x, 3, -((a + b*x^2)^(7/2)/(11*a*x^11)) + (4*b*(a + b*x^2)^(7/2))/(99*a^2*x^9) - (8*b^2*(a + b*x^2)^(7/2))/(693*a^3*x^7)]
    @test_int [(a + b*x^2)^(5/2)/x^14, x, 4, -((a + b*x^2)^(7/2)/(13*a*x^13)) + (6*b*(a + b*x^2)^(7/2))/(143*a^2*x^11) - (8*b^2*(a + b*x^2)^(7/2))/(429*a^3*x^9) + (16*b^3*(a + b*x^2)^(7/2))/(3003*a^4*x^7)]
    @test_int [(a + b*x^2)^(5/2)/x^16, x, 5, -((a + b*x^2)^(7/2)/(15*a*x^15)) + (8*b*(a + b*x^2)^(7/2))/(195*a^2*x^13) - (16*b^2*(a + b*x^2)^(7/2))/(715*a^3*x^11) + (64*b^3*(a + b*x^2)^(7/2))/(6435*a^4*x^9) - (128*b^4*(a + b*x^2)^(7/2))/(45045*a^5*x^7)]
    @test_int [(a + b*x^2)^(5/2)/x^18, x, 6, -((a + b*x^2)^(7/2)/(17*a*x^17)) + (2*b*(a + b*x^2)^(7/2))/(51*a^2*x^15) - (16*b^2*(a + b*x^2)^(7/2))/(663*a^3*x^13) + (32*b^3*(a + b*x^2)^(7/2))/(2431*a^4*x^11) - (128*b^4*(a + b*x^2)^(7/2))/(21879*a^5*x^9) + (256*b^5*(a + b*x^2)^(7/2))/(153153*a^6*x^7)]

    @test_int [x^15*(a + b*x^2)^(9/2), x, 3, -(a^7*(a + b*x^2)^(11/2))/(11*b^8) + (7*a^6*(a + b*x^2)^(13/2))/(13*b^8) - (7*a^5*(a + b*x^2)^(15/2))/(5*b^8) + (35*a^4*(a + b*x^2)^(17/2))/(17*b^8) - (35*a^3*(a + b*x^2)^(19/2))/(19*b^8) + (a^2*(a + b*x^2)^(21/2))/b^8 - (7*a*(a + b*x^2)^(23/2))/(23*b^8) + (a + b*x^2)^(25/2)/(25*b^8)]
    @test_int [x^13*(a + b*x^2)^(9/2), x, 3, (a^6*(a + b*x^2)^(11/2))/(11*b^7) - (6*a^5*(a + b*x^2)^(13/2))/(13*b^7) + (a^4*(a + b*x^2)^(15/2))/b^7 - (20*a^3*(a + b*x^2)^(17/2))/(17*b^7) + (15*a^2*(a + b*x^2)^(19/2))/(19*b^7) - (2*a*(a + b*x^2)^(21/2))/(7*b^7) + (a + b*x^2)^(23/2)/(23*b^7)]
    @test_int [x^11*(a + b*x^2)^(9/2), x, 3, -(a^5*(a + b*x^2)^(11/2))/(11*b^6) + (5*a^4*(a + b*x^2)^(13/2))/(13*b^6) - (2*a^3*(a + b*x^2)^(15/2))/(3*b^6) + (10*a^2*(a + b*x^2)^(17/2))/(17*b^6) - (5*a*(a + b*x^2)^(19/2))/(19*b^6) + (a + b*x^2)^(21/2)/(21*b^6)]
    @test_int [x^9*(a + b*x^2)^(9/2), x, 3, (a^4*(a + b*x^2)^(11/2))/(11*b^5) - (4*a^3*(a + b*x^2)^(13/2))/(13*b^5) + (2*a^2*(a + b*x^2)^(15/2))/(5*b^5) - (4*a*(a + b*x^2)^(17/2))/(17*b^5) + (a + b*x^2)^(19/2)/(19*b^5)]
    @test_int [x^7*(a + b*x^2)^(9/2), x, 3, -(a^3*(a + b*x^2)^(11/2))/(11*b^4) + (3*a^2*(a + b*x^2)^(13/2))/(13*b^4) - (a*(a + b*x^2)^(15/2))/(5*b^4) + (a + b*x^2)^(17/2)/(17*b^4)]
    @test_int [x^5*(a + b*x^2)^(9/2), x, 3, (a^2*(a + b*x^2)^(11/2))/(11*b^3) - (2*a*(a + b*x^2)^(13/2))/(13*b^3) + (a + b*x^2)^(15/2)/(15*b^3)]
    @test_int [x^3*(a + b*x^2)^(9/2), x, 3, -(a*(a + b*x^2)^(11/2))/(11*b^2) + (a + b*x^2)^(13/2)/(13*b^2)]
    @test_int [x^1*(a + b*x^2)^(9/2), x, 1, (a + b*x^2)^(11/2)/(11*b)]
    @test_int [(a + b*x^2)^(9/2)/x^1, x, 8, a^4*sqrt(a + b*x^2) + (a^3*(a + b*x^2)^(3/2))/3 + (a^2*(a + b*x^2)^(5/2))/5 + (a*(a + b*x^2)^(7/2))/7 + (a + b*x^2)^(9/2)/9 - a^(9/2)*atanh(sqrt(a + b*x^2)/sqrt(a))]
    @test_int [(a + b*x^2)^(9/2)/x^3, x, 8, (9*a^3*b*sqrt(a + b*x^2))/2 + (3*a^2*b*(a + b*x^2)^(3/2))/2 + (9*a*b*(a + b*x^2)^(5/2))/10 + (9*b*(a + b*x^2)^(7/2))/14 - (a + b*x^2)^(9/2)/(2*x^2) - (9*a^(7/2)*b*atanh(sqrt(a + b*x^2)/sqrt(a)))/2]
    @test_int [(a + b*x^2)^(9/2)/x^5, x, 8, (63*a^2*b^2*sqrt(a + b*x^2))/8 + (21*a*b^2*(a + b*x^2)^(3/2))/8 + (63*b^2*(a + b*x^2)^(5/2))/40 - (9*b*(a + b*x^2)^(7/2))/(8*x^2) - (a + b*x^2)^(9/2)/(4*x^4) - (63*a^(5/2)*b^2*atanh(sqrt(a + b*x^2)/sqrt(a)))/8]
    @test_int [(a + b*x^2)^(9/2)/x^7, x, 8, (105*a*b^3*sqrt(a + b*x^2))/16 + (35*b^3*(a + b*x^2)^(3/2))/16 - (21*b^2*(a + b*x^2)^(5/2))/(16*x^2) - (3*b*(a + b*x^2)^(7/2))/(8*x^4) - (a + b*x^2)^(9/2)/(6*x^6) - (105*a^(3/2)*b^3*atanh(sqrt(a + b*x^2)/sqrt(a)))/16]
    @test_int [(a + b*x^2)^(9/2)/x^9, x, 8, (315*b^4*sqrt(a + b*x^2))/128 - (105*b^3*(a + b*x^2)^(3/2))/(128*x^2) - (21*b^2*(a + b*x^2)^(5/2))/(64*x^4) - (3*b*(a + b*x^2)^(7/2))/(16*x^6) - (a + b*x^2)^(9/2)/(8*x^8) - (315*sqrt(a)*b^4*atanh(sqrt(a + b*x^2)/sqrt(a)))/128]
    @test_int [(a + b*x^2)^(9/2)/x^11, x, 8, (-63*b^4*sqrt(a + b*x^2))/(256*x^2) - (21*b^3*(a + b*x^2)^(3/2))/(128*x^4) - (21*b^2*(a + b*x^2)^(5/2))/(160*x^6) - (9*b*(a + b*x^2)^(7/2))/(80*x^8) - (a + b*x^2)^(9/2)/(10*x^10) - (63*b^5*atanh(sqrt(a + b*x^2)/sqrt(a)))/(256*sqrt(a))]
    @test_int [(a + b*x^2)^(9/2)/x^13, x, 9, (-21*b^4*sqrt(a + b*x^2))/(512*x^4) - (21*b^5*sqrt(a + b*x^2))/(1024*a*x^2) - (7*b^3*(a + b*x^2)^(3/2))/(128*x^6) - (21*b^2*(a + b*x^2)^(5/2))/(320*x^8) - (3*b*(a + b*x^2)^(7/2))/(40*x^10) - (a + b*x^2)^(9/2)/(12*x^12) + (21*b^6*atanh(sqrt(a + b*x^2)/sqrt(a)))/(1024*a^(3/2))]
    @test_int [(a + b*x^2)^(9/2)/x^15, x, 10, (-3*b^4*sqrt(a + b*x^2))/(256*x^6) - (3*b^5*sqrt(a + b*x^2))/(1024*a*x^4) + (9*b^6*sqrt(a + b*x^2))/(2048*a^2*x^2) - (3*b^3*(a + b*x^2)^(3/2))/(128*x^8) - (3*b^2*(a + b*x^2)^(5/2))/(80*x^10) - (3*b*(a + b*x^2)^(7/2))/(56*x^12) - (a + b*x^2)^(9/2)/(14*x^14) - (9*b^7*atanh(sqrt(a + b*x^2)/sqrt(a)))/(2048*a^(5/2))]

    @test_int [x^6*(a + b*x^2)^(9/2), x, 10, (45*a^7*x*sqrt(a + b*x^2))/(32768*b^3) - (15*a^6*x^3*sqrt(a + b*x^2))/(16384*b^2) + (3*a^5*x^5*sqrt(a + b*x^2))/(4096*b) + (9*a^4*x^7*sqrt(a + b*x^2))/2048 + (3*a^3*x^7*(a + b*x^2)^(3/2))/256 + (3*a^2*x^7*(a + b*x^2)^(5/2))/128 + (9*a*x^7*(a + b*x^2)^(7/2))/224 + (x^7*(a + b*x^2)^(9/2))/16 - (45*a^8*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(32768*b^(7/2))]
    @test_int [x^4*(a + b*x^2)^(9/2), x, 9, (-9*a^6*x*sqrt(a + b*x^2))/(2048*b^2) + (3*a^5*x^3*sqrt(a + b*x^2))/(1024*b) + (3*a^4*x^5*sqrt(a + b*x^2))/256 + (3*a^3*x^5*(a + b*x^2)^(3/2))/128 + (3*a^2*x^5*(a + b*x^2)^(5/2))/80 + (3*a*x^5*(a + b*x^2)^(7/2))/56 + (x^5*(a + b*x^2)^(9/2))/14 + (9*a^7*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(2048*b^(5/2))]
    @test_int [x^2*(a + b*x^2)^(9/2), x, 8, (21*a^5*x*sqrt(a + b*x^2))/(1024*b) + (21*a^4*x^3*sqrt(a + b*x^2))/512 + (7*a^3*x^3*(a + b*x^2)^(3/2))/128 + (21*a^2*x^3*(a + b*x^2)^(5/2))/320 + (3*a*x^3*(a + b*x^2)^(7/2))/40 + (x^3*(a + b*x^2)^(9/2))/12 - (21*a^6*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(1024*b^(3/2))]
    @test_int [x^0*(a + b*x^2)^(9/2), x, 7, (63*a^4*x*sqrt(a + b*x^2))/256 + (21*a^3*x*(a + b*x^2)^(3/2))/128 + (21*a^2*x*(a + b*x^2)^(5/2))/160 + (9*a*x*(a + b*x^2)^(7/2))/80 + (x*(a + b*x^2)^(9/2))/10 + (63*a^5*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(256*sqrt(b))]
    @test_int [(a + b*x^2)^(9/2)/x^2, x, 7, (315*a^3*b*x*sqrt(a + b*x^2))/128 + (105*a^2*b*x*(a + b*x^2)^(3/2))/64 + (21*a*b*x*(a + b*x^2)^(5/2))/16 + (9*b*x*(a + b*x^2)^(7/2))/8 - (a + b*x^2)^(9/2)/x + (315*a^4*sqrt(b)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/128]
    @test_int [(a + b*x^2)^(9/2)/x^4, x, 7, (105*a^2*b^2*x*sqrt(a + b*x^2))/16 + (35*a*b^2*x*(a + b*x^2)^(3/2))/8 + (7*b^2*x*(a + b*x^2)^(5/2))/2 - (3*b*(a + b*x^2)^(7/2))/x - (a + b*x^2)^(9/2)/(3*x^3) + (105*a^3*b^(3/2)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/16]
    @test_int [(a + b*x^2)^(9/2)/x^6, x, 7, (63*a*b^3*x*sqrt(a + b*x^2))/8 + (21*b^3*x*(a + b*x^2)^(3/2))/4 - (21*b^2*(a + b*x^2)^(5/2))/(5*x) - (3*b*(a + b*x^2)^(7/2))/(5*x^3) - (a + b*x^2)^(9/2)/(5*x^5) + (63*a^2*b^(5/2)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/8]
    @test_int [(a + b*x^2)^(9/2)/x^8, x, 7, (9*b^4*x*sqrt(a + b*x^2))/2 - (3*b^3*(a + b*x^2)^(3/2))/x - (3*b^2*(a + b*x^2)^(5/2))/(5*x^3) - (9*b*(a + b*x^2)^(7/2))/(35*x^5) - (a + b*x^2)^(9/2)/(7*x^7) + (9*a*b^(7/2)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/2]
    @test_int [(a + b*x^2)^(9/2)/x^10, x, 7, -((b^4*sqrt(a + b*x^2))/x) - (b^3*(a + b*x^2)^(3/2))/(3*x^3) - (b^2*(a + b*x^2)^(5/2))/(5*x^5) - (b*(a + b*x^2)^(7/2))/(7*x^7) - (a + b*x^2)^(9/2)/(9*x^9) + b^(9/2)*atanh((sqrt(b)*x)/sqrt(a + b*x^2))]
    @test_int [(a + b*x^2)^(9/2)/x^12, x, 1, -(a + b*x^2)^(11/2)/(11*a*x^11)]
    @test_int [(a + b*x^2)^(9/2)/x^14, x, 2, -((a + b*x^2)^(11/2)/(13*a*x^13)) + (2*b*(a + b*x^2)^(11/2))/(143*a^2*x^11)]
    @test_int [(a + b*x^2)^(9/2)/x^16, x, 3, -((a + b*x^2)^(11/2)/(15*a*x^15)) + (4*b*(a + b*x^2)^(11/2))/(195*a^2*x^13) - (8*b^2*(a + b*x^2)^(11/2))/(2145*a^3*x^11)]
    @test_int [(a + b*x^2)^(9/2)/x^18, x, 4, -((a + b*x^2)^(11/2)/(17*a*x^17)) + (2*b*(a + b*x^2)^(11/2))/(85*a^2*x^15) - (8*b^2*(a + b*x^2)^(11/2))/(1105*a^3*x^13) + (16*b^3*(a + b*x^2)^(11/2))/(12155*a^4*x^11)]
    @test_int [(a + b*x^2)^(9/2)/x^20, x, 5, -((a + b*x^2)^(11/2)/(19*a*x^19)) + (8*b*(a + b*x^2)^(11/2))/(323*a^2*x^17) - (16*b^2*(a + b*x^2)^(11/2))/(1615*a^3*x^15) + (64*b^3*(a + b*x^2)^(11/2))/(20995*a^4*x^13) - (128*b^4*(a + b*x^2)^(11/2))/(230945*a^5*x^11)]
    @test_int [(a + b*x^2)^(9/2)/x^22, x, 6, -((a + b*x^2)^(11/2)/(21*a*x^21)) + (10*b*(a + b*x^2)^(11/2))/(399*a^2*x^19) - (80*b^2*(a + b*x^2)^(11/2))/(6783*a^3*x^17) + (32*b^3*(a + b*x^2)^(11/2))/(6783*a^4*x^15) - (128*b^4*(a + b*x^2)^(11/2))/(88179*a^5*x^13) + (256*b^5*(a + b*x^2)^(11/2))/(969969*a^6*x^11)]
    @test_int [(a + b*x^2)^(9/2)/x^24, x, 7, -((a + b*x^2)^(11/2)/(23*a*x^23)) + (4*b*(a + b*x^2)^(11/2))/(161*a^2*x^21) - (40*b^2*(a + b*x^2)^(11/2))/(3059*a^3*x^19) + (320*b^3*(a + b*x^2)^(11/2))/(52003*a^4*x^17) - (128*b^4*(a + b*x^2)^(11/2))/(52003*a^5*x^15) + (512*b^5*(a + b*x^2)^(11/2))/(676039*a^6*x^13) - (1024*b^6*(a + b*x^2)^(11/2))/(7436429*a^7*x^11)]

    @test_int [x^5*sqrt(9 + 4*x^2), x, 3, (27/64)*(9 + 4*x^2)^(3/2) - (9/160)*(9 + 4*x^2)^(5/2) + (1/448)*(9 + 4*x^2)^(7/2)]
    @test_int [x^4*sqrt(9 + 4*x^2), x, 4, (-(81/256))*x*sqrt(9 + 4*x^2) + (3/32)*x^3*sqrt(9 + 4*x^2) + (1/6)*x^5*sqrt(9 + 4*x^2) + (729/512)*asinh((2*x)/3)]
    @test_int [x^3*sqrt(9 + 4*x^2), x, 3, (-(3/16))*(9 + 4*x^2)^(3/2) + (1/80)*(9 + 4*x^2)^(5/2)]
    @test_int [x^2*sqrt(9 + 4*x^2), x, 3, (9/32)*x*sqrt(9 + 4*x^2) + (1/4)*x^3*sqrt(9 + 4*x^2) - (81/64)*asinh((2*x)/3)]
    @test_int [x*sqrt(9 + 4*x^2), x, 1, (9 + 4*x^2)^(3/2)/12]
    @test_int [sqrt(9 + 4*x^2), x, 2, (1/2)*x*sqrt(9 + 4*x^2) + (9/4)*asinh((2*x)/3)]
    @test_int [sqrt(9 + 4*x^2)/x, x, 4, sqrt(9 + 4*x^2) - 3*atanh(sqrt(9 + 4*x^2)/3)]
    @test_int [sqrt(9 + 4*x^2)/x^2, x, 2, -(sqrt(9 + 4*x^2)/x) + 2*asinh((2*x)/3)]
    @test_int [sqrt(9 + 4*x^2)/x^3, x, 4, -sqrt(9 + 4*x^2)/(2*x^2) - (2*atanh(sqrt(9 + 4*x^2)/3))/3]
    @test_int [sqrt(9 + 4*x^2)/x^4, x, 1, -(9 + 4*x^2)^(3/2)/(27*x^3)]
    @test_int [sqrt(9 + 4*x^2)/x^5, x, 5, -(sqrt(9 + 4*x^2)/(4*x^4)) - sqrt(9 + 4*x^2)/(18*x^2) + (2/27)*atanh((1/3)*sqrt(9 + 4*x^2))]

    @test_int [x^5*sqrt(9 - 4*x^2), x, 3, (-(27/64))*(9 - 4*x^2)^(3/2) + (9/160)*(9 - 4*x^2)^(5/2) - (1/448)*(9 - 4*x^2)^(7/2)]
    @test_int [x^4*sqrt(9 - 4*x^2), x, 4, (-(81/256))*x*sqrt(9 - 4*x^2) - (3/32)*x^3*sqrt(9 - 4*x^2) + (1/6)*x^5*sqrt(9 - 4*x^2) + (729/512)*asin((2*x)/3)]
    @test_int [x^3*sqrt(9 - 4*x^2), x, 3, (-(3/16))*(9 - 4*x^2)^(3/2) + (1/80)*(9 - 4*x^2)^(5/2)]
    @test_int [x^2*sqrt(9 - 4*x^2), x, 3, (-(9/32))*x*sqrt(9 - 4*x^2) + (1/4)*x^3*sqrt(9 - 4*x^2) + (81/64)*asin((2*x)/3)]
    @test_int [x*sqrt(9 - 4*x^2), x, 1, -(9 - 4*x^2)^(3/2)/12]
    @test_int [sqrt(9 - 4*x^2), x, 2, (1/2)*x*sqrt(9 - 4*x^2) + (9/4)*asin((2*x)/3)]
    @test_int [sqrt(9 - 4*x^2)/x, x, 4, sqrt(9 - 4*x^2) - 3*atanh(sqrt(9 - 4*x^2)/3)]
    @test_int [sqrt(9 - 4*x^2)/x^2, x, 2, -(sqrt(9 - 4*x^2)/x) - 2*asin((2*x)/3)]
    @test_int [sqrt(9 - 4*x^2)/x^3, x, 4, -sqrt(9 - 4*x^2)/(2*x^2) + (2*atanh(sqrt(9 - 4*x^2)/3))/3]
    @test_int [sqrt(9 - 4*x^2)/x^4, x, 1, -(9 - 4*x^2)^(3/2)/(27*x^3)]
    @test_int [sqrt(9 - 4*x^2)/x^5, x, 5, -(sqrt(9 - 4*x^2)/(4*x^4)) + sqrt(9 - 4*x^2)/(18*x^2) + (2/27)*atanh((1/3)*sqrt(9 - 4*x^2))]

    @test_int [x^5*sqrt(-9 + 4*x^2), x, 3, (27/64)*(-9 + 4*x^2)^(3/2) + (9/160)*(-9 + 4*x^2)^(5/2) + (1/448)*(-9 + 4*x^2)^(7/2)]
    @test_int [x^4*sqrt(-9 + 4*x^2), x, 5, (-(81/256))*x*sqrt(-9 + 4*x^2) - (3/32)*x^3*sqrt(-9 + 4*x^2) + (1/6)*x^5*sqrt(-9 + 4*x^2) - (729/512)*atanh((2*x)/sqrt(-9 + 4*x^2))]
    @test_int [x^3*sqrt(-9 + 4*x^2), x, 3, (3/16)*(-9 + 4*x^2)^(3/2) + (1/80)*(-9 + 4*x^2)^(5/2)]
    @test_int [x^2*sqrt(-9 + 4*x^2), x, 4, (-(9/32))*x*sqrt(-9 + 4*x^2) + (1/4)*x^3*sqrt(-9 + 4*x^2) - (81/64)*atanh((2*x)/sqrt(-9 + 4*x^2))]
    @test_int [x*sqrt(-9 + 4*x^2), x, 1, (-9 + 4*x^2)^(3/2)/12]
    @test_int [sqrt(-9 + 4*x^2), x, 3, (1/2)*x*sqrt(-9 + 4*x^2) - (9/4)*atanh((2*x)/sqrt(-9 + 4*x^2))]
    @test_int [sqrt(-9 + 4*x^2)/x, x, 4, sqrt(-9 + 4*x^2) - 3*atan(sqrt(-9 + 4*x^2)/3)]
    @test_int [sqrt(-9 + 4*x^2)/x^2, x, 3, -(sqrt(-9 + 4*x^2)/x) + 2*atanh((2*x)/sqrt(-9 + 4*x^2))]
    @test_int [sqrt(-9 + 4*x^2)/x^3, x, 4, -sqrt(-9 + 4*x^2)/(2*x^2) + (2*atan(sqrt(-9 + 4*x^2)/3))/3]
    @test_int [sqrt(-9 + 4*x^2)/x^4, x, 1, (-9 + 4*x^2)^(3/2)/(27*x^3)]
    @test_int [sqrt(-9 + 4*x^2)/x^5, x, 5, -(sqrt(-9 + 4*x^2)/(4*x^4)) + sqrt(-9 + 4*x^2)/(18*x^2) + (2/27)*atan((1/3)*sqrt(-9 + 4*x^2))]

    @test_int [x^5*sqrt(-9 - 4*x^2), x, 3, (-(27/64))*(-9 - 4*x^2)^(3/2) - (9/160)*(-9 - 4*x^2)^(5/2) - (1/448)*(-9 - 4*x^2)^(7/2)]
    @test_int [x^4*sqrt(-9 - 4*x^2), x, 5, (-(81/256))*x*sqrt(-9 - 4*x^2) + (3/32)*x^3*sqrt(-9 - 4*x^2) + (1/6)*x^5*sqrt(-9 - 4*x^2) - (729/512)*atan((2*x)/sqrt(-9 - 4*x^2))]
    @test_int [x^3*sqrt(-9 - 4*x^2), x, 3, (3/16)*(-9 - 4*x^2)^(3/2) + (1/80)*(-9 - 4*x^2)^(5/2)]
    @test_int [x^2*sqrt(-9 - 4*x^2), x, 4, (9/32)*x*sqrt(-9 - 4*x^2) + (1/4)*x^3*sqrt(-9 - 4*x^2) + (81/64)*atan((2*x)/sqrt(-9 - 4*x^2))]
    @test_int [x*sqrt(-9 - 4*x^2), x, 1, -(-9 - 4*x^2)^(3/2)/12]
    @test_int [sqrt(-9 - 4*x^2), x, 3, (1/2)*x*sqrt(-9 - 4*x^2) - (9/4)*atan((2*x)/sqrt(-9 - 4*x^2))]
    @test_int [sqrt(-9 - 4*x^2)/x, x, 4, sqrt(-9 - 4*x^2) - 3*atan(sqrt(-9 - 4*x^2)/3)]
    @test_int [sqrt(-9 - 4*x^2)/x^2, x, 3, -(sqrt(-9 - 4*x^2)/x) - 2*atan((2*x)/sqrt(-9 - 4*x^2))]
    @test_int [sqrt(-9 - 4*x^2)/x^3, x, 4, -sqrt(-9 - 4*x^2)/(2*x^2) - (2*atan(sqrt(-9 - 4*x^2)/3))/3]
    @test_int [sqrt(-9 - 4*x^2)/x^4, x, 1, (-9 - 4*x^2)^(3/2)/(27*x^3)]
    @test_int [sqrt(-9 - 4*x^2)/x^5, x, 5, -(sqrt(-9 - 4*x^2)/(4*x^4)) - sqrt(-9 - 4*x^2)/(18*x^2) + (2/27)*atan((1/3)*sqrt(-9 - 4*x^2))]

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

    @test_int [x^5/sqrt(a + b*x^2), x, 3, (a^2*sqrt(a + b*x^2))/b^3 - (2*a*(a + b*x^2)^(3/2))/(3*b^3) + (a + b*x^2)^(5/2)/(5*b^3)]
    @test_int [x^4/sqrt(a + b*x^2), x, 4, (-3*a*x*sqrt(a + b*x^2))/(8*b^2) + (x^3*sqrt(a + b*x^2))/(4*b) + (3*a^2*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(8*b^(5/2))]
    @test_int [x^3/sqrt(a + b*x^2), x, 3, -((a*sqrt(a + b*x^2))/b^2) + (a + b*x^2)^(3/2)/(3*b^2)]
    @test_int [x^2/sqrt(a + b*x^2), x, 3, (x*sqrt(a + b*x^2))/(2*b) - (a*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(2*b^(3/2))]
    @test_int [x/sqrt(a + b*x^2), x, 1, sqrt(a + b*x^2)/b]
    @test_int [1/sqrt(a + b*x^2), x, 2, atanh((sqrt(b)*x)/sqrt(a + b*x^2))/sqrt(b)]
    @test_int [1/(x*sqrt(a + b*x^2)), x, 3, -(atanh(sqrt(a + b*x^2)/sqrt(a))/sqrt(a))]
    @test_int [1/(x^2*sqrt(a + b*x^2)), x, 1, -(sqrt(a + b*x^2)/(a*x))]
    @test_int [1/(x^3*sqrt(a + b*x^2)), x, 4, -sqrt(a + b*x^2)/(2*a*x^2) + (b*atanh(sqrt(a + b*x^2)/sqrt(a)))/(2*a^(3/2))]
    @test_int [1/(x^4*sqrt(a + b*x^2)), x, 2, -sqrt(a + b*x^2)/(3*a*x^3) + (2*b*sqrt(a + b*x^2))/(3*a^2*x)]
    @test_int [1/(x^5*sqrt(a + b*x^2)), x, 5, -sqrt(a + b*x^2)/(4*a*x^4) + (3*b*sqrt(a + b*x^2))/(8*a^2*x^2) - (3*b^2*atanh(sqrt(a + b*x^2)/sqrt(a)))/(8*a^(5/2))]

    @test_int [x^5/(a + b*x^2)^(3/2), x, 3, -(a^2/(b^3*sqrt(a + b*x^2))) - (2*a*sqrt(a + b*x^2))/b^3 + (a + b*x^2)^(3/2)/(3*b^3)]
    @test_int [x^4/(a + b*x^2)^(3/2), x, 4, -(x^3/(b*sqrt(a + b*x^2))) + (3*x*sqrt(a + b*x^2))/(2*b^2) - (3*a*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(2*b^(5/2))]
    @test_int [x^3/(a + b*x^2)^(3/2), x, 3, a/(b^2*sqrt(a + b*x^2)) + sqrt(a + b*x^2)/b^2]
    @test_int [x^2/(a + b*x^2)^(3/2), x, 3, -(x/(b*sqrt(a + b*x^2))) + atanh((sqrt(b)*x)/sqrt(a + b*x^2))/b^(3/2)]
    @test_int [x/(a + b*x^2)^(3/2), x, 1, -(1/(b*sqrt(a + b*x^2)))]
    @test_int [1/(a + b*x^2)^(3/2), x, 1, x/(a*sqrt(a + b*x^2))]
    @test_int [1/(x*(a + b*x^2)^(3/2)), x, 4, 1/(a*sqrt(a + b*x^2)) - atanh(sqrt(a + b*x^2)/sqrt(a))/a^(3/2)]
    @test_int [1/(x^2*(a + b*x^2)^(3/2)), x, 2, -(1/(a*x*sqrt(a + b*x^2))) - (2*b*x)/(a^2*sqrt(a + b*x^2))]
    @test_int [1/(x^3*(a + b*x^2)^(3/2)), x, 5, -((3*b)/(2*a^2*sqrt(a + b*x^2))) - 1/(2*a*x^2*sqrt(a + b*x^2)) + (3*b*atanh(sqrt(a + b*x^2)/sqrt(a)))/(2*a^(5/2))]
    @test_int [1/(x^4*(a + b*x^2)^(3/2)), x, 3, -(1/(3*a*x^3*sqrt(a + b*x^2))) + (4*b)/(3*a^2*x*sqrt(a + b*x^2)) + (8*b^2*x)/(3*a^3*sqrt(a + b*x^2))]

    @test_int [x^6/(a + b*x^2)^(5/2), x, 5, -x^5/(3*b*(a + b*x^2)^(3/2)) - (5*x^3)/(3*b^2*sqrt(a + b*x^2)) + (5*x*sqrt(a + b*x^2))/(2*b^3) - (5*a*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(2*b^(7/2))]
    @test_int [x^5/(a + b*x^2)^(5/2), x, 3, -(a^2/(3*b^3*(a + b*x^2)^(3/2))) + (2*a)/(b^3*sqrt(a + b*x^2)) + sqrt(a + b*x^2)/b^3]
    @test_int [x^4/(a + b*x^2)^(5/2), x, 4, -x^3/(3*b*(a + b*x^2)^(3/2)) - x/(b^2*sqrt(a + b*x^2)) + atanh((sqrt(b)*x)/sqrt(a + b*x^2))/b^(5/2)]
    @test_int [x^3/(a + b*x^2)^(5/2), x, 3, a/(3*b^2*(a + b*x^2)^(3/2)) - 1/(b^2*sqrt(a + b*x^2))]
    @test_int [x^2/(a + b*x^2)^(5/2), x, 1, x^3/(3*a*(a + b*x^2)^(3/2))]
    @test_int [x/(a + b*x^2)^(5/2), x, 1, -1/(3*b*(a + b*x^2)^(3/2))]
    @test_int [1/(a + b*x^2)^(5/2), x, 2, x/(3*a*(a + b*x^2)^(3/2)) + (2*x)/(3*a^2*sqrt(a + b*x^2))]
    @test_int [1/(x*(a + b*x^2)^(5/2)), x, 5, 1/(3*a*(a + b*x^2)^(3/2)) + 1/(a^2*sqrt(a + b*x^2)) - atanh(sqrt(a + b*x^2)/sqrt(a))/a^(5/2)]
    @test_int [1/(x^2*(a + b*x^2)^(5/2)), x, 3, -(1/(a*x*(a + b*x^2)^(3/2))) - (4*b*x)/(3*a^2*(a + b*x^2)^(3/2)) - (8*b*x)/(3*a^3*sqrt(a + b*x^2))]
    @test_int [1/(x^3*(a + b*x^2)^(5/2)), x, 6, -((5*b)/(6*a^2*(a + b*x^2)^(3/2))) - 1/(2*a*x^2*(a + b*x^2)^(3/2)) - (5*b)/(2*a^3*sqrt(a + b*x^2)) + (5*b*atanh(sqrt(a + b*x^2)/sqrt(a)))/(2*a^(7/2))]
    @test_int [1/(x^4*(a + b*x^2)^(5/2)), x, 4, -(1/(3*a*x^3*(a + b*x^2)^(3/2))) + (2*b)/(a^2*x*(a + b*x^2)^(3/2)) + (8*b^2*x)/(3*a^3*(a + b*x^2)^(3/2)) + (16*b^2*x)/(3*a^4*sqrt(a + b*x^2))]

    @test_int [x^10/(a + b*x^2)^(9/2), x, 7, -x^9/(7*b*(a + b*x^2)^(7/2)) - (9*x^7)/(35*b^2*(a + b*x^2)^(5/2)) - (3*x^5)/(5*b^3*(a + b*x^2)^(3/2)) - (3*x^3)/(b^4*sqrt(a + b*x^2)) + (9*x*sqrt(a + b*x^2))/(2*b^5) - (9*a*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(2*b^(11/2))]
    @test_int [x^9/(a + b*x^2)^(9/2), x, 3, -(a^4/(7*b^5*(a + b*x^2)^(7/2))) + (4*a^3)/(5*b^5*(a + b*x^2)^(5/2)) - (2*a^2)/(b^5*(a + b*x^2)^(3/2)) + (4*a)/(b^5*sqrt(a + b*x^2)) + sqrt(a + b*x^2)/b^5]
    @test_int [x^8/(a + b*x^2)^(9/2), x, 6, -x^7/(7*b*(a + b*x^2)^(7/2)) - x^5/(5*b^2*(a + b*x^2)^(5/2)) - x^3/(3*b^3*(a + b*x^2)^(3/2)) - x/(b^4*sqrt(a + b*x^2)) + atanh((sqrt(b)*x)/sqrt(a + b*x^2))/b^(9/2)]
    @test_int [x^7/(a + b*x^2)^(9/2), x, 3, a^3/(7*b^4*(a + b*x^2)^(7/2)) - (3*a^2)/(5*b^4*(a + b*x^2)^(5/2)) + a/(b^4*(a + b*x^2)^(3/2)) - 1/(b^4*sqrt(a + b*x^2))]
    @test_int [x^6/(a + b*x^2)^(9/2), x, 1, x^7/(7*a*(a + b*x^2)^(7/2))]
    @test_int [x^5/(a + b*x^2)^(9/2), x, 3, -(a^2/(7*b^3*(a + b*x^2)^(7/2))) + (2*a)/(5*b^3*(a + b*x^2)^(5/2)) - 1/(3*b^3*(a + b*x^2)^(3/2))]
    @test_int [x^4/(a + b*x^2)^(9/2), x, 2, x^5/(5*a*(a + b*x^2)^(7/2)) + (2*b*x^7)/(35*a^2*(a + b*x^2)^(7/2))]
    @test_int [x^3/(a + b*x^2)^(9/2), x, 3, a/(7*b^2*(a + b*x^2)^(7/2)) - 1/(5*b^2*(a + b*x^2)^(5/2))]
    @test_int [x^2/(a + b*x^2)^(9/2), x, 3, x^3/(3*a*(a + b*x^2)^(7/2)) + (4*b*x^5)/(15*a^2*(a + b*x^2)^(7/2)) + (8*b^2*x^7)/(105*a^3*(a + b*x^2)^(7/2))]
    @test_int [x/(a + b*x^2)^(9/2), x, 1, -1/(7*b*(a + b*x^2)^(7/2))]
    @test_int [1/(a + b*x^2)^(9/2), x, 4, x/(7*a*(a + b*x^2)^(7/2)) + (6*x)/(35*a^2*(a + b*x^2)^(5/2)) + (8*x)/(35*a^3*(a + b*x^2)^(3/2)) + (16*x)/(35*a^4*sqrt(a + b*x^2))]
    @test_int [1/(x*(a + b*x^2)^(9/2)), x, 7, 1/(7*a*(a + b*x^2)^(7/2)) + 1/(5*a^2*(a + b*x^2)^(5/2)) + 1/(3*a^3*(a + b*x^2)^(3/2)) + 1/(a^4*sqrt(a + b*x^2)) - atanh(sqrt(a + b*x^2)/sqrt(a))/a^(9/2)]
    @test_int [1/(x^2*(a + b*x^2)^(9/2)), x, 5, -(1/(a*x*(a + b*x^2)^(7/2))) - (8*b*x)/(7*a^2*(a + b*x^2)^(7/2)) - (48*b*x)/(35*a^3*(a + b*x^2)^(5/2)) - (64*b*x)/(35*a^4*(a + b*x^2)^(3/2)) - (128*b*x)/(35*a^5*sqrt(a + b*x^2))]
    @test_int [1/(x^3*(a + b*x^2)^(9/2)), x, 8, -((9*b)/(14*a^2*(a + b*x^2)^(7/2))) - 1/(2*a*x^2*(a + b*x^2)^(7/2)) - (9*b)/(10*a^3*(a + b*x^2)^(5/2)) - (3*b)/(2*a^4*(a + b*x^2)^(3/2)) - (9*b)/(2*a^5*sqrt(a + b*x^2)) + (9*b*atanh(sqrt(a + b*x^2)/sqrt(a)))/(2*a^(11/2))]
    @test_int [1/(x^4*(a + b*x^2)^(9/2)), x, 6, -(1/(3*a*x^3*(a + b*x^2)^(7/2))) + (10*b)/(3*a^2*x*(a + b*x^2)^(7/2)) + (80*b^2*x)/(21*a^3*(a + b*x^2)^(7/2)) + (32*b^2*x)/(7*a^4*(a + b*x^2)^(5/2)) + (128*b^2*x)/(21*a^5*(a + b*x^2)^(3/2)) + (256*b^2*x)/(21*a^6*sqrt(a + b*x^2))]

    @test_int [x^5/sqrt(9 + 4*x^2), x, 3, (81/64)*sqrt(9 + 4*x^2) - (3/32)*(9 + 4*x^2)^(3/2) + (1/320)*(9 + 4*x^2)^(5/2)]
    @test_int [x^4/sqrt(9 + 4*x^2), x, 3, (-(27/128))*x*sqrt(9 + 4*x^2) + (1/16)*x^3*sqrt(9 + 4*x^2) + (243/256)*asinh((2*x)/3)]
    @test_int [x^3/sqrt(9 + 4*x^2), x, 3, (-(9/16))*sqrt(9 + 4*x^2) + (1/48)*(9 + 4*x^2)^(3/2)]
    @test_int [x^2/sqrt(9 + 4*x^2), x, 2, (1/8)*x*sqrt(9 + 4*x^2) - (9/16)*asinh((2*x)/3)]
    @test_int [x/sqrt(9 + 4*x^2), x, 1, sqrt(9 + 4*x^2)/4]
    @test_int [1/sqrt(9 + 4*x^2), x, 1, asinh((2*x)/3)/2]
    @test_int [1/(x*sqrt(9 + 4*x^2)), x, 3, -atanh(sqrt(9 + 4*x^2)/3)/3]
    @test_int [1/(x^2*sqrt(9 + 4*x^2)), x, 1, -sqrt(9 + 4*x^2)/(9*x)]
    @test_int [1/(x^3*sqrt(9 + 4*x^2)), x, 4, -(sqrt(9 + 4*x^2)/(18*x^2)) + (2/27)*atanh((1/3)*sqrt(9 + 4*x^2))]
    @test_int [1/(x^4*sqrt(9 + 4*x^2)), x, 2, -(sqrt(9 + 4*x^2)/(27*x^3)) + (8*sqrt(9 + 4*x^2))/(243*x)]
    @test_int [1/(x^5*sqrt(9 + 4*x^2)), x, 5, -(sqrt(9 + 4*x^2)/(36*x^4)) + sqrt(9 + 4*x^2)/(54*x^2) - (2/81)*atanh((1/3)*sqrt(9 + 4*x^2))]

    @test_int [x^5/sqrt(9 - 4*x^2), x, 3, (-(81/64))*sqrt(9 - 4*x^2) + (3/32)*(9 - 4*x^2)^(3/2) - (1/320)*(9 - 4*x^2)^(5/2)]
    @test_int [x^4/sqrt(9 - 4*x^2), x, 3, (-(27/128))*x*sqrt(9 - 4*x^2) - (1/16)*x^3*sqrt(9 - 4*x^2) + (243/256)*asin((2*x)/3)]
    @test_int [x^3/sqrt(9 - 4*x^2), x, 3, (-(9/16))*sqrt(9 - 4*x^2) + (1/48)*(9 - 4*x^2)^(3/2)]
    @test_int [x^2/sqrt(9 - 4*x^2), x, 2, (-(1/8))*x*sqrt(9 - 4*x^2) + (9/16)*asin((2*x)/3)]
    @test_int [x/sqrt(9 - 4*x^2), x, 1, -sqrt(9 - 4*x^2)/4]
    @test_int [1/sqrt(9 - 4*x^2), x, 1, asin((2*x)/3)/2]
    @test_int [1/(x*sqrt(9 - 4*x^2)), x, 3, -atanh(sqrt(9 - 4*x^2)/3)/3]
    @test_int [1/(x^2*sqrt(9 - 4*x^2)), x, 1, -sqrt(9 - 4*x^2)/(9*x)]
    @test_int [1/(x^3*sqrt(9 - 4*x^2)), x, 4, -(sqrt(9 - 4*x^2)/(18*x^2)) - (2/27)*atanh((1/3)*sqrt(9 - 4*x^2))]
    @test_int [1/(x^4*sqrt(9 - 4*x^2)), x, 2, -(sqrt(9 - 4*x^2)/(27*x^3)) - (8*sqrt(9 - 4*x^2))/(243*x)]
    @test_int [1/(x^5*sqrt(9 - 4*x^2)), x, 5, -(sqrt(9 - 4*x^2)/(36*x^4)) - sqrt(9 - 4*x^2)/(54*x^2) - (2/81)*atanh((1/3)*sqrt(9 - 4*x^2))]

    @test_int [x^5/sqrt(-9 + 4*x^2), x, 3, (81/64)*sqrt(-9 + 4*x^2) + (3/32)*(-9 + 4*x^2)^(3/2) + (1/320)*(-9 + 4*x^2)^(5/2)]
    @test_int [x^4/sqrt(-9 + 4*x^2), x, 4, (27/128)*x*sqrt(-9 + 4*x^2) + (1/16)*x^3*sqrt(-9 + 4*x^2) + (243/256)*atanh((2*x)/sqrt(-9 + 4*x^2))]
    @test_int [x^3/sqrt(-9 + 4*x^2), x, 3, (9/16)*sqrt(-9 + 4*x^2) + (1/48)*(-9 + 4*x^2)^(3/2)]
    @test_int [x^2/sqrt(-9 + 4*x^2), x, 3, (1/8)*x*sqrt(-9 + 4*x^2) + (9/16)*atanh((2*x)/sqrt(-9 + 4*x^2))]
    @test_int [x/sqrt(-9 + 4*x^2), x, 1, sqrt(-9 + 4*x^2)/4]
    @test_int [1/sqrt(-9 + 4*x^2), x, 2, (1/2)*atanh((2*x)/sqrt(-9 + 4*x^2))]
    @test_int [1/(x*sqrt(-9 + 4*x^2)), x, 3, atan(sqrt(-9 + 4*x^2)/3)/3]
    @test_int [1/(x^2*sqrt(-9 + 4*x^2)), x, 1, sqrt(-9 + 4*x^2)/(9*x)]
    @test_int [1/(x^3*sqrt(-9 + 4*x^2)), x, 4, sqrt(-9 + 4*x^2)/(18*x^2) + (2/27)*atan((1/3)*sqrt(-9 + 4*x^2))]
    @test_int [1/(x^4*sqrt(-9 + 4*x^2)), x, 2, sqrt(-9 + 4*x^2)/(27*x^3) + (8*sqrt(-9 + 4*x^2))/(243*x)]
    @test_int [1/(x^5*sqrt(-9 + 4*x^2)), x, 5, sqrt(-9 + 4*x^2)/(36*x^4) + sqrt(-9 + 4*x^2)/(54*x^2) + (2/81)*atan((1/3)*sqrt(-9 + 4*x^2))]

    @test_int [x^5/sqrt(-9 - 4*x^2), x, 3, (-(81/64))*sqrt(-9 - 4*x^2) - (3/32)*(-9 - 4*x^2)^(3/2) - (1/320)*(-9 - 4*x^2)^(5/2)]
    @test_int [x^4/sqrt(-9 - 4*x^2), x, 4, (27/128)*x*sqrt(-9 - 4*x^2) - (1/16)*x^3*sqrt(-9 - 4*x^2) + (243/256)*atan((2*x)/sqrt(-9 - 4*x^2))]
    @test_int [x^3/sqrt(-9 - 4*x^2), x, 3, (9/16)*sqrt(-9 - 4*x^2) + (1/48)*(-9 - 4*x^2)^(3/2)]
    @test_int [x^2/sqrt(-9 - 4*x^2), x, 3, (-(1/8))*x*sqrt(-9 - 4*x^2) - (9/16)*atan((2*x)/sqrt(-9 - 4*x^2))]
    @test_int [x/sqrt(-9 - 4*x^2), x, 1, -sqrt(-9 - 4*x^2)/4]
    @test_int [1/sqrt(-9 - 4*x^2), x, 2, (1/2)*atan((2*x)/sqrt(-9 - 4*x^2))]
    @test_int [1/(x*sqrt(-9 - 4*x^2)), x, 3, atan(sqrt(-9 - 4*x^2)/3)/3]
    @test_int [1/(x^2*sqrt(-9 - 4*x^2)), x, 1, sqrt(-9 - 4*x^2)/(9*x)]
    @test_int [1/(x^3*sqrt(-9 - 4*x^2)), x, 4, sqrt(-9 - 4*x^2)/(18*x^2) - (2/27)*atan((1/3)*sqrt(-9 - 4*x^2))]
    @test_int [1/(x^4*sqrt(-9 - 4*x^2)), x, 2, sqrt(-9 - 4*x^2)/(27*x^3) - (8*sqrt(-9 - 4*x^2))/(243*x)]
    @test_int [1/(x^5*sqrt(-9 - 4*x^2)), x, 5, sqrt(-9 - 4*x^2)/(36*x^4) - sqrt(-9 - 4*x^2)/(54*x^2) + (2/81)*atan((1/3)*sqrt(-9 - 4*x^2))]

    @test_int [1/sqrt(9 + b*x^2), x, 1, asinh((sqrt(b)*x)/3)/sqrt(b)]
    @test_int [1/sqrt(9 - b*x^2), x, 1, asin((sqrt(b)*x)/3)/sqrt(b)]
    @test_int [1/sqrt(-9 + b*x^2), x, 2, atanh((sqrt(b)*x)/sqrt(-9 + b*x^2))/sqrt(b)]
    @test_int [1/sqrt(-9 - b*x^2), x, 2, atan((sqrt(b)*x)/sqrt(-9 - b*x^2))/sqrt(b)]

    @test_int [1/sqrt(pi + b*x^2), x, 1, asinh((sqrt(b)*x)/sqrt(pi))/sqrt(b)]
    @test_int [1/sqrt(pi - b*x^2), x, 1, asin((sqrt(b)*x)/sqrt(pi))/sqrt(b)]
    @test_int [1/sqrt(-pi + b*x^2), x, 2, atanh((sqrt(b)*x)/sqrt(-pi + b*x^2))/sqrt(b)]
    @test_int [1/sqrt(-pi - b*x^2), x, 2, atan((sqrt(b)*x)/sqrt(-pi - b*x^2))/sqrt(b)]

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

    @test_int [1/sqrt(a^2 - x^2), x, 2, atan(x/sqrt(a^2 - x^2))]

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

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

    @test_int [(c*x)^(7/2)*sqrt(a + b*x^2), x, 5, -((20*a^2*c^3*sqrt(c*x)*sqrt(a + b*x^2))/(231*b^2)) + (4*a*c*(c*x)^(5/2)*sqrt(a + b*x^2))/(77*b) + (2*(c*x)^(9/2)*sqrt(a + b*x^2))/(11*c) + (10*a^(11/4)*c^(7/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(231*b^(9/4)*sqrt(a + b*x^2))]
    @test_int [(c*x)^(5/2)*sqrt(a + b*x^2), x, 6, (4*a*c*(c*x)^(3/2)*sqrt(a + b*x^2))/(45*b) + (2*(c*x)^(7/2)*sqrt(a + b*x^2))/(9*c) - (4*a^2*c^2*sqrt(c*x)*sqrt(a + b*x^2))/(15*b^(3/2)*(sqrt(a) + sqrt(b)*x)) + (4*a^(9/4)*c^(5/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(15*b^(7/4)*sqrt(a + b*x^2)) - (2*a^(9/4)*c^(5/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(15*b^(7/4)*sqrt(a + b*x^2))]
    @test_int [(c*x)^(3/2)*sqrt(a + b*x^2), x, 4, (4*a*c*sqrt(c*x)*sqrt(a + b*x^2))/(21*b) + (2*(c*x)^(5/2)*sqrt(a + b*x^2))/(7*c) - (2*a^(7/4)*c^(3/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(21*b^(5/4)*sqrt(a + b*x^2))]
    @test_int [(c*x)^(1/2)*sqrt(a + b*x^2), x, 5, (2*(c*x)^(3/2)*sqrt(a + b*x^2))/(5*c) + (4*a*sqrt(c*x)*sqrt(a + b*x^2))/(5*sqrt(b)*(sqrt(a) + sqrt(b)*x)) - (4*a^(5/4)*sqrt(c)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(5*b^(3/4)*sqrt(a + b*x^2)) + (2*a^(5/4)*sqrt(c)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(5*b^(3/4)*sqrt(a + b*x^2))]
    @test_int [sqrt(a + b*x^2)/(c*x)^(1/2), x, 3, (2*sqrt(c*x)*sqrt(a + b*x^2))/(3*c) + (2*a^(3/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(3*b^(1/4)*sqrt(c)*sqrt(a + b*x^2))]
    @test_int [sqrt(a + b*x^2)/(c*x)^(3/2), x, 5, -((2*sqrt(a + b*x^2))/(c*sqrt(c*x))) + (4*sqrt(b)*sqrt(c*x)*sqrt(a + b*x^2))/(c^2*(sqrt(a) + sqrt(b)*x)) - (4*a^(1/4)*b^(1/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(c^(3/2)*sqrt(a + b*x^2)) + (2*a^(1/4)*b^(1/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(c^(3/2)*sqrt(a + b*x^2))]
    @test_int [sqrt(a + b*x^2)/(c*x)^(5/2), x, 3, -((2*sqrt(a + b*x^2))/(3*c*(c*x)^(3/2))) + (2*b^(3/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(3*a^(1/4)*c^(5/2)*sqrt(a + b*x^2))]
    @test_int [sqrt(a + b*x^2)/(c*x)^(7/2), x, 6, -((2*sqrt(a + b*x^2))/(5*c*(c*x)^(5/2))) - (4*b*sqrt(a + b*x^2))/(5*a*c^3*sqrt(c*x)) + (4*b^(3/2)*sqrt(c*x)*sqrt(a + b*x^2))/(5*a*c^4*(sqrt(a) + sqrt(b)*x)) - (4*b^(5/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(5*a^(3/4)*c^(7/2)*sqrt(a + b*x^2)) + (2*b^(5/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(5*a^(3/4)*c^(7/2)*sqrt(a + b*x^2))]

    @test_int [(c*x)^(7/2)*(a + b*x^2)^(3/2), x, 6, -((8*a^3*c^3*sqrt(c*x)*sqrt(a + b*x^2))/(231*b^2)) + (8*a^2*c*(c*x)^(5/2)*sqrt(a + b*x^2))/(385*b) + (4*a*(c*x)^(9/2)*sqrt(a + b*x^2))/(55*c) + (2*(c*x)^(9/2)*(a + b*x^2)^(3/2))/(15*c) + (4*a^(15/4)*c^(7/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(231*b^(9/4)*sqrt(a + b*x^2))]
    @test_int [(c*x)^(5/2)*(a + b*x^2)^(3/2), x, 7, (8*a^2*c*(c*x)^(3/2)*sqrt(a + b*x^2))/(195*b) + (4*a*(c*x)^(7/2)*sqrt(a + b*x^2))/(39*c) - (8*a^3*c^2*sqrt(c*x)*sqrt(a + b*x^2))/(65*b^(3/2)*(sqrt(a) + sqrt(b)*x)) + (2*(c*x)^(7/2)*(a + b*x^2)^(3/2))/(13*c) + (8*a^(13/4)*c^(5/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(65*b^(7/4)*sqrt(a + b*x^2)) - (4*a^(13/4)*c^(5/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(65*b^(7/4)*sqrt(a + b*x^2))]
    @test_int [(c*x)^(3/2)*(a + b*x^2)^(3/2), x, 5, (8*a^2*c*sqrt(c*x)*sqrt(a + b*x^2))/(77*b) + (12*a*(c*x)^(5/2)*sqrt(a + b*x^2))/(77*c) + (2*(c*x)^(5/2)*(a + b*x^2)^(3/2))/(11*c) - (4*a^(11/4)*c^(3/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(77*b^(5/4)*sqrt(a + b*x^2))]
    @test_int [(c*x)^(1/2)*(a + b*x^2)^(3/2), x, 6, (4*a*(c*x)^(3/2)*sqrt(a + b*x^2))/(15*c) + (8*a^2*sqrt(c*x)*sqrt(a + b*x^2))/(15*sqrt(b)*(sqrt(a) + sqrt(b)*x)) + (2*(c*x)^(3/2)*(a + b*x^2)^(3/2))/(9*c) - (8*a^(9/4)*sqrt(c)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(15*b^(3/4)*sqrt(a + b*x^2)) + (4*a^(9/4)*sqrt(c)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(15*b^(3/4)*sqrt(a + b*x^2))]
    @test_int [(a + b*x^2)^(3/2)/(c*x)^(1/2), x, 4, (4*a*sqrt(c*x)*sqrt(a + b*x^2))/(7*c) + (2*sqrt(c*x)*(a + b*x^2)^(3/2))/(7*c) + (4*a^(7/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(7*b^(1/4)*sqrt(c)*sqrt(a + b*x^2))]
    @test_int [(a + b*x^2)^(3/2)/(c*x)^(3/2), x, 6, (12*b*(c*x)^(3/2)*sqrt(a + b*x^2))/(5*c^3) + (24*a*sqrt(b)*sqrt(c*x)*sqrt(a + b*x^2))/(5*c^2*(sqrt(a) + sqrt(b)*x)) - (2*(a + b*x^2)^(3/2))/(c*sqrt(c*x)) - (24*a^(5/4)*b^(1/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(5*c^(3/2)*sqrt(a + b*x^2)) + (12*a^(5/4)*b^(1/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(5*c^(3/2)*sqrt(a + b*x^2))]
    @test_int [(a + b*x^2)^(3/2)/(c*x)^(5/2), x, 4, (4*b*sqrt(c*x)*sqrt(a + b*x^2))/(3*c^3) - (2*(a + b*x^2)^(3/2))/(3*c*(c*x)^(3/2)) + (4*a^(3/4)*b^(3/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(3*c^(5/2)*sqrt(a + b*x^2))]
    @test_int [(a + b*x^2)^(3/2)/(c*x)^(7/2), x, 6, -((12*b*sqrt(a + b*x^2))/(5*c^3*sqrt(c*x))) + (24*b^(3/2)*sqrt(c*x)*sqrt(a + b*x^2))/(5*c^4*(sqrt(a) + sqrt(b)*x)) - (2*(a + b*x^2)^(3/2))/(5*c*(c*x)^(5/2)) - (24*a^(1/4)*b^(5/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(5*c^(7/2)*sqrt(a + b*x^2)) + (12*a^(1/4)*b^(5/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(5*c^(7/2)*sqrt(a + b*x^2))]
    @test_int [(a + b*x^2)^(3/2)/(c*x)^(9/2), x, 4, -((4*b*sqrt(a + b*x^2))/(7*c^3*(c*x)^(3/2))) - (2*(a + b*x^2)^(3/2))/(7*c*(c*x)^(7/2)) + (4*b^(7/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(7*a^(1/4)*c^(9/2)*sqrt(a + b*x^2))]
    @test_int [(a + b*x^2)^(3/2)/(c*x)^(11/2), x, 7, -((4*b*sqrt(a + b*x^2))/(15*c^3*(c*x)^(5/2))) - (8*b^2*sqrt(a + b*x^2))/(15*a*c^5*sqrt(c*x)) + (8*b^(5/2)*sqrt(c*x)*sqrt(a + b*x^2))/(15*a*c^6*(sqrt(a) + sqrt(b)*x)) - (2*(a + b*x^2)^(3/2))/(9*c*(c*x)^(9/2)) - (8*b^(9/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(15*a^(3/4)*c^(11/2)*sqrt(a + b*x^2)) + (4*b^(9/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(15*a^(3/4)*c^(11/2)*sqrt(a + b*x^2))]

    @test_int [(c*x)^(5/2)*sqrt(3*a - 2*a*x^2), x, 6, (-2*c*(c*x)^(3/2)*sqrt(3*a - 2*a*x^2))/15 + (2*(c*x)^(7/2)*sqrt(3*a - 2*a*x^2))/(9*c) - (3*6^(1/4)*a*c^2*sqrt(c*x)*sqrt(3 - 2*x^2)*Elliptic.E(asin(sqrt(3 - sqrt(6)*x)/sqrt(6)), 2))/(5*sqrt(x)*sqrt(3*a - 2*a*x^2))]
    @test_int [(c*x)^(3/2)*sqrt(3*a - 2*a*x^2), x, 5, (-(2/7))*c*sqrt(c*x)*sqrt(3*a - 2*a*x^2) + (2*(c*x)^(5/2)*sqrt(3*a - 2*a*x^2))/(7*c) + (6^(3/4)*a*c^(3/2)*sqrt(3 - 2*x^2)*Elliptic.F(asin(((2/3)^(1/4)*sqrt(c*x))/sqrt(c)), -1))/(7*sqrt(a*(3 - 2*x^2)))]
    @test_int [(c*x)^(1/2)*sqrt(3*a - 2*a*x^2), x, 5, (2*(c*x)^(3/2)*sqrt(3*a - 2*a*x^2))/(5*c) - (6*6^(1/4)*a*sqrt(c*x)*sqrt(3 - 2*x^2)*Elliptic.E(asin(sqrt(3 - sqrt(6)*x)/sqrt(6)), 2))/(5*sqrt(x)*sqrt(3*a - 2*a*x^2))]
    @test_int [sqrt(3*a - 2*a*x^2)/(c*x)^(1/2), x, 4, (2*sqrt(c*x)*sqrt(3*a - 2*a*x^2))/(3*c) + (2*2^(3/4)*a*sqrt(3 - 2*x^2)*Elliptic.F(asin(((2/3)^(1/4)*sqrt(c*x))/sqrt(c)), -1))/(3^(1/4)*sqrt(c)*sqrt(a*(3 - 2*x^2)))]
    @test_int [sqrt(3*a - 2*a*x^2)/(c*x)^(3/2), x, 5, (-2*sqrt(3*a - 2*a*x^2))/(c*sqrt(c*x)) + (4*6^(1/4)*a*sqrt(c*x)*sqrt(3 - 2*x^2)*Elliptic.E(asin(sqrt(3 - sqrt(6)*x)/sqrt(6)), 2))/(c^2*sqrt(x)*sqrt(3*a - 2*a*x^2))]
    @test_int [sqrt(3*a - 2*a*x^2)/(c*x)^(5/2), x, 4, -((2*sqrt(3*a - 2*a*x^2))/(3*c*(c*x)^(3/2))) - (4*2^(3/4)*a*sqrt(3 - 2*x^2)*Elliptic.F(asin(((2/3)^(1/4)*sqrt(c*x))/sqrt(c)), -1))/(3*3^(1/4)*c^(5/2)*sqrt(a*(3 - 2*x^2)))]

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

    @test_int [(c*x)^(7/2)/sqrt(a + b*x^2), x, 4, -((10*a*c^3*sqrt(c*x)*sqrt(a + b*x^2))/(21*b^2)) + (2*c*(c*x)^(5/2)*sqrt(a + b*x^2))/(7*b) + (5*a^(7/4)*c^(7/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(21*b^(9/4)*sqrt(a + b*x^2))]
    @test_int [(c*x)^(5/2)/sqrt(a + b*x^2), x, 5, (2*c*(c*x)^(3/2)*sqrt(a + b*x^2))/(5*b) - (6*a*c^2*sqrt(c*x)*sqrt(a + b*x^2))/(5*b^(3/2)*(sqrt(a) + sqrt(b)*x)) + (6*a^(5/4)*c^(5/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(5*b^(7/4)*sqrt(a + b*x^2)) - (3*a^(5/4)*c^(5/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(5*b^(7/4)*sqrt(a + b*x^2))]
    @test_int [(c*x)^(3/2)/sqrt(a + b*x^2), x, 3, (2*c*sqrt(c*x)*sqrt(a + b*x^2))/(3*b) - (a^(3/4)*c^(3/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(3*b^(5/4)*sqrt(a + b*x^2))]
    @test_int [(c*x)^(1/2)/sqrt(a + b*x^2), x, 4, (2*sqrt(c*x)*sqrt(a + b*x^2))/(sqrt(b)*(sqrt(a) + sqrt(b)*x)) - (2*a^(1/4)*sqrt(c)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(b^(3/4)*sqrt(a + b*x^2)) + (a^(1/4)*sqrt(c)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(b^(3/4)*sqrt(a + b*x^2))]
    @test_int [1/((c*x)^(1/2)*sqrt(a + b*x^2)), x, 2, ((sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(a^(1/4)*b^(1/4)*sqrt(c)*sqrt(a + b*x^2))]
    @test_int [1/((c*x)^(3/2)*sqrt(a + b*x^2)), x, 5, -((2*sqrt(a + b*x^2))/(a*c*sqrt(c*x))) + (2*sqrt(b)*sqrt(c*x)*sqrt(a + b*x^2))/(a*c^2*(sqrt(a) + sqrt(b)*x)) - (2*b^(1/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(a^(3/4)*c^(3/2)*sqrt(a + b*x^2)) + (b^(1/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(a^(3/4)*c^(3/2)*sqrt(a + b*x^2))]
    @test_int [1/((c*x)^(5/2)*sqrt(a + b*x^2)), x, 3, -((2*sqrt(a + b*x^2))/(3*a*c*(c*x)^(3/2))) - (b^(3/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(3*a^(5/4)*c^(5/2)*sqrt(a + b*x^2))]
    @test_int [1/((c*x)^(7/2)*sqrt(a + b*x^2)), x, 6, -((2*sqrt(a + b*x^2))/(5*a*c*(c*x)^(5/2))) + (6*b*sqrt(a + b*x^2))/(5*a^2*c^3*sqrt(c*x)) - (6*b^(3/2)*sqrt(c*x)*sqrt(a + b*x^2))/(5*a^2*c^4*(sqrt(a) + sqrt(b)*x)) + (6*b^(5/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(5*a^(7/4)*c^(7/2)*sqrt(a + b*x^2)) - (3*b^(5/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(5*a^(7/4)*c^(7/2)*sqrt(a + b*x^2))]

    @test_int [(c*x)^(7/2)/(a + b*x^2)^(3/2), x, 4, -((c*(c*x)^(5/2))/(b*sqrt(a + b*x^2))) + (5*c^3*sqrt(c*x)*sqrt(a + b*x^2))/(3*b^2) - (5*a^(3/4)*c^(7/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(6*b^(9/4)*sqrt(a + b*x^2))]
    @test_int [(c*x)^(5/2)/(a + b*x^2)^(3/2), x, 5, -((c*(c*x)^(3/2))/(b*sqrt(a + b*x^2))) + (3*c^2*sqrt(c*x)*sqrt(a + b*x^2))/(b^(3/2)*(sqrt(a) + sqrt(b)*x)) - (3*a^(1/4)*c^(5/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(b^(7/4)*sqrt(a + b*x^2)) + (3*a^(1/4)*c^(5/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(2*b^(7/4)*sqrt(a + b*x^2))]
    @test_int [(c*x)^(3/2)/(a + b*x^2)^(3/2), x, 3, -((c*sqrt(c*x))/(b*sqrt(a + b*x^2))) + (c^(3/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(2*a^(1/4)*b^(5/4)*sqrt(a + b*x^2))]
    @test_int [(c*x)^(1/2)/(a + b*x^2)^(3/2), x, 5, (c*x)^(3/2)/(a*c*sqrt(a + b*x^2)) - (sqrt(c*x)*sqrt(a + b*x^2))/(a*sqrt(b)*(sqrt(a) + sqrt(b)*x)) + (sqrt(c)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(a^(3/4)*b^(3/4)*sqrt(a + b*x^2)) - (sqrt(c)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(2*a^(3/4)*b^(3/4)*sqrt(a + b*x^2))]
    @test_int [1/((c*x)^(1/2)*(a + b*x^2)^(3/2)), x, 3, sqrt(c*x)/(a*c*sqrt(a + b*x^2)) + ((sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(2*a^(5/4)*b^(1/4)*sqrt(c)*sqrt(a + b*x^2))]
    @test_int [1/((c*x)^(3/2)*(a + b*x^2)^(3/2)), x, 6, 1/(a*c*sqrt(c*x)*sqrt(a + b*x^2)) - (3*sqrt(a + b*x^2))/(a^2*c*sqrt(c*x)) + (3*sqrt(b)*sqrt(c*x)*sqrt(a + b*x^2))/(a^2*c^2*(sqrt(a) + sqrt(b)*x)) - (3*b^(1/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(a^(7/4)*c^(3/2)*sqrt(a + b*x^2)) + (3*b^(1/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(2*a^(7/4)*c^(3/2)*sqrt(a + b*x^2))]
    @test_int [1/((c*x)^(5/2)*(a + b*x^2)^(3/2)), x, 4, 1/(a*c*(c*x)^(3/2)*sqrt(a + b*x^2)) - (5*sqrt(a + b*x^2))/(3*a^2*c*(c*x)^(3/2)) - (5*b^(3/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(6*a^(9/4)*c^(5/2)*sqrt(a + b*x^2))]
    @test_int [1/((c*x)^(7/2)*(a + b*x^2)^(3/2)), x, 7, 1/(a*c*(c*x)^(5/2)*sqrt(a + b*x^2)) - (7*sqrt(a + b*x^2))/(5*a^2*c*(c*x)^(5/2)) + (21*b*sqrt(a + b*x^2))/(5*a^3*c^3*sqrt(c*x)) - (21*b^(3/2)*sqrt(c*x)*sqrt(a + b*x^2))/(5*a^3*c^4*(sqrt(a) + sqrt(b)*x)) + (21*b^(5/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(5*a^(11/4)*c^(7/2)*sqrt(a + b*x^2)) - (21*b^(5/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(10*a^(11/4)*c^(7/2)*sqrt(a + b*x^2))]

    @test_int [(c*x)^(7/2)/(a + b*x^2)^(5/2), x, 4, -((c*(c*x)^(5/2))/(3*b*(a + b*x^2)^(3/2))) - (5*c^3*sqrt(c*x))/(6*b^2*sqrt(a + b*x^2)) + (5*c^(7/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(12*a^(1/4)*b^(9/4)*sqrt(a + b*x^2))]
    @test_int [(c*x)^(5/2)/(a + b*x^2)^(5/2), x, 6, -((c*(c*x)^(3/2))/(3*b*(a + b*x^2)^(3/2))) + (c*(c*x)^(3/2))/(2*a*b*sqrt(a + b*x^2)) - (c^2*sqrt(c*x)*sqrt(a + b*x^2))/(2*a*b^(3/2)*(sqrt(a) + sqrt(b)*x)) + (c^(5/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(2*a^(3/4)*b^(7/4)*sqrt(a + b*x^2)) - (c^(5/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(4*a^(3/4)*b^(7/4)*sqrt(a + b*x^2))]
    @test_int [(c*x)^(3/2)/(a + b*x^2)^(5/2), x, 4, -((c*sqrt(c*x))/(3*b*(a + b*x^2)^(3/2))) + (c*sqrt(c*x))/(6*a*b*sqrt(a + b*x^2)) + (c^(3/2)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(12*a^(5/4)*b^(5/4)*sqrt(a + b*x^2))]
    @test_int [(c*x)^(1/2)/(a + b*x^2)^(5/2), x, 6, (c*x)^(3/2)/(3*a*c*(a + b*x^2)^(3/2)) + (c*x)^(3/2)/(2*a^2*c*sqrt(a + b*x^2)) - (sqrt(c*x)*sqrt(a + b*x^2))/(2*a^2*sqrt(b)*(sqrt(a) + sqrt(b)*x)) + (sqrt(c)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(2*a^(7/4)*b^(3/4)*sqrt(a + b*x^2)) - (sqrt(c)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(4*a^(7/4)*b^(3/4)*sqrt(a + b*x^2))]
    @test_int [1/((c*x)^(1/2)*(a + b*x^2)^(5/2)), x, 4, sqrt(c*x)/(3*a*c*(a + b*x^2)^(3/2)) + (5*sqrt(c*x))/(6*a^2*c*sqrt(a + b*x^2)) + (5*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(12*a^(9/4)*b^(1/4)*sqrt(c)*sqrt(a + b*x^2))]
    @test_int [1/((c*x)^(3/2)*(a + b*x^2)^(5/2)), x, 7, 1/(3*a*c*sqrt(c*x)*(a + b*x^2)^(3/2)) + 7/(6*a^2*c*sqrt(c*x)*sqrt(a + b*x^2)) - (7*sqrt(a + b*x^2))/(2*a^3*c*sqrt(c*x)) + (7*sqrt(b)*sqrt(c*x)*sqrt(a + b*x^2))/(2*a^3*c^2*(sqrt(a) + sqrt(b)*x)) - (7*b^(1/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(2*a^(11/4)*c^(3/2)*sqrt(a + b*x^2)) + (7*b^(1/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(4*a^(11/4)*c^(3/2)*sqrt(a + b*x^2))]
    @test_int [1/((c*x)^(5/2)*(a + b*x^2)^(5/2)), x, 5, 1/(3*a*c*(c*x)^(3/2)*(a + b*x^2)^(3/2)) + 3/(2*a^2*c*(c*x)^(3/2)*sqrt(a + b*x^2)) - (5*sqrt(a + b*x^2))/(2*a^3*c*(c*x)^(3/2)) - (5*b^(3/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(4*a^(13/4)*c^(5/2)*sqrt(a + b*x^2))]
    @test_int [1/((c*x)^(7/2)*(a + b*x^2)^(5/2)), x, 8, 1/(3*a*c*(c*x)^(5/2)*(a + b*x^2)^(3/2)) + 11/(6*a^2*c*(c*x)^(5/2)*sqrt(a + b*x^2)) - (77*sqrt(a + b*x^2))/(30*a^3*c*(c*x)^(5/2)) + (77*b*sqrt(a + b*x^2))/(10*a^4*c^3*sqrt(c*x)) - (77*b^(3/2)*sqrt(c*x)*sqrt(a + b*x^2))/(10*a^4*c^4*(sqrt(a) + sqrt(b)*x)) + (77*b^(5/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(10*a^(15/4)*c^(7/2)*sqrt(a + b*x^2)) - (77*b^(5/4)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(c*x))/(a^(1/4)*sqrt(c))), 1/2))/(20*a^(15/4)*c^(7/2)*sqrt(a + b*x^2))]

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

    @test_int [(c*x)^(5/2)/(3*a - 2*a*x^2)^(3/2), x, 5, (c*(c*x)^(3/2))/(2*a*sqrt(3*a - 2*a*x^2)) + (3*3^(1/4)*c^2*sqrt(c*x)*sqrt(3 - 2*x^2)*Elliptic.E(asin(sqrt(3 - sqrt(6)*x)/sqrt(6)), 2))/(2*2^(3/4)*a*sqrt(x)*sqrt(3*a - 2*a*x^2))]
    @test_int [(c*x)^(3/2)/(3*a - 2*a*x^2)^(3/2), x, 4, (c*sqrt(c*x))/(2*a*sqrt(3*a - 2*a*x^2)) - (c^(3/2)*sqrt(3 - 2*x^2)*Elliptic.F(asin(((2/3)^(1/4)*sqrt(c*x))/sqrt(c)), -1))/(2*6^(1/4)*a*sqrt(a*(3 - 2*x^2)))]
    @test_int [(c*x)^(1/2)/(3*a - 2*a*x^2)^(3/2), x, 5, (c*x)^(3/2)/(3*a*c*sqrt(3*a - 2*a*x^2)) + (sqrt(c*x)*sqrt(3 - 2*x^2)*Elliptic.E(asin(sqrt(3 - sqrt(6)*x)/sqrt(6)), 2))/(6^(3/4)*a*sqrt(x)*sqrt(3*a - 2*a*x^2))]
    @test_int [1/((c*x)^(1/2)*(3*a - 2*a*x^2)^(3/2)), x, 4, sqrt(c*x)/(3*a*c*sqrt(3*a - 2*a*x^2)) + (sqrt(3 - 2*x^2)*Elliptic.F(asin(((2/3)^(1/4)*sqrt(c*x))/sqrt(c)), -1))/(3*6^(1/4)*a*sqrt(c)*sqrt(a*(3 - 2*x^2)))]
    @test_int [1/((c*x)^(3/2)*(3*a - 2*a*x^2)^(3/2)), x, 6, 1/(3*a*c*sqrt(c*x)*sqrt(3*a - 2*a*x^2)) - sqrt(3*a - 2*a*x^2)/(3*a^2*c*sqrt(c*x)) + (2^(1/4)*sqrt(c*x)*sqrt(3 - 2*x^2)*Elliptic.E(asin(sqrt(3 - sqrt(6)*x)/sqrt(6)), 2))/(3^(3/4)*a*c^2*sqrt(x)*sqrt(3*a - 2*a*x^2))]
    @test_int [1/((c*x)^(5/2)*(3*a - 2*a*x^2)^(3/2)), x, 5, 1/(3*a*c*(c*x)^(3/2)*sqrt(3*a - 2*a*x^2)) - (5*sqrt(3*a - 2*a*x^2))/(27*a^2*c*(c*x)^(3/2)) + (5*2^(3/4)*sqrt(3 - 2*x^2)*Elliptic.F(asin(((2/3)^(1/4)*sqrt(c*x))/sqrt(c)), -1))/(27*3^(1/4)*a*c^(5/2)*sqrt(a*(3 - 2*x^2)))]

    #= Compare*with*Mathematica's*more*complicated*complex*result. =#
    @test_int [1/(sqrt(x)*sqrt(1 - a^2*x^2)), x, 2, (2*Elliptic.F(asin(sqrt(a)*sqrt(x)), -1))/sqrt(a)]
    @test_int [1/(sqrt(x)*sqrt(1 + a*x^2)), x, 2, ((1 + sqrt(a)*x)*sqrt((1 + a*x^2)/(1 + sqrt(a)*x)^2)*Elliptic.F(2*atan(a^(1/4)*sqrt(x)), 1/2))/(a^(1/4)*sqrt(1 + a*x^2))]

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

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

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

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

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

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(c*x)^m*(a+b*x^2)^(p/3)=#

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

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

    @test_int [x^7*(a + b*x^2)^(1/3), x, 3, (-3*a^3*(a + b*x^2)^(4/3))/(8*b^4) + (9*a^2*(a + b*x^2)^(7/3))/(14*b^4) - (9*a*(a + b*x^2)^(10/3))/(20*b^4) + (3*(a + b*x^2)^(13/3))/(26*b^4)]
    @test_int [x^5*(a + b*x^2)^(1/3), x, 3, (3*a^2*(a + b*x^2)^(4/3))/(8*b^3) - (3*a*(a + b*x^2)^(7/3))/(7*b^3) + (3*(a + b*x^2)^(10/3))/(20*b^3)]
    @test_int [x^3*(a + b*x^2)^(1/3), x, 3, (-3*a*(a + b*x^2)^(4/3))/(8*b^2) + (3*(a + b*x^2)^(7/3))/(14*b^2)]
    @test_int [x^1*(a + b*x^2)^(1/3), x, 1, (3*(a + b*x^2)^(4/3))/(8*b)]
    @test_int [(a + b*x^2)^(1/3)/x^1, x, 6, (3*(a + b*x^2)^(1/3))/2 - (sqrt(3)*a^(1/3)*atan((a^(1/3) + 2*(a + b*x^2)^(1/3))/(sqrt(3)*a^(1/3))))/2 - (a^(1/3)*log(x))/2 + (3*a^(1/3)*log(a^(1/3) - (a + b*x^2)^(1/3)))/4]
    @test_int [(a + b*x^2)^(1/3)/x^3, x, 6, -(a + b*x^2)^(1/3)/(2*x^2) - (b*atan((a^(1/3) + 2*(a + b*x^2)^(1/3))/(sqrt(3)*a^(1/3))))/(2*sqrt(3)*a^(2/3)) - (b*log(x))/(6*a^(2/3)) + (b*log(a^(1/3) - (a + b*x^2)^(1/3)))/(4*a^(2/3))]
    @test_int [(a + b*x^2)^(1/3)/x^5, x, 7, -(a + b*x^2)^(1/3)/(4*x^4) - (b*(a + b*x^2)^(1/3))/(12*a*x^2) + (b^2*atan((a^(1/3) + 2*(a + b*x^2)^(1/3))/(sqrt(3)*a^(1/3))))/(6*sqrt(3)*a^(5/3)) + (b^2*log(x))/(18*a^(5/3)) - (b^2*log(a^(1/3) - (a + b*x^2)^(1/3)))/(12*a^(5/3))]

    @test_int [x^4*(a + b*x^2)^(1/3), x, 5, -((54*a^2*x*(a + b*x^2)^(1/3))/(935*b^2)) + (6*a*x^3*(a + b*x^2)^(1/3))/(187*b) + (3/17)*x^5*(a + b*x^2)^(1/3) - (54*3^(3/4)*sqrt(2 - sqrt(3))*a^3*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(935*b^3*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]
    @test_int [x^2*(a + b*x^2)^(1/3), x, 4, (6*a*x*(a + b*x^2)^(1/3))/(55*b) + (3/11)*x^3*(a + b*x^2)^(1/3) + (6*3^(3/4)*sqrt(2 - sqrt(3))*a^2*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(55*b^2*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]
    @test_int [x^0*(a + b*x^2)^(1/3), x, 3, (3/5)*x*(a + b*x^2)^(1/3) - (2*3^(3/4)*sqrt(2 - sqrt(3))*a*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(5*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]
    @test_int [(a + b*x^2)^(1/3)/x^2, x, 3, -((a + b*x^2)^(1/3)/x) - (2*sqrt(2 - sqrt(3))*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(3^(1/4)*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]
    @test_int [(a + b*x^2)^(1/3)/x^4, x, 4, -((a + b*x^2)^(1/3)/(3*x^3)) - (2*b*(a + b*x^2)^(1/3))/(9*a*x) + (2*sqrt(2 - sqrt(3))*b*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(9*3^(1/4)*a*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]

    @test_int [x^7*(a + b*x^2)^(2/3), x, 3, (-3*a^3*(a + b*x^2)^(5/3))/(10*b^4) + (9*a^2*(a + b*x^2)^(8/3))/(16*b^4) - (9*a*(a + b*x^2)^(11/3))/(22*b^4) + (3*(a + b*x^2)^(14/3))/(28*b^4)]
    @test_int [x^5*(a + b*x^2)^(2/3), x, 3, (3*a^2*(a + b*x^2)^(5/3))/(10*b^3) - (3*a*(a + b*x^2)^(8/3))/(8*b^3) + (3*(a + b*x^2)^(11/3))/(22*b^3)]
    @test_int [x^3*(a + b*x^2)^(2/3), x, 3, (-3*a*(a + b*x^2)^(5/3))/(10*b^2) + (3*(a + b*x^2)^(8/3))/(16*b^2)]
    @test_int [x^1*(a + b*x^2)^(2/3), x, 1, (3*(a + b*x^2)^(5/3))/(10*b)]
    @test_int [(a + b*x^2)^(2/3)/x^1, x, 6, (3*(a + b*x^2)^(2/3))/4 + (sqrt(3)*a^(2/3)*atan((a^(1/3) + 2*(a + b*x^2)^(1/3))/(sqrt(3)*a^(1/3))))/2 - (a^(2/3)*log(x))/2 + (3*a^(2/3)*log(a^(1/3) - (a + b*x^2)^(1/3)))/4]
    @test_int [(a + b*x^2)^(2/3)/x^3, x, 6, -(a + b*x^2)^(2/3)/(2*x^2) + (b*atan((a^(1/3) + 2*(a + b*x^2)^(1/3))/(sqrt(3)*a^(1/3))))/(sqrt(3)*a^(1/3)) - (b*log(x))/(3*a^(1/3)) + (b*log(a^(1/3) - (a + b*x^2)^(1/3)))/(2*a^(1/3))]
    @test_int [(a + b*x^2)^(2/3)/x^5, x, 7, -(a + b*x^2)^(2/3)/(4*x^4) - (b*(a + b*x^2)^(2/3))/(6*a*x^2) - (b^2*atan((a^(1/3) + 2*(a + b*x^2)^(1/3))/(sqrt(3)*a^(1/3))))/(6*sqrt(3)*a^(4/3)) + (b^2*log(x))/(18*a^(4/3)) - (b^2*log(a^(1/3) - (a + b*x^2)^(1/3)))/(12*a^(4/3))]

    @test_int [x^4*(a + b*x^2)^(2/3), x, 7, -((108*a^2*x*(a + b*x^2)^(2/3))/(1729*b^2)) + (12*a*x^3*(a + b*x^2)^(2/3))/(247*b) + (3/19)*x^5*(a + b*x^2)^(2/3) - (324*a^3*x)/(1729*b^2*((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))) + (162*3^(1/4)*sqrt(2 + sqrt(3))*a^(10/3)*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(1729*b^3*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2))) - (108*sqrt(2)*3^(3/4)*a^(10/3)*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(1729*b^3*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]
    @test_int [x^2*(a + b*x^2)^(2/3), x, 6, (12*a*x*(a + b*x^2)^(2/3))/(91*b) + (3/13)*x^3*(a + b*x^2)^(2/3) + (36*a^2*x)/(91*b*((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))) - (18*3^(1/4)*sqrt(2 + sqrt(3))*a^(7/3)*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(91*b^2*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2))) + (12*sqrt(2)*3^(3/4)*a^(7/3)*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(91*b^2*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]
    @test_int [x^0*(a + b*x^2)^(2/3), x, 5, (3/7)*x*(a + b*x^2)^(2/3) - (12*a*x)/(7*((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))) + (6*3^(1/4)*sqrt(2 + sqrt(3))*a^(4/3)*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(7*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2))) - (4*sqrt(2)*3^(3/4)*a^(4/3)*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(7*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]
    @test_int [(a + b*x^2)^(2/3)/x^2, x, 5, -((a + b*x^2)^(2/3)/x) - (4*b*x)/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3)) + (2*3^(1/4)*sqrt(2 + sqrt(3))*a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2))) - (4*sqrt(2)*a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(3^(1/4)*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]
    @test_int [(a + b*x^2)^(2/3)/x^4, x, 6, -((a + b*x^2)^(2/3)/(3*x^3)) - (4*b*(a + b*x^2)^(2/3))/(9*a*x) - (4*b^2*x)/(9*a*((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))) + (2*3^(1/4)*sqrt(2 + sqrt(3))*b*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(9*a^(2/3)*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2))) - (4*sqrt(2)*b*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(9*3^(1/4)*a^(2/3)*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]

    @test_int [x^7*(a + b*x^2)^(4/3), x, 3, (-3*a^3*(a + b*x^2)^(7/3))/(14*b^4) + (9*a^2*(a + b*x^2)^(10/3))/(20*b^4) - (9*a*(a + b*x^2)^(13/3))/(26*b^4) + (3*(a + b*x^2)^(16/3))/(32*b^4)]
    @test_int [x^5*(a + b*x^2)^(4/3), x, 3, (3*a^2*(a + b*x^2)^(7/3))/(14*b^3) - (3*a*(a + b*x^2)^(10/3))/(10*b^3) + (3*(a + b*x^2)^(13/3))/(26*b^3)]
    @test_int [x^3*(a + b*x^2)^(4/3), x, 3, (-3*a*(a + b*x^2)^(7/3))/(14*b^2) + (3*(a + b*x^2)^(10/3))/(20*b^2)]
    @test_int [x^1*(a + b*x^2)^(4/3), x, 1, (3*(a + b*x^2)^(7/3))/(14*b)]
    @test_int [(a + b*x^2)^(4/3)/x^1, x, 7, (3*a*(a + b*x^2)^(1/3))/2 + (3*(a + b*x^2)^(4/3))/8 - (sqrt(3)*a^(4/3)*atan((a^(1/3) + 2*(a + b*x^2)^(1/3))/(sqrt(3)*a^(1/3))))/2 - (a^(4/3)*log(x))/2 + (3*a^(4/3)*log(a^(1/3) - (a + b*x^2)^(1/3)))/4]
    @test_int [(a + b*x^2)^(4/3)/x^3, x, 7, 2*b*(a + b*x^2)^(1/3) - (a + b*x^2)^(4/3)/(2*x^2) - (2*a^(1/3)*b*atan((a^(1/3) + 2*(a + b*x^2)^(1/3))/(sqrt(3)*a^(1/3))))/sqrt(3) - (2*a^(1/3)*b*log(x))/3 + a^(1/3)*b*log(a^(1/3) - (a + b*x^2)^(1/3))]
    @test_int [(a + b*x^2)^(4/3)/x^5, x, 7, -(b*(a + b*x^2)^(1/3))/(3*x^2) - (a + b*x^2)^(4/3)/(4*x^4) - (b^2*atan((a^(1/3) + 2*(a + b*x^2)^(1/3))/(sqrt(3)*a^(1/3))))/(3*sqrt(3)*a^(2/3)) - (b^2*log(x))/(9*a^(2/3)) + (b^2*log(a^(1/3) - (a + b*x^2)^(1/3)))/(6*a^(2/3))]

    @test_int [x^4*(a + b*x^2)^(4/3), x, 6, -((432*a^3*x*(a + b*x^2)^(1/3))/(21505*b^2)) + (48*a^2*x^3*(a + b*x^2)^(1/3))/(4301*b) + (24/391)*a*x^5*(a + b*x^2)^(1/3) + (3/23)*x^5*(a + b*x^2)^(4/3) - (432*3^(3/4)*sqrt(2 - sqrt(3))*a^4*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(21505*b^3*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]
    @test_int [x^2*(a + b*x^2)^(4/3), x, 5, (48*a^2*x*(a + b*x^2)^(1/3))/(935*b) + (24/187)*a*x^3*(a + b*x^2)^(1/3) + (3/17)*x^3*(a + b*x^2)^(4/3) + (48*3^(3/4)*sqrt(2 - sqrt(3))*a^3*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(935*b^2*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]
    @test_int [x^0*(a + b*x^2)^(4/3), x, 4, (24/55)*a*x*(a + b*x^2)^(1/3) + (3/11)*x*(a + b*x^2)^(4/3) - (16*3^(3/4)*sqrt(2 - sqrt(3))*a^2*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(55*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]
    @test_int [(a + b*x^2)^(4/3)/x^2, x, 4, (8/5)*b*x*(a + b*x^2)^(1/3) - (a + b*x^2)^(4/3)/x - (16*sqrt(2 - sqrt(3))*a*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(5*3^(1/4)*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]
    @test_int [(a + b*x^2)^(4/3)/x^4, x, 4, -((8*b*(a + b*x^2)^(1/3))/(9*x)) - (a + b*x^2)^(4/3)/(3*x^3) - (16*sqrt(2 - sqrt(3))*b*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(9*3^(1/4)*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]

    @test_int [x*(-1 + x^2)^(7/3), x, 1, (3*(-1 + x^2)^(10/3))/20]

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

    @test_int [x^7/(a + b*x^2)^(1/3), x, 3, -((3*a^3*(a + b*x^2)^(2/3))/(4*b^4)) + (9*a^2*(a + b*x^2)^(5/3))/(10*b^4) - (9*a*(a + b*x^2)^(8/3))/(16*b^4) + (3*(a + b*x^2)^(11/3))/(22*b^4)]
    @test_int [x^5/(a + b*x^2)^(1/3), x, 3, (3*a^2*(a + b*x^2)^(2/3))/(4*b^3) - (3*a*(a + b*x^2)^(5/3))/(5*b^3) + (3*(a + b*x^2)^(8/3))/(16*b^3)]
    @test_int [x^3/(a + b*x^2)^(1/3), x, 3, -((3*a*(a + b*x^2)^(2/3))/(4*b^2)) + (3*(a + b*x^2)^(5/3))/(10*b^2)]
    @test_int [x^1/(a + b*x^2)^(1/3), x, 1, (3*(a + b*x^2)^(2/3))/(4*b)]
    @test_int [1/(x^1*(a + b*x^2)^(1/3)), x, 5, (sqrt(3)*atan((a^(1/3) + 2*(a + b*x^2)^(1/3))/(sqrt(3)*a^(1/3))))/(2*a^(1/3)) - log(x)/(2*a^(1/3)) + (3*log(a^(1/3) - (a + b*x^2)^(1/3)))/(4*a^(1/3))]
    @test_int [1/(x^3*(a + b*x^2)^(1/3)), x, 6, -((a + b*x^2)^(2/3)/(2*a*x^2)) - (b*atan((a^(1/3) + 2*(a + b*x^2)^(1/3))/(sqrt(3)*a^(1/3))))/(2*sqrt(3)*a^(4/3)) + (b*log(x))/(6*a^(4/3)) - (b*log(a^(1/3) - (a + b*x^2)^(1/3)))/(4*a^(4/3))]
    @test_int [1/(x^5*(a + b*x^2)^(1/3)), x, 7, -((a + b*x^2)^(2/3)/(4*a*x^4)) + (b*(a + b*x^2)^(2/3))/(3*a^2*x^2) + (b^2*atan((a^(1/3) + 2*(a + b*x^2)^(1/3))/(sqrt(3)*a^(1/3))))/(3*sqrt(3)*a^(7/3)) - (b^2*log(x))/(9*a^(7/3)) + (b^2*log(a^(1/3) - (a + b*x^2)^(1/3)))/(6*a^(7/3))]

    @test_int [x^4/(a + b*x^2)^(1/3), x, 6, -((27*a*x*(a + b*x^2)^(2/3))/(91*b^2)) + (3*x^3*(a + b*x^2)^(2/3))/(13*b) - (81*a^2*x)/(91*b^2*((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))) + (81*3^(1/4)*sqrt(2 + sqrt(3))*a^(7/3)*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(182*b^3*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2))) - (27*sqrt(2)*3^(3/4)*a^(7/3)*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(91*b^3*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]
    @test_int [x^2/(a + b*x^2)^(1/3), x, 5, (3*x*(a + b*x^2)^(2/3))/(7*b) + (9*a*x)/(7*b*((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))) - (9*3^(1/4)*sqrt(2 + sqrt(3))*a^(4/3)*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(14*b^2*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2))) + (3*sqrt(2)*3^(3/4)*a^(4/3)*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(7*b^2*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]
    @test_int [x^0/(a + b*x^2)^(1/3), x, 4, -((3*x)/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))) + (3*3^(1/4)*sqrt(2 + sqrt(3))*a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(2*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2))) - (sqrt(2)*3^(3/4)*a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]
    @test_int [1/(x^2*(a + b*x^2)^(1/3)), x, 5, -((a + b*x^2)^(2/3)/(a*x)) - (b*x)/(a*((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))) + (3^(1/4)*sqrt(2 + sqrt(3))*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(2*a^(2/3)*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2))) - (sqrt(2)*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(3^(1/4)*a^(2/3)*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]
    @test_int [1/(x^4*(a + b*x^2)^(1/3)), x, 6, -((a + b*x^2)^(2/3)/(3*a*x^3)) + (5*b*(a + b*x^2)^(2/3))/(9*a^2*x) + (5*b^2*x)/(9*a^2*((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))) - (5*3^(1/4)*sqrt(2 + sqrt(3))*b*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(18*a^(5/3)*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2))) + (5*sqrt(2)*b*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(9*3^(1/4)*a^(5/3)*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]

    @test_int [x^7/(a + b*x^2)^(2/3), x, 3, -((3*a^3*(a + b*x^2)^(1/3))/(2*b^4)) + (9*a^2*(a + b*x^2)^(4/3))/(8*b^4) - (9*a*(a + b*x^2)^(7/3))/(14*b^4) + (3*(a + b*x^2)^(10/3))/(20*b^4)]
    @test_int [x^5/(a + b*x^2)^(2/3), x, 3, (3*a^2*(a + b*x^2)^(1/3))/(2*b^3) - (3*a*(a + b*x^2)^(4/3))/(4*b^3) + (3*(a + b*x^2)^(7/3))/(14*b^3)]
    @test_int [x^3/(a + b*x^2)^(2/3), x, 3, -((3*a*(a + b*x^2)^(1/3))/(2*b^2)) + (3*(a + b*x^2)^(4/3))/(8*b^2)]
    @test_int [x^1/(a + b*x^2)^(2/3), x, 1, (3*(a + b*x^2)^(1/3))/(2*b)]
    @test_int [1/(x^1*(a + b*x^2)^(2/3)), x, 5, -((sqrt(3)*atan((a^(1/3) + 2*(a + b*x^2)^(1/3))/(sqrt(3)*a^(1/3))))/(2*a^(2/3))) - log(x)/(2*a^(2/3)) + (3*log(a^(1/3) - (a + b*x^2)^(1/3)))/(4*a^(2/3))]
    @test_int [1/(x^3*(a + b*x^2)^(2/3)), x, 6, -((a + b*x^2)^(1/3)/(2*a*x^2)) + (b*atan((a^(1/3) + 2*(a + b*x^2)^(1/3))/(sqrt(3)*a^(1/3))))/(sqrt(3)*a^(5/3)) + (b*log(x))/(3*a^(5/3)) - (b*log(a^(1/3) - (a + b*x^2)^(1/3)))/(2*a^(5/3))]
    @test_int [1/(x^5*(a + b*x^2)^(2/3)), x, 7, -((a + b*x^2)^(1/3)/(4*a*x^4)) + (5*b*(a + b*x^2)^(1/3))/(12*a^2*x^2) - (5*b^2*atan((a^(1/3) + 2*(a + b*x^2)^(1/3))/(sqrt(3)*a^(1/3))))/(6*sqrt(3)*a^(8/3)) - (5*b^2*log(x))/(18*a^(8/3)) + (5*b^2*log(a^(1/3) - (a + b*x^2)^(1/3)))/(12*a^(8/3))]

    @test_int [x^4/(a + b*x^2)^(2/3), x, 4, -((27*a*x*(a + b*x^2)^(1/3))/(55*b^2)) + (3*x^3*(a + b*x^2)^(1/3))/(11*b) - (27*3^(3/4)*sqrt(2 - sqrt(3))*a^2*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(55*b^3*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]
    @test_int [x^2/(a + b*x^2)^(2/3), x, 3, (3*x*(a + b*x^2)^(1/3))/(5*b) + (3*3^(3/4)*sqrt(2 - sqrt(3))*a*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(5*b^2*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]
    @test_int [x^0/(a + b*x^2)^(2/3), x, 2, -((3^(3/4)*sqrt(2 - sqrt(3))*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2))))]
    @test_int [1/(x^2*(a + b*x^2)^(2/3)), x, 3, -((a + b*x^2)^(1/3)/(a*x)) + (sqrt(2 - sqrt(3))*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(3^(1/4)*a*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]
    @test_int [1/(x^4*(a + b*x^2)^(2/3)), x, 4, -((a + b*x^2)^(1/3)/(3*a*x^3)) + (7*b*(a + b*x^2)^(1/3))/(9*a^2*x) - (7*sqrt(2 - sqrt(3))*b*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(9*3^(1/4)*a^2*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]

    @test_int [x^7/(a + b*x^2)^(4/3), x, 3, (3*a^3)/(2*b^4*(a + b*x^2)^(1/3)) + (9*a^2*(a + b*x^2)^(2/3))/(4*b^4) - (9*a*(a + b*x^2)^(5/3))/(10*b^4) + (3*(a + b*x^2)^(8/3))/(16*b^4)]
    @test_int [x^5/(a + b*x^2)^(4/3), x, 3, -((3*a^2)/(2*b^3*(a + b*x^2)^(1/3))) - (3*a*(a + b*x^2)^(2/3))/(2*b^3) + (3*(a + b*x^2)^(5/3))/(10*b^3)]
    @test_int [x^3/(a + b*x^2)^(4/3), x, 3, (3*a)/(2*b^2*(a + b*x^2)^(1/3)) + (3*(a + b*x^2)^(2/3))/(4*b^2)]
    @test_int [x^1/(a + b*x^2)^(4/3), x, 1, -(3/(2*b*(a + b*x^2)^(1/3)))]
    @test_int [1/(x^1*(a + b*x^2)^(4/3)), x, 6, 3/(2*a*(a + b*x^2)^(1/3)) + (sqrt(3)*atan((a^(1/3) + 2*(a + b*x^2)^(1/3))/(sqrt(3)*a^(1/3))))/(2*a^(4/3)) - log(x)/(2*a^(4/3)) + (3*log(a^(1/3) - (a + b*x^2)^(1/3)))/(4*a^(4/3))]
    @test_int [1/(x^3*(a + b*x^2)^(4/3)), x, 7, -((2*b)/(a^2*(a + b*x^2)^(1/3))) - 1/(2*a*x^2*(a + b*x^2)^(1/3)) - (2*b*atan((a^(1/3) + 2*(a + b*x^2)^(1/3))/(sqrt(3)*a^(1/3))))/(sqrt(3)*a^(7/3)) + (2*b*log(x))/(3*a^(7/3)) - (b*log(a^(1/3) - (a + b*x^2)^(1/3)))/a^(7/3)]
    @test_int [1/(x^5*(a + b*x^2)^(4/3)), x, 8, (7*b^2)/(3*a^3*(a + b*x^2)^(1/3)) - 1/(4*a*x^4*(a + b*x^2)^(1/3)) + (7*b)/(12*a^2*x^2*(a + b*x^2)^(1/3)) + (7*b^2*atan((a^(1/3) + 2*(a + b*x^2)^(1/3))/(sqrt(3)*a^(1/3))))/(3*sqrt(3)*a^(10/3)) - (7*b^2*log(x))/(9*a^(10/3)) + (7*b^2*log(a^(1/3) - (a + b*x^2)^(1/3)))/(6*a^(10/3))]

    @test_int [x^4/(a + b*x^2)^(4/3), x, 6, -((3*x^3)/(2*b*(a + b*x^2)^(1/3))) + (27*x*(a + b*x^2)^(2/3))/(14*b^2) + (81*a*x)/(14*b^2*((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))) - (81*3^(1/4)*sqrt(2 + sqrt(3))*a^(4/3)*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(28*b^3*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2))) + (27*3^(3/4)*a^(4/3)*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(7*sqrt(2)*b^3*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]
    @test_int [x^2/(a + b*x^2)^(4/3), x, 5, -((3*x)/(2*b*(a + b*x^2)^(1/3))) - (9*x)/(2*b*((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))) + (9*3^(1/4)*sqrt(2 + sqrt(3))*a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(4*b^2*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2))) - (3*3^(3/4)*a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(sqrt(2)*b^2*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]
    @test_int [x^0/(a + b*x^2)^(4/3), x, 5, (3*x)/(2*a*(a + b*x^2)^(1/3)) + (3*x)/(2*a*((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))) - (3*3^(1/4)*sqrt(2 + sqrt(3))*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(4*a^(2/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2))) + (3^(3/4)*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(sqrt(2)*a^(2/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]
    @test_int [1/(x^2*(a + b*x^2)^(4/3)), x, 6, 3/(2*a*x*(a + b*x^2)^(1/3)) - (5*(a + b*x^2)^(2/3))/(2*a^2*x) - (5*b*x)/(2*a^2*((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))) + (5*3^(1/4)*sqrt(2 + sqrt(3))*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(4*a^(5/3)*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2))) - (5*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(sqrt(2)*3^(1/4)*a^(5/3)*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]
    @test_int [1/(x^4*(a + b*x^2)^(4/3)), x, 7, 3/(2*a*x^3*(a + b*x^2)^(1/3)) - (11*(a + b*x^2)^(2/3))/(6*a^2*x^3) + (55*b*(a + b*x^2)^(2/3))/(18*a^3*x) + (55*b^2*x)/(18*a^3*((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))) - (55*3^(1/4)*sqrt(2 + sqrt(3))*b*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(36*a^(8/3)*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2))) + (55*b*(a^(1/3) - (a + b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(9*sqrt(2)*3^(1/4)*a^(8/3)*x*sqrt(-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a + b*x^2)^(1/3))^2)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(c*x)^(m/3)*(a+b*x^2)^(p/3)=#

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

    @test_int [(c*x)^(13/3)*(a + b*x^2)^(1/3), x, 6, -((5*a^2*c^3*(c*x)^(4/3)*(a + b*x^2)^(1/3))/(108*b^2)) + (a*c*(c*x)^(10/3)*(a + b*x^2)^(1/3))/(36*b) + ((c*x)^(16/3)*(a + b*x^2)^(1/3))/(6*c) - (5*a^3*c^(13/3)*atan((1 + (2*b^(1/3)*(c*x)^(2/3))/(c^(2/3)*(a + b*x^2)^(1/3)))/sqrt(3)))/(54*sqrt(3)*b^(8/3)) - (5*a^3*c^(13/3)*log(b^(1/3)*(c*x)^(2/3) - c^(2/3)*(a + b*x^2)^(1/3)))/(108*b^(8/3))]
    @test_int [(c*x)^(7/3)*(a + b*x^2)^(1/3), x, 5, (a*c*(c*x)^(4/3)*(a + b*x^2)^(1/3))/(12*b) + ((c*x)^(10/3)*(a + b*x^2)^(1/3))/(4*c) + (a^2*c^(7/3)*atan((1 + (2*b^(1/3)*(c*x)^(2/3))/(c^(2/3)*(a + b*x^2)^(1/3)))/sqrt(3)))/(6*sqrt(3)*b^(5/3)) + (a^2*c^(7/3)*log(b^(1/3)*(c*x)^(2/3) - c^(2/3)*(a + b*x^2)^(1/3)))/(12*b^(5/3))]
    @test_int [(c*x)^(1/3)*(a + b*x^2)^(1/3), x, 4, ((c*x)^(4/3)*(a + b*x^2)^(1/3))/(2*c) - (a*c^(1/3)*atan((1 + (2*b^(1/3)*(c*x)^(2/3))/(c^(2/3)*(a + b*x^2)^(1/3)))/sqrt(3)))/(2*sqrt(3)*b^(2/3)) - (a*c^(1/3)*log(b^(1/3)*(c*x)^(2/3) - c^(2/3)*(a + b*x^2)^(1/3)))/(4*b^(2/3))]
    @test_int [(a + b*x^2)^(1/3)/(c*x)^(5/3), x, 4, -((3*(a + b*x^2)^(1/3))/(2*c*(c*x)^(2/3))) - (sqrt(3)*b^(1/3)*atan((1 + (2*b^(1/3)*(c*x)^(2/3))/(c^(2/3)*(a + b*x^2)^(1/3)))/sqrt(3)))/(2*c^(5/3)) - (3*b^(1/3)*log(b^(1/3)*(c*x)^(2/3) - c^(2/3)*(a + b*x^2)^(1/3)))/(4*c^(5/3))]
    @test_int [(a + b*x^2)^(1/3)/(c*x)^(11/3), x, 1, -((3*(a + b*x^2)^(4/3))/(8*a*c*(c*x)^(8/3)))]
    @test_int [(a + b*x^2)^(1/3)/(c*x)^(17/3), x, 2, -((3*(a + b*x^2)^(4/3))/(8*a*c*(c*x)^(14/3))) + (9*(a + b*x^2)^(7/3))/(56*a^2*c*(c*x)^(14/3))]
    @test_int [(a + b*x^2)^(1/3)/(c*x)^(23/3), x, 3, -((3*(a + b*x^2)^(4/3))/(8*a*c*(c*x)^(20/3))) + (9*(a + b*x^2)^(7/3))/(28*a^2*c*(c*x)^(20/3)) - (27*(a + b*x^2)^(10/3))/(280*a^3*c*(c*x)^(20/3))]
    @test_int [(a + b*x^2)^(1/3)/(c*x)^(29/3), x, 4, -((3*(a + b*x^2)^(4/3))/(8*a*c*(c*x)^(26/3))) + (27*(a + b*x^2)^(7/3))/(56*a^2*c*(c*x)^(26/3)) - (81*(a + b*x^2)^(10/3))/(280*a^3*c*(c*x)^(26/3)) + (243*(a + b*x^2)^(13/3))/(3640*a^4*c*(c*x)^(26/3))]

    @test_int [(c*x)^(10/3)*(a + b*x^2)^(1/3), x, 6, -((14*a^2*c^3*(c*x)^(1/3)*(a + b*x^2)^(1/3))/(135*b^2)) + (2*a*c*(c*x)^(7/3)*(a + b*x^2)^(1/3))/(45*b) + ((c*x)^(13/3)*(a + b*x^2)^(1/3))/(5*c) + (7*a^2*c^(7/3)*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))*sqrt((c^(4/3) + (b^(2/3)*(c*x)^(4/3))/(a + b*x^2)^(2/3) + (b^(1/3)*c^(2/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2)*Elliptic.F(acos((c^(2/3) - ((1 - sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))), (1/4)*(2 + sqrt(3))))/(135*3^(1/4)*b^2*sqrt(-((b^(1/3)*(c*x)^(2/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)))/((a + b*x^2)^(1/3)*(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2))))]
    @test_int [(c*x)^(4/3)*(a + b*x^2)^(1/3), x, 5, (2*a*c*(c*x)^(1/3)*(a + b*x^2)^(1/3))/(9*b) + ((c*x)^(7/3)*(a + b*x^2)^(1/3))/(3*c) - (a*c^(1/3)*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))*sqrt((c^(4/3) + (b^(2/3)*(c*x)^(4/3))/(a + b*x^2)^(2/3) + (b^(1/3)*c^(2/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2)*Elliptic.F(acos((c^(2/3) - ((1 - sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))), (1/4)*(2 + sqrt(3))))/(9*3^(1/4)*b*sqrt(-((b^(1/3)*(c*x)^(2/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)))/((a + b*x^2)^(1/3)*(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2))))]
    @test_int [(a + b*x^2)^(1/3)/(c*x)^(2/3), x, 4, ((c*x)^(1/3)*(a + b*x^2)^(1/3))/c + ((c*x)^(1/3)*(a + b*x^2)^(1/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))*sqrt((c^(4/3) + (b^(2/3)*(c*x)^(4/3))/(a + b*x^2)^(2/3) + (b^(1/3)*c^(2/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2)*Elliptic.F(acos((c^(2/3) - ((1 - sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))), (1/4)*(2 + sqrt(3))))/(3^(1/4)*c^(5/3)*sqrt(-((b^(1/3)*(c*x)^(2/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)))/((a + b*x^2)^(1/3)*(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2))))]
    @test_int [(a + b*x^2)^(1/3)/(c*x)^(8/3), x, 4, -((3*(a + b*x^2)^(1/3))/(5*c*(c*x)^(5/3))) + (3^(3/4)*b*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))*sqrt((c^(4/3) + (b^(2/3)*(c*x)^(4/3))/(a + b*x^2)^(2/3) + (b^(1/3)*c^(2/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2)*Elliptic.F(acos((c^(2/3) - ((1 - sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))), (1/4)*(2 + sqrt(3))))/(5*a*c^(11/3)*sqrt(-((b^(1/3)*(c*x)^(2/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)))/((a + b*x^2)^(1/3)*(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2))))]
    @test_int [(a + b*x^2)^(1/3)/(c*x)^(14/3), x, 5, -((3*(a + b*x^2)^(1/3))/(11*c*(c*x)^(11/3))) - (6*b*(a + b*x^2)^(1/3))/(55*a*c^3*(c*x)^(5/3)) - (3*3^(3/4)*b^2*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))*sqrt((c^(4/3) + (b^(2/3)*(c*x)^(4/3))/(a + b*x^2)^(2/3) + (b^(1/3)*c^(2/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2)*Elliptic.F(acos((c^(2/3) - ((1 - sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))), (1/4)*(2 + sqrt(3))))/(55*a^2*c^(17/3)*sqrt(-((b^(1/3)*(c*x)^(2/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)))/((a + b*x^2)^(1/3)*(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2))))]

    @test_int [(c*x)^(2/3)*(a + b*x^2)^(1/3), x, 2, (3*(c*x)^(5/3)*(a + b*x^2)^(1/3)*HypergeometricFunctions._₂F₁(-(1/3), 5/6, 11/6, -((b*x^2)/a)))/(5*c*(1 + (b*x^2)/a)^(1/3))]
    @test_int [(a + b*x^2)^(1/3)/(c*x)^(1/3), x, 2, (3*(c*x)^(2/3)*(a + b*x^2)^(1/3)*HypergeometricFunctions._₂F₁(-(1/3), 1/3, 4/3, -((b*x^2)/a)))/(2*c*(1 + (b*x^2)/a)^(1/3))]
    @test_int [(a + b*x^2)^(1/3)/(c*x)^(4/3), x, 2, -((3*(a + b*x^2)^(1/3)*HypergeometricFunctions._₂F₁(-(1/3), -(1/6), 5/6, -((b*x^2)/a)))/(c*(c*x)^(1/3)*(1 + (b*x^2)/a)^(1/3)))]

    @test_int [(c*x)^(13/3)*(a + b*x^2)^(4/3), x, 7, -((5*a^3*c^3*(c*x)^(4/3)*(a + b*x^2)^(1/3))/(324*b^2)) + (a^2*c*(c*x)^(10/3)*(a + b*x^2)^(1/3))/(108*b) + (a*(c*x)^(16/3)*(a + b*x^2)^(1/3))/(18*c) + ((c*x)^(16/3)*(a + b*x^2)^(4/3))/(8*c) - (5*a^4*c^(13/3)*atan((1 + (2*b^(1/3)*(c*x)^(2/3))/(c^(2/3)*(a + b*x^2)^(1/3)))/sqrt(3)))/(162*sqrt(3)*b^(8/3)) - (5*a^4*c^(13/3)*log(b^(1/3)*(c*x)^(2/3) - c^(2/3)*(a + b*x^2)^(1/3)))/(324*b^(8/3))]
    @test_int [(c*x)^(7/3)*(a + b*x^2)^(4/3), x, 6, (a^2*c*(c*x)^(4/3)*(a + b*x^2)^(1/3))/(27*b) + (a*(c*x)^(10/3)*(a + b*x^2)^(1/3))/(9*c) + ((c*x)^(10/3)*(a + b*x^2)^(4/3))/(6*c) + (2*a^3*c^(7/3)*atan((1 + (2*b^(1/3)*(c*x)^(2/3))/(c^(2/3)*(a + b*x^2)^(1/3)))/sqrt(3)))/(27*sqrt(3)*b^(5/3)) + (a^3*c^(7/3)*log(b^(1/3)*(c*x)^(2/3) - c^(2/3)*(a + b*x^2)^(1/3)))/(27*b^(5/3))]
    @test_int [(c*x)^(1/3)*(a + b*x^2)^(4/3), x, 5, (a*(c*x)^(4/3)*(a + b*x^2)^(1/3))/(3*c) + ((c*x)^(4/3)*(a + b*x^2)^(4/3))/(4*c) - (a^2*c^(1/3)*atan((1 + (2*b^(1/3)*(c*x)^(2/3))/(c^(2/3)*(a + b*x^2)^(1/3)))/sqrt(3)))/(3*sqrt(3)*b^(2/3)) - (a^2*c^(1/3)*log(b^(1/3)*(c*x)^(2/3) - c^(2/3)*(a + b*x^2)^(1/3)))/(6*b^(2/3))]
    @test_int [(a + b*x^2)^(4/3)/(c*x)^(5/3), x, 5, (2*b*(c*x)^(4/3)*(a + b*x^2)^(1/3))/c^3 - (3*(a + b*x^2)^(4/3))/(2*c*(c*x)^(2/3)) - (2*a*b^(1/3)*atan((1 + (2*b^(1/3)*(c*x)^(2/3))/(c^(2/3)*(a + b*x^2)^(1/3)))/sqrt(3)))/(sqrt(3)*c^(5/3)) - (a*b^(1/3)*log(b^(1/3)*(c*x)^(2/3) - c^(2/3)*(a + b*x^2)^(1/3)))/c^(5/3)]
    @test_int [(a + b*x^2)^(4/3)/(c*x)^(11/3), x, 5, -((3*b*(a + b*x^2)^(1/3))/(2*c^3*(c*x)^(2/3))) - (3*(a + b*x^2)^(4/3))/(8*c*(c*x)^(8/3)) - (sqrt(3)*b^(4/3)*atan((1 + (2*b^(1/3)*(c*x)^(2/3))/(c^(2/3)*(a + b*x^2)^(1/3)))/sqrt(3)))/(2*c^(11/3)) - (3*b^(4/3)*log(b^(1/3)*(c*x)^(2/3) - c^(2/3)*(a + b*x^2)^(1/3)))/(4*c^(11/3))]
    @test_int [(a + b*x^2)^(4/3)/(c*x)^(17/3), x, 1, -((3*(a + b*x^2)^(7/3))/(14*a*c*(c*x)^(14/3)))]
    @test_int [(a + b*x^2)^(4/3)/(c*x)^(23/3), x, 2, -((3*(a + b*x^2)^(7/3))/(14*a*c*(c*x)^(20/3))) + (9*(a + b*x^2)^(10/3))/(140*a^2*c*(c*x)^(20/3))]
    @test_int [(a + b*x^2)^(4/3)/(c*x)^(29/3), x, 3, -((3*(a + b*x^2)^(7/3))/(14*a*c*(c*x)^(26/3))) + (9*(a + b*x^2)^(10/3))/(70*a^2*c*(c*x)^(26/3)) - (27*(a + b*x^2)^(13/3))/(910*a^3*c*(c*x)^(26/3))]

    @test_int [(c*x)^(10/3)*(a + b*x^2)^(4/3), x, 7, -((16*a^3*c^3*(c*x)^(1/3)*(a + b*x^2)^(1/3))/(405*b^2)) + (16*a^2*c*(c*x)^(7/3)*(a + b*x^2)^(1/3))/(945*b) + (8*a*(c*x)^(13/3)*(a + b*x^2)^(1/3))/(105*c) + ((c*x)^(13/3)*(a + b*x^2)^(4/3))/(7*c) + (8*a^3*c^(7/3)*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))*sqrt((c^(4/3) + (b^(2/3)*(c*x)^(4/3))/(a + b*x^2)^(2/3) + (b^(1/3)*c^(2/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2)*Elliptic.F(acos((c^(2/3) - ((1 - sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))), (1/4)*(2 + sqrt(3))))/(405*3^(1/4)*b^2*sqrt(-((b^(1/3)*(c*x)^(2/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)))/((a + b*x^2)^(1/3)*(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2))))]
    @test_int [(c*x)^(4/3)*(a + b*x^2)^(4/3), x, 6, (16*a^2*c*(c*x)^(1/3)*(a + b*x^2)^(1/3))/(135*b) + (8*a*(c*x)^(7/3)*(a + b*x^2)^(1/3))/(45*c) + ((c*x)^(7/3)*(a + b*x^2)^(4/3))/(5*c) - (8*a^2*c^(1/3)*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))*sqrt((c^(4/3) + (b^(2/3)*(c*x)^(4/3))/(a + b*x^2)^(2/3) + (b^(1/3)*c^(2/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2)*Elliptic.F(acos((c^(2/3) - ((1 - sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))), (1/4)*(2 + sqrt(3))))/(135*3^(1/4)*b*sqrt(-((b^(1/3)*(c*x)^(2/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)))/((a + b*x^2)^(1/3)*(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2))))]
    @test_int [(a + b*x^2)^(4/3)/(c*x)^(2/3), x, 5, (8*a*(c*x)^(1/3)*(a + b*x^2)^(1/3))/(9*c) + ((c*x)^(1/3)*(a + b*x^2)^(4/3))/(3*c) + (8*a*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))*sqrt((c^(4/3) + (b^(2/3)*(c*x)^(4/3))/(a + b*x^2)^(2/3) + (b^(1/3)*c^(2/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2)*Elliptic.F(acos((c^(2/3) - ((1 - sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))), (1/4)*(2 + sqrt(3))))/(9*3^(1/4)*c^(5/3)*sqrt(-((b^(1/3)*(c*x)^(2/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)))/((a + b*x^2)^(1/3)*(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2))))]
    @test_int [(a + b*x^2)^(4/3)/(c*x)^(8/3), x, 5, (8*b*(c*x)^(1/3)*(a + b*x^2)^(1/3))/(5*c^3) - (3*(a + b*x^2)^(4/3))/(5*c*(c*x)^(5/3)) + (8*b*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))*sqrt((c^(4/3) + (b^(2/3)*(c*x)^(4/3))/(a + b*x^2)^(2/3) + (b^(1/3)*c^(2/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2)*Elliptic.F(acos((c^(2/3) - ((1 - sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))), (1/4)*(2 + sqrt(3))))/(5*3^(1/4)*c^(11/3)*sqrt(-((b^(1/3)*(c*x)^(2/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)))/((a + b*x^2)^(1/3)*(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2))))]
    @test_int [(a + b*x^2)^(4/3)/(c*x)^(14/3), x, 5, -((24*b*(a + b*x^2)^(1/3))/(55*c^3*(c*x)^(5/3))) - (3*(a + b*x^2)^(4/3))/(11*c*(c*x)^(11/3)) + (8*3^(3/4)*b^2*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))*sqrt((c^(4/3) + (b^(2/3)*(c*x)^(4/3))/(a + b*x^2)^(2/3) + (b^(1/3)*c^(2/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2)*Elliptic.F(acos((c^(2/3) - ((1 - sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))), (1/4)*(2 + sqrt(3))))/(55*a*c^(17/3)*sqrt(-((b^(1/3)*(c*x)^(2/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)))/((a + b*x^2)^(1/3)*(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2))))]
    @test_int [(a + b*x^2)^(4/3)/(c*x)^(20/3), x, 6, -((24*b*(a + b*x^2)^(1/3))/(187*c^3*(c*x)^(11/3))) - (48*b^2*(a + b*x^2)^(1/3))/(935*a*c^5*(c*x)^(5/3)) - (3*(a + b*x^2)^(4/3))/(17*c*(c*x)^(17/3)) - (24*3^(3/4)*b^3*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))*sqrt((c^(4/3) + (b^(2/3)*(c*x)^(4/3))/(a + b*x^2)^(2/3) + (b^(1/3)*c^(2/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2)*Elliptic.F(acos((c^(2/3) - ((1 - sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))), (1/4)*(2 + sqrt(3))))/(935*a^2*c^(23/3)*sqrt(-((b^(1/3)*(c*x)^(2/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)))/((a + b*x^2)^(1/3)*(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2))))]

    @test_int [(c*x)^(2/3)*(a + b*x^2)^(4/3), x, 2, (3*a*(c*x)^(5/3)*(a + b*x^2)^(1/3)*HypergeometricFunctions._₂F₁(-(4/3), 5/6, 11/6, -((b*x^2)/a)))/(5*c*(1 + (b*x^2)/a)^(1/3))]
    @test_int [(a + b*x^2)^(4/3)/(c*x)^(1/3), x, 2, (3*a*(c*x)^(2/3)*(a + b*x^2)^(1/3)*HypergeometricFunctions._₂F₁(-(4/3), 1/3, 4/3, -((b*x^2)/a)))/(2*c*(1 + (b*x^2)/a)^(1/3))]
    @test_int [(a + b*x^2)^(4/3)/(c*x)^(4/3), x, 2, -((3*a*(a + b*x^2)^(1/3)*HypergeometricFunctions._₂F₁(-(4/3), -(1/6), 5/6, -((b*x^2)/a)))/(c*(c*x)^(1/3)*(1 + (b*x^2)/a)^(1/3)))]

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

    @test_int [(c*x)^(19/3)/(a + b*x^2)^(2/3), x, 6, (10*a^2*c^5*(c*x)^(4/3)*(a + b*x^2)^(1/3))/(27*b^3) - (2*a*c^3*(c*x)^(10/3)*(a + b*x^2)^(1/3))/(9*b^2) + (c*(c*x)^(16/3)*(a + b*x^2)^(1/3))/(6*b) + (20*a^3*c^(19/3)*atan((1 + (2*b^(1/3)*(c*x)^(2/3))/(c^(2/3)*(a + b*x^2)^(1/3)))/sqrt(3)))/(27*sqrt(3)*b^(11/3)) + (10*a^3*c^(19/3)*log(b^(1/3)*(c*x)^(2/3) - c^(2/3)*(a + b*x^2)^(1/3)))/(27*b^(11/3))]
    @test_int [(c*x)^(13/3)/(a + b*x^2)^(2/3), x, 5, -((5*a*c^3*(c*x)^(4/3)*(a + b*x^2)^(1/3))/(12*b^2)) + (c*(c*x)^(10/3)*(a + b*x^2)^(1/3))/(4*b) - (5*a^2*c^(13/3)*atan((1 + (2*b^(1/3)*(c*x)^(2/3))/(c^(2/3)*(a + b*x^2)^(1/3)))/sqrt(3)))/(6*sqrt(3)*b^(8/3)) - (5*a^2*c^(13/3)*log(b^(1/3)*(c*x)^(2/3) - c^(2/3)*(a + b*x^2)^(1/3)))/(12*b^(8/3))]
    @test_int [(c*x)^(7/3)/(a + b*x^2)^(2/3), x, 4, (c*(c*x)^(4/3)*(a + b*x^2)^(1/3))/(2*b) + (a*c^(7/3)*atan((1 + (2*b^(1/3)*(c*x)^(2/3))/(c^(2/3)*(a + b*x^2)^(1/3)))/sqrt(3)))/(sqrt(3)*b^(5/3)) + (a*c^(7/3)*log(b^(1/3)*(c*x)^(2/3) - c^(2/3)*(a + b*x^2)^(1/3)))/(2*b^(5/3))]
    @test_int [(c*x)^(1/3)/(a + b*x^2)^(2/3), x, 3, -((sqrt(3)*c^(1/3)*atan((1 + (2*b^(1/3)*(c*x)^(2/3))/(c^(2/3)*(a + b*x^2)^(1/3)))/sqrt(3)))/(2*b^(2/3))) - (3*c^(1/3)*log(b^(1/3)*(c*x)^(2/3) - c^(2/3)*(a + b*x^2)^(1/3)))/(4*b^(2/3))]
    @test_int [1/((c*x)^(5/3)*(a + b*x^2)^(2/3)), x, 1, -((3*(a + b*x^2)^(1/3))/(2*a*c*(c*x)^(2/3)))]
    @test_int [1/((c*x)^(11/3)*(a + b*x^2)^(2/3)), x, 2, -((3*(a + b*x^2)^(1/3))/(2*a*c*(c*x)^(8/3))) + (9*(a + b*x^2)^(4/3))/(8*a^2*c*(c*x)^(8/3))]
    @test_int [1/((c*x)^(17/3)*(a + b*x^2)^(2/3)), x, 3, -((3*(a + b*x^2)^(1/3))/(2*a*c*(c*x)^(14/3))) + (9*(a + b*x^2)^(4/3))/(4*a^2*c*(c*x)^(14/3)) - (27*(a + b*x^2)^(7/3))/(28*a^3*c*(c*x)^(14/3))]
    @test_int [1/((c*x)^(23/3)*(a + b*x^2)^(2/3)), x, 4, -((3*(a + b*x^2)^(1/3))/(2*a*c*(c*x)^(20/3))) + (27*(a + b*x^2)^(4/3))/(8*a^2*c*(c*x)^(20/3)) - (81*(a + b*x^2)^(7/3))/(28*a^3*c*(c*x)^(20/3)) + (243*(a + b*x^2)^(10/3))/(280*a^4*c*(c*x)^(20/3))]

    @test_int [(c*x)^(10/3)/(a + b*x^2)^(2/3), x, 5, -((7*a*c^3*(c*x)^(1/3)*(a + b*x^2)^(1/3))/(9*b^2)) + (c*(c*x)^(7/3)*(a + b*x^2)^(1/3))/(3*b) + (7*a*c^(7/3)*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))*sqrt((c^(4/3) + (b^(2/3)*(c*x)^(4/3))/(a + b*x^2)^(2/3) + (b^(1/3)*c^(2/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2)*Elliptic.F(acos((c^(2/3) - ((1 - sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))), (1/4)*(2 + sqrt(3))))/(18*3^(1/4)*b^2*sqrt(-((b^(1/3)*(c*x)^(2/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)))/((a + b*x^2)^(1/3)*(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2))))]
    @test_int [(c*x)^(4/3)/(a + b*x^2)^(2/3), x, 4, (c*(c*x)^(1/3)*(a + b*x^2)^(1/3))/b - (c^(1/3)*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))*sqrt((c^(4/3) + (b^(2/3)*(c*x)^(4/3))/(a + b*x^2)^(2/3) + (b^(1/3)*c^(2/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2)*Elliptic.F(acos((c^(2/3) - ((1 - sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))), (1/4)*(2 + sqrt(3))))/(2*3^(1/4)*b*sqrt(-((b^(1/3)*(c*x)^(2/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)))/((a + b*x^2)^(1/3)*(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2))))]
    @test_int [1/((c*x)^(2/3)*(a + b*x^2)^(2/3)), x, 3, (3^(3/4)*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))*sqrt((c^(4/3) + (b^(2/3)*(c*x)^(4/3))/(a + b*x^2)^(2/3) + (b^(1/3)*c^(2/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2)*Elliptic.F(acos((c^(2/3) - ((1 - sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))), (1/4)*(2 + sqrt(3))))/(2*a*c^(5/3)*sqrt(-((b^(1/3)*(c*x)^(2/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)))/((a + b*x^2)^(1/3)*(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2))))]
    @test_int [1/((c*x)^(8/3)*(a + b*x^2)^(2/3)), x, 4, -((3*(a + b*x^2)^(1/3))/(5*a*c*(c*x)^(5/3))) - (3*3^(3/4)*b*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))*sqrt((c^(4/3) + (b^(2/3)*(c*x)^(4/3))/(a + b*x^2)^(2/3) + (b^(1/3)*c^(2/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2)*Elliptic.F(acos((c^(2/3) - ((1 - sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))), (1/4)*(2 + sqrt(3))))/(10*a^2*c^(11/3)*sqrt(-((b^(1/3)*(c*x)^(2/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)))/((a + b*x^2)^(1/3)*(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2))))]
    @test_int [1/((c*x)^(14/3)*(a + b*x^2)^(2/3)), x, 5, -((3*(a + b*x^2)^(1/3))/(11*a*c*(c*x)^(11/3))) + (27*b*(a + b*x^2)^(1/3))/(55*a^2*c^3*(c*x)^(5/3)) + (27*3^(3/4)*b^2*(c*x)^(1/3)*(a + b*x^2)^(1/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))*sqrt((c^(4/3) + (b^(2/3)*(c*x)^(4/3))/(a + b*x^2)^(2/3) + (b^(1/3)*c^(2/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2)*Elliptic.F(acos((c^(2/3) - ((1 - sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))), (1/4)*(2 + sqrt(3))))/(110*a^3*c^(17/3)*sqrt(-((b^(1/3)*(c*x)^(2/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3)))/((a + b*x^2)^(1/3)*(c^(2/3) - ((1 + sqrt(3))*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2))))]

    @test_int [(c*x)^(2/3)/(a + b*x^2)^(2/3), x, 2, (3*(c*x)^(5/3)*(1 + (b*x^2)/a)^(2/3)*HypergeometricFunctions._₂F₁(2/3, 5/6, 11/6, -((b*x^2)/a)))/(5*c*(a + b*x^2)^(2/3))]
    @test_int [1/((c*x)^(1/3)*(a + b*x^2)^(2/3)), x, 2, (3*(c*x)^(2/3)*(1 + (b*x^2)/a)^(2/3)*HypergeometricFunctions._₂F₁(1/3, 2/3, 4/3, -((b*x^2)/a)))/(2*c*(a + b*x^2)^(2/3))]
    @test_int [1/((c*x)^(4/3)*(a + b*x^2)^(2/3)), x, 2, -((3*(1 + (b*x^2)/a)^(2/3)*HypergeometricFunctions._₂F₁(-(1/6), 2/3, 5/6, -((b*x^2)/a)))/(c*(c*x)^(1/3)*(a + b*x^2)^(2/3)))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(c*x)^m*(a+b*x^2)^(p/4)=#

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

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

    @test_int [x^4*(a + b*x^2)^(1/4), x, 5, -((4*a^2*x*(a + b*x^2)^(1/4))/(77*b^2)) + (2*a*x^3*(a + b*x^2)^(1/4))/(77*b) + (2/11)*x^5*(a + b*x^2)^(1/4) + (8*a^(7/2)*(1 + (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(77*b^(5/2)*(a + b*x^2)^(3/4))]
    @test_int [x^2*(a + b*x^2)^(1/4), x, 4, (2*a*x*(a + b*x^2)^(1/4))/(21*b) + (2/7)*x^3*(a + b*x^2)^(1/4) - (4*a^(5/2)*(1 + (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(21*b^(3/2)*(a + b*x^2)^(3/4))]
    @test_int [x^0*(a + b*x^2)^(1/4), x, 3, (2/3)*x*(a + b*x^2)^(1/4) + (2*a^(3/2)*(1 + (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(3*sqrt(b)*(a + b*x^2)^(3/4))]
    @test_int [(a + b*x^2)^(1/4)/x^2, x, 3, -((a + b*x^2)^(1/4)/x) + (sqrt(a)*sqrt(b)*(1 + (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(a + b*x^2)^(3/4)]
    @test_int [(a + b*x^2)^(1/4)/x^4, x, 4, -((a + b*x^2)^(1/4)/(3*x^3)) - (b*(a + b*x^2)^(1/4))/(6*a*x) - (b^(3/2)*(1 + (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(6*sqrt(a)*(a + b*x^2)^(3/4))]
    @test_int [(a + b*x^2)^(1/4)/x^6, x, 5, -((a + b*x^2)^(1/4)/(5*x^5)) - (b*(a + b*x^2)^(1/4))/(30*a*x^3) + (b^2*(a + b*x^2)^(1/4))/(12*a^2*x) + (b^(5/2)*(1 + (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(12*a^(3/2)*(a + b*x^2)^(3/4))]

    @test_int [x^4*(a - b*x^2)^(1/4), x, 5, -((4*a^2*x*(a - b*x^2)^(1/4))/(77*b^2)) - (2*a*x^3*(a - b*x^2)^(1/4))/(77*b) + (2/11)*x^5*(a - b*x^2)^(1/4) + (8*a^(7/2)*(1 - (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*asin((sqrt(b)*x)/sqrt(a)), 2))/(77*b^(5/2)*(a - b*x^2)^(3/4))]
    @test_int [x^2*(a - b*x^2)^(1/4), x, 4, -((2*a*x*(a - b*x^2)^(1/4))/(21*b)) + (2/7)*x^3*(a - b*x^2)^(1/4) + (4*a^(5/2)*(1 - (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*asin((sqrt(b)*x)/sqrt(a)), 2))/(21*b^(3/2)*(a - b*x^2)^(3/4))]
    @test_int [x^0*(a - b*x^2)^(1/4), x, 3, (2/3)*x*(a - b*x^2)^(1/4) + (2*a^(3/2)*(1 - (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*asin((sqrt(b)*x)/sqrt(a)), 2))/(3*sqrt(b)*(a - b*x^2)^(3/4))]
    @test_int [(a - b*x^2)^(1/4)/x^2, x, 3, -((a - b*x^2)^(1/4)/x) - (sqrt(a)*sqrt(b)*(1 - (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*asin((sqrt(b)*x)/sqrt(a)), 2))/(a - b*x^2)^(3/4)]
    @test_int [(a - b*x^2)^(1/4)/x^4, x, 4, -((a - b*x^2)^(1/4)/(3*x^3)) + (b*(a - b*x^2)^(1/4))/(6*a*x) - (b^(3/2)*(1 - (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*asin((sqrt(b)*x)/sqrt(a)), 2))/(6*sqrt(a)*(a - b*x^2)^(3/4))]
    @test_int [(a - b*x^2)^(1/4)/x^6, x, 5, -((a - b*x^2)^(1/4)/(5*x^5)) + (b*(a - b*x^2)^(1/4))/(30*a*x^3) + (b^2*(a - b*x^2)^(1/4))/(12*a^2*x) - (b^(5/2)*(1 - (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*asin((sqrt(b)*x)/sqrt(a)), 2))/(12*a^(3/2)*(a - b*x^2)^(3/4))]

    @test_int [x^4*(a + b*x^2)^(3/4), x, 6, (8*a^3*x)/(65*b^2*(a + b*x^2)^(1/4)) - (4*a^2*x*(a + b*x^2)^(3/4))/(65*b^2) + (2*a*x^3*(a + b*x^2)^(3/4))/(39*b) + (2*x^5*(a + b*x^2)^(3/4))/13 - (8*a^(7/2)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E(atan((sqrt(b)*x)/sqrt(a))/2, 2))/(65*b^(5/2)*(a + b*x^2)^(1/4))]
    @test_int [x^2*(a + b*x^2)^(3/4), x, 5, (-4*a^2*x)/(15*b*(a + b*x^2)^(1/4)) + (2*a*x*(a + b*x^2)^(3/4))/(15*b) + (2*x^3*(a + b*x^2)^(3/4))/9 + (4*a^(5/2)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E(atan((sqrt(b)*x)/sqrt(a))/2, 2))/(15*b^(3/2)*(a + b*x^2)^(1/4))]
    @test_int [x^0*(a + b*x^2)^(3/4), x, 4, (6*a*x)/(5*(a + b*x^2)^(1/4)) + (2*x*(a + b*x^2)^(3/4))/5 - (6*a^(3/2)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E(atan((sqrt(b)*x)/sqrt(a))/2, 2))/(5*sqrt(b)*(a + b*x^2)^(1/4))]
    @test_int [(a + b*x^2)^(3/4)/x^2, x, 4, (3*b*x)/(a + b*x^2)^(1/4) - (a + b*x^2)^(3/4)/x - (3*sqrt(a)*sqrt(b)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E(atan((sqrt(b)*x)/sqrt(a))/2, 2))/(a + b*x^2)^(1/4)]
    @test_int [(a + b*x^2)^(3/4)/x^4, x, 5, (b^2*x)/(2*a*(a + b*x^2)^(1/4)) - (a + b*x^2)^(3/4)/(3*x^3) - (b*(a + b*x^2)^(3/4))/(2*a*x) - (b^(3/2)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E(atan((sqrt(b)*x)/sqrt(a))/2, 2))/(2*sqrt(a)*(a + b*x^2)^(1/4))]
    @test_int [(a + b*x^2)^(3/4)/x^6, x, 6, (-3*b^3*x)/(20*a^2*(a + b*x^2)^(1/4)) - (a + b*x^2)^(3/4)/(5*x^5) - (b*(a + b*x^2)^(3/4))/(10*a*x^3) + (3*b^2*(a + b*x^2)^(3/4))/(20*a^2*x) + (3*b^(5/2)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E(atan((sqrt(b)*x)/sqrt(a))/2, 2))/(20*a^(3/2)*(a + b*x^2)^(1/4))]

    @test_int [x^4*(a - b*x^2)^(3/4), x, 5, (-4*a^2*x*(a - b*x^2)^(3/4))/(65*b^2) - (2*a*x^3*(a - b*x^2)^(3/4))/(39*b) + (2*x^5*(a - b*x^2)^(3/4))/13 + (8*a^(7/2)*(1 - (b*x^2)/a)^(1/4)*Elliptic.E(asin((sqrt(b)*x)/sqrt(a))/2, 2))/(65*b^(5/2)*(a - b*x^2)^(1/4))]
    @test_int [x^2*(a - b*x^2)^(3/4), x, 4, (-2*a*x*(a - b*x^2)^(3/4))/(15*b) + (2*x^3*(a - b*x^2)^(3/4))/9 + (4*a^(5/2)*(1 - (b*x^2)/a)^(1/4)*Elliptic.E(asin((sqrt(b)*x)/sqrt(a))/2, 2))/(15*b^(3/2)*(a - b*x^2)^(1/4))]
    @test_int [x^0*(a - b*x^2)^(3/4), x, 3, (2*x*(a - b*x^2)^(3/4))/5 + (6*a^(3/2)*(1 - (b*x^2)/a)^(1/4)*Elliptic.E(asin((sqrt(b)*x)/sqrt(a))/2, 2))/(5*sqrt(b)*(a - b*x^2)^(1/4))]
    @test_int [(a - b*x^2)^(3/4)/x^2, x, 3, -((a - b*x^2)^(3/4)/x) - (3*sqrt(a)*sqrt(b)*(1 - (b*x^2)/a)^(1/4)*Elliptic.E(asin((sqrt(b)*x)/sqrt(a))/2, 2))/(a - b*x^2)^(1/4)]
    @test_int [(a - b*x^2)^(3/4)/x^4, x, 4, -(a - b*x^2)^(3/4)/(3*x^3) + (b*(a - b*x^2)^(3/4))/(2*a*x) + (b^(3/2)*(1 - (b*x^2)/a)^(1/4)*Elliptic.E(asin((sqrt(b)*x)/sqrt(a))/2, 2))/(2*sqrt(a)*(a - b*x^2)^(1/4))]
    @test_int [(a - b*x^2)^(3/4)/x^6, x, 5, -(a - b*x^2)^(3/4)/(5*x^5) + (b*(a - b*x^2)^(3/4))/(10*a*x^3) + (3*b^2*(a - b*x^2)^(3/4))/(20*a^2*x) + (3*b^(5/2)*(1 - (b*x^2)/a)^(1/4)*Elliptic.E(asin((sqrt(b)*x)/sqrt(a))/2, 2))/(20*a^(3/2)*(a - b*x^2)^(1/4))]

    @test_int [(a + b*x^2)^(5/4), x, 4, (10/21)*a*x*(a + b*x^2)^(1/4) + (2/7)*x*(a + b*x^2)^(5/4) + (10*a^(5/2)*(1 + (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(21*sqrt(b)*(a + b*x^2)^(3/4))]

    @test_int [(a - b*x^2)^(5/4), x, 4, (10/21)*a*x*(a - b*x^2)^(1/4) + (2/7)*x*(a - b*x^2)^(5/4) + (10*a^(5/2)*(1 - (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*asin((sqrt(b)*x)/sqrt(a)), 2))/(21*sqrt(b)*(a - b*x^2)^(3/4))]

    @test_int [(a + b*x^2)^(7/4), x, 5, (14*a^2*x)/(15*(a + b*x^2)^(1/4)) + (14*a*x*(a + b*x^2)^(3/4))/45 + (2*x*(a + b*x^2)^(7/4))/9 - (14*a^(5/2)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E(atan((sqrt(b)*x)/sqrt(a))/2, 2))/(15*sqrt(b)*(a + b*x^2)^(1/4))]

    @test_int [(a - b*x^2)^(7/4), x, 4, (14*a*x*(a - b*x^2)^(3/4))/45 + (2*x*(a - b*x^2)^(7/4))/9 + (14*a^(5/2)*(1 - (b*x^2)/a)^(1/4)*Elliptic.E(asin((sqrt(b)*x)/sqrt(a))/2, 2))/(15*sqrt(b)*(a - b*x^2)^(1/4))]

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

    @test_int [x^6/(a + b*x^2)^(1/4), x, 6, (-16*a^3*x)/(39*b^3*(a + b*x^2)^(1/4)) + (8*a^2*x*(a + b*x^2)^(3/4))/(39*b^3) - (20*a*x^3*(a + b*x^2)^(3/4))/(117*b^2) + (2*x^5*(a + b*x^2)^(3/4))/(13*b) + (16*a^(7/2)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E(atan((sqrt(b)*x)/sqrt(a))/2, 2))/(39*b^(7/2)*(a + b*x^2)^(1/4))]
    @test_int [x^4/(a + b*x^2)^(1/4), x, 5, (8*a^2*x)/(15*b^2*(a + b*x^2)^(1/4)) - (4*a*x*(a + b*x^2)^(3/4))/(15*b^2) + (2*x^3*(a + b*x^2)^(3/4))/(9*b) - (8*a^(5/2)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E(atan((sqrt(b)*x)/sqrt(a))/2, 2))/(15*b^(5/2)*(a + b*x^2)^(1/4))]
    @test_int [x^2/(a + b*x^2)^(1/4), x, 4, (-4*a*x)/(5*b*(a + b*x^2)^(1/4)) + (2*x*(a + b*x^2)^(3/4))/(5*b) + (4*a^(3/2)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E(atan((sqrt(b)*x)/sqrt(a))/2, 2))/(5*b^(3/2)*(a + b*x^2)^(1/4))]
    @test_int [x^0/(a + b*x^2)^(1/4), x, 3, (2*x)/(a + b*x^2)^(1/4) - (2*sqrt(a)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E(atan((sqrt(b)*x)/sqrt(a))/2, 2))/(sqrt(b)*(a + b*x^2)^(1/4))]
    @test_int [1/(x^2*(a + b*x^2)^(1/4)), x, 4, (b*x)/(a*(a + b*x^2)^(1/4)) - (a + b*x^2)^(3/4)/(a*x) - (sqrt(b)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(sqrt(a)*(a + b*x^2)^(1/4))]
    @test_int [1/(x^4*(a + b*x^2)^(1/4)), x, 5, -((b^2*x)/(2*a^2*(a + b*x^2)^(1/4))) - (a + b*x^2)^(3/4)/(3*a*x^3) + (b*(a + b*x^2)^(3/4))/(2*a^2*x) + (b^(3/2)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(2*a^(3/2)*(a + b*x^2)^(1/4))]
    @test_int [1/(x^6*(a + b*x^2)^(1/4)), x, 6, (7*b^3*x)/(20*a^3*(a + b*x^2)^(1/4)) - (a + b*x^2)^(3/4)/(5*a*x^5) + (7*b*(a + b*x^2)^(3/4))/(30*a^2*x^3) - (7*b^2*(a + b*x^2)^(3/4))/(20*a^3*x) - (7*b^(5/2)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(20*a^(5/2)*(a + b*x^2)^(1/4))]

    @test_int [x^6/(a - b*x^2)^(1/4), x, 5, (-8*a^2*x*(a - b*x^2)^(3/4))/(39*b^3) - (20*a*x^3*(a - b*x^2)^(3/4))/(117*b^2) - (2*x^5*(a - b*x^2)^(3/4))/(13*b) + (16*a^(7/2)*(1 - (b*x^2)/a)^(1/4)*Elliptic.E(asin((sqrt(b)*x)/sqrt(a))/2, 2))/(39*b^(7/2)*(a - b*x^2)^(1/4))]
    @test_int [x^4/(a - b*x^2)^(1/4), x, 4, (-4*a*x*(a - b*x^2)^(3/4))/(15*b^2) - (2*x^3*(a - b*x^2)^(3/4))/(9*b) + (8*a^(5/2)*(1 - (b*x^2)/a)^(1/4)*Elliptic.E(asin((sqrt(b)*x)/sqrt(a))/2, 2))/(15*b^(5/2)*(a - b*x^2)^(1/4))]
    @test_int [x^2/(a - b*x^2)^(1/4), x, 3, (-2*x*(a - b*x^2)^(3/4))/(5*b) + (4*a^(3/2)*(1 - (b*x^2)/a)^(1/4)*Elliptic.E(asin((sqrt(b)*x)/sqrt(a))/2, 2))/(5*b^(3/2)*(a - b*x^2)^(1/4))]
    @test_int [x^0/(a - b*x^2)^(1/4), x, 2, (2*sqrt(a)*(1 - (b*x^2)/a)^(1/4)*Elliptic.E(asin((sqrt(b)*x)/sqrt(a))/2, 2))/(sqrt(b)*(a - b*x^2)^(1/4))]
    @test_int [1/(x^2*(a - b*x^2)^(1/4)), x, 3, -((a - b*x^2)^(3/4)/(a*x)) - (sqrt(b)*(1 - (b*x^2)/a)^(1/4)*Elliptic.E(asin((sqrt(b)*x)/sqrt(a))/2, 2))/(sqrt(a)*(a - b*x^2)^(1/4))]
    @test_int [1/(x^4*(a - b*x^2)^(1/4)), x, 4, -(a - b*x^2)^(3/4)/(3*a*x^3) - (b*(a - b*x^2)^(3/4))/(2*a^2*x) - (b^(3/2)*(1 - (b*x^2)/a)^(1/4)*Elliptic.E(asin((sqrt(b)*x)/sqrt(a))/2, 2))/(2*a^(3/2)*(a - b*x^2)^(1/4))]
    @test_int [1/(x^6*(a - b*x^2)^(1/4)), x, 5, -(a - b*x^2)^(3/4)/(5*a*x^5) - (7*b*(a - b*x^2)^(3/4))/(30*a^2*x^3) - (7*b^2*(a - b*x^2)^(3/4))/(20*a^3*x) - (7*b^(5/2)*(1 - (b*x^2)/a)^(1/4)*Elliptic.E(asin((sqrt(b)*x)/sqrt(a))/2, 2))/(20*a^(5/2)*(a - b*x^2)^(1/4))]

    @test_int [x^6/(a + b*x^2)^(3/4), x, 5, (40*a^2*x*(a + b*x^2)^(1/4))/(77*b^3) - (20*a*x^3*(a + b*x^2)^(1/4))/(77*b^2) + (2*x^5*(a + b*x^2)^(1/4))/(11*b) - (80*a^(7/2)*(1 + (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(77*b^(7/2)*(a + b*x^2)^(3/4))]
    @test_int [x^4/(a + b*x^2)^(3/4), x, 4, -((4*a*x*(a + b*x^2)^(1/4))/(7*b^2)) + (2*x^3*(a + b*x^2)^(1/4))/(7*b) + (8*a^(5/2)*(1 + (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(7*b^(5/2)*(a + b*x^2)^(3/4))]
    @test_int [x^2/(a + b*x^2)^(3/4), x, 3, (2*x*(a + b*x^2)^(1/4))/(3*b) - (4*a^(3/2)*(1 + (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(3*b^(3/2)*(a + b*x^2)^(3/4))]
    @test_int [x^0/(a + b*x^2)^(3/4), x, 2, (2*sqrt(a)*(1 + (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(sqrt(b)*(a + b*x^2)^(3/4))]
    @test_int [1/(x^2*(a + b*x^2)^(3/4)), x, 3, -((a + b*x^2)^(1/4)/(a*x)) - (sqrt(b)*(1 + (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(sqrt(a)*(a + b*x^2)^(3/4))]
    @test_int [1/(x^4*(a + b*x^2)^(3/4)), x, 4, -((a + b*x^2)^(1/4)/(3*a*x^3)) + (5*b*(a + b*x^2)^(1/4))/(6*a^2*x) + (5*b^(3/2)*(1 + (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(6*a^(3/2)*(a + b*x^2)^(3/4))]
    @test_int [1/(x^6*(a + b*x^2)^(3/4)), x, 5, -((a + b*x^2)^(1/4)/(5*a*x^5)) + (3*b*(a + b*x^2)^(1/4))/(10*a^2*x^3) - (3*b^2*(a + b*x^2)^(1/4))/(4*a^3*x) - (3*b^(5/2)*(1 + (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(4*a^(5/2)*(a + b*x^2)^(3/4))]

    @test_int [x^6/(a - b*x^2)^(3/4), x, 5, -((40*a^2*x*(a - b*x^2)^(1/4))/(77*b^3)) - (20*a*x^3*(a - b*x^2)^(1/4))/(77*b^2) - (2*x^5*(a - b*x^2)^(1/4))/(11*b) + (80*a^(7/2)*(1 - (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*asin((sqrt(b)*x)/sqrt(a)), 2))/(77*b^(7/2)*(a - b*x^2)^(3/4))]
    @test_int [x^4/(a - b*x^2)^(3/4), x, 4, -((4*a*x*(a - b*x^2)^(1/4))/(7*b^2)) - (2*x^3*(a - b*x^2)^(1/4))/(7*b) + (8*a^(5/2)*(1 - (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*asin((sqrt(b)*x)/sqrt(a)), 2))/(7*b^(5/2)*(a - b*x^2)^(3/4))]
    @test_int [x^2/(a - b*x^2)^(3/4), x, 3, -((2*x*(a - b*x^2)^(1/4))/(3*b)) + (4*a^(3/2)*(1 - (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*asin((sqrt(b)*x)/sqrt(a)), 2))/(3*b^(3/2)*(a - b*x^2)^(3/4))]
    @test_int [x^0/(a - b*x^2)^(3/4), x, 2, (2*sqrt(a)*(1 - (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*asin((sqrt(b)*x)/sqrt(a)), 2))/(sqrt(b)*(a - b*x^2)^(3/4))]
    @test_int [1/(x^2*(a - b*x^2)^(3/4)), x, 3, -((a - b*x^2)^(1/4)/(a*x)) + (sqrt(b)*(1 - (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*asin((sqrt(b)*x)/sqrt(a)), 2))/(sqrt(a)*(a - b*x^2)^(3/4))]
    @test_int [1/(x^4*(a - b*x^2)^(3/4)), x, 4, -((a - b*x^2)^(1/4)/(3*a*x^3)) - (5*b*(a - b*x^2)^(1/4))/(6*a^2*x) + (5*b^(3/2)*(1 - (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*asin((sqrt(b)*x)/sqrt(a)), 2))/(6*a^(3/2)*(a - b*x^2)^(3/4))]
    @test_int [1/(x^6*(a - b*x^2)^(3/4)), x, 5, -((a - b*x^2)^(1/4)/(5*a*x^5)) - (3*b*(a - b*x^2)^(1/4))/(10*a^2*x^3) - (3*b^2*(a - b*x^2)^(1/4))/(4*a^3*x) + (3*b^(5/2)*(1 - (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*asin((sqrt(b)*x)/sqrt(a)), 2))/(4*a^(5/2)*(a - b*x^2)^(3/4))]

    @test_int [x^6/(a + b*x^2)^(5/4), x, 5, (8*a^2*x)/(3*b^3*(a + b*x^2)^(1/4)) - (4*a*x^3)/(9*b^2*(a + b*x^2)^(1/4)) + (2*x^5)/(9*b*(a + b*x^2)^(1/4)) - (16*a^(5/2)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(3*b^(7/2)*(a + b*x^2)^(1/4))]
    @test_int [x^4/(a + b*x^2)^(5/4), x, 4, -((12*a*x)/(5*b^2*(a + b*x^2)^(1/4))) + (2*x^3)/(5*b*(a + b*x^2)^(1/4)) + (24*a^(3/2)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(5*b^(5/2)*(a + b*x^2)^(1/4))]
    @test_int [x^2/(a + b*x^2)^(5/4), x, 3, (2*x)/(b*(a + b*x^2)^(1/4)) - (4*sqrt(a)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E(atan((sqrt(b)*x)/sqrt(a))/2, 2))/(b^(3/2)*(a + b*x^2)^(1/4))]
    @test_int [x^0/(a + b*x^2)^(5/4), x, 2, (2*(1 + (b*x^2)/a)^(1/4)*Elliptic.E(atan((sqrt(b)*x)/sqrt(a))/2, 2))/(sqrt(a)*sqrt(b)*(a + b*x^2)^(1/4))]
    @test_int [1/(x^2*(a + b*x^2)^(5/4)), x, 3, -(1/(a*x*(a + b*x^2)^(1/4))) - (3*sqrt(b)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(a^(3/2)*(a + b*x^2)^(1/4))]
    @test_int [1/(x^4*(a + b*x^2)^(5/4)), x, 4, -(1/(3*a*x^3*(a + b*x^2)^(1/4))) + (7*b)/(6*a^2*x*(a + b*x^2)^(1/4)) + (7*b^(3/2)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(2*a^(5/2)*(a + b*x^2)^(1/4))]
    @test_int [1/(x^6*(a + b*x^2)^(5/4)), x, 5, -(1/(5*a*x^5*(a + b*x^2)^(1/4))) + (11*b)/(30*a^2*x^3*(a + b*x^2)^(1/4)) - (77*b^2)/(60*a^3*x*(a + b*x^2)^(1/4)) - (77*b^(5/2)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(20*a^(7/2)*(a + b*x^2)^(1/4))]

    @test_int [x^6/(a - b*x^2)^(5/4), x, 5, (2*x^5)/(b*(a - b*x^2)^(1/4)) + (8*a*x*(a - b*x^2)^(3/4))/(3*b^3) + (20*x^3*(a - b*x^2)^(3/4))/(9*b^2) - (16*a^(5/2)*(1 - (b*x^2)/a)^(1/4)*Elliptic.E(asin((sqrt(b)*x)/sqrt(a))/2, 2))/(3*b^(7/2)*(a - b*x^2)^(1/4))]
    @test_int [x^4/(a - b*x^2)^(5/4), x, 4, (2*x^3)/(b*(a - b*x^2)^(1/4)) + (12*x*(a - b*x^2)^(3/4))/(5*b^2) - (24*a^(3/2)*(1 - (b*x^2)/a)^(1/4)*Elliptic.E(asin((sqrt(b)*x)/sqrt(a))/2, 2))/(5*b^(5/2)*(a - b*x^2)^(1/4))]
    @test_int [x^2/(a - b*x^2)^(5/4), x, 3, (2*x)/(b*(a - b*x^2)^(1/4)) - (4*sqrt(a)*(1 - (b*x^2)/a)^(1/4)*Elliptic.E(asin((sqrt(b)*x)/sqrt(a))/2, 2))/(b^(3/2)*(a - b*x^2)^(1/4))]
    @test_int [x^0/(a - b*x^2)^(5/4), x, 3, (2*x)/(a*(a - b*x^2)^(1/4)) - (2*(1 - (b*x^2)/a)^(1/4)*Elliptic.E(asin((sqrt(b)*x)/sqrt(a))/2, 2))/(sqrt(a)*sqrt(b)*(a - b*x^2)^(1/4))]
    @test_int [1/(x^2*(a - b*x^2)^(5/4)), x, 4, 2/(a*x*(a - b*x^2)^(1/4)) - (3*(a - b*x^2)^(3/4))/(a^2*x) - (3*sqrt(b)*(1 - (b*x^2)/a)^(1/4)*Elliptic.E(asin((sqrt(b)*x)/sqrt(a))/2, 2))/(a^(3/2)*(a - b*x^2)^(1/4))]
    @test_int [1/(x^4*(a - b*x^2)^(5/4)), x, 5, 2/(a*x^3*(a - b*x^2)^(1/4)) - (7*(a - b*x^2)^(3/4))/(3*a^2*x^3) - (7*b*(a - b*x^2)^(3/4))/(2*a^3*x) - (7*b^(3/2)*(1 - (b*x^2)/a)^(1/4)*Elliptic.E(asin((sqrt(b)*x)/sqrt(a))/2, 2))/(2*a^(5/2)*(a - b*x^2)^(1/4))]
    @test_int [1/(x^6*(a - b*x^2)^(5/4)), x, 6, 2/(a*x^5*(a - b*x^2)^(1/4)) - (11*(a - b*x^2)^(3/4))/(5*a^2*x^5) - (77*b*(a - b*x^2)^(3/4))/(30*a^3*x^3) - (77*b^2*(a - b*x^2)^(3/4))/(20*a^4*x) - (77*b^(5/2)*(1 - (b*x^2)/a)^(1/4)*Elliptic.E(asin((sqrt(b)*x)/sqrt(a))/2, 2))/(20*a^(7/2)*(a - b*x^2)^(1/4))]

    @test_int [1/(a + b*x^2)^(7/4), x, 3, (2*x)/(3*a*(a + b*x^2)^(3/4)) + (2*(1 + (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(3*sqrt(a)*sqrt(b)*(a + b*x^2)^(3/4))]
    @test_int [1/(a + b*x^2)^(9/4), x, 3, (2*x)/(5*a*(a + b*x^2)^(5/4)) + (6*(1 + (b*x^2)/a)^(1/4)*Elliptic.E(atan((sqrt(b)*x)/sqrt(a))/2, 2))/(5*a^(3/2)*sqrt(b)*(a + b*x^2)^(1/4))]
    @test_int [1/(a + b*x^2)^(11/4), x, 4, (2*x)/(7*a*(a + b*x^2)^(7/4)) + (10*x)/(21*a^2*(a + b*x^2)^(3/4)) + (10*(1 + (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(21*a^(3/2)*sqrt(b)*(a + b*x^2)^(3/4))]

    @test_int [1/(a - b*x^2)^(7/4), x, 3, (2*x)/(3*a*(a - b*x^2)^(3/4)) + (2*(1 - (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*asin((sqrt(b)*x)/sqrt(a)), 2))/(3*sqrt(a)*sqrt(b)*(a - b*x^2)^(3/4))]
    @test_int [1/(a - b*x^2)^(9/4), x, 4, (2*x)/(5*a*(a - b*x^2)^(5/4)) + (6*x)/(5*a^2*(a - b*x^2)^(1/4)) - (6*(1 - (b*x^2)/a)^(1/4)*Elliptic.E(asin((sqrt(b)*x)/sqrt(a))/2, 2))/(5*a^(3/2)*sqrt(b)*(a - b*x^2)^(1/4))]
    @test_int [1/(a - b*x^2)^(11/4), x, 4, (2*x)/(7*a*(a - b*x^2)^(7/4)) + (10*x)/(21*a^2*(a - b*x^2)^(3/4)) + (10*(1 - (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*asin((sqrt(b)*x)/sqrt(a)), 2))/(21*a^(3/2)*sqrt(b)*(a - b*x^2)^(3/4))]

    @test_int [x^6/(2 + 3*x^2)^(1/4), x, 5, (-128*x)/(1053*(2 + 3*x^2)^(1/4)) + (32*x*(2 + 3*x^2)^(3/4))/1053 - (40*x^3*(2 + 3*x^2)^(3/4))/1053 + (2*x^5*(2 + 3*x^2)^(3/4))/39 + (128*2^(1/4)*Elliptic.E(atan(sqrt(3/2)*x)/2, 2))/(1053*sqrt(3))]
    @test_int [x^4/(2 + 3*x^2)^(1/4), x, 4, (32*x)/(135*(2 + 3*x^2)^(1/4)) - (8*x*(2 + 3*x^2)^(3/4))/135 + (2*x^3*(2 + 3*x^2)^(3/4))/27 - (32*2^(1/4)*Elliptic.E(atan(sqrt(3/2)*x)/2, 2))/(135*sqrt(3))]
    @test_int [x^2/(2 + 3*x^2)^(1/4), x, 3, (-8*x)/(15*(2 + 3*x^2)^(1/4)) + (2*x*(2 + 3*x^2)^(3/4))/15 + (8*2^(1/4)*Elliptic.E(atan(sqrt(3/2)*x)/2, 2))/(15*sqrt(3))]
    @test_int [x^0/(2 + 3*x^2)^(1/4), x, 2, (2*x)/(2 + 3*x^2)^(1/4) - (2*2^(1/4)*Elliptic.E(atan(sqrt(3/2)*x)/2, 2))/sqrt(3)]
    @test_int [1/(x^2*(2 + 3*x^2)^(1/4)), x, 3, (3*x)/(2*(2 + 3*x^2)^(1/4)) - (2 + 3*x^2)^(3/4)/(2*x) - (sqrt(3)*Elliptic.E((1/2)*atan(sqrt(3/2)*x), 2))/2^(3/4)]
    @test_int [1/(x^4*(2 + 3*x^2)^(1/4)), x, 4, -((9*x)/(8*(2 + 3*x^2)^(1/4))) - (2 + 3*x^2)^(3/4)/(6*x^3) + (3*(2 + 3*x^2)^(3/4))/(8*x) + (3*sqrt(3)*Elliptic.E((1/2)*atan(sqrt(3/2)*x), 2))/(4*2^(3/4))]
    @test_int [1/(x^6*(2 + 3*x^2)^(1/4)), x, 5, (189*x)/(160*(2 + 3*x^2)^(1/4)) - (2 + 3*x^2)^(3/4)/(10*x^5) + (7*(2 + 3*x^2)^(3/4))/(40*x^3) - (63*(2 + 3*x^2)^(3/4))/(160*x) - (63*sqrt(3)*Elliptic.E((1/2)*atan(sqrt(3/2)*x), 2))/(80*2^(3/4))]

    @test_int [x^6/(2 - 3*x^2)^(1/4), x, 4, (-32*x*(2 - 3*x^2)^(3/4))/1053 - (40*x^3*(2 - 3*x^2)^(3/4))/1053 - (2*x^5*(2 - 3*x^2)^(3/4))/39 + (128*2^(1/4)*Elliptic.E(asin(sqrt(3/2)*x)/2, 2))/(1053*sqrt(3))]
    @test_int [x^4/(2 - 3*x^2)^(1/4), x, 3, (-8*x*(2 - 3*x^2)^(3/4))/135 - (2*x^3*(2 - 3*x^2)^(3/4))/27 + (32*2^(1/4)*Elliptic.E(asin(sqrt(3/2)*x)/2, 2))/(135*sqrt(3))]
    @test_int [x^2/(2 - 3*x^2)^(1/4), x, 2, (-2*x*(2 - 3*x^2)^(3/4))/15 + (8*2^(1/4)*Elliptic.E(asin(sqrt(3/2)*x)/2, 2))/(15*sqrt(3))]
    @test_int [x^0/(2 - 3*x^2)^(1/4), x, 1, (2*2^(1/4)*Elliptic.E(asin(sqrt(3/2)*x)/2, 2))/sqrt(3)]
    @test_int [1/(x^2*(2 - 3*x^2)^(1/4)), x, 2, -((2 - 3*x^2)^(3/4)/(2*x)) - (sqrt(3)*Elliptic.E((1/2)*asin(sqrt(3/2)*x), 2))/2^(3/4)]
    @test_int [1/(x^4*(2 - 3*x^2)^(1/4)), x, 3, -((2 - 3*x^2)^(3/4)/(6*x^3)) - (3*(2 - 3*x^2)^(3/4))/(8*x) - (3*sqrt(3)*Elliptic.E((1/2)*asin(sqrt(3/2)*x), 2))/(4*2^(3/4))]
    @test_int [1/(x^6*(2 - 3*x^2)^(1/4)), x, 4, -((2 - 3*x^2)^(3/4)/(10*x^5)) - (7*(2 - 3*x^2)^(3/4))/(40*x^3) - (63*(2 - 3*x^2)^(3/4))/(160*x) - (63*sqrt(3)*Elliptic.E((1/2)*asin(sqrt(3/2)*x), 2))/(80*2^(3/4))]

    @test_int [x^6/(2 + 3*x^2)^(3/4), x, 4, (160*x*(2 + 3*x^2)^(1/4))/2079 - (40*x^3*(2 + 3*x^2)^(1/4))/693 + (2*x^5*(2 + 3*x^2)^(1/4))/33 - (320*2^(3/4)*Elliptic.F(atan(sqrt(3/2)*x)/2, 2))/(2079*sqrt(3))]
    @test_int [x^4/(2 + 3*x^2)^(3/4), x, 3, (-8*x*(2 + 3*x^2)^(1/4))/63 + (2*x^3*(2 + 3*x^2)^(1/4))/21 + (16*2^(3/4)*Elliptic.F(atan(sqrt(3/2)*x)/2, 2))/(63*sqrt(3))]
    @test_int [x^2/(2 + 3*x^2)^(3/4), x, 2, (2*x*(2 + 3*x^2)^(1/4))/9 - (4*2^(3/4)*Elliptic.F(atan(sqrt(3/2)*x)/2, 2))/(9*sqrt(3))]
    @test_int [x^0/(2 + 3*x^2)^(3/4), x, 1, (2^(3/4)*Elliptic.F(atan(sqrt(3/2)*x)/2, 2))/sqrt(3)]
    @test_int [1/(x^2*(2 + 3*x^2)^(3/4)), x, 2, -((2 + 3*x^2)^(1/4)/(2*x)) - (sqrt(3)*Elliptic.F((1/2)*atan(sqrt(3/2)*x), 2))/(2*2^(1/4))]
    @test_int [1/(x^4*(2 + 3*x^2)^(3/4)), x, 3, -((2 + 3*x^2)^(1/4)/(6*x^3)) + (5*(2 + 3*x^2)^(1/4))/(8*x) + (5*sqrt(3)*Elliptic.F((1/2)*atan(sqrt(3/2)*x), 2))/(8*2^(1/4))]
    @test_int [1/(x^6*(2 + 3*x^2)^(3/4)), x, 4, -((2 + 3*x^2)^(1/4)/(10*x^5)) + (9*(2 + 3*x^2)^(1/4))/(40*x^3) - (27*(2 + 3*x^2)^(1/4))/(32*x) - (27*sqrt(3)*Elliptic.F((1/2)*atan(sqrt(3/2)*x), 2))/(32*2^(1/4))]

    @test_int [x^6/(2 - 3*x^2)^(3/4), x, 4, (-160*x*(2 - 3*x^2)^(1/4))/2079 - (40*x^3*(2 - 3*x^2)^(1/4))/693 - (2*x^5*(2 - 3*x^2)^(1/4))/33 + (320*2^(3/4)*Elliptic.F(asin(sqrt(3/2)*x)/2, 2))/(2079*sqrt(3))]
    @test_int [x^4/(2 - 3*x^2)^(3/4), x, 3, (-8*x*(2 - 3*x^2)^(1/4))/63 - (2*x^3*(2 - 3*x^2)^(1/4))/21 + (16*2^(3/4)*Elliptic.F(asin(sqrt(3/2)*x)/2, 2))/(63*sqrt(3))]
    @test_int [x^2/(2 - 3*x^2)^(3/4), x, 2, (-2*x*(2 - 3*x^2)^(1/4))/9 + (4*2^(3/4)*Elliptic.F(asin(sqrt(3/2)*x)/2, 2))/(9*sqrt(3))]
    @test_int [x^0/(2 - 3*x^2)^(3/4), x, 1, (2^(3/4)*Elliptic.F(asin(sqrt(3/2)*x)/2, 2))/sqrt(3)]
    @test_int [1/(x^2*(2 - 3*x^2)^(3/4)), x, 2, -((2 - 3*x^2)^(1/4)/(2*x)) + (sqrt(3)*Elliptic.F((1/2)*asin(sqrt(3/2)*x), 2))/(2*2^(1/4))]
    @test_int [1/(x^4*(2 - 3*x^2)^(3/4)), x, 3, -((2 - 3*x^2)^(1/4)/(6*x^3)) - (5*(2 - 3*x^2)^(1/4))/(8*x) + (5*sqrt(3)*Elliptic.F((1/2)*asin(sqrt(3/2)*x), 2))/(8*2^(1/4))]
    @test_int [1/(x^6*(2 - 3*x^2)^(3/4)), x, 4, -((2 - 3*x^2)^(1/4)/(10*x^5)) - (9*(2 - 3*x^2)^(1/4))/(40*x^3) - (27*(2 - 3*x^2)^(1/4))/(32*x) + (27*sqrt(3)*Elliptic.F((1/2)*asin(sqrt(3/2)*x), 2))/(32*2^(1/4))]

    @test_int [x^6/(-2 + 3*x^2)^(1/4), x, 7, (32*x*(-2 + 3*x^2)^(3/4))/1053 + (40*x^3*(-2 + 3*x^2)^(3/4))/1053 + (2/39)*x^5*(-2 + 3*x^2)^(3/4) + (128*x*(-2 + 3*x^2)^(1/4))/(1053*(sqrt(2) + sqrt(-2 + 3*x^2))) - (128*2^(1/4)*sqrt(x^2/(sqrt(2) + sqrt(-2 + 3*x^2))^2)*(sqrt(2) + sqrt(-2 + 3*x^2))*Elliptic.E(2*atan((-2 + 3*x^2)^(1/4)/2^(1/4)), 1/2))/(1053*sqrt(3)*x) + (64*2^(1/4)*sqrt(x^2/(sqrt(2) + sqrt(-2 + 3*x^2))^2)*(sqrt(2) + sqrt(-2 + 3*x^2))*Elliptic.F(2*atan((-2 + 3*x^2)^(1/4)/2^(1/4)), 1/2))/(1053*sqrt(3)*x)]
    @test_int [x^4/(-2 + 3*x^2)^(1/4), x, 6, (8/135)*x*(-2 + 3*x^2)^(3/4) + (2/27)*x^3*(-2 + 3*x^2)^(3/4) + (32*x*(-2 + 3*x^2)^(1/4))/(135*(sqrt(2) + sqrt(-2 + 3*x^2))) - (32*2^(1/4)*sqrt(x^2/(sqrt(2) + sqrt(-2 + 3*x^2))^2)*(sqrt(2) + sqrt(-2 + 3*x^2))*Elliptic.E(2*atan((-2 + 3*x^2)^(1/4)/2^(1/4)), 1/2))/(135*sqrt(3)*x) + (16*2^(1/4)*sqrt(x^2/(sqrt(2) + sqrt(-2 + 3*x^2))^2)*(sqrt(2) + sqrt(-2 + 3*x^2))*Elliptic.F(2*atan((-2 + 3*x^2)^(1/4)/2^(1/4)), 1/2))/(135*sqrt(3)*x)]
    @test_int [x^2/(-2 + 3*x^2)^(1/4), x, 5, (2/15)*x*(-2 + 3*x^2)^(3/4) + (8*x*(-2 + 3*x^2)^(1/4))/(15*(sqrt(2) + sqrt(-2 + 3*x^2))) - (8*2^(1/4)*sqrt(x^2/(sqrt(2) + sqrt(-2 + 3*x^2))^2)*(sqrt(2) + sqrt(-2 + 3*x^2))*Elliptic.E(2*atan((-2 + 3*x^2)^(1/4)/2^(1/4)), 1/2))/(15*sqrt(3)*x) + (4*2^(1/4)*sqrt(x^2/(sqrt(2) + sqrt(-2 + 3*x^2))^2)*(sqrt(2) + sqrt(-2 + 3*x^2))*Elliptic.F(2*atan((-2 + 3*x^2)^(1/4)/2^(1/4)), 1/2))/(15*sqrt(3)*x)]
    @test_int [x^0/(-2 + 3*x^2)^(1/4), x, 4, (2*x*(-2 + 3*x^2)^(1/4))/(sqrt(2) + sqrt(-2 + 3*x^2)) - (2*2^(1/4)*sqrt(x^2/(sqrt(2) + sqrt(-2 + 3*x^2))^2)*(sqrt(2) + sqrt(-2 + 3*x^2))*Elliptic.E(2*atan((-2 + 3*x^2)^(1/4)/2^(1/4)), 1/2))/(sqrt(3)*x) + (2^(1/4)*sqrt(x^2/(sqrt(2) + sqrt(-2 + 3*x^2))^2)*(sqrt(2) + sqrt(-2 + 3*x^2))*Elliptic.F(2*atan((-2 + 3*x^2)^(1/4)/2^(1/4)), 1/2))/(sqrt(3)*x)]
    @test_int [1/(x^2*(-2 + 3*x^2)^(1/4)), x, 5, (-2 + 3*x^2)^(3/4)/(2*x) - (3*x*(-2 + 3*x^2)^(1/4))/(2*(sqrt(2) + sqrt(-2 + 3*x^2))) + (sqrt(3)*sqrt(x^2/(sqrt(2) + sqrt(-2 + 3*x^2))^2)*(sqrt(2) + sqrt(-2 + 3*x^2))*Elliptic.E(2*atan((-2 + 3*x^2)^(1/4)/2^(1/4)), 1/2))/(2^(3/4)*x) - (sqrt(3)*sqrt(x^2/(sqrt(2) + sqrt(-2 + 3*x^2))^2)*(sqrt(2) + sqrt(-2 + 3*x^2))*Elliptic.F(2*atan((-2 + 3*x^2)^(1/4)/2^(1/4)), 1/2))/(2*2^(3/4)*x)]
    @test_int [1/(x^4*(-2 + 3*x^2)^(1/4)), x, 6, (-2 + 3*x^2)^(3/4)/(6*x^3) + (3*(-2 + 3*x^2)^(3/4))/(8*x) - (9*x*(-2 + 3*x^2)^(1/4))/(8*(sqrt(2) + sqrt(-2 + 3*x^2))) + (3*sqrt(3)*sqrt(x^2/(sqrt(2) + sqrt(-2 + 3*x^2))^2)*(sqrt(2) + sqrt(-2 + 3*x^2))*Elliptic.E(2*atan((-2 + 3*x^2)^(1/4)/2^(1/4)), 1/2))/(4*2^(3/4)*x) - (3*sqrt(3)*sqrt(x^2/(sqrt(2) + sqrt(-2 + 3*x^2))^2)*(sqrt(2) + sqrt(-2 + 3*x^2))*Elliptic.F(2*atan((-2 + 3*x^2)^(1/4)/2^(1/4)), 1/2))/(8*2^(3/4)*x)]
    @test_int [1/(x^6*(-2 + 3*x^2)^(1/4)), x, 7, (-2 + 3*x^2)^(3/4)/(10*x^5) + (7*(-2 + 3*x^2)^(3/4))/(40*x^3) + (63*(-2 + 3*x^2)^(3/4))/(160*x) - (189*x*(-2 + 3*x^2)^(1/4))/(160*(sqrt(2) + sqrt(-2 + 3*x^2))) + (63*sqrt(3)*sqrt(x^2/(sqrt(2) + sqrt(-2 + 3*x^2))^2)*(sqrt(2) + sqrt(-2 + 3*x^2))*Elliptic.E(2*atan((-2 + 3*x^2)^(1/4)/2^(1/4)), 1/2))/(80*2^(3/4)*x) - (63*sqrt(3)*sqrt(x^2/(sqrt(2) + sqrt(-2 + 3*x^2))^2)*(sqrt(2) + sqrt(-2 + 3*x^2))*Elliptic.F(2*atan((-2 + 3*x^2)^(1/4)/2^(1/4)), 1/2))/(160*2^(3/4)*x)]

    @test_int [x^6/(-2 - 3*x^2)^(1/4), x, 7, -((32*x*(-2 - 3*x^2)^(3/4))/1053) + (40*x^3*(-2 - 3*x^2)^(3/4))/1053 - (2/39)*x^5*(-2 - 3*x^2)^(3/4) - (128*x*(-2 - 3*x^2)^(1/4))/(1053*(sqrt(2) + sqrt(-2 - 3*x^2))) - (128*2^(1/4)*sqrt(-(x^2/(sqrt(2) + sqrt(-2 - 3*x^2))^2))*(sqrt(2) + sqrt(-2 - 3*x^2))*Elliptic.E(2*atan((-2 - 3*x^2)^(1/4)/2^(1/4)), 1/2))/(1053*sqrt(3)*x) + (64*2^(1/4)*sqrt(-(x^2/(sqrt(2) + sqrt(-2 - 3*x^2))^2))*(sqrt(2) + sqrt(-2 - 3*x^2))*Elliptic.F(2*atan((-2 - 3*x^2)^(1/4)/2^(1/4)), 1/2))/(1053*sqrt(3)*x)]
    @test_int [x^4/(-2 - 3*x^2)^(1/4), x, 6, (8/135)*x*(-2 - 3*x^2)^(3/4) - (2/27)*x^3*(-2 - 3*x^2)^(3/4) + (32*x*(-2 - 3*x^2)^(1/4))/(135*(sqrt(2) + sqrt(-2 - 3*x^2))) + (32*2^(1/4)*sqrt(-(x^2/(sqrt(2) + sqrt(-2 - 3*x^2))^2))*(sqrt(2) + sqrt(-2 - 3*x^2))*Elliptic.E(2*atan((-2 - 3*x^2)^(1/4)/2^(1/4)), 1/2))/(135*sqrt(3)*x) - (16*2^(1/4)*sqrt(-(x^2/(sqrt(2) + sqrt(-2 - 3*x^2))^2))*(sqrt(2) + sqrt(-2 - 3*x^2))*Elliptic.F(2*atan((-2 - 3*x^2)^(1/4)/2^(1/4)), 1/2))/(135*sqrt(3)*x)]
    @test_int [x^2/(-2 - 3*x^2)^(1/4), x, 5, (-(2/15))*x*(-2 - 3*x^2)^(3/4) - (8*x*(-2 - 3*x^2)^(1/4))/(15*(sqrt(2) + sqrt(-2 - 3*x^2))) - (8*2^(1/4)*sqrt(-(x^2/(sqrt(2) + sqrt(-2 - 3*x^2))^2))*(sqrt(2) + sqrt(-2 - 3*x^2))*Elliptic.E(2*atan((-2 - 3*x^2)^(1/4)/2^(1/4)), 1/2))/(15*sqrt(3)*x) + (4*2^(1/4)*sqrt(-(x^2/(sqrt(2) + sqrt(-2 - 3*x^2))^2))*(sqrt(2) + sqrt(-2 - 3*x^2))*Elliptic.F(2*atan((-2 - 3*x^2)^(1/4)/2^(1/4)), 1/2))/(15*sqrt(3)*x)]
    @test_int [x^0/(-2 - 3*x^2)^(1/4), x, 4, (2*x*(-2 - 3*x^2)^(1/4))/(sqrt(2) + sqrt(-2 - 3*x^2)) + (2*2^(1/4)*sqrt(-(x^2/(sqrt(2) + sqrt(-2 - 3*x^2))^2))*(sqrt(2) + sqrt(-2 - 3*x^2))*Elliptic.E(2*atan((-2 - 3*x^2)^(1/4)/2^(1/4)), 1/2))/(sqrt(3)*x) - (2^(1/4)*sqrt(-(x^2/(sqrt(2) + sqrt(-2 - 3*x^2))^2))*(sqrt(2) + sqrt(-2 - 3*x^2))*Elliptic.F(2*atan((-2 - 3*x^2)^(1/4)/2^(1/4)), 1/2))/(sqrt(3)*x)]
    @test_int [1/(x^2*(-2 - 3*x^2)^(1/4)), x, 5, (-2 - 3*x^2)^(3/4)/(2*x) + (3*x*(-2 - 3*x^2)^(1/4))/(2*(sqrt(2) + sqrt(-2 - 3*x^2))) + (sqrt(3)*sqrt(-(x^2/(sqrt(2) + sqrt(-2 - 3*x^2))^2))*(sqrt(2) + sqrt(-2 - 3*x^2))*Elliptic.E(2*atan((-2 - 3*x^2)^(1/4)/2^(1/4)), 1/2))/(2^(3/4)*x) - (sqrt(3)*sqrt(-(x^2/(sqrt(2) + sqrt(-2 - 3*x^2))^2))*(sqrt(2) + sqrt(-2 - 3*x^2))*Elliptic.F(2*atan((-2 - 3*x^2)^(1/4)/2^(1/4)), 1/2))/(2*2^(3/4)*x)]
    @test_int [1/(x^4*(-2 - 3*x^2)^(1/4)), x, 6, (-2 - 3*x^2)^(3/4)/(6*x^3) - (3*(-2 - 3*x^2)^(3/4))/(8*x) - (9*x*(-2 - 3*x^2)^(1/4))/(8*(sqrt(2) + sqrt(-2 - 3*x^2))) - (3*sqrt(3)*sqrt(-(x^2/(sqrt(2) + sqrt(-2 - 3*x^2))^2))*(sqrt(2) + sqrt(-2 - 3*x^2))*Elliptic.E(2*atan((-2 - 3*x^2)^(1/4)/2^(1/4)), 1/2))/(4*2^(3/4)*x) + (3*sqrt(3)*sqrt(-(x^2/(sqrt(2) + sqrt(-2 - 3*x^2))^2))*(sqrt(2) + sqrt(-2 - 3*x^2))*Elliptic.F(2*atan((-2 - 3*x^2)^(1/4)/2^(1/4)), 1/2))/(8*2^(3/4)*x)]
    @test_int [1/(x^6*(-2 - 3*x^2)^(1/4)), x, 7, (-2 - 3*x^2)^(3/4)/(10*x^5) - (7*(-2 - 3*x^2)^(3/4))/(40*x^3) + (63*(-2 - 3*x^2)^(3/4))/(160*x) + (189*x*(-2 - 3*x^2)^(1/4))/(160*(sqrt(2) + sqrt(-2 - 3*x^2))) + (63*sqrt(3)*sqrt(-(x^2/(sqrt(2) + sqrt(-2 - 3*x^2))^2))*(sqrt(2) + sqrt(-2 - 3*x^2))*Elliptic.E(2*atan((-2 - 3*x^2)^(1/4)/2^(1/4)), 1/2))/(80*2^(3/4)*x) - (63*sqrt(3)*sqrt(-(x^2/(sqrt(2) + sqrt(-2 - 3*x^2))^2))*(sqrt(2) + sqrt(-2 - 3*x^2))*Elliptic.F(2*atan((-2 - 3*x^2)^(1/4)/2^(1/4)), 1/2))/(160*2^(3/4)*x)]

    @test_int [x^6/(-2 + 3*x^2)^(3/4), x, 5, (160*x*(-2 + 3*x^2)^(1/4))/2079 + (40/693)*x^3*(-2 + 3*x^2)^(1/4) + (2/33)*x^5*(-2 + 3*x^2)^(1/4) + (160*2^(3/4)*sqrt(x^2/(sqrt(2) + sqrt(-2 + 3*x^2))^2)*(sqrt(2) + sqrt(-2 + 3*x^2))*Elliptic.F(2*atan((-2 + 3*x^2)^(1/4)/2^(1/4)), 1/2))/(2079*sqrt(3)*x)]
    @test_int [x^4/(-2 + 3*x^2)^(3/4), x, 4, (8/63)*x*(-2 + 3*x^2)^(1/4) + (2/21)*x^3*(-2 + 3*x^2)^(1/4) + (8*2^(3/4)*sqrt(x^2/(sqrt(2) + sqrt(-2 + 3*x^2))^2)*(sqrt(2) + sqrt(-2 + 3*x^2))*Elliptic.F(2*atan((-2 + 3*x^2)^(1/4)/2^(1/4)), 1/2))/(63*sqrt(3)*x)]
    @test_int [x^2/(-2 + 3*x^2)^(3/4), x, 3, (2/9)*x*(-2 + 3*x^2)^(1/4) + (2*2^(3/4)*sqrt(x^2/(sqrt(2) + sqrt(-2 + 3*x^2))^2)*(sqrt(2) + sqrt(-2 + 3*x^2))*Elliptic.F(2*atan((-2 + 3*x^2)^(1/4)/2^(1/4)), 1/2))/(9*sqrt(3)*x)]
    @test_int [x^0/(-2 + 3*x^2)^(3/4), x, 2, (sqrt(x^2/(sqrt(2) + sqrt(-2 + 3*x^2))^2)*(sqrt(2) + sqrt(-2 + 3*x^2))*Elliptic.F(2*atan((-2 + 3*x^2)^(1/4)/2^(1/4)), 1/2))/(2^(1/4)*sqrt(3)*x)]
    @test_int [1/(x^2*(-2 + 3*x^2)^(3/4)), x, 3, (-2 + 3*x^2)^(1/4)/(2*x) + (sqrt(3)*sqrt(x^2/(sqrt(2) + sqrt(-2 + 3*x^2))^2)*(sqrt(2) + sqrt(-2 + 3*x^2))*Elliptic.F(2*atan((-2 + 3*x^2)^(1/4)/2^(1/4)), 1/2))/(4*2^(1/4)*x)]
    @test_int [1/(x^4*(-2 + 3*x^2)^(3/4)), x, 4, (-2 + 3*x^2)^(1/4)/(6*x^3) + (5*(-2 + 3*x^2)^(1/4))/(8*x) + (5*sqrt(3)*sqrt(x^2/(sqrt(2) + sqrt(-2 + 3*x^2))^2)*(sqrt(2) + sqrt(-2 + 3*x^2))*Elliptic.F(2*atan((-2 + 3*x^2)^(1/4)/2^(1/4)), 1/2))/(16*2^(1/4)*x)]
    @test_int [1/(x^6*(-2 + 3*x^2)^(3/4)), x, 5, (-2 + 3*x^2)^(1/4)/(10*x^5) + (9*(-2 + 3*x^2)^(1/4))/(40*x^3) + (27*(-2 + 3*x^2)^(1/4))/(32*x) + (27*sqrt(3)*sqrt(x^2/(sqrt(2) + sqrt(-2 + 3*x^2))^2)*(sqrt(2) + sqrt(-2 + 3*x^2))*Elliptic.F(2*atan((-2 + 3*x^2)^(1/4)/2^(1/4)), 1/2))/(64*2^(1/4)*x)]

    @test_int [x^6/(-2 - 3*x^2)^(3/4), x, 5, -((160*x*(-2 - 3*x^2)^(1/4))/2079) + (40/693)*x^3*(-2 - 3*x^2)^(1/4) - (2/33)*x^5*(-2 - 3*x^2)^(1/4) + (160*2^(3/4)*sqrt(-(x^2/(sqrt(2) + sqrt(-2 - 3*x^2))^2))*(sqrt(2) + sqrt(-2 - 3*x^2))*Elliptic.F(2*atan((-2 - 3*x^2)^(1/4)/2^(1/4)), 1/2))/(2079*sqrt(3)*x)]
    @test_int [x^4/(-2 - 3*x^2)^(3/4), x, 4, (8/63)*x*(-2 - 3*x^2)^(1/4) - (2/21)*x^3*(-2 - 3*x^2)^(1/4) - (8*2^(3/4)*sqrt(-(x^2/(sqrt(2) + sqrt(-2 - 3*x^2))^2))*(sqrt(2) + sqrt(-2 - 3*x^2))*Elliptic.F(2*atan((-2 - 3*x^2)^(1/4)/2^(1/4)), 1/2))/(63*sqrt(3)*x)]
    @test_int [x^2/(-2 - 3*x^2)^(3/4), x, 3, (-(2/9))*x*(-2 - 3*x^2)^(1/4) + (2*2^(3/4)*sqrt(-(x^2/(sqrt(2) + sqrt(-2 - 3*x^2))^2))*(sqrt(2) + sqrt(-2 - 3*x^2))*Elliptic.F(2*atan((-2 - 3*x^2)^(1/4)/2^(1/4)), 1/2))/(9*sqrt(3)*x)]
    @test_int [x^0/(-2 - 3*x^2)^(3/4), x, 2, -((sqrt(-(x^2/(sqrt(2) + sqrt(-2 - 3*x^2))^2))*(sqrt(2) + sqrt(-2 - 3*x^2))*Elliptic.F(2*atan((-2 - 3*x^2)^(1/4)/2^(1/4)), 1/2))/(2^(1/4)*sqrt(3)*x))]
    @test_int [1/(x^2*(-2 - 3*x^2)^(3/4)), x, 3, (-2 - 3*x^2)^(1/4)/(2*x) + (sqrt(3)*sqrt(-(x^2/(sqrt(2) + sqrt(-2 - 3*x^2))^2))*(sqrt(2) + sqrt(-2 - 3*x^2))*Elliptic.F(2*atan((-2 - 3*x^2)^(1/4)/2^(1/4)), 1/2))/(4*2^(1/4)*x)]
    @test_int [1/(x^4*(-2 - 3*x^2)^(3/4)), x, 4, (-2 - 3*x^2)^(1/4)/(6*x^3) - (5*(-2 - 3*x^2)^(1/4))/(8*x) - (5*sqrt(3)*sqrt(-(x^2/(sqrt(2) + sqrt(-2 - 3*x^2))^2))*(sqrt(2) + sqrt(-2 - 3*x^2))*Elliptic.F(2*atan((-2 - 3*x^2)^(1/4)/2^(1/4)), 1/2))/(16*2^(1/4)*x)]
    @test_int [1/(x^6*(-2 - 3*x^2)^(3/4)), x, 5, (-2 - 3*x^2)^(1/4)/(10*x^5) - (9*(-2 - 3*x^2)^(1/4))/(40*x^3) + (27*(-2 - 3*x^2)^(1/4))/(32*x) + (27*sqrt(3)*sqrt(-(x^2/(sqrt(2) + sqrt(-2 - 3*x^2))^2))*(sqrt(2) + sqrt(-2 - 3*x^2))*Elliptic.F(2*atan((-2 - 3*x^2)^(1/4)/2^(1/4)), 1/2))/(64*2^(1/4)*x)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(c*x)^(m/2)*(a+b*x^2)^(p/4)=#

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

    @test_int [(c*x)^(7/2)*(a + b*x^2)^(1/4), x, 8, -((a^2*c^3*sqrt(c*x)*(a + b*x^2)^(1/4))/(12*b^2)) + (a*c*(c*x)^(5/2)*(a + b*x^2)^(1/4))/(30*b) + ((c*x)^(9/2)*(a + b*x^2)^(1/4))/(5*c) - (a^(5/2)*c^2*(1 + a/(b*x^2))^(3/4)*(c*x)^(3/2)*Elliptic.F((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(12*b^(3/2)*(a + b*x^2)^(3/4))]
    @test_int [(c*x)^(3/2)*(a + b*x^2)^(1/4), x, 7, (a*c*sqrt(c*x)*(a + b*x^2)^(1/4))/(6*b) + ((c*x)^(5/2)*(a + b*x^2)^(1/4))/(3*c) + (a^(3/2)*(1 + a/(b*x^2))^(3/4)*(c*x)^(3/2)*Elliptic.F(acot((sqrt(b)*x)/sqrt(a))/2, 2))/(6*sqrt(b)*(a + b*x^2)^(3/4))]
    @test_int [(a + b*x^2)^(1/4)/(c*x)^(1/2), x, 6, (sqrt(c*x)*(a + b*x^2)^(1/4))/c - (sqrt(a)*sqrt(b)*(1 + a/(b*x^2))^(3/4)*(c*x)^(3/2)*Elliptic.F(acot((sqrt(b)*x)/sqrt(a))/2, 2))/(c^2*(a + b*x^2)^(3/4))]
    @test_int [(a + b*x^2)^(1/4)/(c*x)^(5/2), x, 6, (-2*(a + b*x^2)^(1/4))/(3*c*(c*x)^(3/2)) - (2*b^(3/2)*(1 + a/(b*x^2))^(3/4)*(c*x)^(3/2)*Elliptic.F(acot((sqrt(b)*x)/sqrt(a))/2, 2))/(3*sqrt(a)*c^4*(a + b*x^2)^(3/4))]
    @test_int [(a + b*x^2)^(1/4)/(c*x)^(9/2), x, 7, (-2*(a + b*x^2)^(1/4))/(7*c*(c*x)^(7/2)) - (2*b*(a + b*x^2)^(1/4))/(21*a*c^3*(c*x)^(3/2)) + (4*b^(5/2)*(1 + a/(b*x^2))^(3/4)*(c*x)^(3/2)*Elliptic.F(acot((sqrt(b)*x)/sqrt(a))/2, 2))/(21*a^(3/2)*c^6*(a + b*x^2)^(3/4))]
    @test_int [(a + b*x^2)^(1/4)/(c*x)^(13/2), x, 8, (-2*(a + b*x^2)^(1/4))/(11*c*(c*x)^(11/2)) - (2*b*(a + b*x^2)^(1/4))/(77*a*c^3*(c*x)^(7/2)) + (4*b^2*(a + b*x^2)^(1/4))/(77*a^2*c^5*(c*x)^(3/2)) - (8*b^(7/2)*(1 + a/(b*x^2))^(3/4)*(c*x)^(3/2)*Elliptic.F(acot((sqrt(b)*x)/sqrt(a))/2, 2))/(77*a^(5/2)*c^8*(a + b*x^2)^(3/4))]

    @test_int [(c*x)^(5/2)*(a + b*x^2)^(1/4), x, 7, (a*c*(c*x)^(3/2)*(a + b*x^2)^(1/4))/(16*b) + ((c*x)^(7/2)*(a + b*x^2)^(1/4))/(4*c) + (3*a^2*c^(5/2)*atan((b^(1/4)*sqrt(c*x))/(sqrt(c)*(a + b*x^2)^(1/4))))/(32*b^(7/4)) - (3*a^2*c^(5/2)*atanh((b^(1/4)*sqrt(c*x))/(sqrt(c)*(a + b*x^2)^(1/4))))/(32*b^(7/4))]
    @test_int [(c*x)^(1/2)*(a + b*x^2)^(1/4), x, 6, ((c*x)^(3/2)*(a + b*x^2)^(1/4))/(2*c) - (a*sqrt(c)*atan((b^(1/4)*sqrt(c*x))/(sqrt(c)*(a + b*x^2)^(1/4))))/(4*b^(3/4)) + (a*sqrt(c)*atanh((b^(1/4)*sqrt(c*x))/(sqrt(c)*(a + b*x^2)^(1/4))))/(4*b^(3/4))]
    @test_int [(a + b*x^2)^(1/4)/(c*x)^(3/2), x, 6, (-2*(a + b*x^2)^(1/4))/(c*sqrt(c*x)) - (b^(1/4)*atan((b^(1/4)*sqrt(c*x))/(sqrt(c)*(a + b*x^2)^(1/4))))/c^(3/2) + (b^(1/4)*atanh((b^(1/4)*sqrt(c*x))/(sqrt(c)*(a + b*x^2)^(1/4))))/c^(3/2)]
    @test_int [(a + b*x^2)^(1/4)/(c*x)^(7/2), x, 1, (-2*(a + b*x^2)^(5/4))/(5*a*c*(c*x)^(5/2))]
    @test_int [(a + b*x^2)^(1/4)/(c*x)^(11/2), x, 2, (-2*(a + b*x^2)^(5/4))/(5*a*c*(c*x)^(9/2)) + (8*(a + b*x^2)^(9/4))/(45*a^2*c*(c*x)^(9/2))]
    @test_int [(a + b*x^2)^(1/4)/(c*x)^(15/2), x, 3, -((2*(a + b*x^2)^(5/4))/(5*a*c*(c*x)^(13/2))) + (16*(a + b*x^2)^(9/4))/(45*a^2*c*(c*x)^(13/2)) - (64*(a + b*x^2)^(13/4))/(585*a^3*c*(c*x)^(13/2))]
    @test_int [(a + b*x^2)^(1/4)/(c*x)^(19/2), x, 4, -((2*(a + b*x^2)^(5/4))/(5*a*c*(c*x)^(17/2))) + (8*(a + b*x^2)^(9/4))/(15*a^2*c*(c*x)^(17/2)) - (64*(a + b*x^2)^(13/4))/(195*a^3*c*(c*x)^(17/2)) + (256*(a + b*x^2)^(17/4))/(3315*a^4*c*(c*x)^(17/2))]

    @test_int [(c*x)^(3/2)*(a - b*x^2)^(1/4), x, 7, -(a*c*sqrt(c*x)*(a - b*x^2)^(1/4))/(6*b) + ((c*x)^(5/2)*(a - b*x^2)^(1/4))/(3*c) - (a^(3/2)*(1 - a/(b*x^2))^(3/4)*(c*x)^(3/2)*Elliptic.F(acsc((sqrt(b)*x)/sqrt(a))/2, 2))/(6*sqrt(b)*(a - b*x^2)^(3/4))]
    @test_int [(a - b*x^2)^(1/4)/(c*x)^(1/2), x, 6, (sqrt(c*x)*(a - b*x^2)^(1/4))/c - (sqrt(a)*sqrt(b)*(1 - a/(b*x^2))^(3/4)*(c*x)^(3/2)*Elliptic.F(acsc((sqrt(b)*x)/sqrt(a))/2, 2))/(c^2*(a - b*x^2)^(3/4))]
    @test_int [(a - b*x^2)^(1/4)/(c*x)^(5/2), x, 6, (-2*(a - b*x^2)^(1/4))/(3*c*(c*x)^(3/2)) + (2*b^(3/2)*(1 - a/(b*x^2))^(3/4)*(c*x)^(3/2)*Elliptic.F(acsc((sqrt(b)*x)/sqrt(a))/2, 2))/(3*sqrt(a)*c^4*(a - b*x^2)^(3/4))]
    @test_int [(a - b*x^2)^(1/4)/(c*x)^(9/2), x, 7, (-2*(a - b*x^2)^(1/4))/(7*c*(c*x)^(7/2)) + (2*b*(a - b*x^2)^(1/4))/(21*a*c^3*(c*x)^(3/2)) + (4*b^(5/2)*(1 - a/(b*x^2))^(3/4)*(c*x)^(3/2)*Elliptic.F(acsc((sqrt(b)*x)/sqrt(a))/2, 2))/(21*a^(3/2)*c^6*(a - b*x^2)^(3/4))]
    @test_int [(a - b*x^2)^(1/4)/(c*x)^(13/2), x, 8, (-2*(a - b*x^2)^(1/4))/(11*c*(c*x)^(11/2)) + (2*b*(a - b*x^2)^(1/4))/(77*a*c^3*(c*x)^(7/2)) + (4*b^2*(a - b*x^2)^(1/4))/(77*a^2*c^5*(c*x)^(3/2)) + (8*b^(7/2)*(1 - a/(b*x^2))^(3/4)*(c*x)^(3/2)*Elliptic.F(acsc((sqrt(b)*x)/sqrt(a))/2, 2))/(77*a^(5/2)*c^8*(a - b*x^2)^(3/4))]

    @test_int [(c*x)^(5/2)*(a - b*x^2)^(1/4), x, 13, -(a*c*(c*x)^(3/2)*(a - b*x^2)^(1/4))/(16*b) + ((c*x)^(7/2)*(a - b*x^2)^(1/4))/(4*c) - (3*a^2*c^(5/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(c*x))/(sqrt(c)*(a - b*x^2)^(1/4))))/(32*sqrt(2)*b^(7/4)) + (3*a^2*c^(5/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(c*x))/(sqrt(c)*(a - b*x^2)^(1/4))))/(32*sqrt(2)*b^(7/4)) + (3*a^2*c^(5/2)*log(sqrt(c) + (sqrt(b)*sqrt(c)*x)/sqrt(a - b*x^2) - (sqrt(2)*b^(1/4)*sqrt(c*x))/(a - b*x^2)^(1/4)))/(64*sqrt(2)*b^(7/4)) - (3*a^2*c^(5/2)*log(sqrt(c) + (sqrt(b)*sqrt(c)*x)/sqrt(a - b*x^2) + (sqrt(2)*b^(1/4)*sqrt(c*x))/(a - b*x^2)^(1/4)))/(64*sqrt(2)*b^(7/4))]
    @test_int [(c*x)^(1/2)*(a - b*x^2)^(1/4), x, 12, ((c*x)^(3/2)*(a - b*x^2)^(1/4))/(2*c) - (a*sqrt(c)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(c*x))/(sqrt(c)*(a - b*x^2)^(1/4))))/(4*sqrt(2)*b^(3/4)) + (a*sqrt(c)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(c*x))/(sqrt(c)*(a - b*x^2)^(1/4))))/(4*sqrt(2)*b^(3/4)) + (a*sqrt(c)*log(sqrt(c) + (sqrt(b)*sqrt(c)*x)/sqrt(a - b*x^2) - (sqrt(2)*b^(1/4)*sqrt(c*x))/(a - b*x^2)^(1/4)))/(8*sqrt(2)*b^(3/4)) - (a*sqrt(c)*log(sqrt(c) + (sqrt(b)*sqrt(c)*x)/sqrt(a - b*x^2) + (sqrt(2)*b^(1/4)*sqrt(c*x))/(a - b*x^2)^(1/4)))/(8*sqrt(2)*b^(3/4))]
    @test_int [(a - b*x^2)^(1/4)/(c*x)^(3/2), x, 12, (-2*(a - b*x^2)^(1/4))/(c*sqrt(c*x)) + (b^(1/4)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(c*x))/(sqrt(c)*(a - b*x^2)^(1/4))))/(sqrt(2)*c^(3/2)) - (b^(1/4)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(c*x))/(sqrt(c)*(a - b*x^2)^(1/4))))/(sqrt(2)*c^(3/2)) - (b^(1/4)*log(sqrt(c) + (sqrt(b)*sqrt(c)*x)/sqrt(a - b*x^2) - (sqrt(2)*b^(1/4)*sqrt(c*x))/(a - b*x^2)^(1/4)))/(2*sqrt(2)*c^(3/2)) + (b^(1/4)*log(sqrt(c) + (sqrt(b)*sqrt(c)*x)/sqrt(a - b*x^2) + (sqrt(2)*b^(1/4)*sqrt(c*x))/(a - b*x^2)^(1/4)))/(2*sqrt(2)*c^(3/2))]
    @test_int [(a - b*x^2)^(1/4)/(c*x)^(7/2), x, 1, (-2*(a - b*x^2)^(5/4))/(5*a*c*(c*x)^(5/2))]
    @test_int [(a - b*x^2)^(1/4)/(c*x)^(11/2), x, 2, (-2*(a - b*x^2)^(5/4))/(5*a*c*(c*x)^(9/2)) + (8*(a - b*x^2)^(9/4))/(45*a^2*c*(c*x)^(9/2))]
    @test_int [(a - b*x^2)^(1/4)/(c*x)^(15/2), x, 3, -((2*(a - b*x^2)^(5/4))/(5*a*c*(c*x)^(13/2))) + (16*(a - b*x^2)^(9/4))/(45*a^2*c*(c*x)^(13/2)) - (64*(a - b*x^2)^(13/4))/(585*a^3*c*(c*x)^(13/2))]
    @test_int [(a - b*x^2)^(1/4)/(c*x)^(19/2), x, 4, -((2*(a - b*x^2)^(5/4))/(5*a*c*(c*x)^(17/2))) + (8*(a - b*x^2)^(9/4))/(15*a^2*c*(c*x)^(17/2)) - (64*(a - b*x^2)^(13/4))/(195*a^3*c*(c*x)^(17/2)) + (256*(a - b*x^2)^(17/4))/(3315*a^4*c*(c*x)^(17/2))]

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

    @test_int [(c*x)^(3/2)/(a + b*x^2)^(1/4), x, 6, (c*sqrt(c*x)*(a + b*x^2)^(3/4))/(2*b) - (a*c^(3/2)*atan((b^(1/4)*sqrt(c*x))/(sqrt(c)*(a + b*x^2)^(1/4))))/(4*b^(5/4)) - (a*c^(3/2)*atanh((b^(1/4)*sqrt(c*x))/(sqrt(c)*(a + b*x^2)^(1/4))))/(4*b^(5/4))]
    @test_int [1/((c*x)^(1/2)*(a + b*x^2)^(1/4)), x, 5, atan((b^(1/4)*sqrt(c*x))/(sqrt(c)*(a + b*x^2)^(1/4)))/(b^(1/4)*sqrt(c)) + atanh((b^(1/4)*sqrt(c*x))/(sqrt(c)*(a + b*x^2)^(1/4)))/(b^(1/4)*sqrt(c))]
    @test_int [1/((c*x)^(5/2)*(a + b*x^2)^(1/4)), x, 1, -((2*(a + b*x^2)^(3/4))/(3*a*c*(c*x)^(3/2)))]
    @test_int [1/((c*x)^(9/2)*(a + b*x^2)^(1/4)), x, 2, -((2*(a + b*x^2)^(3/4))/(3*a*c*(c*x)^(7/2))) + (8*(a + b*x^2)^(7/4))/(21*a^2*c*(c*x)^(7/2))]
    @test_int [1/((c*x)^(13/2)*(a + b*x^2)^(1/4)), x, 3, -((2*(a + b*x^2)^(3/4))/(3*a*c*(c*x)^(11/2))) + (16*(a + b*x^2)^(7/4))/(21*a^2*c*(c*x)^(11/2)) - (64*(a + b*x^2)^(11/4))/(231*a^3*c*(c*x)^(11/2))]

    @test_int [(c*x)^(9/2)/(a + b*x^2)^(1/4), x, 6, (7*a^2*c^4*x*sqrt(c*x))/(20*b^2*(a + b*x^2)^(1/4)) - (7*a*c^3*(c*x)^(3/2)*(a + b*x^2)^(3/4))/(30*b^2) + (c*(c*x)^(7/2)*(a + b*x^2)^(3/4))/(5*b) + (7*a^(5/2)*c^4*(1 + a/(b*x^2))^(1/4)*sqrt(c*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(20*b^(5/2)*(a + b*x^2)^(1/4))]
    @test_int [(c*x)^(5/2)/(a + b*x^2)^(1/4), x, 5, -((a*c^2*x*sqrt(c*x))/(2*b*(a + b*x^2)^(1/4))) + (c*(c*x)^(3/2)*(a + b*x^2)^(3/4))/(3*b) - (a^(3/2)*c^2*(1 + a/(b*x^2))^(1/4)*sqrt(c*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(2*b^(3/2)*(a + b*x^2)^(1/4))]
    @test_int [(c*x)^(1/2)/(a + b*x^2)^(1/4), x, 4, (x*sqrt(c*x))/(a + b*x^2)^(1/4) + (sqrt(a)*(1 + a/(b*x^2))^(1/4)*sqrt(c*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(sqrt(b)*(a + b*x^2)^(1/4))]
    @test_int [1/((c*x)^(3/2)*(a + b*x^2)^(1/4)), x, 4, -(2/(c*sqrt(c*x)*(a + b*x^2)^(1/4))) + (2*sqrt(b)*(1 + a/(b*x^2))^(1/4)*sqrt(c*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(sqrt(a)*c^2*(a + b*x^2)^(1/4))]
    @test_int [1/((c*x)^(7/2)*(a + b*x^2)^(1/4)), x, 5, (4*b)/(5*a*c^3*sqrt(c*x)*(a + b*x^2)^(1/4)) - (2*(a + b*x^2)^(3/4))/(5*a*c*(c*x)^(5/2)) - (4*b^(3/2)*(1 + a/(b*x^2))^(1/4)*sqrt(c*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(5*a^(3/2)*c^4*(a + b*x^2)^(1/4))]
    @test_int [1/((c*x)^(11/2)*(a + b*x^2)^(1/4)), x, 6, -((8*b^2)/(15*a^2*c^5*sqrt(c*x)*(a + b*x^2)^(1/4))) - (2*(a + b*x^2)^(3/4))/(9*a*c*(c*x)^(9/2)) + (4*b*(a + b*x^2)^(3/4))/(15*a^2*c^3*(c*x)^(5/2)) + (8*b^(5/2)*(1 + a/(b*x^2))^(1/4)*sqrt(c*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(15*a^(5/2)*c^6*(a + b*x^2)^(1/4))]

    @test_int [(c*x)^(3/2)/(a - b*x^2)^(1/4), x, 12, -((c*sqrt(c*x)*(a - b*x^2)^(3/4))/(2*b)) - (a*c^(3/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(c*x))/(sqrt(c)*(a - b*x^2)^(1/4))))/(4*sqrt(2)*b^(5/4)) + (a*c^(3/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(c*x))/(sqrt(c)*(a - b*x^2)^(1/4))))/(4*sqrt(2)*b^(5/4)) - (a*c^(3/2)*log(sqrt(c) + (sqrt(b)*sqrt(c)*x)/sqrt(a - b*x^2) - (sqrt(2)*b^(1/4)*sqrt(c*x))/(a - b*x^2)^(1/4)))/(8*sqrt(2)*b^(5/4)) + (a*c^(3/2)*log(sqrt(c) + (sqrt(b)*sqrt(c)*x)/sqrt(a - b*x^2) + (sqrt(2)*b^(1/4)*sqrt(c*x))/(a - b*x^2)^(1/4)))/(8*sqrt(2)*b^(5/4))]
    @test_int [1/((c*x)^(1/2)*(a - b*x^2)^(1/4)), x, 11, -(atan(1 - (sqrt(2)*b^(1/4)*sqrt(c*x))/(sqrt(c)*(a - b*x^2)^(1/4)))/(sqrt(2)*b^(1/4)*sqrt(c))) + atan(1 + (sqrt(2)*b^(1/4)*sqrt(c*x))/(sqrt(c)*(a - b*x^2)^(1/4)))/(sqrt(2)*b^(1/4)*sqrt(c)) - log(sqrt(c) + (sqrt(b)*sqrt(c)*x)/sqrt(a - b*x^2) - (sqrt(2)*b^(1/4)*sqrt(c*x))/(a - b*x^2)^(1/4))/(2*sqrt(2)*b^(1/4)*sqrt(c)) + log(sqrt(c) + (sqrt(b)*sqrt(c)*x)/sqrt(a - b*x^2) + (sqrt(2)*b^(1/4)*sqrt(c*x))/(a - b*x^2)^(1/4))/(2*sqrt(2)*b^(1/4)*sqrt(c))]
    @test_int [1/((c*x)^(5/2)*(a - b*x^2)^(1/4)), x, 1, -((2*(a - b*x^2)^(3/4))/(3*a*c*(c*x)^(3/2)))]
    @test_int [1/((c*x)^(9/2)*(a - b*x^2)^(1/4)), x, 2, -((2*(a - b*x^2)^(3/4))/(3*a*c*(c*x)^(7/2))) + (8*(a - b*x^2)^(7/4))/(21*a^2*c*(c*x)^(7/2))]
    @test_int [1/((c*x)^(13/2)*(a - b*x^2)^(1/4)), x, 3, -((2*(a - b*x^2)^(3/4))/(3*a*c*(c*x)^(11/2))) + (16*(a - b*x^2)^(7/4))/(21*a^2*c*(c*x)^(11/2)) - (64*(a - b*x^2)^(11/4))/(231*a^3*c*(c*x)^(11/2))]

    @test_int [(c*x)^(5/2)/(a - b*x^2)^(1/4), x, 5, -((a*c^3*(a - b*x^2)^(3/4))/(2*b^2*sqrt(c*x))) - (c*(c*x)^(3/2)*(a - b*x^2)^(3/4))/(3*b) + (a^(3/2)*c^2*(1 - a/(b*x^2))^(1/4)*sqrt(c*x)*Elliptic.E((1/2)*acsc((sqrt(b)*x)/sqrt(a)), 2))/(2*b^(3/2)*(a - b*x^2)^(1/4))]
    @test_int [(c*x)^(1/2)/(a - b*x^2)^(1/4), x, 4, -((c*(a - b*x^2)^(3/4))/(b*sqrt(c*x))) + (sqrt(a)*(1 - a/(b*x^2))^(1/4)*sqrt(c*x)*Elliptic.E((1/2)*acsc((sqrt(b)*x)/sqrt(a)), 2))/(sqrt(b)*(a - b*x^2)^(1/4))]
    @test_int [1/((c*x)^(3/2)*(a - b*x^2)^(1/4)), x, 3, -((2*sqrt(b)*(1 - a/(b*x^2))^(1/4)*sqrt(c*x)*Elliptic.E((1/2)*acsc((sqrt(b)*x)/sqrt(a)), 2))/(sqrt(a)*c^2*(a - b*x^2)^(1/4)))]
    @test_int [1/((c*x)^(7/2)*(a - b*x^2)^(1/4)), x, 4, -((2*(a - b*x^2)^(3/4))/(5*a*c*(c*x)^(5/2))) - (4*b^(3/2)*(1 - a/(b*x^2))^(1/4)*sqrt(c*x)*Elliptic.E((1/2)*acsc((sqrt(b)*x)/sqrt(a)), 2))/(5*a^(3/2)*c^4*(a - b*x^2)^(1/4))]
    @test_int [1/((c*x)^(11/2)*(a - b*x^2)^(1/4)), x, 5, -((2*(a - b*x^2)^(3/4))/(9*a*c*(c*x)^(9/2))) - (4*b*(a - b*x^2)^(3/4))/(15*a^2*c^3*(c*x)^(5/2)) - (8*b^(5/2)*(1 - a/(b*x^2))^(1/4)*sqrt(c*x)*Elliptic.E((1/2)*acsc((sqrt(b)*x)/sqrt(a)), 2))/(15*a^(5/2)*c^6*(a - b*x^2)^(1/4))]

    @test_int [(c*x)^(3/2)/(a + b*x^2)^(3/4), x, 6, (c*sqrt(c*x)*(a + b*x^2)^(1/4))/b + (sqrt(a)*(1 + a/(b*x^2))^(3/4)*(c*x)^(3/2)*Elliptic.F(acot((sqrt(b)*x)/sqrt(a))/2, 2))/(sqrt(b)*(a + b*x^2)^(3/4))]
    @test_int [1/((c*x)^(1/2)*(a + b*x^2)^(3/4)), x, 5, (-2*sqrt(b)*(1 + a/(b*x^2))^(3/4)*(c*x)^(3/2)*Elliptic.F(acot((sqrt(b)*x)/sqrt(a))/2, 2))/(sqrt(a)*c^2*(a + b*x^2)^(3/4))]
    @test_int [1/((c*x)^(5/2)*(a + b*x^2)^(3/4)), x, 6, (-2*(a + b*x^2)^(1/4))/(3*a*c*(c*x)^(3/2)) + (4*b^(3/2)*(1 + a/(b*x^2))^(3/4)*(c*x)^(3/2)*Elliptic.F(acot((sqrt(b)*x)/sqrt(a))/2, 2))/(3*a^(3/2)*c^4*(a + b*x^2)^(3/4))]
    @test_int [1/((c*x)^(9/2)*(a + b*x^2)^(3/4)), x, 7, (-2*(a + b*x^2)^(1/4))/(7*a*c*(c*x)^(7/2)) + (4*b*(a + b*x^2)^(1/4))/(7*a^2*c^3*(c*x)^(3/2)) - (8*b^(5/2)*(1 + a/(b*x^2))^(3/4)*(c*x)^(3/2)*Elliptic.F(acot((sqrt(b)*x)/sqrt(a))/2, 2))/(7*a^(5/2)*c^6*(a + b*x^2)^(3/4))]
    @test_int [1/((c*x)^(13/2)*(a + b*x^2)^(3/4)), x, 8, (-2*(a + b*x^2)^(1/4))/(11*a*c*(c*x)^(11/2)) + (20*b*(a + b*x^2)^(1/4))/(77*a^2*c^3*(c*x)^(7/2)) - (40*b^2*(a + b*x^2)^(1/4))/(77*a^3*c^5*(c*x)^(3/2)) + (80*b^(7/2)*(1 + a/(b*x^2))^(3/4)*(c*x)^(3/2)*Elliptic.F(acot((sqrt(b)*x)/sqrt(a))/2, 2))/(77*a^(7/2)*c^8*(a + b*x^2)^(3/4))]

    @test_int [(c*x)^(5/2)/(a + b*x^2)^(3/4), x, 6, (c*(c*x)^(3/2)*(a + b*x^2)^(1/4))/(2*b) + (3*a*c^(5/2)*atan((b^(1/4)*sqrt(c*x))/(sqrt(c)*(a + b*x^2)^(1/4))))/(4*b^(7/4)) - (3*a*c^(5/2)*atanh((b^(1/4)*sqrt(c*x))/(sqrt(c)*(a + b*x^2)^(1/4))))/(4*b^(7/4))]
    @test_int [(c*x)^(1/2)/(a + b*x^2)^(3/4), x, 5, -((sqrt(c)*atan((b^(1/4)*sqrt(c*x))/(sqrt(c)*(a + b*x^2)^(1/4))))/b^(3/4)) + (sqrt(c)*atanh((b^(1/4)*sqrt(c*x))/(sqrt(c)*(a + b*x^2)^(1/4))))/b^(3/4)]
    @test_int [1/((c*x)^(3/2)*(a + b*x^2)^(3/4)), x, 1, (-2*(a + b*x^2)^(1/4))/(a*c*sqrt(c*x))]
    @test_int [1/((c*x)^(7/2)*(a + b*x^2)^(3/4)), x, 2, (-2*(a + b*x^2)^(1/4))/(a*c*(c*x)^(5/2)) + (8*(a + b*x^2)^(5/4))/(5*a^2*c*(c*x)^(5/2))]
    @test_int [1/((c*x)^(11/2)*(a + b*x^2)^(3/4)), x, 3, (-2*(a + b*x^2)^(1/4))/(a*c*(c*x)^(9/2)) + (16*(a + b*x^2)^(5/4))/(5*a^2*c*(c*x)^(9/2)) - (64*(a + b*x^2)^(9/4))/(45*a^3*c*(c*x)^(9/2))]

    @test_int [(c*x)^(3/2)/(a - b*x^2)^(3/4), x, 6, -((c*sqrt(c*x)*(a - b*x^2)^(1/4))/b) - (sqrt(a)*(1 - a/(b*x^2))^(3/4)*(c*x)^(3/2)*Elliptic.F(acsc((sqrt(b)*x)/sqrt(a))/2, 2))/(sqrt(b)*(a - b*x^2)^(3/4))]
    @test_int [1/((c*x)^(1/2)*(a - b*x^2)^(3/4)), x, 5, (-2*sqrt(b)*(1 - a/(b*x^2))^(3/4)*(c*x)^(3/2)*Elliptic.F(acsc((sqrt(b)*x)/sqrt(a))/2, 2))/(sqrt(a)*c^2*(a - b*x^2)^(3/4))]
    @test_int [1/((c*x)^(5/2)*(a - b*x^2)^(3/4)), x, 6, (-2*(a - b*x^2)^(1/4))/(3*a*c*(c*x)^(3/2)) - (4*b^(3/2)*(1 - a/(b*x^2))^(3/4)*(c*x)^(3/2)*Elliptic.F(acsc((sqrt(b)*x)/sqrt(a))/2, 2))/(3*a^(3/2)*c^4*(a - b*x^2)^(3/4))]
    @test_int [1/((c*x)^(9/2)*(a - b*x^2)^(3/4)), x, 7, (-2*(a - b*x^2)^(1/4))/(7*a*c*(c*x)^(7/2)) - (4*b*(a - b*x^2)^(1/4))/(7*a^2*c^3*(c*x)^(3/2)) - (8*b^(5/2)*(1 - a/(b*x^2))^(3/4)*(c*x)^(3/2)*Elliptic.F(acsc((sqrt(b)*x)/sqrt(a))/2, 2))/(7*a^(5/2)*c^6*(a - b*x^2)^(3/4))]
    @test_int [1/((c*x)^(13/2)*(a - b*x^2)^(3/4)), x, 8, (-2*(a - b*x^2)^(1/4))/(11*a*c*(c*x)^(11/2)) - (20*b*(a - b*x^2)^(1/4))/(77*a^2*c^3*(c*x)^(7/2)) - (40*b^2*(a - b*x^2)^(1/4))/(77*a^3*c^5*(c*x)^(3/2)) - (80*b^(7/2)*(1 - a/(b*x^2))^(3/4)*(c*x)^(3/2)*Elliptic.F(acsc((sqrt(b)*x)/sqrt(a))/2, 2))/(77*a^(7/2)*c^8*(a - b*x^2)^(3/4))]

    @test_int [(c*x)^(5/2)/(a - b*x^2)^(3/4), x, 12, -(c*(c*x)^(3/2)*(a - b*x^2)^(1/4))/(2*b) - (3*a*c^(5/2)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(c*x))/(sqrt(c)*(a - b*x^2)^(1/4))))/(4*sqrt(2)*b^(7/4)) + (3*a*c^(5/2)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(c*x))/(sqrt(c)*(a - b*x^2)^(1/4))))/(4*sqrt(2)*b^(7/4)) + (3*a*c^(5/2)*log(sqrt(c) + (sqrt(b)*sqrt(c)*x)/sqrt(a - b*x^2) - (sqrt(2)*b^(1/4)*sqrt(c*x))/(a - b*x^2)^(1/4)))/(8*sqrt(2)*b^(7/4)) - (3*a*c^(5/2)*log(sqrt(c) + (sqrt(b)*sqrt(c)*x)/sqrt(a - b*x^2) + (sqrt(2)*b^(1/4)*sqrt(c*x))/(a - b*x^2)^(1/4)))/(8*sqrt(2)*b^(7/4))]
    @test_int [(c*x)^(1/2)/(a - b*x^2)^(3/4), x, 11, -((sqrt(c)*atan(1 - (sqrt(2)*b^(1/4)*sqrt(c*x))/(sqrt(c)*(a - b*x^2)^(1/4))))/(sqrt(2)*b^(3/4))) + (sqrt(c)*atan(1 + (sqrt(2)*b^(1/4)*sqrt(c*x))/(sqrt(c)*(a - b*x^2)^(1/4))))/(sqrt(2)*b^(3/4)) + (sqrt(c)*log(sqrt(c) + (sqrt(b)*sqrt(c)*x)/sqrt(a - b*x^2) - (sqrt(2)*b^(1/4)*sqrt(c*x))/(a - b*x^2)^(1/4)))/(2*sqrt(2)*b^(3/4)) - (sqrt(c)*log(sqrt(c) + (sqrt(b)*sqrt(c)*x)/sqrt(a - b*x^2) + (sqrt(2)*b^(1/4)*sqrt(c*x))/(a - b*x^2)^(1/4)))/(2*sqrt(2)*b^(3/4))]
    @test_int [1/((c*x)^(3/2)*(a - b*x^2)^(3/4)), x, 1, (-2*(a - b*x^2)^(1/4))/(a*c*sqrt(c*x))]
    @test_int [1/((c*x)^(7/2)*(a - b*x^2)^(3/4)), x, 2, (-2*(a - b*x^2)^(1/4))/(a*c*(c*x)^(5/2)) + (8*(a - b*x^2)^(5/4))/(5*a^2*c*(c*x)^(5/2))]
    @test_int [1/((c*x)^(11/2)*(a - b*x^2)^(3/4)), x, 3, (-2*(a - b*x^2)^(1/4))/(a*c*(c*x)^(9/2)) + (16*(a - b*x^2)^(5/4))/(5*a^2*c*(c*x)^(9/2)) - (64*(a - b*x^2)^(9/4))/(45*a^3*c*(c*x)^(9/2))]

    @test_int [(c*x)^(7/2)/(a + b*x^2)^(5/4), x, 7, (5*a*c^3*sqrt(c*x))/(2*b^2*(a + b*x^2)^(1/4)) + (c*(c*x)^(5/2))/(2*b*(a + b*x^2)^(1/4)) - (5*a*c^(7/2)*atan((b^(1/4)*sqrt(c*x))/(sqrt(c)*(a + b*x^2)^(1/4))))/(4*b^(9/4)) - (5*a*c^(7/2)*atanh((b^(1/4)*sqrt(c*x))/(sqrt(c)*(a + b*x^2)^(1/4))))/(4*b^(9/4))]
    @test_int [(c*x)^(3/2)/(a + b*x^2)^(5/4), x, 6, -((2*c*sqrt(c*x))/(b*(a + b*x^2)^(1/4))) + (c^(3/2)*atan((b^(1/4)*sqrt(c*x))/(sqrt(c)*(a + b*x^2)^(1/4))))/b^(5/4) + (c^(3/2)*atanh((b^(1/4)*sqrt(c*x))/(sqrt(c)*(a + b*x^2)^(1/4))))/b^(5/4)]
    @test_int [1/((c*x)^(1/2)*(a + b*x^2)^(5/4)), x, 1, (2*sqrt(c*x))/(a*c*(a + b*x^2)^(1/4))]
    @test_int [1/((c*x)^(5/2)*(a + b*x^2)^(5/4)), x, 2, 2/(a*c*(c*x)^(3/2)*(a + b*x^2)^(1/4)) - (8*(a + b*x^2)^(3/4))/(3*a^2*c*(c*x)^(3/2))]
    @test_int [1/((c*x)^(9/2)*(a + b*x^2)^(5/4)), x, 3, 2/(a*c*(c*x)^(7/2)*(a + b*x^2)^(1/4)) - (16*(a + b*x^2)^(3/4))/(3*a^2*c*(c*x)^(7/2)) + (64*(a + b*x^2)^(7/4))/(21*a^3*c*(c*x)^(7/2))]
    @test_int [1/((c*x)^(13/2)*(a + b*x^2)^(5/4)), x, 4, 2/(a*c*(c*x)^(11/2)*(a + b*x^2)^(1/4)) - (8*(a + b*x^2)^(3/4))/(a^2*c*(c*x)^(11/2)) + (64*(a + b*x^2)^(7/4))/(7*a^3*c*(c*x)^(11/2)) - (256*(a + b*x^2)^(11/4))/(77*a^4*c*(c*x)^(11/2))]

    @test_int [(c*x)^(13/2)/(a + b*x^2)^(5/4), x, 6, (77*a^2*c^5*(c*x)^(3/2))/(60*b^3*(a + b*x^2)^(1/4)) - (11*a*c^3*(c*x)^(7/2))/(30*b^2*(a + b*x^2)^(1/4)) + (c*(c*x)^(11/2))/(5*b*(a + b*x^2)^(1/4)) + (77*a^(5/2)*c^6*(1 + a/(b*x^2))^(1/4)*sqrt(c*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(20*b^(7/2)*(a + b*x^2)^(1/4))]
    @test_int [(c*x)^(9/2)/(a + b*x^2)^(5/4), x, 5, -((7*a*c^3*(c*x)^(3/2))/(6*b^2*(a + b*x^2)^(1/4))) + (c*(c*x)^(7/2))/(3*b*(a + b*x^2)^(1/4)) - (7*a^(3/2)*c^4*(1 + a/(b*x^2))^(1/4)*sqrt(c*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(2*b^(5/2)*(a + b*x^2)^(1/4))]
    @test_int [(c*x)^(5/2)/(a + b*x^2)^(5/4), x, 4, (c*(c*x)^(3/2))/(b*(a + b*x^2)^(1/4)) + (3*sqrt(a)*c^2*(1 + a/(b*x^2))^(1/4)*sqrt(c*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(b^(3/2)*(a + b*x^2)^(1/4))]
    @test_int [(c*x)^(1/2)/(a + b*x^2)^(5/4), x, 3, -((2*(1 + a/(b*x^2))^(1/4)*sqrt(c*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(sqrt(a)*sqrt(b)*(a + b*x^2)^(1/4)))]
    @test_int [1/((c*x)^(3/2)*(a + b*x^2)^(5/4)), x, 4, -(2/(a*c*sqrt(c*x)*(a + b*x^2)^(1/4))) + (4*sqrt(b)*(1 + a/(b*x^2))^(1/4)*sqrt(c*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(a^(3/2)*c^2*(a + b*x^2)^(1/4))]
    @test_int [1/((c*x)^(7/2)*(a + b*x^2)^(5/4)), x, 5, -(2/(5*a*c*(c*x)^(5/2)*(a + b*x^2)^(1/4))) + (12*b)/(5*a^2*c^3*sqrt(c*x)*(a + b*x^2)^(1/4)) - (24*b^(3/2)*(1 + a/(b*x^2))^(1/4)*sqrt(c*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(5*a^(5/2)*c^4*(a + b*x^2)^(1/4))]
    @test_int [1/((c*x)^(11/2)*(a + b*x^2)^(5/4)), x, 6, -(2/(9*a*c*(c*x)^(9/2)*(a + b*x^2)^(1/4))) + (4*b)/(9*a^2*c^3*(c*x)^(5/2)*(a + b*x^2)^(1/4)) - (8*b^2)/(3*a^3*c^5*sqrt(c*x)*(a + b*x^2)^(1/4)) + (16*b^(5/2)*(1 + a/(b*x^2))^(1/4)*sqrt(c*x)*Elliptic.E((1/2)*acot((sqrt(b)*x)/sqrt(a)), 2))/(3*a^(7/2)*c^6*(a + b*x^2)^(1/4))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(c*x)^(m/4)*(a+b*x^2)^(p/4)=#

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

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

    @test_int [(c*x)^(5/4)/(a + b*x^2)^(1/4), x, 2, (4*(c*x)^(9/4)*(1 + (b*x^2)/a)^(1/4)*HypergeometricFunctions._₂F₁(1/4, 9/8, 17/8, -((b*x^2)/a)))/(9*c*(a + b*x^2)^(1/4))]
    @test_int [(c*x)^(3/4)/(a + b*x^2)^(1/4), x, 2, (4*(c*x)^(7/4)*(1 + (b*x^2)/a)^(1/4)*HypergeometricFunctions._₂F₁(1/4, 7/8, 15/8, -((b*x^2)/a)))/(7*c*(a + b*x^2)^(1/4))]
    @test_int [(c*x)^(1/4)/(a + b*x^2)^(1/4), x, 2, (4*(c*x)^(5/4)*(1 + (b*x^2)/a)^(1/4)*HypergeometricFunctions._₂F₁(1/4, 5/8, 13/8, -((b*x^2)/a)))/(5*c*(a + b*x^2)^(1/4))]
    @test_int [1/((c*x)^(1/4)*(a + b*x^2)^(1/4)), x, 2, (4*(c*x)^(3/4)*(1 + (b*x^2)/a)^(1/4)*HypergeometricFunctions._₂F₁(1/4, 3/8, 11/8, -((b*x^2)/a)))/(3*c*(a + b*x^2)^(1/4))]
    @test_int [1/((c*x)^(3/4)*(a + b*x^2)^(1/4)), x, 2, (4*(c*x)^(1/4)*(1 + (b*x^2)/a)^(1/4)*HypergeometricFunctions._₂F₁(1/8, 1/4, 9/8, -((b*x^2)/a)))/(c*(a + b*x^2)^(1/4))]
    @test_int [1/((c*x)^(5/4)*(a + b*x^2)^(1/4)), x, 2, -((4*(1 + (b*x^2)/a)^(1/4)*HypergeometricFunctions._₂F₁(-(1/8), 1/4, 7/8, -((b*x^2)/a)))/(c*(c*x)^(1/4)*(a + b*x^2)^(1/4)))]

    @test_int [(c*x)^(5/4)/(a + b*x^2)^(7/4), x, 2, (4*(c*x)^(9/4)*(1 + (b*x^2)/a)^(3/4)*HypergeometricFunctions._₂F₁(9/8, 7/4, 17/8, -((b*x^2)/a)))/(9*a*c*(a + b*x^2)^(3/4))]
    @test_int [(c*x)^(3/4)/(a + b*x^2)^(7/4), x, 2, (4*(c*x)^(7/4)*(1 + (b*x^2)/a)^(3/4)*HypergeometricFunctions._₂F₁(7/8, 7/4, 15/8, -((b*x^2)/a)))/(7*a*c*(a + b*x^2)^(3/4))]
    @test_int [(c*x)^(1/4)/(a + b*x^2)^(7/4), x, 2, (4*(c*x)^(5/4)*(1 + (b*x^2)/a)^(3/4)*HypergeometricFunctions._₂F₁(5/8, 7/4, 13/8, -((b*x^2)/a)))/(5*a*c*(a + b*x^2)^(3/4))]
    @test_int [1/((c*x)^(1/4)*(a + b*x^2)^(7/4)), x, 2, (4*(c*x)^(3/4)*(1 + (b*x^2)/a)^(3/4)*HypergeometricFunctions._₂F₁(3/8, 7/4, 11/8, -((b*x^2)/a)))/(3*a*c*(a + b*x^2)^(3/4))]
    @test_int [1/((c*x)^(3/4)*(a + b*x^2)^(7/4)), x, 2, (4*(c*x)^(1/4)*(1 + (b*x^2)/a)^(3/4)*HypergeometricFunctions._₂F₁(1/8, 7/4, 9/8, -((b*x^2)/a)))/(a*c*(a + b*x^2)^(3/4))]
    @test_int [1/((c*x)^(5/4)*(a + b*x^2)^(7/4)), x, 2, -((4*(1 + (b*x^2)/a)^(3/4)*HypergeometricFunctions._₂F₁(-(1/8), 7/4, 7/8, -((b*x^2)/a)))/(a*c*(c*x)^(1/4)*(a + b*x^2)^(3/4)))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(c*x)^m*(a+b*x^2)^(p/6)=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a+b*x^2)^(p/6)=#

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

    @test_int [x^6*(a + b*x^2)^(1/6), x, 7, (81*a^3*x*(a + b*x^2)^(1/6))/(2816*b^3) - (9*a^2*x^3*(a + b*x^2)^(1/6))/(704*b^2) + (3*a*x^5*(a + b*x^2)^(1/6))/(352*b) + (3/22)*x^7*(a + b*x^2)^(1/6) - (81*3^(3/4)*sqrt(2 - sqrt(3))*a^4*(a + b*x^2)^(1/6)*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(2816*b^4*x*(a/(a + b*x^2))^(1/3)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [x^4*(a + b*x^2)^(1/6), x, 6, -((27*a^2*x*(a + b*x^2)^(1/6))/(640*b^2)) + (3*a*x^3*(a + b*x^2)^(1/6))/(160*b) + (3/16)*x^5*(a + b*x^2)^(1/6) + (27*3^(3/4)*sqrt(2 - sqrt(3))*a^3*(a + b*x^2)^(1/6)*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(640*b^3*x*(a/(a + b*x^2))^(1/3)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [x^2*(a + b*x^2)^(1/6), x, 5, (3*a*x*(a + b*x^2)^(1/6))/(40*b) + (3/10)*x^3*(a + b*x^2)^(1/6) - (3*3^(3/4)*sqrt(2 - sqrt(3))*a^2*(a + b*x^2)^(1/6)*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(40*b^2*x*(a/(a + b*x^2))^(1/3)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [x^0*(a + b*x^2)^(1/6), x, 4, (3/4)*x*(a + b*x^2)^(1/6) + (3^(3/4)*sqrt(2 - sqrt(3))*a*(a + b*x^2)^(1/6)*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(4*b*x*(a/(a + b*x^2))^(1/3)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [(a + b*x^2)^(1/6)/x^2, x, 4, -((a + b*x^2)^(1/6)/x) + (sqrt(2 - sqrt(3))*(a + b*x^2)^(1/6)*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(3^(1/4)*x*(a/(a + b*x^2))^(1/3)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [(a + b*x^2)^(1/6)/x^4, x, 5, -((a + b*x^2)^(1/6)/(3*x^3)) - (b*(a + b*x^2)^(1/6))/(9*a*x) - (2*sqrt(2 - sqrt(3))*b*(a + b*x^2)^(1/6)*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(9*3^(1/4)*a*x*(a/(a + b*x^2))^(1/3)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [(a + b*x^2)^(1/6)/x^6, x, 6, -((a + b*x^2)^(1/6)/(5*x^5)) - (b*(a + b*x^2)^(1/6))/(45*a*x^3) + (8*b^2*(a + b*x^2)^(1/6))/(135*a^2*x) + (16*sqrt(2 - sqrt(3))*b^2*(a + b*x^2)^(1/6)*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(135*3^(1/4)*a^2*x*(a/(a + b*x^2))^(1/3)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [(a + b*x^2)^(1/6)/x^8, x, 7, -((a + b*x^2)^(1/6)/(7*x^7)) - (b*(a + b*x^2)^(1/6))/(105*a*x^5) + (2*b^2*(a + b*x^2)^(1/6))/(135*a^2*x^3) - (16*b^3*(a + b*x^2)^(1/6))/(405*a^3*x) - (32*sqrt(2 - sqrt(3))*b^3*(a + b*x^2)^(1/6)*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(405*3^(1/4)*a^3*x*(a/(a + b*x^2))^(1/3)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]

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

    @test_int [x^6/(a + b*x^2)^(1/6), x, 9, -((243*a^3*x)/(896*b^3*(a + b*x^2)^(1/6))) + (81*a^2*x*(a + b*x^2)^(5/6))/(448*b^3) - (9*a*x^3*(a + b*x^2)^(5/6))/(56*b^2) + (3*x^5*(a + b*x^2)^(5/6))/(20*b) - (243*a^4*x)/(896*b^3*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(7/6)*(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))) - (243*3^(1/4)*sqrt(2 + sqrt(3))*a^4*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(1792*b^4*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2))) + (81*3^(3/4)*a^4*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(448*sqrt(2)*b^4*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [x^4/(a + b*x^2)^(1/6), x, 8, (81*a^2*x)/(224*b^2*(a + b*x^2)^(1/6)) - (27*a*x*(a + b*x^2)^(5/6))/(112*b^2) + (3*x^3*(a + b*x^2)^(5/6))/(14*b) + (81*a^3*x)/(224*b^2*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(7/6)*(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))) + (81*3^(1/4)*sqrt(2 + sqrt(3))*a^3*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(448*b^3*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2))) - (27*3^(3/4)*a^3*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(112*sqrt(2)*b^3*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [x^2/(a + b*x^2)^(1/6), x, 7, -((9*a*x)/(16*b*(a + b*x^2)^(1/6))) + (3*x*(a + b*x^2)^(5/6))/(8*b) - (9*a^2*x)/(16*b*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(7/6)*(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))) - (9*3^(1/4)*sqrt(2 + sqrt(3))*a^2*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(32*b^2*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2))) + (3*3^(3/4)*a^2*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(8*sqrt(2)*b^2*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [x^0/(a + b*x^2)^(1/6), x, 6, (3*x)/(2*(a + b*x^2)^(1/6)) + (3*a*x)/(2*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(7/6)*(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))) + (3*3^(1/4)*sqrt(2 + sqrt(3))*a*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(4*b*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2))) - (3^(3/4)*a*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(sqrt(2)*b*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [1/(x^2*(a + b*x^2)^(1/6)), x, 7, (b*x)/(a*(a + b*x^2)^(1/6)) - (a + b*x^2)^(5/6)/(a*x) + (b*x)/((a/(a + b*x^2))^(2/3)*(a + b*x^2)^(7/6)*(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))) + (3^(1/4)*sqrt(2 + sqrt(3))*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(2*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2))) - (sqrt(2)*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(3^(1/4)*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [1/(x^4*(a + b*x^2)^(1/6)), x, 8, -((4*b^2*x)/(9*a^2*(a + b*x^2)^(1/6))) - (a + b*x^2)^(5/6)/(3*a*x^3) + (4*b*(a + b*x^2)^(5/6))/(9*a^2*x) - (4*b^2*x)/(9*a*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(7/6)*(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))) - (2*sqrt(2 + sqrt(3))*b*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(3*3^(3/4)*a*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2))) + (4*sqrt(2)*b*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(9*3^(1/4)*a*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [1/(x^6*(a + b*x^2)^(1/6)), x, 9, (8*b^3*x)/(27*a^3*(a + b*x^2)^(1/6)) - (a + b*x^2)^(5/6)/(5*a*x^5) + (2*b*(a + b*x^2)^(5/6))/(9*a^2*x^3) - (8*b^2*(a + b*x^2)^(5/6))/(27*a^3*x) + (8*b^3*x)/(27*a^2*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(7/6)*(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))) + (4*sqrt(2 + sqrt(3))*b^2*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(9*3^(3/4)*a^2*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2))) - (8*sqrt(2)*b^2*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(27*3^(1/4)*a^2*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]

    @test_int [x^6/(a + b*x^2)^(5/6), x, 6, (81*a^2*x*(a + b*x^2)^(1/6))/(128*b^3) - (9*a*x^3*(a + b*x^2)^(1/6))/(32*b^2) + (3*x^5*(a + b*x^2)^(1/6))/(16*b) - (81*3^(3/4)*sqrt(2 - sqrt(3))*a^3*(a + b*x^2)^(1/6)*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(128*b^4*x*(a/(a + b*x^2))^(1/3)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [x^4/(a + b*x^2)^(5/6), x, 5, -((27*a*x*(a + b*x^2)^(1/6))/(40*b^2)) + (3*x^3*(a + b*x^2)^(1/6))/(10*b) + (27*3^(3/4)*sqrt(2 - sqrt(3))*a^2*(a + b*x^2)^(1/6)*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(40*b^3*x*(a/(a + b*x^2))^(1/3)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [x^2/(a + b*x^2)^(5/6), x, 4, (3*x*(a + b*x^2)^(1/6))/(4*b) - (3*3^(3/4)*sqrt(2 - sqrt(3))*a*(a + b*x^2)^(1/6)*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(4*b^2*x*(a/(a + b*x^2))^(1/3)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [x^0/(a + b*x^2)^(5/6), x, 3, (3^(3/4)*sqrt(2 - sqrt(3))*(a + b*x^2)^(1/6)*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(b*x*(a/(a + b*x^2))^(1/3)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [1/(x^2*(a + b*x^2)^(5/6)), x, 4, -((a + b*x^2)^(1/6)/(a*x)) - (2*sqrt(2 - sqrt(3))*(a + b*x^2)^(1/6)*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(3^(1/4)*a*x*(a/(a + b*x^2))^(1/3)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [1/(x^4*(a + b*x^2)^(5/6)), x, 5, -((a + b*x^2)^(1/6)/(3*a*x^3)) + (8*b*(a + b*x^2)^(1/6))/(9*a^2*x) + (16*sqrt(2 - sqrt(3))*b*(a + b*x^2)^(1/6)*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(9*3^(1/4)*a^2*x*(a/(a + b*x^2))^(1/3)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [1/(x^6*(a + b*x^2)^(5/6)), x, 6, -((a + b*x^2)^(1/6)/(5*a*x^5)) + (14*b*(a + b*x^2)^(1/6))/(45*a^2*x^3) - (112*b^2*(a + b*x^2)^(1/6))/(135*a^3*x) - (224*sqrt(2 - sqrt(3))*b^2*(a + b*x^2)^(1/6)*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(135*3^(1/4)*a^3*x*(a/(a + b*x^2))^(1/3)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]

    @test_int [x^6/(a + b*x^2)^(7/6), x, 9, (1215*a^2*x)/(224*b^3*(a + b*x^2)^(1/6)) - (3*x^5)/(b*(a + b*x^2)^(1/6)) - (405*a*x*(a + b*x^2)^(5/6))/(112*b^3) + (45*x^3*(a + b*x^2)^(5/6))/(14*b^2) + (1215*a^3*x)/(224*b^3*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(7/6)*(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))) + (1215*3^(1/4)*sqrt(2 + sqrt(3))*a^3*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(448*b^4*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2))) - (405*3^(3/4)*a^3*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(112*sqrt(2)*b^4*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [x^4/(a + b*x^2)^(7/6), x, 8, -((81*a*x)/(16*b^2*(a + b*x^2)^(1/6))) - (3*x^3)/(b*(a + b*x^2)^(1/6)) + (27*x*(a + b*x^2)^(5/6))/(8*b^2) - (81*a^2*x)/(16*b^2*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(7/6)*(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))) - (81*3^(1/4)*sqrt(2 + sqrt(3))*a^2*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(32*b^3*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2))) + (27*3^(3/4)*a^2*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(8*sqrt(2)*b^3*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [x^2/(a + b*x^2)^(7/6), x, 7, (3*x)/(2*b*(a + b*x^2)^(1/6)) + (9*a*x)/(2*b*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(7/6)*(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))) + (9*3^(1/4)*sqrt(2 + sqrt(3))*a*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(4*b^2*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2))) - (3*3^(3/4)*a*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(sqrt(2)*b^2*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [x^0/(a + b*x^2)^(7/6), x, 5, -((3*x)/((a/(a + b*x^2))^(2/3)*(a + b*x^2)^(7/6)*(1 - sqrt(3) - (a/(a + b*x^2))^(1/3)))) - (3*3^(1/4)*sqrt(2 + sqrt(3))*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(2*b*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2))) + (sqrt(2)*3^(3/4)*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(b*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [1/(x^2*(a + b*x^2)^(7/6)), x, 8, 3/(a*x*(a + b*x^2)^(1/6)) + (4*b*x)/(a^2*(a + b*x^2)^(1/6)) - (4*(a + b*x^2)^(5/6))/(a^2*x) + (4*b*x)/(a*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(7/6)*(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))) + (2*3^(1/4)*sqrt(2 + sqrt(3))*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(a*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2))) - (4*sqrt(2)*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(3^(1/4)*a*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [1/(x^4*(a + b*x^2)^(7/6)), x, 9, 3/(a*x^3*(a + b*x^2)^(1/6)) - (40*b^2*x)/(9*a^3*(a + b*x^2)^(1/6)) - (10*(a + b*x^2)^(5/6))/(3*a^2*x^3) + (40*b*(a + b*x^2)^(5/6))/(9*a^3*x) - (40*b^2*x)/(9*a^2*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(7/6)*(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))) - (20*sqrt(2 + sqrt(3))*b*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(3*3^(3/4)*a^2*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2))) + (40*sqrt(2)*b*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(9*3^(1/4)*a^2*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]
    @test_int [1/(x^6*(a + b*x^2)^(7/6)), x, 10, 3/(a*x^5*(a + b*x^2)^(1/6)) + (128*b^3*x)/(27*a^4*(a + b*x^2)^(1/6)) - (16*(a + b*x^2)^(5/6))/(5*a^2*x^5) + (32*b*(a + b*x^2)^(5/6))/(9*a^3*x^3) - (128*b^2*(a + b*x^2)^(5/6))/(27*a^4*x) + (128*b^3*x)/(27*a^3*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(7/6)*(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))) + (64*sqrt(2 + sqrt(3))*b^2*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.E(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(9*3^(3/4)*a^3*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2))) - (128*sqrt(2)*b^2*(1 - (a/(a + b*x^2))^(1/3))*sqrt((1 + (a/(a + b*x^2))^(1/3) + (a/(a + b*x^2))^(2/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)*Elliptic.F(asin((1 + sqrt(3) - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))), -7 + 4*sqrt(3)))/(27*3^(1/4)*a^3*x*(a/(a + b*x^2))^(2/3)*(a + b*x^2)^(1/6)*sqrt(-((1 - (a/(a + b*x^2))^(1/3))/(1 - sqrt(3) - (a/(a + b*x^2))^(1/3))^2)))]

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

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

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

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

    @test_int [x^(7/2)*(a + b*x^2)^p, x, 2, (2*x^(9/2)*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 13/4 + p, 13/4, -((b*x^2)/a)))/(9*a), ((2/9)*x^(9/2)*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(9/4, -p, 13/4, -((b*x^2)/a)))/(1 + (b*x^2)/a)^p]
    @test_int [x^(5/2)*(a + b*x^2)^p, x, 2, (2*x^(7/2)*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 11/4 + p, 11/4, -((b*x^2)/a)))/(7*a), ((2/7)*x^(7/2)*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(7/4, -p, 11/4, -((b*x^2)/a)))/(1 + (b*x^2)/a)^p]
    @test_int [x^(3/2)*(a + b*x^2)^p, x, 2, (2*x^(5/2)*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 9/4 + p, 9/4, -((b*x^2)/a)))/(5*a), ((2/5)*x^(5/2)*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(5/4, -p, 9/4, -((b*x^2)/a)))/(1 + (b*x^2)/a)^p]
    @test_int [x^(1/2)*(a + b*x^2)^p, x, 2, (2*x^(3/2)*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 7/4 + p, 7/4, -((b*x^2)/a)))/(3*a), ((2/3)*x^(3/2)*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(3/4, -p, 7/4, -((b*x^2)/a)))/(1 + (b*x^2)/a)^p]
    @test_int [(a + b*x^2)^p/x^(1/2), x, 2, (2*sqrt(x)*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 5/4 + p, 5/4, -((b*x^2)/a)))/a, (2*sqrt(x)*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/4, -p, 5/4, -((b*x^2)/a)))/(1 + (b*x^2)/a)^p]
    @test_int [(a + b*x^2)^p/x^(3/2), x, 2, -((2*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 3/4 + p, 3/4, -((b*x^2)/a)))/(a*sqrt(x))), -((2*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(-(1/4), -p, 3/4, -((b*x^2)/a)))/((1 + (b*x^2)/a)^p*sqrt(x)))]
    @test_int [(a + b*x^2)^p/x^(5/2), x, 2, -((2*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, 1/4 + p, 1/4, -((b*x^2)/a)))/(3*a*x^(3/2))), -((2*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(-(3/4), -p, 1/4, -((b*x^2)/a)))/((1 + (b*x^2)/a)^p*(3*x^(3/2))))]
    @test_int [(a + b*x^2)^p/x^(7/2), x, 2, -((2*(a + b*x^2)^(1 + p)*HypergeometricFunctions._₂F₁(1, -(1/4) + p, -(1/4), -((b*x^2)/a)))/(5*a*x^(5/2))), -((2*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(-(5/4), -p, -(1/4), -((b*x^2)/a)))/((1 + (b*x^2)/a)^p*(5*x^(5/2))))]

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

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

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

max_stars_repo_path: test/1 Algebraic functions/1.1 Binomial products/1.1.2 Quadratic/1.1.2.2 (c x)^m (a+b x^2)^p.jl