text:
@testset "1.1.4.2 (c x)^m (a x^j+b x^n)^p" begin
    (a, b, c, j, m, n, p, q, x, ) = @variables a b c j m n p q x

    #= ::Package:: =#

    #= ::Title::Closed:: =#
    #=Integration*problems*of*the*form*(c*x)^m*(a*x+b*x^n)^p=#

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

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

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

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

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

    @test_int [(-4*x + 3*x^3)^6, x, 3, (4096*x^7)/7 - 2048*x^9 + (34560*x^11)/11 - (34560*x^13)/13 + 1296*x^15 - (5832*x^17)/17 + (729*x^19)/19]

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

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

    #= In*some*of*the*following*examples*gcd*cancellation*should*occur*without*also*partial*fraction
    	expansion, since*that*will*result*in*unnecessary*expansion. =#
    @test_int [x^5/(x - x^3), x, 4, -x - x^3/3 + atanh(x)]
    @test_int [x^4/(x - x^3), x, 4, -(x^2/2) - (1/2)*log(1 - x^2)]
    @test_int [x^3/(x - x^3), x, 3, -x + atanh(x)]
    @test_int [x^2/(x - x^3), x, 2, (-(1/2))*log(1 - x^2)]
    @test_int [x/(x - x^3), x, 2, atanh(x)]
    @test_int [1/(x - x^3), x, 5, log(x) - (1/2)*log(1 - x^2)]
    @test_int [1/(x*(x - x^3)), x, 3, -(1/x) + atanh(x)]
    @test_int [1/(x^2*(x - x^3)), x, 4, -(1/(2*x^2)) + log(x) - (1/2)*log(1 - x^2)]
    @test_int [1/(x^3*(x - x^3)), x, 4, -(1/(3*x^3)) - 1/x + atanh(x)]
    @test_int [1/(x^4*(x - x^3)), x, 4, -(1/(4*x^4)) - 1/(2*x^2) + log(x) - (1/2)*log(1 - x^2)]

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

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

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

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

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

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

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

    @test_int [x^7/(a*x + b*x^3)^(3/2), x, 6, -(x^5/(b*sqrt(a*x + b*x^3))) - (15*a*sqrt(a*x + b*x^3))/(7*b^3) + (9*x^2*sqrt(a*x + b*x^3))/(7*b^2) + (15*a^(7/4)*sqrt(x)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(x))/a^(1/4)), 1/2))/(14*b^(13/4)*sqrt(a*x + b*x^3))]
    @test_int [x^6/(a*x + b*x^3)^(3/2), x, 7, -(x^4/(b*sqrt(a*x + b*x^3))) - (21*a*x*(a + b*x^2))/(5*b^(5/2)*(sqrt(a) + sqrt(b)*x)*sqrt(a*x + b*x^3)) + (7*x*sqrt(a*x + b*x^3))/(5*b^2) + (21*a^(5/4)*sqrt(x)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(x))/a^(1/4)), 1/2))/(5*b^(11/4)*sqrt(a*x + b*x^3)) - (21*a^(5/4)*sqrt(x)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(x))/a^(1/4)), 1/2))/(10*b^(11/4)*sqrt(a*x + b*x^3))]
    @test_int [x^5/(a*x + b*x^3)^(3/2), x, 5, -(x^3/(b*sqrt(a*x + b*x^3))) + (5*sqrt(a*x + b*x^3))/(3*b^2) - (5*a^(3/4)*sqrt(x)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(x))/a^(1/4)), 1/2))/(6*b^(9/4)*sqrt(a*x + b*x^3))]
    @test_int [x^4/(a*x + b*x^3)^(3/2), x, 6, -(x^2/(b*sqrt(a*x + b*x^3))) + (3*x*(a + b*x^2))/(b^(3/2)*(sqrt(a) + sqrt(b)*x)*sqrt(a*x + b*x^3)) - (3*a^(1/4)*sqrt(x)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(x))/a^(1/4)), 1/2))/(b^(7/4)*sqrt(a*x + b*x^3)) + (3*a^(1/4)*sqrt(x)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(x))/a^(1/4)), 1/2))/(2*b^(7/4)*sqrt(a*x + b*x^3))]
    @test_int [x^3/(a*x + b*x^3)^(3/2), x, 4, -(x/(b*sqrt(a*x + b*x^3))) + (sqrt(x)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(x))/a^(1/4)), 1/2))/(2*a^(1/4)*b^(5/4)*sqrt(a*x + b*x^3))]
    @test_int [x^2/(a*x + b*x^3)^(3/2), x, 6, x^2/(a*sqrt(a*x + b*x^3)) - (x*(a + b*x^2))/(a*sqrt(b)*(sqrt(a) + sqrt(b)*x)*sqrt(a*x + b*x^3)) + (sqrt(x)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(x))/a^(1/4)), 1/2))/(a^(3/4)*b^(3/4)*sqrt(a*x + b*x^3)) - (sqrt(x)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(x))/a^(1/4)), 1/2))/(2*a^(3/4)*b^(3/4)*sqrt(a*x + b*x^3))]
    @test_int [x/(a*x + b*x^3)^(3/2), x, 4, x/(a*sqrt(a*x + b*x^3)) + (sqrt(x)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(x))/a^(1/4)), 1/2))/(2*a^(5/4)*b^(1/4)*sqrt(a*x + b*x^3))]
    @test_int [1/(a*x + b*x^3)^(3/2), x, 7, 1/(a*sqrt(a*x + b*x^3)) + (3*sqrt(b)*x*(a + b*x^2))/(a^2*(sqrt(a) + sqrt(b)*x)*sqrt(a*x + b*x^3)) - (3*sqrt(a*x + b*x^3))/(a^2*x) - (3*b^(1/4)*sqrt(x)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(x))/a^(1/4)), 1/2))/(a^(7/4)*sqrt(a*x + b*x^3)) + (3*b^(1/4)*sqrt(x)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(x))/a^(1/4)), 1/2))/(2*a^(7/4)*sqrt(a*x + b*x^3))]
    @test_int [1/(x*(a*x + b*x^3)^(3/2)), x, 5, 1/(a*x*sqrt(a*x + b*x^3)) - (5*sqrt(a*x + b*x^3))/(3*a^2*x^2) - (5*b^(3/4)*sqrt(x)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(x))/a^(1/4)), 1/2))/(6*a^(9/4)*sqrt(a*x + b*x^3))]
    @test_int [1/(x^2*(a*x + b*x^3)^(3/2)), x, 8, 1/(a*x^2*sqrt(a*x + b*x^3)) - (21*b^(3/2)*x*(a + b*x^2))/(5*a^3*(sqrt(a) + sqrt(b)*x)*sqrt(a*x + b*x^3)) - (7*sqrt(a*x + b*x^3))/(5*a^2*x^3) + (21*b*sqrt(a*x + b*x^3))/(5*a^3*x) + (21*b^(5/4)*sqrt(x)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.E(2*atan((b^(1/4)*sqrt(x))/a^(1/4)), 1/2))/(5*a^(11/4)*sqrt(a*x + b*x^3)) - (21*b^(5/4)*sqrt(x)*(sqrt(a) + sqrt(b)*x)*sqrt((a + b*x^2)/(sqrt(a) + sqrt(b)*x)^2)*Elliptic.F(2*atan((b^(1/4)*sqrt(x))/a^(1/4)), 1/2))/(10*a^(11/4)*sqrt(a*x + b*x^3))]

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

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

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

    @test_int [x^(29/2)/(a*x + b*x^3)^(9/2), x, 7, -(x^(25/2)/(7*b*(a*x + b*x^3)^(7/2))) - (9*x^(19/2))/(35*b^2*(a*x + b*x^3)^(5/2)) - (3*x^(13/2))/(5*b^3*(a*x + b*x^3)^(3/2)) - (3*x^(7/2))/(b^4*sqrt(a*x + b*x^3)) + (9*sqrt(x)*sqrt(a*x + b*x^3))/(2*b^5) - (9*a*atanh((sqrt(b)*x^(3/2))/sqrt(a*x + b*x^3)))/(2*b^(11/2))]
    @test_int [x^(27/2)/(a*x + b*x^3)^(9/2), x, 5, -(x^(23/2)/(7*b*(a*x + b*x^3)^(7/2))) - (8*x^(17/2))/(35*b^2*(a*x + b*x^3)^(5/2)) - (16*x^(11/2))/(35*b^3*(a*x + b*x^3)^(3/2)) - (64*x^(5/2))/(35*b^4*sqrt(a*x + b*x^3)) + (128*sqrt(a*x + b*x^3))/(35*b^5*sqrt(x))]
    @test_int [x^(25/2)/(a*x + b*x^3)^(9/2), x, 6, -(x^(21/2)/(7*b*(a*x + b*x^3)^(7/2))) - x^(15/2)/(5*b^2*(a*x + b*x^3)^(5/2)) - x^(9/2)/(3*b^3*(a*x + b*x^3)^(3/2)) - x^(3/2)/(b^4*sqrt(a*x + b*x^3)) + atanh((sqrt(b)*x^(3/2))/sqrt(a*x + b*x^3))/b^(9/2)]
    @test_int [x^(23/2)/(a*x + b*x^3)^(9/2), x, 4, -(x^(19/2)/(7*b*(a*x + b*x^3)^(7/2))) - (6*x^(13/2))/(35*b^2*(a*x + b*x^3)^(5/2)) - (8*x^(7/2))/(35*b^3*(a*x + b*x^3)^(3/2)) - (16*sqrt(x))/(35*b^4*sqrt(a*x + b*x^3))]
    @test_int [x^(21/2)/(a*x + b*x^3)^(9/2), x, 1, x^(21/2)/(7*a*(a*x + b*x^3)^(7/2))]
    @test_int [x^(19/2)/(a*x + b*x^3)^(9/2), x, 3, -(x^(15/2)/(7*b*(a*x + b*x^3)^(7/2))) - (4*x^(9/2))/(35*b^2*(a*x + b*x^3)^(5/2)) - (8*x^(3/2))/(105*b^3*(a*x + b*x^3)^(3/2))]
    @test_int [x^(17/2)/(a*x + b*x^3)^(9/2), x, 2, x^(17/2)/(7*a*(a*x + b*x^3)^(7/2)) + (2*x^(15/2))/(35*a^2*(a*x + b*x^3)^(5/2))]
    @test_int [x^(15/2)/(a*x + b*x^3)^(9/2), x, 2, -(x^(11/2)/(7*b*(a*x + b*x^3)^(7/2))) - (2*x^(5/2))/(35*b^2*(a*x + b*x^3)^(5/2))]
    @test_int [x^(13/2)/(a*x + b*x^3)^(9/2), x, 3, x^(13/2)/(7*a*(a*x + b*x^3)^(7/2)) + (4*x^(11/2))/(35*a^2*(a*x + b*x^3)^(5/2)) + (8*x^(9/2))/(105*a^3*(a*x + b*x^3)^(3/2))]
    @test_int [x^(11/2)/(a*x + b*x^3)^(9/2), x, 1, -(x^(7/2)/(7*b*(a*x + b*x^3)^(7/2)))]
    @test_int [x^(9/2)/(a*x + b*x^3)^(9/2), x, 4, x^(9/2)/(7*a*(a*x + b*x^3)^(7/2)) + (6*x^(7/2))/(35*a^2*(a*x + b*x^3)^(5/2)) + (8*x^(5/2))/(35*a^3*(a*x + b*x^3)^(3/2)) + (16*x^(3/2))/(35*a^4*sqrt(a*x + b*x^3))]
    @test_int [x^(7/2)/(a*x + b*x^3)^(9/2), x, 6, x^(7/2)/(7*a*(a*x + b*x^3)^(7/2)) + x^(5/2)/(5*a^2*(a*x + b*x^3)^(5/2)) + x^(3/2)/(3*a^3*(a*x + b*x^3)^(3/2)) + sqrt(x)/(a^4*sqrt(a*x + b*x^3)) - atanh((sqrt(a)*sqrt(x))/sqrt(a*x + b*x^3))/a^(9/2)]
    @test_int [x^(5/2)/(a*x + b*x^3)^(9/2), x, 5, x^(5/2)/(7*a*(a*x + b*x^3)^(7/2)) + (8*x^(3/2))/(35*a^2*(a*x + b*x^3)^(5/2)) + (16*sqrt(x))/(35*a^3*(a*x + b*x^3)^(3/2)) + 64/(35*a^4*sqrt(x)*sqrt(a*x + b*x^3)) - (128*sqrt(a*x + b*x^3))/(35*a^5*x^(3/2))]
    @test_int [x^(3/2)/(a*x + b*x^3)^(9/2), x, 7, x^(3/2)/(7*a*(a*x + b*x^3)^(7/2)) + (9*sqrt(x))/(35*a^2*(a*x + b*x^3)^(5/2)) + 3/(5*a^3*sqrt(x)*(a*x + b*x^3)^(3/2)) + 3/(a^4*x^(3/2)*sqrt(a*x + b*x^3)) - (9*sqrt(a*x + b*x^3))/(2*a^5*x^(5/2)) + (9*b*atanh((sqrt(a)*sqrt(x))/sqrt(a*x + b*x^3)))/(2*a^(11/2))]
    @test_int [x^(1/2)/(a*x + b*x^3)^(9/2), x, 6, sqrt(x)/(7*a*(a*x + b*x^3)^(7/2)) + 2/(7*a^2*sqrt(x)*(a*x + b*x^3)^(5/2)) + 16/(21*a^3*x^(3/2)*(a*x + b*x^3)^(3/2)) + 32/(7*a^4*x^(5/2)*sqrt(a*x + b*x^3)) - (128*sqrt(a*x + b*x^3))/(21*a^5*x^(7/2)) + (256*b*sqrt(a*x + b*x^3))/(21*a^6*x^(3/2))]
    @test_int [1/(x^(1/2)*(a*x + b*x^3)^(9/2)), x, 8, 1/(7*a*sqrt(x)*(a*x + b*x^3)^(7/2)) + 11/(35*a^2*x^(3/2)*(a*x + b*x^3)^(5/2)) + 33/(35*a^3*x^(5/2)*(a*x + b*x^3)^(3/2)) + 33/(5*a^4*x^(7/2)*sqrt(a*x + b*x^3)) - (33*sqrt(a*x + b*x^3))/(4*a^5*x^(9/2)) + (99*b*sqrt(a*x + b*x^3))/(8*a^6*x^(5/2)) - (99*b^2*atanh((sqrt(a)*sqrt(x))/sqrt(a*x + b*x^3)))/(8*a^(13/2))]
    @test_int [1/(x^(3/2)*(a*x + b*x^3)^(9/2)), x, 7, 1/(7*a*x^(3/2)*(a*x + b*x^3)^(7/2)) + 12/(35*a^2*x^(5/2)*(a*x + b*x^3)^(5/2)) + 8/(7*a^3*x^(7/2)*(a*x + b*x^3)^(3/2)) + 64/(7*a^4*x^(9/2)*sqrt(a*x + b*x^3)) - (384*sqrt(a*x + b*x^3))/(35*a^5*x^(11/2)) + (512*b*sqrt(a*x + b*x^3))/(35*a^6*x^(7/2)) - (1024*b^2*sqrt(a*x + b*x^3))/(35*a^7*x^(3/2))]

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

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

    @test_int [x^4/sqrt(a*x + b*x^4), x, 3, (x*sqrt(a*x + b*x^4))/(3*b) - (a*atanh((sqrt(b)*x^2)/sqrt(a*x + b*x^4)))/(3*b^(3/2))]
    @test_int [x^1/sqrt(a*x + b*x^4), x, 2, (2*atanh((sqrt(b)*x^2)/sqrt(a*x + b*x^4)))/(3*sqrt(b))]
    @test_int [1/(x^2*sqrt(a*x + b*x^4)), x, 1, -((2*sqrt(a*x + b*x^4))/(3*a*x^2))]
    @test_int [1/(x^5*sqrt(a*x + b*x^4)), x, 2, -((2*sqrt(a*x + b*x^4))/(9*a*x^5)) + (4*b*sqrt(a*x + b*x^4))/(9*a^2*x^2)]
    @test_int [1/(x^8*sqrt(a*x + b*x^4)), x, 3, -((2*sqrt(a*x + b*x^4))/(15*a*x^8)) + (8*b*sqrt(a*x + b*x^4))/(45*a^2*x^5) - (16*b^2*sqrt(a*x + b*x^4))/(45*a^3*x^2)]

    @test_int [x^3/sqrt(a*x + b*x^4), x, 4, sqrt(a*x + b*x^4)/(2*b) - (a^(2/3)*x*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*Elliptic.F(acos((a^(1/3) + (1 - sqrt(3))*b^(1/3)*x)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)), (1/4)*(2 + sqrt(3))))/(4*3^(1/4)*b*sqrt((b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*sqrt(a*x + b*x^4))]
    @test_int [x^0/sqrt(a*x + b*x^4), x, 3, (x*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*Elliptic.F(acos((a^(1/3) + (1 - sqrt(3))*b^(1/3)*x)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)), (1/4)*(2 + sqrt(3))))/(3^(1/4)*a^(1/3)*sqrt((b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*sqrt(a*x + b*x^4))]
    @test_int [1/(x^3*sqrt(a*x + b*x^4)), x, 4, -((2*sqrt(a*x + b*x^4))/(5*a*x^3)) - (2*b*x*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*Elliptic.F(acos((a^(1/3) + (1 - sqrt(3))*b^(1/3)*x)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)), (1/4)*(2 + sqrt(3))))/(5*3^(1/4)*a^(4/3)*sqrt((b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*sqrt(a*x + b*x^4))]

    @test_int [x^5/sqrt(a*x + b*x^4), x, 6, -((5*(1 + sqrt(3))*a*x*(a + b*x^3))/(8*b^(5/3)*(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)*sqrt(a*x + b*x^4))) + (x^2*sqrt(a*x + b*x^4))/(4*b) + (5*3^(1/4)*a^(4/3)*x*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*Elliptic.E(acos((a^(1/3) + (1 - sqrt(3))*b^(1/3)*x)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)), (1/4)*(2 + sqrt(3))))/(8*b^(5/3)*sqrt((b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*sqrt(a*x + b*x^4)) + (5*(1 - sqrt(3))*a^(4/3)*x*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*Elliptic.F(acos((a^(1/3) + (1 - sqrt(3))*b^(1/3)*x)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)), (1/4)*(2 + sqrt(3))))/(16*3^(1/4)*b^(5/3)*sqrt((b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*sqrt(a*x + b*x^4))]
    @test_int [x^2/sqrt(a*x + b*x^4), x, 5, ((1 + sqrt(3))*x*(a + b*x^3))/(b^(2/3)*(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)*sqrt(a*x + b*x^4)) - (3^(1/4)*a^(1/3)*x*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*Elliptic.E(acos((a^(1/3) + (1 - sqrt(3))*b^(1/3)*x)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)), (1/4)*(2 + sqrt(3))))/(b^(2/3)*sqrt((b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*sqrt(a*x + b*x^4)) - ((1 - sqrt(3))*a^(1/3)*x*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*Elliptic.F(acos((a^(1/3) + (1 - sqrt(3))*b^(1/3)*x)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)), (1/4)*(2 + sqrt(3))))/(2*3^(1/4)*b^(2/3)*sqrt((b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*sqrt(a*x + b*x^4))]
    @test_int [1/(x^1*sqrt(a*x + b*x^4)), x, 6, (2*(1 + sqrt(3))*b^(1/3)*x*(a + b*x^3))/(a*(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)*sqrt(a*x + b*x^4)) - (2*sqrt(a*x + b*x^4))/(a*x) - (2*3^(1/4)*b^(1/3)*x*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*Elliptic.E(acos((a^(1/3) + (1 - sqrt(3))*b^(1/3)*x)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)), (1/4)*(2 + sqrt(3))))/(a^(2/3)*sqrt((b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*sqrt(a*x + b*x^4)) - ((1 - sqrt(3))*b^(1/3)*x*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*Elliptic.F(acos((a^(1/3) + (1 - sqrt(3))*b^(1/3)*x)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)), (1/4)*(2 + sqrt(3))))/(3^(1/4)*a^(2/3)*sqrt((b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*sqrt(a*x + b*x^4))]

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

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*x^m*(a*x+b*x^5)^p=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^m*(a*x+b*x^5)^(p/2)=#

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^(m/2)*(a*x+b*x^5)^(p/2)=#

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

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

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

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

    @test_int [x^2/(a*x + b*x^(1/2))^(1/2), x, 8, (63*b^4*sqrt(b*sqrt(x) + a*x))/(64*a^5) - (21*b^3*sqrt(x)*sqrt(b*sqrt(x) + a*x))/(32*a^4) + (21*b^2*x*sqrt(b*sqrt(x) + a*x))/(40*a^3) - (9*b*x^(3/2)*sqrt(b*sqrt(x) + a*x))/(20*a^2) + (2*x^2*sqrt(b*sqrt(x) + a*x))/(5*a) - (63*b^5*atanh((sqrt(a)*sqrt(x))/sqrt(b*sqrt(x) + a*x)))/(64*a^(11/2))]
    @test_int [x^1/(a*x + b*x^(1/2))^(1/2), x, 6, (5*b^2*sqrt(b*sqrt(x) + a*x))/(4*a^3) - (5*b*sqrt(x)*sqrt(b*sqrt(x) + a*x))/(6*a^2) + (2*x*sqrt(b*sqrt(x) + a*x))/(3*a) - (5*b^3*atanh((sqrt(a)*sqrt(x))/sqrt(b*sqrt(x) + a*x)))/(4*a^(7/2))]
    @test_int [x^0/(a*x + b*x^(1/2))^(1/2), x, 4, (2*sqrt(b*sqrt(x) + a*x))/a - (2*b*atanh((sqrt(a)*sqrt(x))/sqrt(b*sqrt(x) + a*x)))/a^(3/2)]
    @test_int [1/(x^1*(a*x + b*x^(1/2))^(1/2)), x, 1, -((4*sqrt(b*sqrt(x) + a*x))/(b*sqrt(x)))]
    @test_int [1/(x^2*(a*x + b*x^(1/2))^(1/2)), x, 3, -((4*sqrt(b*sqrt(x) + a*x))/(5*b*x^(3/2))) + (16*a*sqrt(b*sqrt(x) + a*x))/(15*b^2*x) - (32*a^2*sqrt(b*sqrt(x) + a*x))/(15*b^3*sqrt(x))]
    @test_int [1/(x^3*(a*x + b*x^(1/2))^(1/2)), x, 5, -((4*sqrt(b*sqrt(x) + a*x))/(9*b*x^(5/2))) + (32*a*sqrt(b*sqrt(x) + a*x))/(63*b^2*x^2) - (64*a^2*sqrt(b*sqrt(x) + a*x))/(105*b^3*x^(3/2)) + (256*a^3*sqrt(b*sqrt(x) + a*x))/(315*b^4*x) - (512*a^4*sqrt(b*sqrt(x) + a*x))/(315*b^5*sqrt(x))]
    @test_int [1/(x^4*(a*x + b*x^(1/2))^(1/2)), x, 7, -((4*sqrt(b*sqrt(x) + a*x))/(13*b*x^(7/2))) + (48*a*sqrt(b*sqrt(x) + a*x))/(143*b^2*x^3) - (160*a^2*sqrt(b*sqrt(x) + a*x))/(429*b^3*x^(5/2)) + (1280*a^3*sqrt(b*sqrt(x) + a*x))/(3003*b^4*x^2) - (512*a^4*sqrt(b*sqrt(x) + a*x))/(1001*b^5*x^(3/2)) + (2048*a^5*sqrt(b*sqrt(x) + a*x))/(3003*b^6*x) - (4096*a^6*sqrt(b*sqrt(x) + a*x))/(3003*b^7*sqrt(x))]

    @test_int [x^3/(a*x + b*x^(1/2))^(3/2), x, 9, -((4*x^3)/(a*sqrt(b*sqrt(x) + a*x))) + (693*b^4*sqrt(b*sqrt(x) + a*x))/(64*a^6) - (231*b^3*sqrt(x)*sqrt(b*sqrt(x) + a*x))/(32*a^5) + (231*b^2*x*sqrt(b*sqrt(x) + a*x))/(40*a^4) - (99*b*x^(3/2)*sqrt(b*sqrt(x) + a*x))/(20*a^3) + (22*x^2*sqrt(b*sqrt(x) + a*x))/(5*a^2) - (693*b^5*atanh((sqrt(a)*sqrt(x))/sqrt(b*sqrt(x) + a*x)))/(64*a^(13/2))]
    @test_int [x^2/(a*x + b*x^(1/2))^(3/2), x, 7, -((4*x^2)/(a*sqrt(b*sqrt(x) + a*x))) + (35*b^2*sqrt(b*sqrt(x) + a*x))/(4*a^4) - (35*b*sqrt(x)*sqrt(b*sqrt(x) + a*x))/(6*a^3) + (14*x*sqrt(b*sqrt(x) + a*x))/(3*a^2) - (35*b^3*atanh((sqrt(a)*sqrt(x))/sqrt(b*sqrt(x) + a*x)))/(4*a^(9/2))]
    @test_int [x^1/(a*x + b*x^(1/2))^(3/2), x, 5, -((4*x)/(a*sqrt(b*sqrt(x) + a*x))) + (6*sqrt(b*sqrt(x) + a*x))/a^2 - (6*b*atanh((sqrt(a)*sqrt(x))/sqrt(b*sqrt(x) + a*x)))/a^(5/2)]
    @test_int [x^0/(a*x + b*x^(1/2))^(3/2), x, 1, (4*sqrt(x))/(b*sqrt(b*sqrt(x) + a*x))]
    @test_int [1/(x^1*(a*x + b*x^(1/2))^(3/2)), x, 3, 4/(b*sqrt(x)*sqrt(b*sqrt(x) + a*x)) - (16*sqrt(b*sqrt(x) + a*x))/(3*b^2*x) + (32*a*sqrt(b*sqrt(x) + a*x))/(3*b^3*sqrt(x))]
    @test_int [1/(x^2*(a*x + b*x^(1/2))^(3/2)), x, 5, 4/(b*x^(3/2)*sqrt(b*sqrt(x) + a*x)) - (32*sqrt(b*sqrt(x) + a*x))/(7*b^2*x^2) + (192*a*sqrt(b*sqrt(x) + a*x))/(35*b^3*x^(3/2)) - (256*a^2*sqrt(b*sqrt(x) + a*x))/(35*b^4*x) + (512*a^3*sqrt(b*sqrt(x) + a*x))/(35*b^5*sqrt(x))]
    @test_int [1/(x^3*(a*x + b*x^(1/2))^(3/2)), x, 7, 4/(b*x^(5/2)*sqrt(b*sqrt(x) + a*x)) - (48*sqrt(b*sqrt(x) + a*x))/(11*b^2*x^3) + (160*a*sqrt(b*sqrt(x) + a*x))/(33*b^3*x^(5/2)) - (1280*a^2*sqrt(b*sqrt(x) + a*x))/(231*b^4*x^2) + (512*a^3*sqrt(b*sqrt(x) + a*x))/(77*b^5*x^(3/2)) - (2048*a^4*sqrt(b*sqrt(x) + a*x))/(231*b^6*x) + (4096*a^5*sqrt(b*sqrt(x) + a*x))/(231*b^7*sqrt(x))]

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

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

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

    @test_int [x^(5/2)/(a*x + b*x^(1/2))^(1/2), x, 9, -((231*b^5*sqrt(b*sqrt(x) + a*x))/(256*a^6)) + (77*b^4*sqrt(x)*sqrt(b*sqrt(x) + a*x))/(128*a^5) - (77*b^3*x*sqrt(b*sqrt(x) + a*x))/(160*a^4) + (33*b^2*x^(3/2)*sqrt(b*sqrt(x) + a*x))/(80*a^3) - (11*b*x^2*sqrt(b*sqrt(x) + a*x))/(30*a^2) + (x^(5/2)*sqrt(b*sqrt(x) + a*x))/(3*a) + (231*b^6*atanh((sqrt(a)*sqrt(x))/sqrt(b*sqrt(x) + a*x)))/(256*a^(13/2))]
    @test_int [x^(3/2)/(a*x + b*x^(1/2))^(1/2), x, 7, -((35*b^3*sqrt(b*sqrt(x) + a*x))/(32*a^4)) + (35*b^2*sqrt(x)*sqrt(b*sqrt(x) + a*x))/(48*a^3) - (7*b*x*sqrt(b*sqrt(x) + a*x))/(12*a^2) + (x^(3/2)*sqrt(b*sqrt(x) + a*x))/(2*a) + (35*b^4*atanh((sqrt(a)*sqrt(x))/sqrt(b*sqrt(x) + a*x)))/(32*a^(9/2))]
    @test_int [x^(1/2)/(a*x + b*x^(1/2))^(1/2), x, 5, -((3*b*sqrt(b*sqrt(x) + a*x))/(2*a^2)) + (sqrt(x)*sqrt(b*sqrt(x) + a*x))/a + (3*b^2*atanh((sqrt(a)*sqrt(x))/sqrt(b*sqrt(x) + a*x)))/(2*a^(5/2))]
    @test_int [1/(x^(1/2)*(a*x + b*x^(1/2))^(1/2)), x, 3, (4*atanh((sqrt(a)*sqrt(x))/sqrt(b*sqrt(x) + a*x)))/sqrt(a)]
    @test_int [1/(x^(3/2)*(a*x + b*x^(1/2))^(1/2)), x, 2, -((4*sqrt(b*sqrt(x) + a*x))/(3*b*x)) + (8*a*sqrt(b*sqrt(x) + a*x))/(3*b^2*sqrt(x))]
    @test_int [1/(x^(5/2)*(a*x + b*x^(1/2))^(1/2)), x, 4, -((4*sqrt(b*sqrt(x) + a*x))/(7*b*x^2)) + (24*a*sqrt(b*sqrt(x) + a*x))/(35*b^2*x^(3/2)) - (32*a^2*sqrt(b*sqrt(x) + a*x))/(35*b^3*x) + (64*a^3*sqrt(b*sqrt(x) + a*x))/(35*b^4*sqrt(x))]
    @test_int [1/(x^(7/2)*(a*x + b*x^(1/2))^(1/2)), x, 6, -((4*sqrt(b*sqrt(x) + a*x))/(11*b*x^3)) + (40*a*sqrt(b*sqrt(x) + a*x))/(99*b^2*x^(5/2)) - (320*a^2*sqrt(b*sqrt(x) + a*x))/(693*b^3*x^2) + (128*a^3*sqrt(b*sqrt(x) + a*x))/(231*b^4*x^(3/2)) - (512*a^4*sqrt(b*sqrt(x) + a*x))/(693*b^5*x) + (1024*a^5*sqrt(b*sqrt(x) + a*x))/(693*b^6*sqrt(x))]

    @test_int [x^(5/2)/(a*x + b*x^(1/2))^(3/2), x, 8, -((4*x^(5/2))/(a*sqrt(b*sqrt(x) + a*x))) - (315*b^3*sqrt(b*sqrt(x) + a*x))/(32*a^5) + (105*b^2*sqrt(x)*sqrt(b*sqrt(x) + a*x))/(16*a^4) - (21*b*x*sqrt(b*sqrt(x) + a*x))/(4*a^3) + (9*x^(3/2)*sqrt(b*sqrt(x) + a*x))/(2*a^2) + (315*b^4*atanh((sqrt(a)*sqrt(x))/sqrt(b*sqrt(x) + a*x)))/(32*a^(11/2))]
    @test_int [x^(3/2)/(a*x + b*x^(1/2))^(3/2), x, 6, -((4*x^(3/2))/(a*sqrt(b*sqrt(x) + a*x))) - (15*b*sqrt(b*sqrt(x) + a*x))/(2*a^3) + (5*sqrt(x)*sqrt(b*sqrt(x) + a*x))/a^2 + (15*b^2*atanh((sqrt(a)*sqrt(x))/sqrt(b*sqrt(x) + a*x)))/(2*a^(7/2))]
    @test_int [x^(1/2)/(a*x + b*x^(1/2))^(3/2), x, 4, -((4*sqrt(x))/(a*sqrt(b*sqrt(x) + a*x))) + (4*atanh((sqrt(a)*sqrt(x))/sqrt(b*sqrt(x) + a*x)))/a^(3/2)]
    @test_int [1/(x^(1/2)*(a*x + b*x^(1/2))^(3/2)), x, 2, -((4*(b + 2*a*sqrt(x)))/(b^2*sqrt(b*sqrt(x) + a*x)))]
    @test_int [1/(x^(3/2)*(a*x + b*x^(1/2))^(3/2)), x, 4, 4/(b*x*sqrt(b*sqrt(x) + a*x)) - (24*sqrt(b*sqrt(x) + a*x))/(5*b^2*x^(3/2)) + (32*a*sqrt(b*sqrt(x) + a*x))/(5*b^3*x) - (64*a^2*sqrt(b*sqrt(x) + a*x))/(5*b^4*sqrt(x))]
    @test_int [1/(x^(5/2)*(a*x + b*x^(1/2))^(3/2)), x, 6, 4/(b*x^2*sqrt(b*sqrt(x) + a*x)) - (40*sqrt(b*sqrt(x) + a*x))/(9*b^2*x^(5/2)) + (320*a*sqrt(b*sqrt(x) + a*x))/(63*b^3*x^2) - (128*a^2*sqrt(b*sqrt(x) + a*x))/(21*b^4*x^(3/2)) + (512*a^3*sqrt(b*sqrt(x) + a*x))/(63*b^5*x) - (1024*a^4*sqrt(b*sqrt(x) + a*x))/(63*b^6*sqrt(x))]
    @test_int [1/(x^(7/2)*(a*x + b*x^(1/2))^(3/2)), x, 8, 4/(b*x^3*sqrt(b*sqrt(x) + a*x)) - (56*sqrt(b*sqrt(x) + a*x))/(13*b^2*x^(7/2)) + (672*a*sqrt(b*sqrt(x) + a*x))/(143*b^3*x^3) - (2240*a^2*sqrt(b*sqrt(x) + a*x))/(429*b^4*x^(5/2)) + (2560*a^3*sqrt(b*sqrt(x) + a*x))/(429*b^5*x^2) - (1024*a^4*sqrt(b*sqrt(x) + a*x))/(143*b^6*x^(3/2)) + (4096*a^5*sqrt(b*sqrt(x) + a*x))/(429*b^7*x) - (8192*a^6*sqrt(b*sqrt(x) + a*x))/(429*b^8*sqrt(x))]

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a*x+b*x^(1/3))^(p/2)=#

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

    @test_int [x^3*sqrt(b*x^(1/3) + a*x), x, 11, -((884*b^6*sqrt(b*x^(1/3) + a*x))/(14421*a^6)) + (884*b^5*x^(2/3)*sqrt(b*x^(1/3) + a*x))/(24035*a^5) - (6188*b^4*x^(4/3)*sqrt(b*x^(1/3) + a*x))/(216315*a^4) + (476*b^3*x^2*sqrt(b*x^(1/3) + a*x))/(19665*a^3) - (28*b^2*x^(8/3)*sqrt(b*x^(1/3) + a*x))/(1311*a^2) + (4*b*x^(10/3)*sqrt(b*x^(1/3) + a*x))/(207*a) + (2/9)*x^4*sqrt(b*x^(1/3) + a*x) + (442*b^(27/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(14421*a^(25/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [x^2*sqrt(b*x^(1/3) + a*x), x, 11, (44*b^5*(b + a*x^(2/3))*x^(1/3))/(221*a^(9/2)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt(b*x^(1/3) + a*x)) - (44*b^4*x^(1/3)*sqrt(b*x^(1/3) + a*x))/(663*a^4) + (220*b^3*x*sqrt(b*x^(1/3) + a*x))/(4641*a^3) - (60*b^2*x^(5/3)*sqrt(b*x^(1/3) + a*x))/(1547*a^2) + (4*b*x^(7/3)*sqrt(b*x^(1/3) + a*x))/(119*a) + (2/7)*x^3*sqrt(b*x^(1/3) + a*x) - (44*b^(21/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.E(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(221*a^(19/4)*sqrt(b*x^(1/3) + a*x)) + (22*b^(21/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(221*a^(19/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [x^1*sqrt(b*x^(1/3) + a*x), x, 8, (12*b^3*sqrt(b*x^(1/3) + a*x))/(77*a^3) - (36*b^2*x^(2/3)*sqrt(b*x^(1/3) + a*x))/(385*a^2) + (4*b*x^(4/3)*sqrt(b*x^(1/3) + a*x))/(55*a) + (2/5)*x^2*sqrt(b*x^(1/3) + a*x) - (6*b^(15/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(77*a^(13/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [x^0*sqrt(b*x^(1/3) + a*x), x, 8, -((4*b^2*(b + a*x^(2/3))*x^(1/3))/(5*a^(3/2)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt(b*x^(1/3) + a*x))) + (4*b*x^(1/3)*sqrt(b*x^(1/3) + a*x))/(15*a) + (2/3)*x*sqrt(b*x^(1/3) + a*x) + (4*b^(9/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.E(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(5*a^(7/4)*sqrt(b*x^(1/3) + a*x)) - (2*b^(9/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(5*a^(7/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [sqrt(b*x^(1/3) + a*x)/x^1, x, 5, 2*sqrt(b*x^(1/3) + a*x) + (2*b^(3/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(a^(1/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [sqrt(b*x^(1/3) + a*x)/x^2, x, 8, (12*a^(3/2)*(b + a*x^(2/3))*x^(1/3))/(5*b*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt(b*x^(1/3) + a*x)) - (6*sqrt(b*x^(1/3) + a*x))/(5*x) - (12*a*sqrt(b*x^(1/3) + a*x))/(5*b*x^(1/3)) - (12*a^(5/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.E(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(5*b^(3/4)*sqrt(b*x^(1/3) + a*x)) + (6*a^(5/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(5*b^(3/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [sqrt(b*x^(1/3) + a*x)/x^3, x, 7, -((6*sqrt(b*x^(1/3) + a*x))/(11*x^2)) - (12*a*sqrt(b*x^(1/3) + a*x))/(77*b*x^(4/3)) + (20*a^2*sqrt(b*x^(1/3) + a*x))/(77*b^2*x^(2/3)) + (10*a^(11/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(77*b^(9/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [sqrt(b*x^(1/3) + a*x)/x^4, x, 11, -((308*a^(9/2)*(b + a*x^(2/3))*x^(1/3))/(1105*b^4*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt(b*x^(1/3) + a*x))) - (6*sqrt(b*x^(1/3) + a*x))/(17*x^3) - (12*a*sqrt(b*x^(1/3) + a*x))/(221*b*x^(7/3)) + (44*a^2*sqrt(b*x^(1/3) + a*x))/(663*b^2*x^(5/3)) - (308*a^3*sqrt(b*x^(1/3) + a*x))/(3315*b^3*x) + (308*a^4*sqrt(b*x^(1/3) + a*x))/(1105*b^4*x^(1/3)) + (308*a^(17/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.E(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(1105*b^(15/4)*sqrt(b*x^(1/3) + a*x)) - (154*a^(17/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(1105*b^(15/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [sqrt(b*x^(1/3) + a*x)/x^5, x, 10, -((6*sqrt(b*x^(1/3) + a*x))/(23*x^4)) - (12*a*sqrt(b*x^(1/3) + a*x))/(437*b*x^(10/3)) + (68*a^2*sqrt(b*x^(1/3) + a*x))/(2185*b^2*x^(8/3)) - (884*a^3*sqrt(b*x^(1/3) + a*x))/(24035*b^3*x^2) + (7956*a^4*sqrt(b*x^(1/3) + a*x))/(168245*b^4*x^(4/3)) - (2652*a^5*sqrt(b*x^(1/3) + a*x))/(33649*b^5*x^(2/3)) - (1326*a^(23/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(33649*b^(21/4)*sqrt(b*x^(1/3) + a*x))]

    @test_int [x^2*(b*x^(1/3) + a*x)^(3/2), x, 11, (1768*b^6*sqrt(b*x^(1/3) + a*x))/(100947*a^5) - (1768*b^5*x^(2/3)*sqrt(b*x^(1/3) + a*x))/(168245*a^4) + (1768*b^4*x^(4/3)*sqrt(b*x^(1/3) + a*x))/(216315*a^3) - (136*b^3*x^2*sqrt(b*x^(1/3) + a*x))/(19665*a^2) + (8*b^2*x^(8/3)*sqrt(b*x^(1/3) + a*x))/(1311*a) + (4/69)*b*x^(10/3)*sqrt(b*x^(1/3) + a*x) + (2/9)*x^3*(b*x^(1/3) + a*x)^(3/2) - (884*b^(27/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(100947*a^(21/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [x^1*(b*x^(1/3) + a*x)^(3/2), x, 11, -((88*b^5*(b + a*x^(2/3))*x^(1/3))/(1105*a^(7/2)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt(b*x^(1/3) + a*x))) + (88*b^4*x^(1/3)*sqrt(b*x^(1/3) + a*x))/(3315*a^3) - (88*b^3*x*sqrt(b*x^(1/3) + a*x))/(4641*a^2) + (24*b^2*x^(5/3)*sqrt(b*x^(1/3) + a*x))/(1547*a) + (12/119)*b*x^(7/3)*sqrt(b*x^(1/3) + a*x) + (2/7)*x^2*(b*x^(1/3) + a*x)^(3/2) + (88*b^(21/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.E(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(1105*a^(15/4)*sqrt(b*x^(1/3) + a*x)) - (44*b^(21/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(1105*a^(15/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [x^0*(b*x^(1/3) + a*x)^(3/2), x, 8, -((8*b^3*sqrt(b*x^(1/3) + a*x))/(77*a^2)) + (24*b^2*x^(2/3)*sqrt(b*x^(1/3) + a*x))/(385*a) + (12/55)*b*x^(4/3)*sqrt(b*x^(1/3) + a*x) + (2/5)*x*(b*x^(1/3) + a*x)^(3/2) + (4*b^(15/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(77*a^(9/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [(b*x^(1/3) + a*x)^(3/2)/x^1, x, 8, (8*b^2*(b + a*x^(2/3))*x^(1/3))/(5*sqrt(a)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt(b*x^(1/3) + a*x)) + (4/5)*b*x^(1/3)*sqrt(b*x^(1/3) + a*x) + (2/3)*(b*x^(1/3) + a*x)^(3/2) - (8*b^(9/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.E(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(5*a^(3/4)*sqrt(b*x^(1/3) + a*x)) + (4*b^(9/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(5*a^(3/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [(b*x^(1/3) + a*x)^(3/2)/x^2, x, 6, 4*a*sqrt(b*x^(1/3) + a*x) - (2*(b*x^(1/3) + a*x)^(3/2))/x + (4*a^(3/4)*b^(3/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/sqrt(b*x^(1/3) + a*x)]
    @test_int [(b*x^(1/3) + a*x)^(3/2)/x^3, x, 9, (8*a^(5/2)*(b + a*x^(2/3))*x^(1/3))/(5*b*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt(b*x^(1/3) + a*x)) - (4*a*sqrt(b*x^(1/3) + a*x))/(5*x) - (8*a^2*sqrt(b*x^(1/3) + a*x))/(5*b*x^(1/3)) - (2*(b*x^(1/3) + a*x)^(3/2))/(3*x^2) - (8*a^(9/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.E(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(5*b^(3/4)*sqrt(b*x^(1/3) + a*x)) + (4*a^(9/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(5*b^(3/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [(b*x^(1/3) + a*x)^(3/2)/x^4, x, 8, -((12*a*sqrt(b*x^(1/3) + a*x))/(55*x^2)) - (24*a^2*sqrt(b*x^(1/3) + a*x))/(385*b*x^(4/3)) + (8*a^3*sqrt(b*x^(1/3) + a*x))/(77*b^2*x^(2/3)) - (2*(b*x^(1/3) + a*x)^(3/2))/(5*x^3) + (4*a^(15/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(77*b^(9/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [(b*x^(1/3) + a*x)^(3/2)/x^5, x, 12, -((88*a^(11/2)*(b + a*x^(2/3))*x^(1/3))/(1105*b^4*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt(b*x^(1/3) + a*x))) - (12*a*sqrt(b*x^(1/3) + a*x))/(119*x^3) - (24*a^2*sqrt(b*x^(1/3) + a*x))/(1547*b*x^(7/3)) + (88*a^3*sqrt(b*x^(1/3) + a*x))/(4641*b^2*x^(5/3)) - (88*a^4*sqrt(b*x^(1/3) + a*x))/(3315*b^3*x) + (88*a^5*sqrt(b*x^(1/3) + a*x))/(1105*b^4*x^(1/3)) - (2*(b*x^(1/3) + a*x)^(3/2))/(7*x^4) + (88*a^(21/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.E(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(1105*b^(15/4)*sqrt(b*x^(1/3) + a*x)) - (44*a^(21/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(1105*b^(15/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [(b*x^(1/3) + a*x)^(3/2)/x^6, x, 11, -((4*a*sqrt(b*x^(1/3) + a*x))/(69*x^4)) - (8*a^2*sqrt(b*x^(1/3) + a*x))/(1311*b*x^(10/3)) + (136*a^3*sqrt(b*x^(1/3) + a*x))/(19665*b^2*x^(8/3)) - (1768*a^4*sqrt(b*x^(1/3) + a*x))/(216315*b^3*x^2) + (1768*a^5*sqrt(b*x^(1/3) + a*x))/(168245*b^4*x^(4/3)) - (1768*a^6*sqrt(b*x^(1/3) + a*x))/(100947*b^5*x^(2/3)) - (2*(b*x^(1/3) + a*x)^(3/2))/(9*x^5) - (884*a^(27/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(100947*b^(21/4)*sqrt(b*x^(1/3) + a*x))]

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

    @test_int [x^4/sqrt(b*x^(1/3) + a*x), x, 11, (11050*b^6*sqrt(b*x^(1/3) + a*x))/(14421*a^7) - (2210*b^5*x^(2/3)*sqrt(b*x^(1/3) + a*x))/(4807*a^6) + (15470*b^4*x^(4/3)*sqrt(b*x^(1/3) + a*x))/(43263*a^5) - (1190*b^3*x^2*sqrt(b*x^(1/3) + a*x))/(3933*a^4) + (350*b^2*x^(8/3)*sqrt(b*x^(1/3) + a*x))/(1311*a^3) - (50*b*x^(10/3)*sqrt(b*x^(1/3) + a*x))/(207*a^2) + (2*x^4*sqrt(b*x^(1/3) + a*x))/(9*a) - (5525*b^(27/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(14421*a^(29/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [x^3/sqrt(b*x^(1/3) + a*x), x, 11, -((418*b^5*(b + a*x^(2/3))*x^(1/3))/(221*a^(11/2)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt(b*x^(1/3) + a*x))) + (418*b^4*x^(1/3)*sqrt(b*x^(1/3) + a*x))/(663*a^5) - (2090*b^3*x*sqrt(b*x^(1/3) + a*x))/(4641*a^4) + (570*b^2*x^(5/3)*sqrt(b*x^(1/3) + a*x))/(1547*a^3) - (38*b*x^(7/3)*sqrt(b*x^(1/3) + a*x))/(119*a^2) + (2*x^3*sqrt(b*x^(1/3) + a*x))/(7*a) + (418*b^(21/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.E(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(221*a^(23/4)*sqrt(b*x^(1/3) + a*x)) - (209*b^(21/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(221*a^(23/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [x^2/sqrt(b*x^(1/3) + a*x), x, 8, -((78*b^3*sqrt(b*x^(1/3) + a*x))/(77*a^4)) + (234*b^2*x^(2/3)*sqrt(b*x^(1/3) + a*x))/(385*a^3) - (26*b*x^(4/3)*sqrt(b*x^(1/3) + a*x))/(55*a^2) + (2*x^2*sqrt(b*x^(1/3) + a*x))/(5*a) + (39*b^(15/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(77*a^(17/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [x^1/sqrt(b*x^(1/3) + a*x), x, 8, (14*b^2*(b + a*x^(2/3))*x^(1/3))/(5*a^(5/2)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt(b*x^(1/3) + a*x)) - (14*b*x^(1/3)*sqrt(b*x^(1/3) + a*x))/(15*a^2) + (2*x*sqrt(b*x^(1/3) + a*x))/(3*a) - (14*b^(9/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.E(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(5*a^(11/4)*sqrt(b*x^(1/3) + a*x)) + (7*b^(9/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(5*a^(11/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [x^0/sqrt(b*x^(1/3) + a*x), x, 5, (2*sqrt(b*x^(1/3) + a*x))/a - (b^(3/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(a^(5/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [1/(x^1*sqrt(b*x^(1/3) + a*x)), x, 7, (6*sqrt(a)*(b + a*x^(2/3))*x^(1/3))/(b*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt(b*x^(1/3) + a*x)) - (6*sqrt(b*x^(1/3) + a*x))/(b*x^(1/3)) - (6*a^(1/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.E(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(b^(3/4)*sqrt(b*x^(1/3) + a*x)) + (3*a^(1/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(b^(3/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [1/(x^2*sqrt(b*x^(1/3) + a*x)), x, 6, -((6*sqrt(b*x^(1/3) + a*x))/(7*b*x^(4/3))) + (10*a*sqrt(b*x^(1/3) + a*x))/(7*b^2*x^(2/3)) + (5*a^(7/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(7*b^(9/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [1/(x^3*sqrt(b*x^(1/3) + a*x)), x, 10, -((154*a^(7/2)*(b + a*x^(2/3))*x^(1/3))/(65*b^4*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt(b*x^(1/3) + a*x))) - (6*sqrt(b*x^(1/3) + a*x))/(13*b*x^(7/3)) + (22*a*sqrt(b*x^(1/3) + a*x))/(39*b^2*x^(5/3)) - (154*a^2*sqrt(b*x^(1/3) + a*x))/(195*b^3*x) + (154*a^3*sqrt(b*x^(1/3) + a*x))/(65*b^4*x^(1/3)) + (154*a^(13/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.E(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(65*b^(15/4)*sqrt(b*x^(1/3) + a*x)) - (77*a^(13/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(65*b^(15/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [1/(x^4*sqrt(b*x^(1/3) + a*x)), x, 9, -((6*sqrt(b*x^(1/3) + a*x))/(19*b*x^(10/3))) + (34*a*sqrt(b*x^(1/3) + a*x))/(95*b^2*x^(8/3)) - (442*a^2*sqrt(b*x^(1/3) + a*x))/(1045*b^3*x^2) + (3978*a^3*sqrt(b*x^(1/3) + a*x))/(7315*b^4*x^(4/3)) - (1326*a^4*sqrt(b*x^(1/3) + a*x))/(1463*b^5*x^(2/3)) - (663*a^(19/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(1463*b^(21/4)*sqrt(b*x^(1/3) + a*x))]

    @test_int [x^4/(b*x^(1/3) + a*x)^(3/2), x, 12, -((4807*b^5*(b + a*x^(2/3))*x^(1/3))/(221*a^(13/2)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt(b*x^(1/3) + a*x))) - (3*x^4)/(a*sqrt(b*x^(1/3) + a*x)) + (4807*b^4*x^(1/3)*sqrt(b*x^(1/3) + a*x))/(663*a^6) - (24035*b^3*x*sqrt(b*x^(1/3) + a*x))/(4641*a^5) + (6555*b^2*x^(5/3)*sqrt(b*x^(1/3) + a*x))/(1547*a^4) - (437*b*x^(7/3)*sqrt(b*x^(1/3) + a*x))/(119*a^3) + (23*x^3*sqrt(b*x^(1/3) + a*x))/(7*a^2) + (4807*b^(21/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.E(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(221*a^(27/4)*sqrt(b*x^(1/3) + a*x)) - (4807*b^(21/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(442*a^(27/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [x^3/(b*x^(1/3) + a*x)^(3/2), x, 9, -((3*x^3)/(a*sqrt(b*x^(1/3) + a*x))) - (663*b^3*sqrt(b*x^(1/3) + a*x))/(77*a^5) + (1989*b^2*x^(2/3)*sqrt(b*x^(1/3) + a*x))/(385*a^4) - (221*b*x^(4/3)*sqrt(b*x^(1/3) + a*x))/(55*a^3) + (17*x^2*sqrt(b*x^(1/3) + a*x))/(5*a^2) + (663*b^(15/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(154*a^(21/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [x^2/(b*x^(1/3) + a*x)^(3/2), x, 9, (77*b^2*(b + a*x^(2/3))*x^(1/3))/(5*a^(7/2)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt(b*x^(1/3) + a*x)) - (3*x^2)/(a*sqrt(b*x^(1/3) + a*x)) - (77*b*x^(1/3)*sqrt(b*x^(1/3) + a*x))/(15*a^3) + (11*x*sqrt(b*x^(1/3) + a*x))/(3*a^2) - (77*b^(9/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.E(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(5*a^(15/4)*sqrt(b*x^(1/3) + a*x)) + (77*b^(9/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(10*a^(15/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [x^1/(b*x^(1/3) + a*x)^(3/2), x, 6, -((3*x)/(a*sqrt(b*x^(1/3) + a*x))) + (5*sqrt(b*x^(1/3) + a*x))/a^2 - (5*b^(3/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(2*a^(9/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [x^0/(b*x^(1/3) + a*x)^(3/2), x, 7, -((3*(b + a*x^(2/3))*x^(1/3))/(sqrt(a)*b*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt(b*x^(1/3) + a*x))) + (3*x^(2/3))/(b*sqrt(b*x^(1/3) + a*x)) + (3*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.E(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(a^(3/4)*b^(3/4)*sqrt(b*x^(1/3) + a*x)) - (3*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(2*a^(3/4)*b^(3/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [1/(x^1*(b*x^(1/3) + a*x)^(3/2)), x, 6, 3/(b*x^(1/3)*sqrt(b*x^(1/3) + a*x)) - (5*sqrt(b*x^(1/3) + a*x))/(b^2*x^(2/3)) - (5*a^(3/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(2*b^(9/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [1/(x^2*(b*x^(1/3) + a*x)^(3/2)), x, 10, 3/(b*x^(4/3)*sqrt(b*x^(1/3) + a*x)) + (77*a^(5/2)*(b + a*x^(2/3))*x^(1/3))/(5*b^4*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt(b*x^(1/3) + a*x)) - (11*sqrt(b*x^(1/3) + a*x))/(3*b^2*x^(5/3)) + (77*a*sqrt(b*x^(1/3) + a*x))/(15*b^3*x) - (77*a^2*sqrt(b*x^(1/3) + a*x))/(5*b^4*x^(1/3)) - (77*a^(9/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.E(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(5*b^(15/4)*sqrt(b*x^(1/3) + a*x)) + (77*a^(9/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(10*b^(15/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [1/(x^3*(b*x^(1/3) + a*x)^(3/2)), x, 9, 3/(b*x^(7/3)*sqrt(b*x^(1/3) + a*x)) - (17*sqrt(b*x^(1/3) + a*x))/(5*b^2*x^(8/3)) + (221*a*sqrt(b*x^(1/3) + a*x))/(55*b^3*x^2) - (1989*a^2*sqrt(b*x^(1/3) + a*x))/(385*b^4*x^(4/3)) + (663*a^3*sqrt(b*x^(1/3) + a*x))/(77*b^5*x^(2/3)) + (663*a^(15/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(154*b^(21/4)*sqrt(b*x^(1/3) + a*x))]
    @test_int [1/(x^4*(b*x^(1/3) + a*x)^(3/2)), x, 13, 3/(b*x^(10/3)*sqrt(b*x^(1/3) + a*x)) - (4807*a^(11/2)*(b + a*x^(2/3))*x^(1/3))/(221*b^7*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt(b*x^(1/3) + a*x)) - (23*sqrt(b*x^(1/3) + a*x))/(7*b^2*x^(11/3)) + (437*a*sqrt(b*x^(1/3) + a*x))/(119*b^3*x^3) - (6555*a^2*sqrt(b*x^(1/3) + a*x))/(1547*b^4*x^(7/3)) + (24035*a^3*sqrt(b*x^(1/3) + a*x))/(4641*b^5*x^(5/3)) - (4807*a^4*sqrt(b*x^(1/3) + a*x))/(663*b^6*x) + (4807*a^5*sqrt(b*x^(1/3) + a*x))/(221*b^7*x^(1/3)) + (4807*a^(21/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.E(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(221*b^(27/4)*sqrt(b*x^(1/3) + a*x)) - (4807*a^(21/4)*(sqrt(b) + sqrt(a)*x^(1/3))*sqrt((b + a*x^(2/3))/(sqrt(b) + sqrt(a)*x^(1/3))^2)*x^(1/6)*Elliptic.F(2*atan((a^(1/4)*x^(1/6))/b^(1/4)), 1/2))/(442*b^(27/4)*sqrt(b*x^(1/3) + a*x))]

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

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

    @test_int [x^3*sqrt(b*x^(2/3) + a*x), x, 13, (-524288*b^9*(b*x^(2/3) + a*x)^(3/2))/(4345965*a^10) + (8388608*b^12*(b*x^(2/3) + a*x)^(3/2))/(152108775*a^13*x) - (4194304*b^11*(b*x^(2/3) + a*x)^(3/2))/(50702925*a^12*x^(2/3)) + (1048576*b^10*(b*x^(2/3) + a*x)^(3/2))/(10140585*a^11*x^(1/3)) + (65536*b^8*x^(1/3)*(b*x^(2/3) + a*x)^(3/2))/(482885*a^9) - (360448*b^7*x^(2/3)*(b*x^(2/3) + a*x)^(3/2))/(2414425*a^8) + (90112*b^6*x*(b*x^(2/3) + a*x)^(3/2))/(557175*a^7) - (45056*b^5*x^(4/3)*(b*x^(2/3) + a*x)^(3/2))/(260015*a^6) + (2816*b^4*x^(5/3)*(b*x^(2/3) + a*x)^(3/2))/(15295*a^5) - (1408*b^3*x^2*(b*x^(2/3) + a*x)^(3/2))/(7245*a^4) + (352*b^2*x^(7/3)*(b*x^(2/3) + a*x)^(3/2))/(1725*a^3) - (16*b*x^(8/3)*(b*x^(2/3) + a*x)^(3/2))/(75*a^2) + (2*x^3*(b*x^(2/3) + a*x)^(3/2))/(9*a)]
    @test_int [x^2*sqrt(b*x^(2/3) + a*x), x, 10, (8192*b^6*(b*x^(2/3) + a*x)^(3/2))/(46189*a^7) - (131072*b^9*(b*x^(2/3) + a*x)^(3/2))/(1616615*a^10*x) + (196608*b^8*(b*x^(2/3) + a*x)^(3/2))/(1616615*a^9*x^(2/3)) - (49152*b^7*(b*x^(2/3) + a*x)^(3/2))/(323323*a^8*x^(1/3)) - (9216*b^5*x^(1/3)*(b*x^(2/3) + a*x)^(3/2))/(46189*a^6) + (4608*b^4*x^(2/3)*(b*x^(2/3) + a*x)^(3/2))/(20995*a^5) - (384*b^3*x*(b*x^(2/3) + a*x)^(3/2))/(1615*a^4) + (576*b^2*x^(4/3)*(b*x^(2/3) + a*x)^(3/2))/(2261*a^3) - (36*b*x^(5/3)*(b*x^(2/3) + a*x)^(3/2))/(133*a^2) + (2*x^2*(b*x^(2/3) + a*x)^(3/2))/(7*a)]
    @test_int [x^1*sqrt(b*x^(2/3) + a*x), x, 7, (-128*b^3*(b*x^(2/3) + a*x)^(3/2))/(429*a^4) + (2048*b^6*(b*x^(2/3) + a*x)^(3/2))/(15015*a^7*x) - (1024*b^5*(b*x^(2/3) + a*x)^(3/2))/(5005*a^6*x^(2/3)) + (256*b^4*(b*x^(2/3) + a*x)^(3/2))/(1001*a^5*x^(1/3)) + (48*b^2*x^(1/3)*(b*x^(2/3) + a*x)^(3/2))/(143*a^3) - (24*b*x^(2/3)*(b*x^(2/3) + a*x)^(3/2))/(65*a^2) + (2*x*(b*x^(2/3) + a*x)^(3/2))/(5*a)]
    @test_int [x^0*sqrt(b*x^(2/3) + a*x), x, 4, (2*(b*x^(2/3) + a*x)^(3/2))/(3*a) - (32*b^3*(b*x^(2/3) + a*x)^(3/2))/(105*a^4*x) + (16*b^2*(b*x^(2/3) + a*x)^(3/2))/(35*a^3*x^(2/3)) - (4*b*(b*x^(2/3) + a*x)^(3/2))/(7*a^2*x^(1/3))]
    @test_int [sqrt(b*x^(2/3) + a*x)/x^1, x, 1, (2*(b*x^(2/3) + a*x)^(3/2))/(a*x)]
    @test_int [sqrt(b*x^(2/3) + a*x)/x^2, x, 4, -((3*sqrt(b*x^(2/3) + a*x))/(2*x)) - (3*a*sqrt(b*x^(2/3) + a*x))/(4*b*x^(2/3)) + (3*a^2*atanh((sqrt(b)*x^(1/3))/sqrt(b*x^(2/3) + a*x)))/(4*b^(3/2))]
    @test_int [sqrt(b*x^(2/3) + a*x)/x^3, x, 7, -((3*sqrt(b*x^(2/3) + a*x))/(5*x^2)) - (3*a*sqrt(b*x^(2/3) + a*x))/(40*b*x^(5/3)) + (7*a^2*sqrt(b*x^(2/3) + a*x))/(80*b^2*x^(4/3)) - (7*a^3*sqrt(b*x^(2/3) + a*x))/(64*b^3*x) + (21*a^4*sqrt(b*x^(2/3) + a*x))/(128*b^4*x^(2/3)) - (21*a^5*atanh((sqrt(b)*x^(1/3))/sqrt(b*x^(2/3) + a*x)))/(128*b^(9/2))]
    @test_int [sqrt(b*x^(2/3) + a*x)/x^4, x, 10, -((3*sqrt(b*x^(2/3) + a*x))/(8*x^3)) - (3*a*sqrt(b*x^(2/3) + a*x))/(112*b*x^(8/3)) + (13*a^2*sqrt(b*x^(2/3) + a*x))/(448*b^2*x^(7/3)) - (143*a^3*sqrt(b*x^(2/3) + a*x))/(4480*b^3*x^2) + (1287*a^4*sqrt(b*x^(2/3) + a*x))/(35840*b^4*x^(5/3)) - (429*a^5*sqrt(b*x^(2/3) + a*x))/(10240*b^5*x^(4/3)) + (429*a^6*sqrt(b*x^(2/3) + a*x))/(8192*b^6*x) - (1287*a^7*sqrt(b*x^(2/3) + a*x))/(16384*b^7*x^(2/3)) + (1287*a^8*atanh((sqrt(b)*x^(1/3))/sqrt(b*x^(2/3) + a*x)))/(16384*b^(15/2))]
    @test_int [sqrt(b*x^(2/3) + a*x)/x^5, x, 13, -((3*sqrt(b*x^(2/3) + a*x))/(11*x^4)) - (3*a*sqrt(b*x^(2/3) + a*x))/(220*b*x^(11/3)) + (19*a^2*sqrt(b*x^(2/3) + a*x))/(1320*b^2*x^(10/3)) - (323*a^3*sqrt(b*x^(2/3) + a*x))/(21120*b^3*x^3) + (323*a^4*sqrt(b*x^(2/3) + a*x))/(19712*b^4*x^(8/3)) - (4199*a^5*sqrt(b*x^(2/3) + a*x))/(236544*b^5*x^(7/3)) + (4199*a^6*sqrt(b*x^(2/3) + a*x))/(215040*b^6*x^2) - (12597*a^7*sqrt(b*x^(2/3) + a*x))/(573440*b^7*x^(5/3)) + (4199*a^8*sqrt(b*x^(2/3) + a*x))/(163840*b^8*x^(4/3)) - (4199*a^9*sqrt(b*x^(2/3) + a*x))/(131072*b^9*x) + (12597*a^10*sqrt(b*x^(2/3) + a*x))/(262144*b^10*x^(2/3)) - (12597*a^11*atanh((sqrt(b)*x^(1/3))/sqrt(b*x^(2/3) + a*x)))/(262144*b^(21/2))]

    @test_int [x^2*(b*x^(2/3) + a*x)^(3/2), x, 12, (45056*b^6*(b*x^(2/3) + a*x)^(5/2))/(557175*a^7) - (1048576*b^11*(b*x^(2/3) + a*x)^(5/2))/(152108775*a^12*x^(5/3)) + (524288*b^10*(b*x^(2/3) + a*x)^(5/2))/(30421755*a^11*x^(4/3)) - (131072*b^9*(b*x^(2/3) + a*x)^(5/2))/(4345965*a^10*x) + (65536*b^8*(b*x^(2/3) + a*x)^(5/2))/(1448655*a^9*x^(2/3)) - (90112*b^7*(b*x^(2/3) + a*x)^(5/2))/(1448655*a^8*x^(1/3)) - (11264*b^5*x^(1/3)*(b*x^(2/3) + a*x)^(5/2))/(111435*a^6) + (5632*b^4*x^(2/3)*(b*x^(2/3) + a*x)^(5/2))/(45885*a^5) - (352*b^3*x*(b*x^(2/3) + a*x)^(5/2))/(2415*a^4) + (176*b^2*x^(4/3)*(b*x^(2/3) + a*x)^(5/2))/(1035*a^3) - (44*b*x^(5/3)*(b*x^(2/3) + a*x)^(5/2))/(225*a^2) + (2*x^2*(b*x^(2/3) + a*x)^(5/2))/(9*a)]
    @test_int [x^1*(b*x^(2/3) + a*x)^(3/2), x, 9, (-256*b^3*(b*x^(2/3) + a*x)^(5/2))/(1615*a^4) + (65536*b^8*(b*x^(2/3) + a*x)^(5/2))/(4849845*a^9*x^(5/3)) - (32768*b^7*(b*x^(2/3) + a*x)^(5/2))/(969969*a^8*x^(4/3)) + (8192*b^6*(b*x^(2/3) + a*x)^(5/2))/(138567*a^7*x) - (4096*b^5*(b*x^(2/3) + a*x)^(5/2))/(46189*a^6*x^(2/3)) + (512*b^4*(b*x^(2/3) + a*x)^(5/2))/(4199*a^5*x^(1/3)) + (64*b^2*x^(1/3)*(b*x^(2/3) + a*x)^(5/2))/(323*a^3) - (32*b*x^(2/3)*(b*x^(2/3) + a*x)^(5/2))/(133*a^2) + (2*x*(b*x^(2/3) + a*x)^(5/2))/(7*a)]
    @test_int [x^0*(b*x^(2/3) + a*x)^(3/2), x, 6, (2*(b*x^(2/3) + a*x)^(5/2))/(5*a) - (512*b^5*(b*x^(2/3) + a*x)^(5/2))/(15015*a^6*x^(5/3)) + (256*b^4*(b*x^(2/3) + a*x)^(5/2))/(3003*a^5*x^(4/3)) - (64*b^3*(b*x^(2/3) + a*x)^(5/2))/(429*a^4*x) + (32*b^2*(b*x^(2/3) + a*x)^(5/2))/(143*a^3*x^(2/3)) - (4*b*(b*x^(2/3) + a*x)^(5/2))/(13*a^2*x^(1/3))]
    @test_int [(b*x^(2/3) + a*x)^(3/2)/x^1, x, 3, (16*b^2*(b*x^(2/3) + a*x)^(5/2))/(105*a^3*x^(5/3)) - (8*b*(b*x^(2/3) + a*x)^(5/2))/(21*a^2*x^(4/3)) + (2*(b*x^(2/3) + a*x)^(5/2))/(3*a*x)]
    @test_int [(b*x^(2/3) + a*x)^(3/2)/x^2, x, 4, (6*b*sqrt(b*x^(2/3) + a*x))/x^(1/3) + (2*(b*x^(2/3) + a*x)^(3/2))/x - 6*b^(3/2)*atanh((sqrt(b)*x^(1/3))/sqrt(b*x^(2/3) + a*x))]
    @test_int [(b*x^(2/3) + a*x)^(3/2)/x^3, x, 5, -((3*a*sqrt(b*x^(2/3) + a*x))/(4*x)) - (3*a^2*sqrt(b*x^(2/3) + a*x))/(8*b*x^(2/3)) - (b*x^(2/3) + a*x)^(3/2)/x^2 + (3*a^3*atanh((sqrt(b)*x^(1/3))/sqrt(b*x^(2/3) + a*x)))/(8*b^(3/2))]
    @test_int [(b*x^(2/3) + a*x)^(3/2)/x^4, x, 8, -((3*a*sqrt(b*x^(2/3) + a*x))/(20*x^2)) - (3*a^2*sqrt(b*x^(2/3) + a*x))/(160*b*x^(5/3)) + (7*a^3*sqrt(b*x^(2/3) + a*x))/(320*b^2*x^(4/3)) - (7*a^4*sqrt(b*x^(2/3) + a*x))/(256*b^3*x) + (21*a^5*sqrt(b*x^(2/3) + a*x))/(512*b^4*x^(2/3)) - (b*x^(2/3) + a*x)^(3/2)/(2*x^3) - (21*a^6*atanh((sqrt(b)*x^(1/3))/sqrt(b*x^(2/3) + a*x)))/(512*b^(9/2))]
    @test_int [(b*x^(2/3) + a*x)^(3/2)/x^5, x, 11, -((a*sqrt(b*x^(2/3) + a*x))/(16*x^3)) - (a^2*sqrt(b*x^(2/3) + a*x))/(224*b*x^(8/3)) + (13*a^3*sqrt(b*x^(2/3) + a*x))/(2688*b^2*x^(7/3)) - (143*a^4*sqrt(b*x^(2/3) + a*x))/(26880*b^3*x^2) + (429*a^5*sqrt(b*x^(2/3) + a*x))/(71680*b^4*x^(5/3)) - (143*a^6*sqrt(b*x^(2/3) + a*x))/(20480*b^5*x^(4/3)) + (143*a^7*sqrt(b*x^(2/3) + a*x))/(16384*b^6*x) - (429*a^8*sqrt(b*x^(2/3) + a*x))/(32768*b^7*x^(2/3)) - (b*x^(2/3) + a*x)^(3/2)/(3*x^4) + (429*a^9*atanh((sqrt(b)*x^(1/3))/sqrt(b*x^(2/3) + a*x)))/(32768*b^(15/2))]
    @test_int [(b*x^(2/3) + a*x)^(3/2)/x^6, x, 14, -((3*a*sqrt(b*x^(2/3) + a*x))/(88*x^4)) - (3*a^2*sqrt(b*x^(2/3) + a*x))/(1760*b*x^(11/3)) + (19*a^3*sqrt(b*x^(2/3) + a*x))/(10560*b^2*x^(10/3)) - (323*a^4*sqrt(b*x^(2/3) + a*x))/(168960*b^3*x^3) + (323*a^5*sqrt(b*x^(2/3) + a*x))/(157696*b^4*x^(8/3)) - (4199*a^6*sqrt(b*x^(2/3) + a*x))/(1892352*b^5*x^(7/3)) + (4199*a^7*sqrt(b*x^(2/3) + a*x))/(1720320*b^6*x^2) - (12597*a^8*sqrt(b*x^(2/3) + a*x))/(4587520*b^7*x^(5/3)) + (4199*a^9*sqrt(b*x^(2/3) + a*x))/(1310720*b^8*x^(4/3)) - (4199*a^10*sqrt(b*x^(2/3) + a*x))/(1048576*b^9*x) + (12597*a^11*sqrt(b*x^(2/3) + a*x))/(2097152*b^10*x^(2/3)) - (b*x^(2/3) + a*x)^(3/2)/(4*x^5) - (12597*a^12*atanh((sqrt(b)*x^(1/3))/sqrt(b*x^(2/3) + a*x)))/(2097152*b^(21/2))]

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

    @test_int [x^4/sqrt(b*x^(2/3) + a*x), x, 14, (8388608*b^12*sqrt(b*x^(2/3) + a*x))/(11700675*a^13) - (16777216*b^13*sqrt(b*x^(2/3) + a*x))/(11700675*a^14*x^(1/3)) - (2097152*b^11*x^(1/3)*sqrt(b*x^(2/3) + a*x))/(3900225*a^12) + (1048576*b^10*x^(2/3)*sqrt(b*x^(2/3) + a*x))/(2340135*a^11) - (131072*b^9*x*sqrt(b*x^(2/3) + a*x))/(334305*a^10) + (65536*b^8*x^(4/3)*sqrt(b*x^(2/3) + a*x))/(185725*a^9) - (180224*b^7*x^(5/3)*sqrt(b*x^(2/3) + a*x))/(557175*a^8) + (1171456*b^6*x^2*sqrt(b*x^(2/3) + a*x))/(3900225*a^7) - (73216*b^5*x^(7/3)*sqrt(b*x^(2/3) + a*x))/(260015*a^6) + (36608*b^4*x^(8/3)*sqrt(b*x^(2/3) + a*x))/(137655*a^5) - (9152*b^3*x^3*sqrt(b*x^(2/3) + a*x))/(36225*a^4) + (416*b^2*x^(10/3)*sqrt(b*x^(2/3) + a*x))/(1725*a^3) - (52*b*x^(11/3)*sqrt(b*x^(2/3) + a*x))/(225*a^2) + (2*x^4*sqrt(b*x^(2/3) + a*x))/(9*a)]
    @test_int [x^3/sqrt(b*x^(2/3) + a*x), x, 11, (-262144*b^9*sqrt(b*x^(2/3) + a*x))/(323323*a^10) + (524288*b^10*sqrt(b*x^(2/3) + a*x))/(323323*a^11*x^(1/3)) + (196608*b^8*x^(1/3)*sqrt(b*x^(2/3) + a*x))/(323323*a^9) - (163840*b^7*x^(2/3)*sqrt(b*x^(2/3) + a*x))/(323323*a^8) + (20480*b^6*x*sqrt(b*x^(2/3) + a*x))/(46189*a^7) - (18432*b^5*x^(4/3)*sqrt(b*x^(2/3) + a*x))/(46189*a^6) + (1536*b^4*x^(5/3)*sqrt(b*x^(2/3) + a*x))/(4199*a^5) - (768*b^3*x^2*sqrt(b*x^(2/3) + a*x))/(2261*a^4) + (720*b^2*x^(7/3)*sqrt(b*x^(2/3) + a*x))/(2261*a^3) - (40*b*x^(8/3)*sqrt(b*x^(2/3) + a*x))/(133*a^2) + (2*x^3*sqrt(b*x^(2/3) + a*x))/(7*a)]
    @test_int [x^2/sqrt(b*x^(2/3) + a*x), x, 8, (2048*b^6*sqrt(b*x^(2/3) + a*x))/(2145*a^7) - (4096*b^7*sqrt(b*x^(2/3) + a*x))/(2145*a^8*x^(1/3)) - (512*b^5*x^(1/3)*sqrt(b*x^(2/3) + a*x))/(715*a^6) + (256*b^4*x^(2/3)*sqrt(b*x^(2/3) + a*x))/(429*a^5) - (224*b^3*x*sqrt(b*x^(2/3) + a*x))/(429*a^4) + (336*b^2*x^(4/3)*sqrt(b*x^(2/3) + a*x))/(715*a^3) - (28*b*x^(5/3)*sqrt(b*x^(2/3) + a*x))/(65*a^2) + (2*x^2*sqrt(b*x^(2/3) + a*x))/(5*a)]
    @test_int [x^1/sqrt(b*x^(2/3) + a*x), x, 5, (-128*b^3*sqrt(b*x^(2/3) + a*x))/(105*a^4) + (256*b^4*sqrt(b*x^(2/3) + a*x))/(105*a^5*x^(1/3)) + (32*b^2*x^(1/3)*sqrt(b*x^(2/3) + a*x))/(35*a^3) - (16*b*x^(2/3)*sqrt(b*x^(2/3) + a*x))/(21*a^2) + (2*x*sqrt(b*x^(2/3) + a*x))/(3*a)]
    @test_int [x^0/sqrt(b*x^(2/3) + a*x), x, 2, (2*sqrt(b*x^(2/3) + a*x))/a - (4*b*sqrt(b*x^(2/3) + a*x))/(a^2*x^(1/3))]
    @test_int [1/(x^1*sqrt(b*x^(2/3) + a*x)), x, 3, -((3*sqrt(b*x^(2/3) + a*x))/(b*x^(2/3))) + (3*a*atanh((sqrt(b)*x^(1/3))/sqrt(b*x^(2/3) + a*x)))/b^(3/2)]
    @test_int [1/(x^2*sqrt(b*x^(2/3) + a*x)), x, 6, -((3*sqrt(b*x^(2/3) + a*x))/(4*b*x^(5/3))) + (7*a*sqrt(b*x^(2/3) + a*x))/(8*b^2*x^(4/3)) - (35*a^2*sqrt(b*x^(2/3) + a*x))/(32*b^3*x) + (105*a^3*sqrt(b*x^(2/3) + a*x))/(64*b^4*x^(2/3)) - (105*a^4*atanh((sqrt(b)*x^(1/3))/sqrt(b*x^(2/3) + a*x)))/(64*b^(9/2))]
    @test_int [1/(x^3*sqrt(b*x^(2/3) + a*x)), x, 9, -((3*sqrt(b*x^(2/3) + a*x))/(7*b*x^(8/3))) + (13*a*sqrt(b*x^(2/3) + a*x))/(28*b^2*x^(7/3)) - (143*a^2*sqrt(b*x^(2/3) + a*x))/(280*b^3*x^2) + (1287*a^3*sqrt(b*x^(2/3) + a*x))/(2240*b^4*x^(5/3)) - (429*a^4*sqrt(b*x^(2/3) + a*x))/(640*b^5*x^(4/3)) + (429*a^5*sqrt(b*x^(2/3) + a*x))/(512*b^6*x) - (1287*a^6*sqrt(b*x^(2/3) + a*x))/(1024*b^7*x^(2/3)) + (1287*a^7*atanh((sqrt(b)*x^(1/3))/sqrt(b*x^(2/3) + a*x)))/(1024*b^(15/2))]
    @test_int [1/(x^4*sqrt(b*x^(2/3) + a*x)), x, 12, -((3*sqrt(b*x^(2/3) + a*x))/(10*b*x^(11/3))) + (19*a*sqrt(b*x^(2/3) + a*x))/(60*b^2*x^(10/3)) - (323*a^2*sqrt(b*x^(2/3) + a*x))/(960*b^3*x^3) + (323*a^3*sqrt(b*x^(2/3) + a*x))/(896*b^4*x^(8/3)) - (4199*a^4*sqrt(b*x^(2/3) + a*x))/(10752*b^5*x^(7/3)) + (46189*a^5*sqrt(b*x^(2/3) + a*x))/(107520*b^6*x^2) - (138567*a^6*sqrt(b*x^(2/3) + a*x))/(286720*b^7*x^(5/3)) + (46189*a^7*sqrt(b*x^(2/3) + a*x))/(81920*b^8*x^(4/3)) - (46189*a^8*sqrt(b*x^(2/3) + a*x))/(65536*b^9*x) + (138567*a^9*sqrt(b*x^(2/3) + a*x))/(131072*b^10*x^(2/3)) - (138567*a^10*atanh((sqrt(b)*x^(1/3))/sqrt(b*x^(2/3) + a*x)))/(131072*b^(21/2))]

    @test_int [x^4/(b*x^(2/3) + a*x)^(3/2), x, 12, (-6*x^4)/(a*sqrt(b*x^(2/3) + a*x)) - (524288*b^9*sqrt(b*x^(2/3) + a*x))/(29393*a^11) + (1048576*b^10*sqrt(b*x^(2/3) + a*x))/(29393*a^12*x^(1/3)) + (393216*b^8*x^(1/3)*sqrt(b*x^(2/3) + a*x))/(29393*a^10) - (327680*b^7*x^(2/3)*sqrt(b*x^(2/3) + a*x))/(29393*a^9) + (40960*b^6*x*sqrt(b*x^(2/3) + a*x))/(4199*a^8) - (36864*b^5*x^(4/3)*sqrt(b*x^(2/3) + a*x))/(4199*a^7) + (33792*b^4*x^(5/3)*sqrt(b*x^(2/3) + a*x))/(4199*a^6) - (16896*b^3*x^2*sqrt(b*x^(2/3) + a*x))/(2261*a^5) + (15840*b^2*x^(7/3)*sqrt(b*x^(2/3) + a*x))/(2261*a^4) - (880*b*x^(8/3)*sqrt(b*x^(2/3) + a*x))/(133*a^3) + (44*x^3*sqrt(b*x^(2/3) + a*x))/(7*a^2)]
    @test_int [x^3/(b*x^(2/3) + a*x)^(3/2), x, 9, (-6*x^3)/(a*sqrt(b*x^(2/3) + a*x)) + (32768*b^6*sqrt(b*x^(2/3) + a*x))/(2145*a^8) - (65536*b^7*sqrt(b*x^(2/3) + a*x))/(2145*a^9*x^(1/3)) - (8192*b^5*x^(1/3)*sqrt(b*x^(2/3) + a*x))/(715*a^7) + (4096*b^4*x^(2/3)*sqrt(b*x^(2/3) + a*x))/(429*a^6) - (3584*b^3*x*sqrt(b*x^(2/3) + a*x))/(429*a^5) + (5376*b^2*x^(4/3)*sqrt(b*x^(2/3) + a*x))/(715*a^4) - (448*b*x^(5/3)*sqrt(b*x^(2/3) + a*x))/(65*a^3) + (32*x^2*sqrt(b*x^(2/3) + a*x))/(5*a^2)]
    @test_int [x^2/(b*x^(2/3) + a*x)^(3/2), x, 6, (-6*x^2)/(a*sqrt(b*x^(2/3) + a*x)) - (256*b^3*sqrt(b*x^(2/3) + a*x))/(21*a^5) + (512*b^4*sqrt(b*x^(2/3) + a*x))/(21*a^6*x^(1/3)) + (64*b^2*x^(1/3)*sqrt(b*x^(2/3) + a*x))/(7*a^4) - (160*b*x^(2/3)*sqrt(b*x^(2/3) + a*x))/(21*a^3) + (20*x*sqrt(b*x^(2/3) + a*x))/(3*a^2)]
    @test_int [x^1/(b*x^(2/3) + a*x)^(3/2), x, 3, (-6*x)/(a*sqrt(b*x^(2/3) + a*x)) + (8*sqrt(b*x^(2/3) + a*x))/a^2 - (16*b*sqrt(b*x^(2/3) + a*x))/(a^3*x^(1/3))]
    @test_int [x^0/(b*x^(2/3) + a*x)^(3/2), x, 3, (6*x^(1/3))/(b*sqrt(b*x^(2/3) + a*x)) - (6*atanh((sqrt(b)*x^(1/3))/sqrt(b*x^(2/3) + a*x)))/b^(3/2)]
    @test_int [1/(x^1*(b*x^(2/3) + a*x)^(3/2)), x, 6, 6/(b*x^(2/3)*sqrt(b*x^(2/3) + a*x)) - (7*sqrt(b*x^(2/3) + a*x))/(b^2*x^(4/3)) + (35*a*sqrt(b*x^(2/3) + a*x))/(4*b^3*x) - (105*a^2*sqrt(b*x^(2/3) + a*x))/(8*b^4*x^(2/3)) + (105*a^3*atanh((sqrt(b)*x^(1/3))/sqrt(b*x^(2/3) + a*x)))/(8*b^(9/2))]
    @test_int [1/(x^2*(b*x^(2/3) + a*x)^(3/2)), x, 9, 6/(b*x^(5/3)*sqrt(b*x^(2/3) + a*x)) - (13*sqrt(b*x^(2/3) + a*x))/(2*b^2*x^(7/3)) + (143*a*sqrt(b*x^(2/3) + a*x))/(20*b^3*x^2) - (1287*a^2*sqrt(b*x^(2/3) + a*x))/(160*b^4*x^(5/3)) + (3003*a^3*sqrt(b*x^(2/3) + a*x))/(320*b^5*x^(4/3)) - (3003*a^4*sqrt(b*x^(2/3) + a*x))/(256*b^6*x) + (9009*a^5*sqrt(b*x^(2/3) + a*x))/(512*b^7*x^(2/3)) - (9009*a^6*atanh((sqrt(b)*x^(1/3))/sqrt(b*x^(2/3) + a*x)))/(512*b^(15/2))]
    @test_int [1/(x^3*(b*x^(2/3) + a*x)^(3/2)), x, 12, 6/(b*x^(8/3)*sqrt(b*x^(2/3) + a*x)) - (19*sqrt(b*x^(2/3) + a*x))/(3*b^2*x^(10/3)) + (323*a*sqrt(b*x^(2/3) + a*x))/(48*b^3*x^3) - (1615*a^2*sqrt(b*x^(2/3) + a*x))/(224*b^4*x^(8/3)) + (20995*a^3*sqrt(b*x^(2/3) + a*x))/(2688*b^5*x^(7/3)) - (46189*a^4*sqrt(b*x^(2/3) + a*x))/(5376*b^6*x^2) + (138567*a^5*sqrt(b*x^(2/3) + a*x))/(14336*b^7*x^(5/3)) - (46189*a^6*sqrt(b*x^(2/3) + a*x))/(4096*b^8*x^(4/3)) + (230945*a^7*sqrt(b*x^(2/3) + a*x))/(16384*b^9*x) - (692835*a^8*sqrt(b*x^(2/3) + a*x))/(32768*b^10*x^(2/3)) + (692835*a^9*atanh((sqrt(b)*x^(1/3))/sqrt(b*x^(2/3) + a*x)))/(32768*b^(21/2))]
    @test_int [1/(x^4*(b*x^(2/3) + a*x)^(3/2)), x, 15, 6/(b*x^(11/3)*sqrt(b*x^(2/3) + a*x)) - (25*sqrt(b*x^(2/3) + a*x))/(4*b^2*x^(13/3)) + (575*a*sqrt(b*x^(2/3) + a*x))/(88*b^3*x^4) - (2415*a^2*sqrt(b*x^(2/3) + a*x))/(352*b^4*x^(11/3)) + (15295*a^3*sqrt(b*x^(2/3) + a*x))/(2112*b^5*x^(10/3)) - (260015*a^4*sqrt(b*x^(2/3) + a*x))/(33792*b^6*x^3) + (185725*a^5*sqrt(b*x^(2/3) + a*x))/(22528*b^7*x^(8/3)) - (2414425*a^6*sqrt(b*x^(2/3) + a*x))/(270336*b^8*x^(7/3)) + (482885*a^7*sqrt(b*x^(2/3) + a*x))/(49152*b^9*x^2) - (1448655*a^8*sqrt(b*x^(2/3) + a*x))/(131072*b^10*x^(5/3)) + (3380195*a^9*sqrt(b*x^(2/3) + a*x))/(262144*b^11*x^(4/3)) - (16900975*a^10*sqrt(b*x^(2/3) + a*x))/(1048576*b^12*x) + (50702925*a^11*sqrt(b*x^(2/3) + a*x))/(2097152*b^13*x^(2/3)) - (50702925*a^12*atanh((sqrt(b)*x^(1/3))/sqrt(b*x^(2/3) + a*x)))/(2097152*b^(27/2))]

    #= ::Title::Closed:: =#
    #=Integration*problems*of*the*form*(c*x)^m*(a*x^2+b*x^n)^p=#

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

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

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

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

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

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

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

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

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

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

    @test_int [x^2*sqrt(a*x^2 + b*x^3), x, 4, (2*(a*x^2 + b*x^3)^(3/2))/(9*b) - (32*a^3*(a*x^2 + b*x^3)^(3/2))/(315*b^4*x^3) + (16*a^2*(a*x^2 + b*x^3)^(3/2))/(105*b^3*x^2) - (4*a*(a*x^2 + b*x^3)^(3/2))/(21*b^2*x)]
    @test_int [x*sqrt(a*x^2 + b*x^3), x, 3, (16*a^2*(a*x^2 + b*x^3)^(3/2))/(105*b^3*x^3) - (8*a*(a*x^2 + b*x^3)^(3/2))/(35*b^2*x^2) + (2*(a*x^2 + b*x^3)^(3/2))/(7*b*x)]
    @test_int [sqrt(a*x^2 + b*x^3), x, 2, -((4*a*(a*x^2 + b*x^3)^(3/2))/(15*b^2*x^3)) + (2*(a*x^2 + b*x^3)^(3/2))/(5*b*x^2)]
    @test_int [sqrt(a*x^2 + b*x^3)/x, x, 1, (2*(a*x^2 + b*x^3)^(3/2))/(3*b*x^3)]
    @test_int [sqrt(a*x^2 + b*x^3)/x^2, x, 3, (2*sqrt(a*x^2 + b*x^3))/x - 2*sqrt(a)*atanh((sqrt(a)*x)/sqrt(a*x^2 + b*x^3))]
    @test_int [sqrt(a*x^2 + b*x^3)/x^3, x, 3, -(sqrt(a*x^2 + b*x^3)/x^2) - (b*atanh((sqrt(a)*x)/sqrt(a*x^2 + b*x^3)))/sqrt(a)]
    @test_int [sqrt(a*x^2 + b*x^3)/x^4, x, 4, -(sqrt(a*x^2 + b*x^3)/(2*x^3)) - (b*sqrt(a*x^2 + b*x^3))/(4*a*x^2) + (b^2*atanh((sqrt(a)*x)/sqrt(a*x^2 + b*x^3)))/(4*a^(3/2))]
    @test_int [sqrt(a*x^2 + b*x^3)/x^5, x, 5, -(sqrt(a*x^2 + b*x^3)/(3*x^4)) - (b*sqrt(a*x^2 + b*x^3))/(12*a*x^3) + (b^2*sqrt(a*x^2 + b*x^3))/(8*a^2*x^2) - (b^3*atanh((sqrt(a)*x)/sqrt(a*x^2 + b*x^3)))/(8*a^(5/2))]

    @test_int [x^2*(a*x^2 + b*x^3)^(3/2), x, 6, (2*(a*x^2 + b*x^3)^(5/2))/(15*b) - (512*a^5*(a*x^2 + b*x^3)^(5/2))/(45045*b^6*x^5) + (256*a^4*(a*x^2 + b*x^3)^(5/2))/(9009*b^5*x^4) - (64*a^3*(a*x^2 + b*x^3)^(5/2))/(1287*b^4*x^3) + (32*a^2*(a*x^2 + b*x^3)^(5/2))/(429*b^3*x^2) - (4*a*(a*x^2 + b*x^3)^(5/2))/(39*b^2*x)]
    @test_int [x*(a*x^2 + b*x^3)^(3/2), x, 5, (256*a^4*(a*x^2 + b*x^3)^(5/2))/(15015*b^5*x^5) - (128*a^3*(a*x^2 + b*x^3)^(5/2))/(3003*b^4*x^4) + (32*a^2*(a*x^2 + b*x^3)^(5/2))/(429*b^3*x^3) - (16*a*(a*x^2 + b*x^3)^(5/2))/(143*b^2*x^2) + (2*(a*x^2 + b*x^3)^(5/2))/(13*b*x)]
    @test_int [(a*x^2 + b*x^3)^(3/2), x, 4, -((32*a^3*(a*x^2 + b*x^3)^(5/2))/(1155*b^4*x^5)) + (16*a^2*(a*x^2 + b*x^3)^(5/2))/(231*b^3*x^4) - (4*a*(a*x^2 + b*x^3)^(5/2))/(33*b^2*x^3) + (2*(a*x^2 + b*x^3)^(5/2))/(11*b*x^2)]
    @test_int [(a*x^2 + b*x^3)^(3/2)/x, x, 3, (16*a^2*(a*x^2 + b*x^3)^(5/2))/(315*b^3*x^5) - (8*a*(a*x^2 + b*x^3)^(5/2))/(63*b^2*x^4) + (2*(a*x^2 + b*x^3)^(5/2))/(9*b*x^3)]
    @test_int [(a*x^2 + b*x^3)^(3/2)/x^2, x, 2, -((4*a*(a*x^2 + b*x^3)^(5/2))/(35*b^2*x^5)) + (2*(a*x^2 + b*x^3)^(5/2))/(7*b*x^4)]
    @test_int [(a*x^2 + b*x^3)^(3/2)/x^3, x, 1, (2*(a*x^2 + b*x^3)^(5/2))/(5*b*x^5)]
    @test_int [(a*x^2 + b*x^3)^(3/2)/x^4, x, 4, (2*a*sqrt(a*x^2 + b*x^3))/x + (2*(a*x^2 + b*x^3)^(3/2))/(3*x^3) - 2*a^(3/2)*atanh((sqrt(a)*x)/sqrt(a*x^2 + b*x^3))]
    @test_int [(a*x^2 + b*x^3)^(3/2)/x^5, x, 4, (3*b*sqrt(a*x^2 + b*x^3))/x - (a*x^2 + b*x^3)^(3/2)/x^4 - 3*sqrt(a)*b*atanh((sqrt(a)*x)/sqrt(a*x^2 + b*x^3))]
    @test_int [(a*x^2 + b*x^3)^(3/2)/x^6, x, 4, -((3*b*sqrt(a*x^2 + b*x^3))/(4*x^2)) - (a*x^2 + b*x^3)^(3/2)/(2*x^5) - (3*b^2*atanh((sqrt(a)*x)/sqrt(a*x^2 + b*x^3)))/(4*sqrt(a))]
    @test_int [(a*x^2 + b*x^3)^(3/2)/x^7, x, 5, -((b*sqrt(a*x^2 + b*x^3))/(4*x^3)) - (b^2*sqrt(a*x^2 + b*x^3))/(8*a*x^2) - (a*x^2 + b*x^3)^(3/2)/(3*x^6) + (b^3*atanh((sqrt(a)*x)/sqrt(a*x^2 + b*x^3)))/(8*a^(3/2))]
    @test_int [(a*x^2 + b*x^3)^(3/2)/x^8, x, 6, -((b*sqrt(a*x^2 + b*x^3))/(8*x^4)) - (b^2*sqrt(a*x^2 + b*x^3))/(32*a*x^3) + (3*b^3*sqrt(a*x^2 + b*x^3))/(64*a^2*x^2) - (a*x^2 + b*x^3)^(3/2)/(4*x^7) - (3*b^4*atanh((sqrt(a)*x)/sqrt(a*x^2 + b*x^3)))/(64*a^(5/2))]
    @test_int [(a*x^2 + b*x^3)^(3/2)/x^9, x, 7, -((3*b*sqrt(a*x^2 + b*x^3))/(40*x^5)) - (b^2*sqrt(a*x^2 + b*x^3))/(80*a*x^4) + (b^3*sqrt(a*x^2 + b*x^3))/(64*a^2*x^3) - (3*b^4*sqrt(a*x^2 + b*x^3))/(128*a^3*x^2) - (a*x^2 + b*x^3)^(3/2)/(5*x^8) + (3*b^5*atanh((sqrt(a)*x)/sqrt(a*x^2 + b*x^3)))/(128*a^(7/2))]

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

    @test_int [x^4/sqrt(a*x^2 + b*x^3), x, 4, (16*a^2*sqrt(a*x^2 + b*x^3))/(35*b^3) - (32*a^3*sqrt(a*x^2 + b*x^3))/(35*b^4*x) - (12*a*x*sqrt(a*x^2 + b*x^3))/(35*b^2) + (2*x^2*sqrt(a*x^2 + b*x^3))/(7*b)]
    @test_int [x^3/sqrt(a*x^2 + b*x^3), x, 3, -((8*a*sqrt(a*x^2 + b*x^3))/(15*b^2)) + (16*a^2*sqrt(a*x^2 + b*x^3))/(15*b^3*x) + (2*x*sqrt(a*x^2 + b*x^3))/(5*b)]
    @test_int [x^2/sqrt(a*x^2 + b*x^3), x, 2, (2*sqrt(a*x^2 + b*x^3))/(3*b) - (4*a*sqrt(a*x^2 + b*x^3))/(3*b^2*x)]
    @test_int [x^1/sqrt(a*x^2 + b*x^3), x, 1, (2*sqrt(a*x^2 + b*x^3))/(b*x)]
    @test_int [x^0/sqrt(a*x^2 + b*x^3), x, 2, -((2*atanh((sqrt(a)*x)/sqrt(a*x^2 + b*x^3)))/sqrt(a))]
    @test_int [1/(x^1*sqrt(a*x^2 + b*x^3)), x, 3, -(sqrt(a*x^2 + b*x^3)/(a*x^2)) + (b*atanh((sqrt(a)*x)/sqrt(a*x^2 + b*x^3)))/a^(3/2)]
    @test_int [1/(x^2*sqrt(a*x^2 + b*x^3)), x, 4, (3*b*sqrt(a*x^2 + b*x^3))/(4*a^2*x^2) - sqrt(a*x^2 + b*x^3)/(2*a*x^3) - (3*b^2*atanh((sqrt(a)*x)/sqrt(a*x^2 + b*x^3)))/(4*a^(5/2))]
    @test_int [1/(x^3*sqrt(a*x^2 + b*x^3)), x, 5, -(sqrt(a*x^2 + b*x^3)/(3*a*x^4)) + (5*b*sqrt(a*x^2 + b*x^3))/(12*a^2*x^3) - (5*b^2*sqrt(a*x^2 + b*x^3))/(8*a^3*x^2) + (5*b^3*atanh((sqrt(a)*x)/sqrt(a*x^2 + b*x^3)))/(8*a^(7/2))]

    @test_int [x^6/(a*x^2 + b*x^3)^(3/2), x, 4, -((2*x^4)/(b*sqrt(a*x^2 + b*x^3))) - (16*a*sqrt(a*x^2 + b*x^3))/(5*b^3) + (32*a^2*sqrt(a*x^2 + b*x^3))/(5*b^4*x) + (12*x*sqrt(a*x^2 + b*x^3))/(5*b^2)]
    @test_int [x^5/(a*x^2 + b*x^3)^(3/2), x, 3, -((2*x^3)/(b*sqrt(a*x^2 + b*x^3))) + (8*sqrt(a*x^2 + b*x^3))/(3*b^2) - (16*a*sqrt(a*x^2 + b*x^3))/(3*b^3*x)]
    @test_int [x^4/(a*x^2 + b*x^3)^(3/2), x, 2, -((2*x^2)/(b*sqrt(a*x^2 + b*x^3))) + (4*sqrt(a*x^2 + b*x^3))/(b^2*x)]
    @test_int [x^3/(a*x^2 + b*x^3)^(3/2), x, 1, -((2*x)/(b*sqrt(a*x^2 + b*x^3)))]
    @test_int [x^2/(a*x^2 + b*x^3)^(3/2), x, 3, (2*x)/(a*sqrt(a*x^2 + b*x^3)) - (2*atanh((sqrt(a)*x)/sqrt(a*x^2 + b*x^3)))/a^(3/2)]
    @test_int [x^1/(a*x^2 + b*x^3)^(3/2), x, 4, 2/(a*sqrt(a*x^2 + b*x^3)) - (3*sqrt(a*x^2 + b*x^3))/(a^2*x^2) + (3*b*atanh((sqrt(a)*x)/sqrt(a*x^2 + b*x^3)))/a^(5/2)]
    @test_int [x^0/(a*x^2 + b*x^3)^(3/2), x, 5, 2/(a*x*sqrt(a*x^2 + b*x^3)) - (5*sqrt(a*x^2 + b*x^3))/(2*a^2*x^3) + (15*b*sqrt(a*x^2 + b*x^3))/(4*a^3*x^2) - (15*b^2*atanh((sqrt(a)*x)/sqrt(a*x^2 + b*x^3)))/(4*a^(7/2))]
    @test_int [1/(x^1*(a*x^2 + b*x^3)^(3/2)), x, 6, 2/(a*x^2*sqrt(a*x^2 + b*x^3)) - (7*sqrt(a*x^2 + b*x^3))/(3*a^2*x^4) + (35*b*sqrt(a*x^2 + b*x^3))/(12*a^3*x^3) - (35*b^2*sqrt(a*x^2 + b*x^3))/(8*a^4*x^2) + (35*b^3*atanh((sqrt(a)*x)/sqrt(a*x^2 + b*x^3)))/(8*a^(9/2))]
    @test_int [1/(x^2*(a*x^2 + b*x^3)^(3/2)), x, 7, 2/(a*x^3*sqrt(a*x^2 + b*x^3)) - (9*sqrt(a*x^2 + b*x^3))/(4*a^2*x^5) + (21*b*sqrt(a*x^2 + b*x^3))/(8*a^3*x^4) - (105*b^2*sqrt(a*x^2 + b*x^3))/(32*a^4*x^3) + (315*b^3*sqrt(a*x^2 + b*x^3))/(64*a^5*x^2) - (315*b^4*atanh((sqrt(a)*x)/sqrt(a*x^2 + b*x^3)))/(64*a^(11/2))]

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

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

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

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

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

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

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

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(c*x)^m*(a*x^2+b*x^5)^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a*x^2+b*x^5)^p=#

    @test_int [x^11/(a*x^2 + b*x^5)^3, x, 2, x^6/(6*a*(a + b*x^3)^2)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a*x^2+b*x^5)^(p/2)=#

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

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

    @test_int [x^9/sqrt(a*x^2 + b*x^5), x, 3, (16*a^2*sqrt(a*x^2 + b*x^5))/(45*b^3*x) - (8*a*x^2*sqrt(a*x^2 + b*x^5))/(45*b^2) + (2*x^5*sqrt(a*x^2 + b*x^5))/(15*b)]
    @test_int [x^6/sqrt(a*x^2 + b*x^5), x, 2, -((4*a*sqrt(a*x^2 + b*x^5))/(9*b^2*x)) + (2*x^2*sqrt(a*x^2 + b*x^5))/(9*b)]
    @test_int [x^3/sqrt(a*x^2 + b*x^5), x, 1, (2*sqrt(a*x^2 + b*x^5))/(3*b*x)]
    @test_int [x^0/sqrt(a*x^2 + b*x^5), x, 2, -((2*atanh((sqrt(a)*x)/sqrt(a*x^2 + b*x^5)))/(3*sqrt(a)))]
    @test_int [1/(x^3*sqrt(a*x^2 + b*x^5)), x, 3, -(sqrt(a*x^2 + b*x^5)/(3*a*x^4)) + (b*atanh((sqrt(a)*x)/sqrt(a*x^2 + b*x^5)))/(3*a^(3/2))]

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

    @test_int [x^5/sqrt(a*x^2 + b*x^5), x, 5, -((8*a*x*(a + b*x^3))/(7*b^(5/3)*((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)*sqrt(a*x^2 + b*x^5))) + (2*x*sqrt(a*x^2 + b*x^5))/(7*b) + (4*3^(1/4)*sqrt(2 - sqrt(3))*a^(4/3)*x*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*Elliptic.E(asin(((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)), -7 - 4*sqrt(3)))/(7*b^(5/3)*sqrt((a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*sqrt(a*x^2 + b*x^5)) - (8*sqrt(2)*a^(4/3)*x*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)), -7 - 4*sqrt(3)))/(7*3^(1/4)*b^(5/3)*sqrt((a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*sqrt(a*x^2 + b*x^5))]
    @test_int [x^2/sqrt(a*x^2 + b*x^5), x, 4, (2*x*(a + b*x^3))/(b^(2/3)*((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)*sqrt(a*x^2 + b*x^5)) - (3^(1/4)*sqrt(2 - sqrt(3))*a^(1/3)*x*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*Elliptic.E(asin(((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)), -7 - 4*sqrt(3)))/(b^(2/3)*sqrt((a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*sqrt(a*x^2 + b*x^5)) + (2*sqrt(2)*a^(1/3)*x*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)), -7 - 4*sqrt(3)))/(3^(1/4)*b^(2/3)*sqrt((a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*sqrt(a*x^2 + b*x^5))]
    @test_int [1/(x^1*sqrt(a*x^2 + b*x^5)), x, 5, (b^(1/3)*x*(a + b*x^3))/(a*((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)*sqrt(a*x^2 + b*x^5)) - sqrt(a*x^2 + b*x^5)/(a*x^2) - (3^(1/4)*sqrt(2 - sqrt(3))*b^(1/3)*x*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*Elliptic.E(asin(((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)), -7 - 4*sqrt(3)))/(2*a^(2/3)*sqrt((a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*sqrt(a*x^2 + b*x^5)) + (sqrt(2)*b^(1/3)*x*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)), -7 - 4*sqrt(3)))/(3^(1/4)*a^(2/3)*sqrt((a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*sqrt(a*x^2 + b*x^5))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^(m/2)*(a*x^2+b*x^5)^(p/2)=#

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

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

    @test_int [x^(13/2)/(a*x^2 + b*x^5)^(1/2), x, 5, -((7*a*sqrt(a*x^2 + b*x^5))/(20*b^2*sqrt(x))) + (x^(5/2)*sqrt(a*x^2 + b*x^5))/(5*b) + (7*a^(5/3)*x^(3/2)*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*Elliptic.F(acos((a^(1/3) + (1 - sqrt(3))*b^(1/3)*x)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)), (1/4)*(2 + sqrt(3))))/(40*3^(1/4)*b^2*sqrt((b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*sqrt(a*x^2 + b*x^5))]
    @test_int [x^(11/2)/(a*x^2 + b*x^5)^(1/2), x, 6, -((5*(1 + sqrt(3))*a*x^(3/2)*(a + b*x^3))/(8*b^(5/3)*(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)*sqrt(a*x^2 + b*x^5))) + (x^(3/2)*sqrt(a*x^2 + b*x^5))/(4*b) + (5*3^(1/4)*a^(4/3)*x^(3/2)*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*Elliptic.E(acos((a^(1/3) + (1 - sqrt(3))*b^(1/3)*x)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)), (1/4)*(2 + sqrt(3))))/(8*b^(5/3)*sqrt((b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*sqrt(a*x^2 + b*x^5)) + (5*(1 - sqrt(3))*a^(4/3)*x^(3/2)*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*Elliptic.F(acos((a^(1/3) + (1 - sqrt(3))*b^(1/3)*x)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)), (1/4)*(2 + sqrt(3))))/(16*3^(1/4)*b^(5/3)*sqrt((b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*sqrt(a*x^2 + b*x^5))]
    @test_int [x^(9/2)/(a*x^2 + b*x^5)^(1/2), x, 3, (sqrt(x)*sqrt(a*x^2 + b*x^5))/(3*b) - (a*atanh((sqrt(b)*x^(5/2))/sqrt(a*x^2 + b*x^5)))/(3*b^(3/2))]
    @test_int [x^(7/2)/(a*x^2 + b*x^5)^(1/2), x, 4, sqrt(a*x^2 + b*x^5)/(2*b*sqrt(x)) - (a^(2/3)*x^(3/2)*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*Elliptic.F(acos((a^(1/3) + (1 - sqrt(3))*b^(1/3)*x)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)), (1/4)*(2 + sqrt(3))))/(4*3^(1/4)*b*sqrt((b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*sqrt(a*x^2 + b*x^5))]
    @test_int [x^(5/2)/(a*x^2 + b*x^5)^(1/2), x, 5, ((1 + sqrt(3))*x^(3/2)*(a + b*x^3))/(b^(2/3)*(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)*sqrt(a*x^2 + b*x^5)) - (3^(1/4)*a^(1/3)*x^(3/2)*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*Elliptic.E(acos((a^(1/3) + (1 - sqrt(3))*b^(1/3)*x)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)), (1/4)*(2 + sqrt(3))))/(b^(2/3)*sqrt((b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*sqrt(a*x^2 + b*x^5)) - ((1 - sqrt(3))*a^(1/3)*x^(3/2)*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*Elliptic.F(acos((a^(1/3) + (1 - sqrt(3))*b^(1/3)*x)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)), (1/4)*(2 + sqrt(3))))/(2*3^(1/4)*b^(2/3)*sqrt((b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*sqrt(a*x^2 + b*x^5))]
    @test_int [x^(3/2)/(a*x^2 + b*x^5)^(1/2), x, 2, (2*atanh((sqrt(b)*x^(5/2))/sqrt(a*x^2 + b*x^5)))/(3*sqrt(b))]
    @test_int [x^(1/2)/(a*x^2 + b*x^5)^(1/2), x, 3, (x^(3/2)*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*Elliptic.F(acos((a^(1/3) + (1 - sqrt(3))*b^(1/3)*x)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)), (1/4)*(2 + sqrt(3))))/(3^(1/4)*a^(1/3)*sqrt((b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*sqrt(a*x^2 + b*x^5))]
    @test_int [1/(x^(1/2)*(a*x^2 + b*x^5)^(1/2)), x, 6, (2*(1 + sqrt(3))*b^(1/3)*x^(3/2)*(a + b*x^3))/(a*(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)*sqrt(a*x^2 + b*x^5)) - (2*sqrt(a*x^2 + b*x^5))/(a*x^(3/2)) - (2*3^(1/4)*b^(1/3)*x^(3/2)*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*Elliptic.E(acos((a^(1/3) + (1 - sqrt(3))*b^(1/3)*x)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)), (1/4)*(2 + sqrt(3))))/(a^(2/3)*sqrt((b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*sqrt(a*x^2 + b*x^5)) - ((1 - sqrt(3))*b^(1/3)*x^(3/2)*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*Elliptic.F(acos((a^(1/3) + (1 - sqrt(3))*b^(1/3)*x)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)), (1/4)*(2 + sqrt(3))))/(3^(1/4)*a^(2/3)*sqrt((b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*sqrt(a*x^2 + b*x^5))]
    @test_int [1/(x^(3/2)*(a*x^2 + b*x^5)^(1/2)), x, 1, -((2*sqrt(a*x^2 + b*x^5))/(3*a*x^(5/2)))]
    @test_int [1/(x^(5/2)*(a*x^2 + b*x^5)^(1/2)), x, 4, -((2*sqrt(a*x^2 + b*x^5))/(5*a*x^(7/2))) - (2*b*x^(3/2)*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*Elliptic.F(acos((a^(1/3) + (1 - sqrt(3))*b^(1/3)*x)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)), (1/4)*(2 + sqrt(3))))/(5*3^(1/4)*a^(4/3)*sqrt((b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*sqrt(a*x^2 + b*x^5))]
    @test_int [1/(x^(7/2)*(a*x^2 + b*x^5)^(1/2)), x, 7, -((8*(1 + sqrt(3))*b^(4/3)*x^(3/2)*(a + b*x^3))/(7*a^2*(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)*sqrt(a*x^2 + b*x^5))) - (2*sqrt(a*x^2 + b*x^5))/(7*a*x^(9/2)) + (8*b*sqrt(a*x^2 + b*x^5))/(7*a^2*x^(3/2)) + (8*3^(1/4)*b^(4/3)*x^(3/2)*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*Elliptic.E(acos((a^(1/3) + (1 - sqrt(3))*b^(1/3)*x)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)), (1/4)*(2 + sqrt(3))))/(7*a^(5/3)*sqrt((b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*sqrt(a*x^2 + b*x^5)) + (4*(1 - sqrt(3))*b^(4/3)*x^(3/2)*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*Elliptic.F(acos((a^(1/3) + (1 - sqrt(3))*b^(1/3)*x)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)), (1/4)*(2 + sqrt(3))))/(7*3^(1/4)*a^(5/3)*sqrt((b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*sqrt(a*x^2 + b*x^5))]
    @test_int [1/(x^(9/2)*(a*x^2 + b*x^5)^(1/2)), x, 2, -((2*sqrt(a*x^2 + b*x^5))/(9*a*x^(11/2))) + (4*b*sqrt(a*x^2 + b*x^5))/(9*a^2*x^(5/2))]
    @test_int [1/(x^(11/2)*(a*x^2 + b*x^5)^(1/2)), x, 5, -((2*sqrt(a*x^2 + b*x^5))/(11*a*x^(13/2))) + (16*b*sqrt(a*x^2 + b*x^5))/(55*a^2*x^(7/2)) + (16*b^2*x^(3/2)*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*Elliptic.F(acos((a^(1/3) + (1 - sqrt(3))*b^(1/3)*x)/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)), (1/4)*(2 + sqrt(3))))/(55*3^(1/4)*a^(7/3)*sqrt((b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + sqrt(3))*b^(1/3)*x)^2)*sqrt(a*x^2 + b*x^5))]

    #= ::Title::Closed:: =#
    #=Integration*problems*of*the*form*(c*x)^m*(a*x^3+b*x^n)^p=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(c*x)^m*(a*x^3+b*x^4)^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a*x^3+b*x^4)^n=#

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a*x^3+b*x^4)^(p/2)=#

    @test_int [x^4/sqrt(a*x^3 + b*x^4), x, 5, -((5*a*sqrt(a*x^3 + b*x^4))/(12*b^2)) + (5*a^2*sqrt(a*x^3 + b*x^4))/(8*b^3*x) + (x*sqrt(a*x^3 + b*x^4))/(3*b) - (5*a^3*atanh((sqrt(b)*x^2)/sqrt(a*x^3 + b*x^4)))/(8*b^(7/2))]
    @test_int [x^3/sqrt(a*x^3 + b*x^4), x, 4, sqrt(a*x^3 + b*x^4)/(2*b) - (3*a*sqrt(a*x^3 + b*x^4))/(4*b^2*x) + (3*a^2*atanh((sqrt(b)*x^2)/sqrt(a*x^3 + b*x^4)))/(4*b^(5/2))]
    @test_int [x^2/sqrt(a*x^3 + b*x^4), x, 3, sqrt(a*x^3 + b*x^4)/(b*x) - (a*atanh((sqrt(b)*x^2)/sqrt(a*x^3 + b*x^4)))/b^(3/2)]
    @test_int [x^1/sqrt(a*x^3 + b*x^4), x, 2, (2*atanh((sqrt(b)*x^2)/sqrt(a*x^3 + b*x^4)))/sqrt(b)]
    @test_int [x^0/sqrt(a*x^3 + b*x^4), x, 1, -((2*sqrt(a*x^3 + b*x^4))/(a*x^2))]
    @test_int [1/(x^1*sqrt(a*x^3 + b*x^4)), x, 2, -((2*sqrt(a*x^3 + b*x^4))/(3*a*x^3)) + (4*b*sqrt(a*x^3 + b*x^4))/(3*a^2*x^2)]
    @test_int [1/(x^2*sqrt(a*x^3 + b*x^4)), x, 3, -((2*sqrt(a*x^3 + b*x^4))/(5*a*x^4)) + (8*b*sqrt(a*x^3 + b*x^4))/(15*a^2*x^3) - (16*b^2*sqrt(a*x^3 + b*x^4))/(15*a^3*x^2)]
    @test_int [1/(x^3*sqrt(a*x^3 + b*x^4)), x, 4, -((2*sqrt(a*x^3 + b*x^4))/(7*a*x^5)) + (12*b*sqrt(a*x^3 + b*x^4))/(35*a^2*x^4) - (16*b^2*sqrt(a*x^3 + b*x^4))/(35*a^3*x^3) + (32*b^3*sqrt(a*x^3 + b*x^4))/(35*a^4*x^2)]
    @test_int [1/(x^4*sqrt(a*x^3 + b*x^4)), x, 5, -((2*sqrt(a*x^3 + b*x^4))/(9*a*x^6)) + (16*b*sqrt(a*x^3 + b*x^4))/(63*a^2*x^5) - (32*b^2*sqrt(a*x^3 + b*x^4))/(105*a^3*x^4) + (128*b^3*sqrt(a*x^3 + b*x^4))/(315*a^4*x^3) - (256*b^4*sqrt(a*x^3 + b*x^4))/(315*a^5*x^2)]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(c*x)^m*(a*x^3+b*x^5)^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a*x^3+b*x^5)^n=#

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

    #= ::Subsection:: =#
    #=Integrands*of*the*form*x^m*(a*x^3+b*x^5)^(p/2)=#

    #= ::Title::Closed:: =#
    #=Integration*problems*of*the*form*(c*x)^m*(a*x^j+b*x^n)^p=#

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

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

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

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

    #= Integrands*of*the*form*x^(12*(m-1))*(a*x + b*x^(12*m+2))^12 =#
    @test_int [(a*x + b*x^14)^12, x, 2, (a + b*x^13)^13/(169*b)]
    @test_int [x^12*(a*x + b*x^26)^12, x, 2, (a + b*x^25)^13/(325*b)]
    @test_int [x^24*(a*x + b*x^38)^12, x, 2, (a + b*x^37)^13/(481*b)]
    @test_int [x^(12*(m-1))*(a*x + b*x^(12*m + 2))^12, x, 2, (a + b*x^(1 + 12*m))^13/(13*b*(1 + 12*m))]

    #= Integrands*of*the*form*(a*x^m + b*x^(12*m+m+1))^12 =#
    @test_int [(a*x + b*x^14)^12, x, 2, (a + b*x^13)^13/(169*b)]
    @test_int [(a*x^2 + b*x^27)^12, x, 2, (a + b*x^25)^13/(325*b)]
    @test_int [(a*x^3 + b*x^40)^12, x, 2, (a + b*x^37)^13/(481*b)]
    @test_int [(a*x^m + b*x^(12*m + m + 1))^12, x, 2, (a + b*x^(1 + 12*m))^13/(13*b*(1 + 12*m))]

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

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

    @test_int [1/(a*x^2 + b/x^3)^3, x, 2, x^10/(10*b*(b + a*x^5)^2)]
    @test_int [1/(a*x^3 + b/x^5)^3, x, 2, x^16/(16*b*(b + a*x^8)^2)]

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

    @test_int [1/(1/x^2 + x^3), x, 7, (-(1/5))*sqrt((1/2)*(5 - sqrt(5)))*atan(sqrt((1/5)*(5 - 2*sqrt(5))) + 2*sqrt(2/(5 + sqrt(5)))*x) - (1/5)*sqrt((1/2)*(5 + sqrt(5)))*atan(sqrt((1/5)*(5 + 2*sqrt(5))) - sqrt((2/5)*(5 + sqrt(5)))*x) + (1/5)*log(1 + x) - (1/20)*(1 + sqrt(5))*log(1 - (1/2)*(1 - sqrt(5))*x + x^2) - (1/20)*(1 - sqrt(5))*log(1 - (1/2)*(1 + sqrt(5))*x + x^2)]

    #= Integrands*of*the*form*x^p*(a*x^n+b*x^(m*n+n+p+1))^m =#
    @test_int [x^p*(a*x^n + b*x^(12*n + n + p + 1))^12, x, 2, (a + b*x^(1 + 12*n + p))^13/(13*b*(1 + 12*n + p))]

    @test_int [x^12*(a + b*x^13)^12, x, 1, (a + b*x^13)^13/(169*b)]
    @test_int [x^12*(a*x + b*x^26)^12, x, 2, (a + b*x^25)^13/(325*b)]
    @test_int [x^12*(a*x^2 + b*x^39)^12, x, 2, (a + b*x^37)^13/(481*b)]

    @test_int [x^24*(a + b*x^25)^12, x, 1, (a + b*x^25)^13/(325*b)]
    @test_int [x^24*(a*x + b*x^38)^12, x, 2, (a + b*x^37)^13/(481*b)]

    @test_int [x^36*(a + b*x^37)^12, x, 1, (a + b*x^37)^13/(481*b)]

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

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

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(c*x)^m*(a*x^j+b*x^n)^(p/2)*with*m+j*p+1=0=#

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

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

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

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

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

    @test_int [(a*x^3 + b*x^n)^(3/2)/(c*x)^(1 + 3*3/2), x, 5, -((2*a*sqrt(a*x^3 + b*x^n))/(c^4*(3 - n)*(c*x)^(3/2))) - (2*(a*x^3 + b*x^n)^(3/2))/(3*c*(3 - n)*(c*x)^(9/2)) + (2*a^(3/2)*sqrt(c*x)*atanh((sqrt(a)*x^(3/2))/sqrt(a*x^3 + b*x^n)))/(c^6*(3 - n)*sqrt(x))]
    @test_int [(a*x^2 + b*x^n)^(3/2)/(c*x)^(1 + 3*2/2), x, 5, -((2*a*sqrt(a*x^2 + b*x^n))/(c^4*(2 - n)*x)) - (2*(a*x^2 + b*x^n)^(3/2))/(3*c^4*(2 - n)*x^3) + (2*a^(3/2)*atanh((sqrt(a)*x)/sqrt(a*x^2 + b*x^n)))/(c^4*(2 - n))]
    @test_int [(a*x^1 + b*x^n)^(3/2)/(c*x)^(1 + 3*1/2), x, 5, -((2*a*sqrt(a*x + b*x^n))/(c^2*(1 - n)*sqrt(c*x))) - (2*(a*x + b*x^n)^(3/2))/(3*c*(1 - n)*(c*x)^(3/2)) + (2*a^(3/2)*sqrt(x)*atanh((sqrt(a)*sqrt(x))/sqrt(a*x + b*x^n)))/(c^2*(1 - n)*sqrt(c*x))]
    @test_int [(a*x^0 + b*x^n)^(3/2)/(c*x)^(1 + 3*0/2), x, 6, (2*a*sqrt(a + b*x^n))/(c*n) + (2*(a + b*x^n)^(3/2))/(3*c*n) - (2*a^(3/2)*atanh(sqrt(a + b*x^n)/sqrt(a)))/(c*n)]
    @test_int [(a/x^1 + b*x^n)^(3/2)/(c*x)^(1 - 3*1/2), x, 5, (2*a*sqrt(c*x)*sqrt(a/x + b*x^n))/(1 + n) + (2*(c*x)^(3/2)*(a/x + b*x^n)^(3/2))/(3*c*(1 + n)) - (2*a^(3/2)*c*sqrt(x)*atanh(sqrt(a)/(sqrt(x)*sqrt(a/x + b*x^n))))/((1 + n)*sqrt(c*x))]
    @test_int [(a/x^2 + b*x^n)^(3/2)/(c*x)^(1 - 3*2/2), x, 5, (2*a*c^2*x*sqrt(a/x^2 + b*x^n))/(2 + n) + (2*c^2*x^3*(a/x^2 + b*x^n)^(3/2))/(3*(2 + n)) - (2*a^(3/2)*c^2*atanh(sqrt(a)/(x*sqrt(a/x^2 + b*x^n))))/(2 + n)]
    @test_int [(a/x^3 + b*x^n)^(3/2)/(c*x)^(1 - 3*3/2), x, 5, (2*a*c^2*(c*x)^(3/2)*sqrt(a/x^3 + b*x^n))/(3 + n) + (2*(c*x)^(9/2)*(a/x^3 + b*x^n)^(3/2))/(3*c*(3 + n)) - (2*a^(3/2)*c^4*sqrt(x)*atanh(sqrt(a)/(x^(3/2)*sqrt(a/x^3 + b*x^n))))/((3 + n)*sqrt(c*x))]
    @test_int [(a/x^4 + b*x^n)^(3/2)/(c*x)^(1 - 3*4/2), x, 5, (2*a*c^5*x^2*sqrt(a/x^4 + b*x^n))/(4 + n) + (2*c^5*x^6*(a/x^4 + b*x^n)^(3/2))/(3*(4 + n)) - (2*a^(3/2)*c^5*atanh(sqrt(a)/(x^2*sqrt(a/x^4 + b*x^n))))/(4 + n)]

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

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

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

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

    @test_int [1/sqrt(a*x^3 + b*x^n)/(c*x)^(1 - 3/2), x, 3, (2*sqrt(c*x)*atanh((sqrt(a)*x^(3/2))/sqrt(a*x^3 + b*x^n)))/(sqrt(a)*(3 - n)*sqrt(x))]
    @test_int [1/sqrt(a*x^2 + b*x^n)/(c*x)^(1 - 2/2), x, 2, (2*atanh((sqrt(a)*x)/sqrt(a*x^2 + b*x^n)))/(sqrt(a)*(2 - n))]
    @test_int [1/sqrt(a*x^1 + b*x^n)/(c*x)^(1 - 1/2), x, 3, (2*sqrt(x)*atanh((sqrt(a)*sqrt(x))/sqrt(a*x + b*x^n)))/(sqrt(a)*(1 - n)*sqrt(c*x))]
    @test_int [1/sqrt(a*x^0 + b*x^n)/(c*x)^(1 - 0/2), x, 4, -((2*atanh(sqrt(a + b*x^n)/sqrt(a)))/(sqrt(a)*c*n))]
    @test_int [1/sqrt(a/x^1 + b*x^n)/(c*x)^(1 + 1/2), x, 3, -((2*sqrt(x)*atanh(sqrt(a)/(sqrt(x)*sqrt(a/x + b*x^n))))/(sqrt(a)*c*(1 + n)*sqrt(c*x)))]
    @test_int [1/sqrt(a/x^2 + b*x^n)/(c*x)^(1 + 2/2), x, 3, -((2*atanh(sqrt(a)/(x*sqrt(a/x^2 + b*x^n))))/(sqrt(a)*c^2*(2 + n)))]
    @test_int [1/sqrt(a/x^3 + b*x^n)/(c*x)^(1 + 3/2), x, 3, -((2*sqrt(x)*atanh(sqrt(a)/(x^(3/2)*sqrt(a/x^3 + b*x^n))))/(sqrt(a)*c^2*(3 + n)*sqrt(c*x)))]

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

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

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

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

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

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

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

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

    @test_int [(a*x^j + b*x^n)^(3/2), x, 3, (2*b*x^(1 + n)*sqrt(a*x^j + b*x^n)*HypergeometricFunctions._₂F₁(-(3/2), (1 + (3*n)/2)/(j - n), (2 + 2*j + n)/(2*(j - n)), -((a*x^(j - n))/b)))/((2 + 3*n)*sqrt(1 + (a*x^(j - n))/b))]
    @test_int [(a*x^j + b*x^n)^(1/2), x, 3, (2*x*sqrt(a*x^j + b*x^n)*HypergeometricFunctions._₂F₁(-(1/2), (2 + n)/(2*(j - n)), 1 + (2 + n)/(2*j - 2*n), -((a*x^(j - n))/b)))/((2 + n)*sqrt(1 + (a*x^(j - n))/b))]
    @test_int [1/(a*x^j + b*x^n)^(1/2), x, 3, If(13<11, (2*x*sqrt(1 + (a*x^(j - n))/b)*HypergeometricFunctions._₂F₁(1/2, (2 - n)/(2*(j - n)), (1/2)*(2 + (2 - n)/(j - n)), -((a*x^(j - n))/b)))/((2 - n)*sqrt(a*x^j + b*x^n)), (2*x*sqrt(1 + (a*x^(j - n))/b)*HypergeometricFunctions._₂F₁(1/2, (2 - n)/(2*(j - n)), 1 + (1 - n/2)/(j - n), -((a*x^(j - n))/b)))/((2 - n)*sqrt(a*x^j + b*x^n)))]
    @test_int [1/(a*x^j + b*x^n)^(3/2), x, 3, If(13<11, (2*x^(1 - n)*sqrt(1 + (a*x^(j - n))/b)*HypergeometricFunctions._₂F₁(3/2, (1 - (3*n)/2)/(j - n), 1 + (2 - 3*n)/(2*j - 2*n), -((a*x^(j - n))/b)))/(b*(2 - 3*n)*sqrt(a*x^j + b*x^n)), (2*x^(1 - n)*sqrt(1 + (a*x^(j - n))/b)*HypergeometricFunctions._₂F₁(3/2, (1 - (3*n)/2)/(j - n), 1 + (1 - (3*n)/2)/(j - n), -((a*x^(j - n))/b)))/(b*(2 - 3*n)*sqrt(a*x^j + b*x^n)))]
    @test_int [1/(a*x^j + b*x^n)^(5/2), x, 3, If(13<11, (2*x^(1 - 2*n)*sqrt(1 + (a*x^(j - n))/b)*HypergeometricFunctions._₂F₁(5/2, (1 - (5*n)/2)/(j - n), 1 + (2 - 5*n)/(2*j - 2*n), -((a*x^(j - n))/b)))/(b^2*(2 - 5*n)*sqrt(a*x^j + b*x^n)), (2*x^(1 - 2*n)*sqrt(1 + (a*x^(j - n))/b)*HypergeometricFunctions._₂F₁(5/2, (1 - (5*n)/2)/(j - n), 1 + (1 - (5*n)/2)/(j - n), -((a*x^(j - n))/b)))/(b^2*(2 - 5*n)*sqrt(a*x^j + b*x^n)))]

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

    @test_int [x*sqrt(x^2*(a + b*x^3)), x, 1, (2*(x^2*(a + b*x^3))^(3/2))/(9*b*x^3)]
    @test_int [x*sqrt(a*x^2 + b*x^5), x, 1, (2*(a*x^2 + b*x^5)^(3/2))/(9*b*x^3)]

    @test_int [sqrt(x^4*(a + b*x^3)), x, 2, (2*(a*x^4 + b*x^7)^(3/2))/(9*b*x^6)]

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

    @test_int [1/(a*x^(1/3) + b*x^(2/3))^(1/3), x, 11, -((45*a^2*(a + 2*b*x^(1/3))*(-((b*(a*x^(1/3) + b*x^(2/3)))/a^2))^(1/3))/(14*2^(1/3)*b^3*(1 - sqrt(3) - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))*(a*x^(1/3) + b*x^(2/3))^(1/3))) - (45*a*(a + b*x^(1/3))*x^(1/3))/(28*b^2*(a*x^(1/3) + b*x^(2/3))^(1/3)) + (9*(a + b*x^(1/3))*x^(2/3))/(7*b*(a*x^(1/3) + b*x^(2/3))^(1/3)) - (45*3^(1/4)*sqrt(2 + sqrt(3))*a^4*(1 - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))*sqrt((1 + 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3) + 2*2^(1/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(2/3))/(1 - sqrt(3) - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))^2)*(-((b*(a*x^(1/3) + b*x^(2/3)))/a^2))^(1/3)*Elliptic.E(asin((1 + sqrt(3) - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))/(1 - sqrt(3) - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))), -7 + 4*sqrt(3)))/(28*2^(1/3)*b^3*sqrt(-((1 - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))/(1 - sqrt(3) - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))^2))*(a + 2*b*x^(1/3))*(a*x^(1/3) + b*x^(2/3))^(1/3)) + (15*3^(3/4)*a^4*(1 - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))*sqrt((1 + 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3) + 2*2^(1/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(2/3))/(1 - sqrt(3) - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))^2)*(-((b*(a*x^(1/3) + b*x^(2/3)))/a^2))^(1/3)*Elliptic.F(asin((1 + sqrt(3) - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))/(1 - sqrt(3) - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))), -7 + 4*sqrt(3)))/(7*2^(5/6)*b^3*sqrt(-((1 - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))/(1 - sqrt(3) - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))^2))*(a + 2*b*x^(1/3))*(a*x^(1/3) + b*x^(2/3))^(1/3))]
    @test_int [1/(a*x^(1/3) + b*x^(2/3))^(2/3), x, 9, -((18*a*(a + b*x^(1/3))*x^(1/3))/(5*b^2*(a*x^(1/3) + b*x^(2/3))^(2/3))) + (9*(a + b*x^(1/3))*x^(2/3))/(5*b*(a*x^(1/3) + b*x^(2/3))^(2/3)) + (6*2^(1/3)*3^(3/4)*sqrt(2 - sqrt(3))*a^4*(1 - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))*sqrt((1 + 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3) + 2*2^(1/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(2/3))/(1 - sqrt(3) - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))^2)*(-((b*(a*x^(1/3) + b*x^(2/3)))/a^2))^(2/3)*Elliptic.F(asin((1 + sqrt(3) - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))/(1 - sqrt(3) - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))), -7 + 4*sqrt(3)))/(5*b^3*sqrt(-((1 - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))/(1 - sqrt(3) - 2^(2/3)*(-((b*(a + b*x^(1/3))*x^(1/3))/a^2))^(1/3))^2))*(a + 2*b*x^(1/3))*(a*x^(1/3) + b*x^(2/3))^(2/3))]

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

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

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

    @test_int [(a*x^m + b*x^(m*p + m + 1))^p, x, 1, (a*x^m + b*x^(1 + m + m*p))^(1 + p)/(x^(m*(1 + p))*(b*(1 + p)*(1 + m*p)))]
    @test_int [(x^m*(a + b*x^(m*p + 1)))^p, x, 2, (a*x^m + b*x^(1 + m + m*p))^(1 + p)/(x^(m*(1 + p))*(b*(1 + p)*(1 + m*p)))]

    @test_int [x^n*(x^m*(a + b*x^(m*p + n + 1)))^p, x, 2, (a*x^m + b*x^(1 + m + n + m*p))^(1 + p)/(x^(m*(1 + p))*(b*(1 + p)*(1 + n + m*p)))]
    @test_int [x^n*(a*x^m + b*x^(m*p + m + n + 1))^p, x, 1, (a*x^m + b*x^(1 + m + n + m*p))^(1 + p)/(x^(m*(1 + p))*(b*(1 + p)*(1 + n + m*p)))]

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

    @test_int [(x^((n - 1)/p)*(a + b*x^n))^p, x, 2, (x^(((1 - n)*(1 + p))/p)*(b*x^(n - (1 - n)/p) + a/x^((1 - n)/p))^(1 + p))/(b*n*(1 + p))]

    @test_int [x^(-1 + n - p*(1 + q))*(a*x^n + b*x^p)^q, x, 1, (a*x^n + b*x^p)^(1 + q)/(x^(p*(1 + q))*(a*(n - p)*(1 + q)))]
    @test_int [x^(-1 - n*q - p*(1 + q))*(x^n*(a + b*x^p))^q, x, 2, -((a*x^n + b*x^(n + p))^(1 + q)/(x^((n + p)*(1 + q))*(a*p*(1 + q))))]
    end

max_stars_repo_path: test/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.2 (c x)^m (a x^j+b x^n)^p.jl