3.28 \(\int \frac {\sqrt [3]{a+b x^3}}{(c+d x)^2} \, dx\)

Optimal. Leaf size=818 \[ -\frac {d^3 \sqrt [3]{b x^3+a} F_1\left (\frac {4}{3};-\frac {1}{3},2;\frac {7}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right ) x^4}{2 c^5 \sqrt [3]{\frac {b x^3}{a}+1}}-\frac {d \sqrt [3]{b x^3+a} x^2}{c^3+d^3 x^3}+\frac {\sqrt [3]{b x^3+a} F_1\left (\frac {1}{3};-\frac {1}{3},2;\frac {4}{3};-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right ) x}{c^2 \sqrt [3]{\frac {b x^3}{a}+1}}-\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{b x^3+a}}+1}{\sqrt {3}}\right )}{\sqrt {3} d^2}+\frac {2 a d \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{b x^3+a}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c \left (b c^3-a d^3\right )^{2/3}}+\frac {\left (3 b c^3-2 a d^3\right ) \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{b x^3+a}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c d^2 \left (b c^3-a d^3\right )^{2/3}}-\frac {b c^2 \tan ^{-1}\left (\frac {1-\frac {2 d \sqrt [3]{b x^3+a}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{\sqrt {3} d^2 \left (b c^3-a d^3\right )^{2/3}}-\frac {a d \log \left (c^3+d^3 x^3\right )}{9 c \left (b c^3-a d^3\right )^{2/3}}-\frac {\left (3 b c^3-2 a d^3\right ) \log \left (c^3+d^3 x^3\right )}{18 c d^2 \left (b c^3-a d^3\right )^{2/3}}-\frac {b c^2 \log \left (c^3+d^3 x^3\right )}{6 d^2 \left (b c^3-a d^3\right )^{2/3}}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{b} x-\sqrt [3]{b x^3+a}\right )}{2 d^2}+\frac {a d \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{b x^3+a}\right )}{3 c \left (b c^3-a d^3\right )^{2/3}}+\frac {\left (3 b c^3-2 a d^3\right ) \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{b x^3+a}\right )}{6 c d^2 \left (b c^3-a d^3\right )^{2/3}}+\frac {b c^2 \log \left (\sqrt [3]{b x^3+a} d+\sqrt [3]{b c^3-a d^3}\right )}{2 d^2 \left (b c^3-a d^3\right )^{2/3}}-\frac {c^2 \sqrt [3]{b x^3+a}}{d \left (c^3+d^3 x^3\right )} \]

[Out]

-c^2*(b*x^3+a)^(1/3)/d/(d^3*x^3+c^3)-d*x^2*(b*x^3+a)^(1/3)/(d^3*x^3+c^3)+x*(b*x^3+a)^(1/3)*AppellF1(1/3,-1/3,2
,4/3,-b*x^3/a,-d^3*x^3/c^3)/c^2/(1+b*x^3/a)^(1/3)-1/2*d^3*x^4*(b*x^3+a)^(1/3)*AppellF1(4/3,-1/3,2,7/3,-b*x^3/a
,-d^3*x^3/c^3)/c^5/(1+b*x^3/a)^(1/3)-1/6*b*c^2*ln(d^3*x^3+c^3)/d^2/(-a*d^3+b*c^3)^(2/3)-1/9*a*d*ln(d^3*x^3+c^3
)/c/(-a*d^3+b*c^3)^(2/3)-1/18*(-2*a*d^3+3*b*c^3)*ln(d^3*x^3+c^3)/c/d^2/(-a*d^3+b*c^3)^(2/3)-1/2*b^(1/3)*ln(b^(
1/3)*x-(b*x^3+a)^(1/3))/d^2+1/3*a*d*ln((-a*d^3+b*c^3)^(1/3)*x/c-(b*x^3+a)^(1/3))/c/(-a*d^3+b*c^3)^(2/3)+1/6*(-
2*a*d^3+3*b*c^3)*ln((-a*d^3+b*c^3)^(1/3)*x/c-(b*x^3+a)^(1/3))/c/d^2/(-a*d^3+b*c^3)^(2/3)+1/2*b*c^2*ln((-a*d^3+
b*c^3)^(1/3)+d*(b*x^3+a)^(1/3))/d^2/(-a*d^3+b*c^3)^(2/3)-1/3*b^(1/3)*arctan(1/3*(1+2*b^(1/3)*x/(b*x^3+a)^(1/3)
)*3^(1/2))/d^2*3^(1/2)+2/9*a*d*arctan(1/3*(1+2*(-a*d^3+b*c^3)^(1/3)*x/c/(b*x^3+a)^(1/3))*3^(1/2))/c/(-a*d^3+b*
c^3)^(2/3)*3^(1/2)+1/9*(-2*a*d^3+3*b*c^3)*arctan(1/3*(1+2*(-a*d^3+b*c^3)^(1/3)*x/c/(b*x^3+a)^(1/3))*3^(1/2))/c
/d^2/(-a*d^3+b*c^3)^(2/3)*3^(1/2)-1/3*b*c^2*arctan(1/3*(1-2*d*(b*x^3+a)^(1/3)/(-a*d^3+b*c^3)^(1/3))*3^(1/2))/d
^2/(-a*d^3+b*c^3)^(2/3)*3^(1/2)

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Rubi [F]  time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt [3]{a+b x^3}}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*x^3)^(1/3)/(c + d*x)^2,x]

[Out]

Defer[Int][(a + b*x^3)^(1/3)/(c + d*x)^2, x]

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{a+b x^3}}{(c+d x)^2} \, dx &=\int \frac {\sqrt [3]{a+b x^3}}{(c+d x)^2} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt [3]{a+b x^3}}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*x^3)^(1/3)/(c + d*x)^2,x]

[Out]

Integrate[(a + b*x^3)^(1/3)/(c + d*x)^2, x]

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(1/3)/(d*x+c)^2,x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(1/3)/(d*x+c)^2,x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(1/3)/(d*x + c)^2, x)

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{\left (d x +c \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(1/3)/(d*x+c)^2,x)

[Out]

int((b*x^3+a)^(1/3)/(d*x+c)^2,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(1/3)/(d*x+c)^2,x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(1/3)/(d*x + c)^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (b\,x^3+a\right )}^{1/3}}{{\left (c+d\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^(1/3)/(c + d*x)^2,x)

[Out]

int((a + b*x^3)^(1/3)/(c + d*x)^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt [3]{a + b x^{3}}}{\left (c + d x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**(1/3)/(d*x+c)**2,x)

[Out]

Integral((a + b*x**3)**(1/3)/(c + d*x)**2, x)

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