Optimal. Leaf size=761 \[ \frac{d^4 x^5 \sqrt [3]{\frac{b x^3}{a}+1} F_1\left (\frac{5}{3};\frac{1}{3},2;\frac{8}{3};-\frac{b x^3}{a},-\frac{d^3 x^3}{c^3}\right )}{5 c^6 \sqrt [3]{a+b x^3}}-\frac{d x^2 \sqrt [3]{\frac{b x^3}{a}+1} F_1\left (\frac{2}{3};\frac{1}{3},2;\frac{5}{3};-\frac{b x^3}{a},-\frac{d^3 x^3}{c^3}\right )}{c^3 \sqrt [3]{a+b x^3}}-\frac{c d^3 x \left (a+b x^3\right )^{2/3}}{\left (c^3+d^3 x^3\right ) \left (b c^3-a d^3\right )}+\frac{c^2 d^2 \left (a+b x^3\right )^{2/3}}{\left (c^3+d^3 x^3\right ) \left (b c^3-a d^3\right )}+\frac{a d^3 \log \left (c^3+d^3 x^3\right )}{9 c \left (b c^3-a d^3\right )^{4/3}}+\frac{b c^2 \log \left (c^3+d^3 x^3\right )}{6 \left (b c^3-a d^3\right )^{4/3}}+\frac{\left (3 b c^3-2 a d^3\right ) \log \left (c^3+d^3 x^3\right )}{18 c \left (b c^3-a d^3\right )^{4/3}}-\frac{a d^3 \log \left (\frac{x \sqrt [3]{b c^3-a d^3}}{c}-\sqrt [3]{a+b x^3}\right )}{3 c \left (b c^3-a d^3\right )^{4/3}}-\frac{\left (3 b c^3-2 a d^3\right ) \log \left (\frac{x \sqrt [3]{b c^3-a d^3}}{c}-\sqrt [3]{a+b x^3}\right )}{6 c \left (b c^3-a d^3\right )^{4/3}}-\frac{b c^2 \log \left (\sqrt [3]{b c^3-a d^3}+d \sqrt [3]{a+b x^3}\right )}{2 \left (b c^3-a d^3\right )^{4/3}}+\frac{2 a d^3 \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c^3-a d^3}}{c \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} c \left (b c^3-a d^3\right )^{4/3}}+\frac{\left (3 b c^3-2 a d^3\right ) \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c^3-a d^3}}{c \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} c \left (b c^3-a d^3\right )^{4/3}}-\frac{b c^2 \tan ^{-1}\left (\frac{1-\frac{2 d \sqrt [3]{a+b x^3}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt{3}}\right )}{\sqrt{3} \left (b c^3-a d^3\right )^{4/3}} \]
[Out]
________________________________________________________________________________________
Rubi [F] time = 0.0830621, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{(c+d x)^2 \sqrt [3]{a+b x^3}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin{align*} \int \frac{1}{(c+d x)^2 \sqrt [3]{a+b x^3}} \, dx &=\int \frac{1}{(c+d x)^2 \sqrt [3]{a+b x^3}} \, dx\\ \end{align*}
Mathematica [F] time = 0.336747, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x)^2 \sqrt [3]{a+b x^3}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dx+c \right ) ^{2}}{\frac{1}{\sqrt [3]{b{x}^{3}+a}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{a + b x^{3}} \left (c + d x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]