Optimal. Leaf size=332 \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} F_1\left (\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d^3 x^3}{c^3}\right )}{c \left (a+b x^3\right )^{2/3}}-\frac{d \log \left (c^3+d^3 x^3\right )}{3 \left (b c^3-a d^3\right )^{2/3}}+\frac{d \log \left (\frac{x \sqrt [3]{b c^3-a d^3}}{c}-\sqrt [3]{a+b x^3}\right )}{2 \left (b c^3-a d^3\right )^{2/3}}+\frac{d \log \left (\sqrt [3]{b c^3-a d^3}+d \sqrt [3]{a+b x^3}\right )}{2 \left (b c^3-a d^3\right )^{2/3}}+\frac{d \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c^3-a d^3}}{c \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} \left (b c^3-a d^3\right )^{2/3}}-\frac{d \tan ^{-1}\left (\frac{1-\frac{2 d \sqrt [3]{a+b x^3}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt{3}}\right )}{\sqrt{3} \left (b c^3-a d^3\right )^{2/3}} \]
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Rubi [F] time = 0.0913542, 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 = {} \[ \int \frac{1}{(c+d x) \left (a+b x^3\right )^{2/3}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{(c+d x) \left (a+b x^3\right )^{2/3}} \, dx &=\int \frac{1}{(c+d x) \left (a+b x^3\right )^{2/3}} \, dx\\ \end{align*}
Mathematica [F] time = 0.041249, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x) \left (a+b x^3\right )^{2/3}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.032, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{dx+c} \left ( b{x}^{3}+a \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x^{3}\right )^{\frac{2}{3}} \left (c + d x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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