Optimal. Leaf size=38 \[ -\frac{\sqrt [3]{a+b x^3} \, _2F_1\left (-\frac{1}{3},1;\frac{1}{3};-\frac{b x^3}{a}\right )}{2 a x^2} \]
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Rubi [A] time = 0.0170219, antiderivative size = 51, normalized size of antiderivative = 1.34, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {365, 364} \[ -\frac{\left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (-\frac{2}{3},\frac{2}{3};\frac{1}{3};-\frac{b x^3}{a}\right )}{2 x^2 \left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a+b x^3\right )^{2/3}} \, dx &=\frac{\left (1+\frac{b x^3}{a}\right )^{2/3} \int \frac{1}{x^3 \left (1+\frac{b x^3}{a}\right )^{2/3}} \, dx}{\left (a+b x^3\right )^{2/3}}\\ &=-\frac{\left (1+\frac{b x^3}{a}\right )^{2/3} \, _2F_1\left (-\frac{2}{3},\frac{2}{3};\frac{1}{3};-\frac{b x^3}{a}\right )}{2 x^2 \left (a+b x^3\right )^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.0090244, size = 51, normalized size = 1.34 \[ -\frac{\left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (-\frac{2}{3},\frac{2}{3};\frac{1}{3};-\frac{b x^3}{a}\right )}{2 x^2 \left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \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}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{b x^{6} + a x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.14335, size = 41, normalized size = 1.08 \begin{align*} \frac{\Gamma \left (- \frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{2}{3} \\ \frac{1}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} x^{2} \Gamma \left (\frac{1}{3}\right )} \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}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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