Optimal. Leaf size=36 \[ -\frac{\left (a+b x^3\right )^{2/3} \, _2F_1\left (\frac{1}{3},1;\frac{2}{3};-\frac{b x^3}{a}\right )}{a x} \]
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Rubi [A] time = 0.0158883, antiderivative size = 49, normalized size of antiderivative = 1.36, 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} \[ -\frac{\sqrt [3]{\frac{b x^3}{a}+1} \, _2F_1\left (-\frac{1}{3},\frac{1}{3};\frac{2}{3};-\frac{b x^3}{a}\right )}{x \sqrt [3]{a+b x^3}} \]
Antiderivative was successfully verified.
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Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{1}{x^2 \sqrt [3]{a+b x^3}} \, dx &=\frac{\sqrt [3]{1+\frac{b x^3}{a}} \int \frac{1}{x^2 \sqrt [3]{1+\frac{b x^3}{a}}} \, dx}{\sqrt [3]{a+b x^3}}\\ &=-\frac{\sqrt [3]{1+\frac{b x^3}{a}} \, _2F_1\left (-\frac{1}{3},\frac{1}{3};\frac{2}{3};-\frac{b x^3}{a}\right )}{x \sqrt [3]{a+b x^3}}\\ \end{align*}
Mathematica [A] time = 0.0085581, size = 49, normalized size = 1.36 \[ -\frac{\sqrt [3]{\frac{b x^3}{a}+1} \, _2F_1\left (-\frac{1}{3},\frac{1}{3};\frac{2}{3};-\frac{b x^3}{a}\right )}{x \sqrt [3]{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}{\frac{1}{\sqrt [3]{b{x}^{3}+a}}}}\, 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{1}{3}} x^{2}}\,{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{2}{3}}}{b x^{5} + a x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.713758, size = 39, normalized size = 1.08 \begin{align*} \frac{\Gamma \left (- \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{1}{3} \\ \frac{2}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} x \Gamma \left (\frac{2}{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{1}{3}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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