Optimal. Leaf size=88 \[ -\frac{1}{2} b^{2/3} \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )+\frac{b^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\left (a+b x^3\right )^{2/3}}{2 x^2} \]
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Rubi [A] time = 0.0191102, antiderivative size = 88, normalized size of antiderivative = 1., 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 = {277, 239} \[ -\frac{1}{2} b^{2/3} \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )+\frac{b^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\left (a+b x^3\right )^{2/3}}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 277
Rule 239
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3\right )^{2/3}}{x^3} \, dx &=-\frac{\left (a+b x^3\right )^{2/3}}{2 x^2}+b \int \frac{1}{\sqrt [3]{a+b x^3}} \, dx\\ &=-\frac{\left (a+b x^3\right )^{2/3}}{2 x^2}+\frac{b^{2/3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{1}{2} b^{2/3} \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )\\ \end{align*}
Mathematica [C] time = 0.0099165, size = 51, normalized size = 0.58 \[ -\frac{\left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac{2}{3},-\frac{2}{3};\frac{1}{3};-\frac{b x^3}{a}\right )}{2 x^2 \left (\frac{b x^3}{a}+1\right )^{2/3}} \]
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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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 [C] time = 1.62442, size = 42, normalized size = 0.48 \begin{align*} \frac{a^{\frac{2}{3}} \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 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{{\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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