Optimal. Leaf size=33 \[ \frac{x \sqrt [3]{a+b x^3} \, _2F_1\left (\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a}\right )}{a} \]
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Rubi [A] time = 0.0090233, antiderivative size = 46, normalized size of antiderivative = 1.39, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {246, 245} \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 246
Rule 245
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^3\right )^{2/3}} \, dx &=\frac{\left (1+\frac{b x^3}{a}\right )^{2/3} \int \frac{1}{\left (1+\frac{b x^3}{a}\right )^{2/3}} \, dx}{\left (a+b x^3\right )^{2/3}}\\ &=\frac{x \left (1+\frac{b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.184317, size = 177, normalized size = 5.36 \[ \frac{3 \sqrt [3]{2} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}\right )^{2/3} \sqrt [3]{\frac{i \left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{\sqrt{3}+3 i}} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{\left (1-i \sqrt{3}\right ) \sqrt [3]{b} x+\left (1+i \sqrt{3}\right ) \sqrt [3]{a}}{2 \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}\right )}{\sqrt [3]{b} \left (a+b x^3\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int \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}}}\,{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{1}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.87613, size = 36, normalized size = 1.09 \begin{align*} \frac{x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{2}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} \Gamma \left (\frac{4}{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}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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