Optimal. Leaf size=89 \[ \frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x^{n/3}}{\sqrt [3]{a+b x^n}}+1}{\sqrt{3}}\right )}{\sqrt [3]{b} n}-\frac{3 \log \left (\sqrt [3]{b} x^{n/3}-\sqrt [3]{a+b x^n}\right )}{2 \sqrt [3]{b} n} \]
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Rubi [A] time = 0.0401447, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {345, 239} \[ \frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x^{n/3}}{\sqrt [3]{a+b x^n}}+1}{\sqrt{3}}\right )}{\sqrt [3]{b} n}-\frac{3 \log \left (\sqrt [3]{b} x^{n/3}-\sqrt [3]{a+b x^n}\right )}{2 \sqrt [3]{b} n} \]
Antiderivative was successfully verified.
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Rule 345
Rule 239
Rubi steps
\begin{align*} \int \frac{x^{-1+\frac{n}{3}}}{\sqrt [3]{a+b x^n}} \, dx &=\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a+b x^3}} \, dx,x,x^{n/3}\right )}{n}\\ &=\frac{\sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b} x^{n/3}}{\sqrt [3]{a+b x^n}}}{\sqrt{3}}\right )}{\sqrt [3]{b} n}-\frac{3 \log \left (\sqrt [3]{b} x^{n/3}-\sqrt [3]{a+b x^n}\right )}{2 \sqrt [3]{b} n}\\ \end{align*}
Mathematica [C] time = 0.0162502, size = 56, normalized size = 0.63 \[ \frac{3 x^{n/3} \sqrt [3]{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^n}{a}\right )}{n \sqrt [3]{a+b x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.115, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1+{\frac{n}{3}}}{\frac{1}{\sqrt [3]{a+b{x}^{n}}}}}\, 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{x^{\frac{1}{3} \, n - 1}}{{\left (b x^{n} + a\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 16.9825, size = 39, normalized size = 0.44 \begin{align*} \frac{x^{\frac{n}{3}} \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt [3]{a} n \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{x^{\frac{1}{3} \, n - 1}}{{\left (b x^{n} + a\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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