Optimal. Leaf size=114 \[ -\frac{3 b^{2/3} \log \left (\sqrt [3]{b} x^{n/3}-\sqrt [3]{a+b x^n}\right )}{2 n}+\frac{\sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x^{n/3}}{\sqrt [3]{a+b x^n}}+1}{\sqrt{3}}\right )}{n}-\frac{3 x^{-2 n/3} \left (a+b x^n\right )^{2/3}}{2 n} \]
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Rubi [A] time = 0.0541963, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {349, 345, 239} \[ -\frac{3 b^{2/3} \log \left (\sqrt [3]{b} x^{n/3}-\sqrt [3]{a+b x^n}\right )}{2 n}+\frac{\sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x^{n/3}}{\sqrt [3]{a+b x^n}}+1}{\sqrt{3}}\right )}{n}-\frac{3 x^{-2 n/3} \left (a+b x^n\right )^{2/3}}{2 n} \]
Antiderivative was successfully verified.
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Rule 349
Rule 345
Rule 239
Rubi steps
\begin{align*} \int x^{-1-\frac{2 n}{3}} \left (a+b x^n\right )^{2/3} \, dx &=-\frac{3 x^{-2 n/3} \left (a+b x^n\right )^{2/3}}{2 n}+b \int \frac{x^{-1+\frac{n}{3}}}{\sqrt [3]{a+b x^n}} \, dx\\ &=-\frac{3 x^{-2 n/3} \left (a+b x^n\right )^{2/3}}{2 n}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a+b x^3}} \, dx,x,x^{n/3}\right )}{n}\\ &=-\frac{3 x^{-2 n/3} \left (a+b x^n\right )^{2/3}}{2 n}+\frac{\sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b} x^{n/3}}{\sqrt [3]{a+b x^n}}}{\sqrt{3}}\right )}{n}-\frac{3 b^{2/3} \log \left (\sqrt [3]{b} x^{n/3}-\sqrt [3]{a+b x^n}\right )}{2 n}\\ \end{align*}
Mathematica [C] time = 0.0186392, size = 58, normalized size = 0.51 \[ -\frac{3 x^{-2 n/3} \left (a+b x^n\right )^{2/3} \, _2F_1\left (-\frac{2}{3},-\frac{2}{3};\frac{1}{3};-\frac{b x^n}{a}\right )}{2 n \left (\frac{b x^n}{a}+1\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.078, size = 0, normalized size = 0. \begin{align*} \int{x}^{-1-{\frac{2\,n}{3}}} \left ( a+b{x}^{n} \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{\left (b x^{n} + a\right )}^{\frac{2}{3}} x^{-\frac{2}{3} \, n - 1}\,{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 [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{n} + a\right )}^{\frac{2}{3}} x^{-\frac{2}{3} \, n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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