Optimal. Leaf size=106 \[ \frac{3 \sqrt [3]{a+b x^n}}{n}+\frac{3 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^n}\right )}{2 n}-\frac{\sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^n}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{n}-\frac{1}{2} \sqrt [3]{a} \log (x) \]
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Rubi [A] time = 0.0673738, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {266, 50, 57, 617, 204, 31} \[ \frac{3 \sqrt [3]{a+b x^n}}{n}+\frac{3 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^n}\right )}{2 n}-\frac{\sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^n}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{n}-\frac{1}{2} \sqrt [3]{a} \log (x) \]
Antiderivative was successfully verified.
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Rule 266
Rule 50
Rule 57
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{a+b x^n}}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt [3]{a+b x}}{x} \, dx,x,x^n\right )}{n}\\ &=\frac{3 \sqrt [3]{a+b x^n}}{n}+\frac{a \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{2/3}} \, dx,x,x^n\right )}{n}\\ &=\frac{3 \sqrt [3]{a+b x^n}}{n}-\frac{1}{2} \sqrt [3]{a} \log (x)-\frac{\left (3 \sqrt [3]{a}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^n}\right )}{2 n}-\frac{\left (3 a^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^n}\right )}{2 n}\\ &=\frac{3 \sqrt [3]{a+b x^n}}{n}-\frac{1}{2} \sqrt [3]{a} \log (x)+\frac{3 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^n}\right )}{2 n}+\frac{\left (3 \sqrt [3]{a}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b x^n}}{\sqrt [3]{a}}\right )}{n}\\ &=\frac{3 \sqrt [3]{a+b x^n}}{n}-\frac{\sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x^n}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{n}-\frac{1}{2} \sqrt [3]{a} \log (x)+\frac{3 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^n}\right )}{2 n}\\ \end{align*}
Mathematica [A] time = 0.0586595, size = 129, normalized size = 1.22 \[ \frac{-\sqrt [3]{a} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^n}+\left (a+b x^n\right )^{2/3}\right )+6 \sqrt [3]{a+b x^n}+2 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^n}\right )-2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b x^n}}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )}{2 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 107, normalized size = 1. \begin{align*} 3\,{\frac{\sqrt [3]{a+b{x}^{n}}}{n}}+{\frac{1}{n}\sqrt [3]{a}\ln \left ( \sqrt [3]{a+b{x}^{n}}-\sqrt [3]{a} \right ) }-{\frac{1}{2\,n}\sqrt [3]{a}\ln \left ( \left ( a+b{x}^{n} \right ) ^{{\frac{2}{3}}}+\sqrt [3]{a}\sqrt [3]{a+b{x}^{n}}+{a}^{{\frac{2}{3}}} \right ) }-{\frac{\sqrt{3}}{n}\sqrt [3]{a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{a+b{x}^{n}}}{\sqrt [3]{a}}}+1 \right ) } \right ) } \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 [A] time = 0.989489, size = 311, normalized size = 2.93 \begin{align*} -\frac{2 \, \sqrt{3} a^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3}{\left (b x^{n} + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} + \sqrt{3} a}{3 \, a}\right ) + a^{\frac{1}{3}} \log \left ({\left (b x^{n} + a\right )}^{\frac{2}{3}} +{\left (b x^{n} + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right ) - 2 \, a^{\frac{1}{3}} \log \left ({\left (b x^{n} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}}\right ) - 6 \,{\left (b x^{n} + a\right )}^{\frac{1}{3}}}{2 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.77144, size = 46, normalized size = 0.43 \begin{align*} - \frac{\sqrt [3]{b} x^{\frac{n}{3}} \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{a x^{- n} e^{i \pi }}{b}} \right )}}{n \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{{\left (b x^{n} + a\right )}^{\frac{1}{3}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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