Optimal. Leaf size=92 \[ \frac{3}{2} a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )+\sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )-\frac{1}{2} a^{2/3} \log (x)+\frac{3}{2} (a+b x)^{2/3} \]
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Rubi [A] time = 0.0320364, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {50, 55, 617, 204, 31} \[ \frac{3}{2} a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )+\sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )-\frac{1}{2} a^{2/3} \log (x)+\frac{3}{2} (a+b x)^{2/3} \]
Antiderivative was successfully verified.
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Rule 50
Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{(a+b x)^{2/3}}{x} \, dx &=\frac{3}{2} (a+b x)^{2/3}+a \int \frac{1}{x \sqrt [3]{a+b x}} \, dx\\ &=\frac{3}{2} (a+b x)^{2/3}-\frac{1}{2} a^{2/3} \log (x)-\frac{1}{2} \left (3 a^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )+\frac{1}{2} (3 a) \operatorname{Subst}\left (\int \frac{1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x}\right )\\ &=\frac{3}{2} (a+b x)^{2/3}-\frac{1}{2} a^{2/3} \log (x)+\frac{3}{2} a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-\left (3 a^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}\right )\\ &=\frac{3}{2} (a+b x)^{2/3}+\sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )-\frac{1}{2} a^{2/3} \log (x)+\frac{3}{2} a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )\\ \end{align*}
Mathematica [A] time = 0.0761639, size = 86, normalized size = 0.93 \[ \frac{3}{2} \left (a^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )+(a+b x)^{2/3}\right )+\sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )-\frac{1}{2} a^{2/3} \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 84, normalized size = 0.9 \begin{align*}{\frac{3}{2} \left ( bx+a \right ) ^{{\frac{2}{3}}}}+{a}^{{\frac{2}{3}}}\ln \left ( \sqrt [3]{bx+a}-\sqrt [3]{a} \right ) -{\frac{1}{2}{a}^{{\frac{2}{3}}}\ln \left ( \left ( bx+a \right ) ^{{\frac{2}{3}}}+\sqrt [3]{a}\sqrt [3]{bx+a}+{a}^{{\frac{2}{3}}} \right ) }+{a}^{{\frac{2}{3}}}\sqrt{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{bx+a}}{\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 = 1.9108, size = 333, normalized size = 3.62 \begin{align*} \sqrt{3}{\left (a^{2}\right )}^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3} a + 2 \, \sqrt{3}{\left (a^{2}\right )}^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}}{3 \, a}\right ) - \frac{1}{2} \,{\left (a^{2}\right )}^{\frac{1}{3}} \log \left ({\left (b x + a\right )}^{\frac{2}{3}} a +{\left (a^{2}\right )}^{\frac{1}{3}} a +{\left (a^{2}\right )}^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{1}{3}}\right ) +{\left (a^{2}\right )}^{\frac{1}{3}} \log \left ({\left (b x + a\right )}^{\frac{1}{3}} a -{\left (a^{2}\right )}^{\frac{2}{3}}\right ) + \frac{3}{2} \,{\left (b x + a\right )}^{\frac{2}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.77062, size = 182, normalized size = 1.98 \begin{align*} \frac{5 a^{\frac{2}{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{5}{3}\right )}{3 \Gamma \left (\frac{8}{3}\right )} + \frac{5 a^{\frac{2}{3}} e^{\frac{2 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x} e^{\frac{2 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{5}{3}\right )}{3 \Gamma \left (\frac{8}{3}\right )} + \frac{5 a^{\frac{2}{3}} e^{- \frac{2 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x} e^{\frac{4 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{5}{3}\right )}{3 \Gamma \left (\frac{8}{3}\right )} + \frac{5 b^{\frac{2}{3}} \left (\frac{a}{b} + x\right )^{\frac{2}{3}} \Gamma \left (\frac{5}{3}\right )}{2 \Gamma \left (\frac{8}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.73107, size = 116, normalized size = 1.26 \begin{align*} \sqrt{3} a^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right ) - \frac{1}{2} \, a^{\frac{2}{3}} \log \left ({\left (b x + a\right )}^{\frac{2}{3}} +{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right ) + a^{\frac{2}{3}} \log \left ({\left |{\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right ) + \frac{3}{2} \,{\left (b x + a\right )}^{\frac{2}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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