Optimal. Leaf size=80 \[ \frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{2/3}}-\frac{\log (x)}{2 a^{2/3}} \]
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Rubi [A] time = 0.0238317, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {57, 617, 204, 31} \[ \frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{2/3}}-\frac{\log (x)}{2 a^{2/3}} \]
Antiderivative was successfully verified.
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Rule 57
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x (a+b x)^{2/3}} \, dx &=-\frac{\log (x)}{2 a^{2/3}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )}{2 a^{2/3}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}}\\ &=-\frac{\log (x)}{2 a^{2/3}}+\frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}\right )}{a^{2/3}}\\ &=-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{2/3}}-\frac{\log (x)}{2 a^{2/3}}+\frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.136673, size = 93, normalized size = 1.16 \[ -\frac{\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )-2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )}{2 a^{2/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 76, normalized size = 1. \begin{align*}{\ln \left ( \sqrt [3]{bx+a}-\sqrt [3]{a} \right ){a}^{-{\frac{2}{3}}}}-{\frac{1}{2}\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 ){a}^{-{\frac{2}{3}}}} \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 [B] time = 1.55426, size = 346, normalized size = 4.32 \begin{align*} -\frac{2 \, \sqrt{3}{\left (a^{2}\right )}^{\frac{1}{6}} a \arctan \left (\frac{\sqrt{3}{\left (a^{2}\right )}^{\frac{1}{6}}{\left ({\left (a^{2}\right )}^{\frac{1}{3}} a + 2 \,{\left (a^{2}\right )}^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{1}{3}}\right )}}{3 \, a^{2}}\right ) +{\left (a^{2}\right )}^{\frac{2}{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 ) - 2 \,{\left (a^{2}\right )}^{\frac{2}{3}} \log \left ({\left (b x + a\right )}^{\frac{1}{3}} a -{\left (a^{2}\right )}^{\frac{2}{3}}\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.08918, size = 150, normalized size = 1.88 \begin{align*} \frac{\log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{1}{3}\right )}{3 a^{\frac{2}{3}} \Gamma \left (\frac{4}{3}\right )} + \frac{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{1}{3}\right )}{3 a^{\frac{2}{3}} \Gamma \left (\frac{4}{3}\right )} + \frac{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{1}{3}\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 [A] time = 1.87057, size = 105, normalized size = 1.31 \begin{align*} -\frac{\sqrt{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 )}{a^{\frac{2}{3}}} - \frac{\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 )}{2 \, a^{\frac{2}{3}}} + \frac{\log \left ({\left |{\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{2}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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