Optimal. Leaf size=91 \[ 3 \sqrt [3]{a+b x}+\frac{3}{2} \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-\sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )-\frac{1}{2} \sqrt [3]{a} \log (x) \]
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Rubi [A] time = 0.0470565, antiderivative size = 91, 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, 57, 617, 204, 31} \[ 3 \sqrt [3]{a+b x}+\frac{3}{2} \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-\sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )-\frac{1}{2} \sqrt [3]{a} \log (x) \]
Antiderivative was successfully verified.
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Rule 50
Rule 57
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{a+b x}}{x} \, dx &=3 \sqrt [3]{a+b x}+a \int \frac{1}{x (a+b x)^{2/3}} \, dx\\ &=3 \sqrt [3]{a+b x}-\frac{1}{2} \sqrt [3]{a} \log (x)-\frac{1}{2} \left (3 \sqrt [3]{a}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )-\frac{1}{2} \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}\right )\\ &=3 \sqrt [3]{a+b x}-\frac{1}{2} \sqrt [3]{a} \log (x)+\frac{3}{2} \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )+\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}}{\sqrt [3]{a}}\right )\\ &=3 \sqrt [3]{a+b x}-\sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )-\frac{1}{2} \sqrt [3]{a} \log (x)+\frac{3}{2} \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )\\ \end{align*}
Mathematica [A] time = 0.191845, size = 113, normalized size = 1.24 \[ -\frac{1}{2} \sqrt [3]{a} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )+3 \sqrt [3]{a+b x}+\sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-\sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 85, normalized size = 0.9 \begin{align*} 3\,\sqrt [3]{bx+a}+\sqrt [3]{a}\ln \left ( \sqrt [3]{bx+a}-\sqrt [3]{a} \right ) -{\frac{1}{2}\sqrt [3]{a}\ln \left ( \left ( bx+a \right ) ^{{\frac{2}{3}}}+\sqrt [3]{a}\sqrt [3]{bx+a}+{a}^{{\frac{2}{3}}} \right ) }-\sqrt [3]{a}\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 = 1.89735, size = 286, normalized size = 3.14 \begin{align*} -\sqrt{3} a^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3}{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} + \sqrt{3} a}{3 \, a}\right ) - \frac{1}{2} \, a^{\frac{1}{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{1}{3}} \log \left ({\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}}\right ) + 3 \,{\left (b x + a\right )}^{\frac{1}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.78096, size = 180, normalized size = 1.98 \begin{align*} \frac{4 \sqrt [3]{a} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac{4}{3}\right )}{3 \Gamma \left (\frac{7}{3}\right )} + \frac{4 \sqrt [3]{a} 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{4}{3}\right )}{3 \Gamma \left (\frac{7}{3}\right )} + \frac{4 \sqrt [3]{a} 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{4}{3}\right )}{3 \Gamma \left (\frac{7}{3}\right )} + \frac{4 \sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x} \Gamma \left (\frac{4}{3}\right )}{\Gamma \left (\frac{7}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.28773, size = 117, normalized size = 1.29 \begin{align*} -\sqrt{3} a^{\frac{1}{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{1}{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{1}{3}} \log \left ({\left |{\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right ) + 3 \,{\left (b x + a\right )}^{\frac{1}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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