Optimal. Leaf size=100 \[ \frac{3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} \sqrt [3]{b}}-\frac{\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{2/3} \sqrt [3]{b}} \]
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Rubi [A] time = 0.02787, antiderivative size = 100, 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 = {58, 617, 204, 31} \[ \frac{3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} \sqrt [3]{b}}-\frac{\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{2/3} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
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Rule 58
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x^{2/3} (a+b x)} \, dx &=-\frac{\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\frac{a^{2/3}}{b^{2/3}}-\frac{\sqrt [3]{a} x}{\sqrt [3]{b}}+x^2} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{x}\right )}{2 a^{2/3} \sqrt [3]{b}}\\ &=\frac{3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} \sqrt [3]{b}}-\frac{\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{a^{2/3} \sqrt [3]{b}}\\ &=-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{2/3} \sqrt [3]{b}}+\frac{3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{2/3} \sqrt [3]{b}}-\frac{\log (a+b x)}{2 a^{2/3} \sqrt [3]{b}}\\ \end{align*}
Mathematica [A] time = 0.0229514, size = 103, normalized size = 1.03 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{2 a^{2/3} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 95, normalized size = 1. \begin{align*}{\frac{1}{b}\ln \left ( \sqrt [3]{x}+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{2\,b}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{{\frac{a}{b}}}\sqrt [3]{x}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{x}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\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 [A] time = 1.9316, size = 807, normalized size = 8.07 \begin{align*} \left [\frac{\sqrt{3} a b \sqrt{-\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} \log \left (\frac{2 \, a b x - a^{2} + \sqrt{3}{\left (2 \, a b x^{\frac{2}{3}} - \left (a^{2} b\right )^{\frac{1}{3}} a + \left (a^{2} b\right )^{\frac{2}{3}} x^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} - 3 \, \left (a^{2} b\right )^{\frac{1}{3}} a x^{\frac{1}{3}}}{b x + a}\right ) - \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{\frac{2}{3}} + \left (a^{2} b\right )^{\frac{1}{3}} a - \left (a^{2} b\right )^{\frac{2}{3}} x^{\frac{1}{3}}\right ) + 2 \, \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{\frac{1}{3}} + \left (a^{2} b\right )^{\frac{2}{3}}\right )}{2 \, a^{2} b}, \frac{2 \, \sqrt{3} a b \sqrt{\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} \arctan \left (-\frac{\sqrt{3}{\left (\left (a^{2} b\right )^{\frac{1}{3}} a - 2 \, \left (a^{2} b\right )^{\frac{2}{3}} x^{\frac{1}{3}}\right )} \sqrt{\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}}}{3 \, a^{2}}\right ) - \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{\frac{2}{3}} + \left (a^{2} b\right )^{\frac{1}{3}} a - \left (a^{2} b\right )^{\frac{2}{3}} x^{\frac{1}{3}}\right ) + 2 \, \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{\frac{1}{3}} + \left (a^{2} b\right )^{\frac{2}{3}}\right )}{2 \, a^{2} b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 16.8926, size = 218, normalized size = 2.18 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{2}{3}}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{3 \sqrt [3]{x}}{a} & \text{for}\: b = 0 \\- \frac{3}{2 b x^{\frac{2}{3}}} & \text{for}\: a = 0 \\- \frac{\sqrt [3]{-1} \log{\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac{1}{b}} + \sqrt [3]{x} \right )}}{a^{\frac{2}{3}} b^{3} \left (\frac{1}{b}\right )^{\frac{8}{3}}} + \frac{\sqrt [3]{-1} \log{\left (4 \left (-1\right )^{\frac{2}{3}} a^{\frac{2}{3}} \left (\frac{1}{b}\right )^{\frac{2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{\frac{1}{b}} + 4 x^{\frac{2}{3}} \right )}}{2 a^{\frac{2}{3}} b^{3} \left (\frac{1}{b}\right )^{\frac{8}{3}}} + \frac{\sqrt [3]{-1} \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} - \frac{2 \left (-1\right )^{\frac{2}{3}} \sqrt{3} \sqrt [3]{x}}{3 \sqrt [3]{a} \sqrt [3]{\frac{1}{b}}} \right )}}{a^{\frac{2}{3}} b^{3} \left (\frac{1}{b}\right )^{\frac{8}{3}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07621, size = 158, normalized size = 1.58 \begin{align*} -\frac{\left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x^{\frac{1}{3}} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{a} + \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{a b} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}} \log \left (x^{\frac{2}{3}} + x^{\frac{1}{3}} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{2 \, a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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