Optimal. Leaf size=114 \[ \frac{\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b}} \]
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Rubi [A] time = 0.046836, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {200, 31, 634, 617, 204, 628} \[ \frac{\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
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Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{a-b x^3} \, dx &=\frac{\int \frac{1}{\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{3 a^{2/3}}+\frac{\int \frac{2 \sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{2/3}}\\ &=-\frac{\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac{\int \frac{1}{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 \sqrt [3]{a}}+\frac{\int \frac{\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{2/3} \sqrt [3]{b}}\\ &=-\frac{\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac{\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{2/3} \sqrt [3]{b}}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}+\frac{\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{b}}\\ \end{align*}
Mathematica [A] time = 0.0094201, size = 89, normalized size = 0.78 \[ \frac{\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )}{6 a^{2/3} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 92, normalized size = 0.8 \begin{align*} -{\frac{1}{3\,b}\ln \left ( x-\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{1}{6\,b}\ln \left ({x}^{2}+\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{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 = 1.56342, size = 775, normalized size = 6.8 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{3}} a b \sqrt{\frac{\left (-a^{2} b\right )^{\frac{1}{3}}}{b}} \log \left (\frac{2 \, a b x^{3} + 3 \, \left (-a^{2} b\right )^{\frac{1}{3}} a x + a^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a b x^{2} - \left (-a^{2} b\right )^{\frac{2}{3}} x + \left (-a^{2} b\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (-a^{2} b\right )^{\frac{1}{3}}}{b}}}{b x^{3} - a}\right ) + \left (-a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{2} + \left (-a^{2} b\right )^{\frac{2}{3}} x - \left (-a^{2} b\right )^{\frac{1}{3}} a\right ) - 2 \, \left (-a^{2} b\right )^{\frac{2}{3}} \log \left (a b x - \left (-a^{2} b\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b}, \frac{6 \, \sqrt{\frac{1}{3}} a b \sqrt{-\frac{\left (-a^{2} b\right )^{\frac{1}{3}}}{b}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, \left (-a^{2} b\right )^{\frac{2}{3}} x - \left (-a^{2} b\right )^{\frac{1}{3}} a\right )} \sqrt{-\frac{\left (-a^{2} b\right )^{\frac{1}{3}}}{b}}}{a^{2}}\right ) + \left (-a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{2} + \left (-a^{2} b\right )^{\frac{2}{3}} x - \left (-a^{2} b\right )^{\frac{1}{3}} a\right ) - 2 \, \left (-a^{2} b\right )^{\frac{2}{3}} \log \left (a b x - \left (-a^{2} b\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.210526, size = 22, normalized size = 0.19 \begin{align*} - \operatorname{RootSum}{\left (27 t^{3} a^{2} b - 1, \left ( t \mapsto t \log{\left (- 3 t a + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12345, size = 140, normalized size = 1.23 \begin{align*} -\frac{\left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a} + \frac{\sqrt{3} \left (a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b} + \frac{\left (a b^{2}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (\frac{a}{b}\right )^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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