Optimal. Leaf size=124 \[ \frac{b^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3}}-\frac{b^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{5/3}}+\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3}}-\frac{1}{2 a x^2} \]
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Rubi [A] time = 0.0625653, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {325, 200, 31, 634, 617, 204, 628} \[ \frac{b^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3}}-\frac{b^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{5/3}}+\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3}}-\frac{1}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 325
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a-b x^3\right )} \, dx &=-\frac{1}{2 a x^2}+\frac{b \int \frac{1}{a-b x^3} \, dx}{a}\\ &=-\frac{1}{2 a x^2}+\frac{b \int \frac{1}{\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{3 a^{5/3}}+\frac{b \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^{5/3}}\\ &=-\frac{1}{2 a x^2}-\frac{b^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{5/3}}+\frac{b^{2/3} \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^{5/3}}+\frac{b \int \frac{1}{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a^{4/3}}\\ &=-\frac{1}{2 a x^2}-\frac{b^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{5/3}}+\frac{b^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3}}-\frac{b^{2/3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{5/3}}\\ &=-\frac{1}{2 a x^2}+\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3}}-\frac{b^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{5/3}}+\frac{b^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3}}\\ \end{align*}
Mathematica [A] time = 0.0203802, size = 119, normalized size = 0.96 \[ \frac{b^{2/3} x^2 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-3 a^{2/3}-2 b^{2/3} x^2 \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )+2 \sqrt{3} b^{2/3} x^2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )}{6 a^{5/3} x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 100, normalized size = 0.8 \begin{align*} -{\frac{1}{3\,a}\ln \left ( x-\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{1}{6\,a}\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\,a}\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}}}}-{\frac{1}{2\,a{x}^{2}}} \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 = 1.47793, size = 336, normalized size = 2.71 \begin{align*} -\frac{2 \, \sqrt{3} x^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} a x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}} + \sqrt{3} b}{3 \, b}\right ) + x^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{2} - a b x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 2 \, x^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x + a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 3}{6 \, a x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.477562, size = 34, normalized size = 0.27 \begin{align*} - \operatorname{RootSum}{\left (27 t^{3} a^{5} - b^{2}, \left ( t \mapsto t \log{\left (- \frac{3 t a^{2}}{b} + x \right )} \right )\right )} - \frac{1}{2 a x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16356, size = 144, normalized size = 1.16 \begin{align*} -\frac{b \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{2}} + \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^{2}} + \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^{2}} - \frac{1}{2 \, a x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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