Optimal. Leaf size=123 \[ \frac{\sqrt [3]{b} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3}}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{4/3}}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3}}-\frac{1}{a x} \]
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Rubi [A] time = 0.0608481, antiderivative size = 123, 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, 292, 31, 634, 617, 204, 628} \[ \frac{\sqrt [3]{b} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3}}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{4/3}}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3}}-\frac{1}{a x} \]
Antiderivative was successfully verified.
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Rule 325
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a-b x^3\right )} \, dx &=-\frac{1}{a x}+\frac{b \int \frac{x}{a-b x^3} \, dx}{a}\\ &=-\frac{1}{a x}+\frac{b^{2/3} \int \frac{1}{\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{3 a^{4/3}}-\frac{b^{2/3} \int \frac{\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^{4/3}}\\ &=-\frac{1}{a x}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{4/3}}+\frac{\sqrt [3]{b} \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^{4/3}}-\frac{b^{2/3} \int \frac{1}{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a}\\ &=-\frac{1}{a x}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{4/3}}+\frac{\sqrt [3]{b} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3}}+\frac{\sqrt [3]{b} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{4/3}}\\ &=-\frac{1}{a x}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3}}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{4/3}}+\frac{\sqrt [3]{b} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3}}\\ \end{align*}
Mathematica [A] time = 0.0249691, size = 114, normalized size = 0.93 \[ -\frac{-\sqrt [3]{b} x \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \sqrt [3]{b} x \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )+2 \sqrt{3} \sqrt [3]{b} x \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )+6 \sqrt [3]{a}}{6 a^{4/3} x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 100, normalized size = 0.8 \begin{align*} -{\frac{1}{3\,a}\ln \left ( x-\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{1}{6\,a}\ln \left ({x}^{2}+\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{\sqrt{3}}{3\,a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{1}{ax}} \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 = 1.52165, size = 271, normalized size = 2.2 \begin{align*} -\frac{2 \, \sqrt{3} x \left (-\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2}{3} \, \sqrt{3} x \left (-\frac{b}{a}\right )^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right ) + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} + a x \left (-\frac{b}{a}\right )^{\frac{2}{3}} - a \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 2 \, x \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x - a \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right ) + 6}{6 \, a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.443201, size = 31, normalized size = 0.25 \begin{align*} - \operatorname{RootSum}{\left (27 t^{3} a^{4} - b, \left ( t \mapsto t \log{\left (- \frac{9 t^{2} a^{3}}{b} + x \right )} \right )\right )} - \frac{1}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11237, size = 153, normalized size = 1.24 \begin{align*} -\frac{b \left (\frac{a}{b}\right )^{\frac{2}{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{2}{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} b} + \frac{\left (a b^{2}\right )^{\frac{2}{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} b} - \frac{1}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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