Optimal. Leaf size=122 \[ -\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3}}+\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3}}+\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3}}-\frac{1}{a x} \]
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Rubi [A] time = 0.0619413, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, 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 = 0.0180317, size = 114, normalized size = 0.93 \[ \frac{-\sqrt [3]{b} x \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \sqrt [3]{b} x \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 \sqrt{3} \sqrt [3]{b} x \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )-6 \sqrt [3]{a}}{6 a^{4/3} x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 99, 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 = 1.45847, size = 262, normalized size = 2.15 \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 = 0.457542, size = 29, normalized size = 0.24 \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 = 1.13272, size = 163, normalized size = 1.34 \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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