Optimal. Leaf size=146 \[ -\frac{5 b^{2/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{18 a^{8/3}}+\frac{5 b^{2/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 a^{8/3}}+\frac{5 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{3 \sqrt{3} a^{8/3}}+\frac{5 x^2}{6 a^2}-\frac{x^5}{3 a \left (a x^3+b\right )} \]
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Rubi [A] time = 0.0758377, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.818, Rules used = {263, 288, 321, 292, 31, 634, 617, 204, 628} \[ -\frac{5 b^{2/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{18 a^{8/3}}+\frac{5 b^{2/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 a^{8/3}}+\frac{5 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{3 \sqrt{3} a^{8/3}}+\frac{5 x^2}{6 a^2}-\frac{x^5}{3 a \left (a x^3+b\right )} \]
Antiderivative was successfully verified.
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Rule 263
Rule 288
Rule 321
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x}{\left (a+\frac{b}{x^3}\right )^2} \, dx &=\int \frac{x^7}{\left (b+a x^3\right )^2} \, dx\\ &=-\frac{x^5}{3 a \left (b+a x^3\right )}+\frac{5 \int \frac{x^4}{b+a x^3} \, dx}{3 a}\\ &=\frac{5 x^2}{6 a^2}-\frac{x^5}{3 a \left (b+a x^3\right )}-\frac{(5 b) \int \frac{x}{b+a x^3} \, dx}{3 a^2}\\ &=\frac{5 x^2}{6 a^2}-\frac{x^5}{3 a \left (b+a x^3\right )}+\frac{\left (5 b^{2/3}\right ) \int \frac{1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{9 a^{7/3}}-\frac{\left (5 b^{2/3}\right ) \int \frac{\sqrt [3]{b}+\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{9 a^{7/3}}\\ &=\frac{5 x^2}{6 a^2}-\frac{x^5}{3 a \left (b+a x^3\right )}+\frac{5 b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 a^{8/3}}-\frac{\left (5 b^{2/3}\right ) \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{18 a^{8/3}}-\frac{(5 b) \int \frac{1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{6 a^{7/3}}\\ &=\frac{5 x^2}{6 a^2}-\frac{x^5}{3 a \left (b+a x^3\right )}+\frac{5 b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 a^{8/3}}-\frac{5 b^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{18 a^{8/3}}-\frac{\left (5 b^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{3 a^{8/3}}\\ &=\frac{5 x^2}{6 a^2}-\frac{x^5}{3 a \left (b+a x^3\right )}+\frac{5 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{3 \sqrt{3} a^{8/3}}+\frac{5 b^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 a^{8/3}}-\frac{5 b^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{18 a^{8/3}}\\ \end{align*}
Mathematica [A] time = 0.078319, size = 131, normalized size = 0.9 \[ \frac{-5 b^{2/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )+\frac{6 a^{2/3} b x^2}{a x^3+b}+9 a^{2/3} x^2+10 b^{2/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )+10 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{18 a^{8/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 120, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{2\,{a}^{2}}}+{\frac{b{x}^{2}}{3\,{a}^{2} \left ( a{x}^{3}+b \right ) }}+{\frac{5\,b}{9\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-{\frac{5\,b}{18\,{a}^{3}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{b}{a}}}x+ \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-{\frac{5\,b\sqrt{3}}{9\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}} \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 = 1.4309, size = 393, normalized size = 2.69 \begin{align*} \frac{9 \, a x^{5} + 15 \, b x^{2} - 10 \, \sqrt{3}{\left (a x^{3} + b\right )} \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{1}{3}} - \sqrt{3} b}{3 \, b}\right ) - 5 \,{\left (a x^{3} + b\right )} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}} + b \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 10 \,{\left (a x^{3} + b\right )} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right )}{18 \,{\left (a^{3} x^{3} + a^{2} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.484293, size = 58, normalized size = 0.4 \begin{align*} \frac{b x^{2}}{3 a^{3} x^{3} + 3 a^{2} b} + \operatorname{RootSum}{\left (729 t^{3} a^{8} - 125 b^{2}, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a^{5}}{25 b} + x \right )} \right )\right )} + \frac{x^{2}}{2 a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16221, size = 178, normalized size = 1.22 \begin{align*} \frac{x^{2}}{2 \, a^{2}} + \frac{b x^{2}}{3 \,{\left (a x^{3} + b\right )} a^{2}} + \frac{5 \, \left (-\frac{b}{a}\right )^{\frac{2}{3}} \log \left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{2}} + \frac{5 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right )}{9 \, a^{4}} - \frac{5 \, \left (-a^{2} b\right )^{\frac{2}{3}} \log \left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{18 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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