Optimal. Leaf size=136 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} \sqrt [3]{a} b^{5/3}}-\frac{x^2}{3 b \left (a+b x^3\right )} \]
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Rubi [A] time = 0.0651125, antiderivative size = 136, 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 = {288, 292, 31, 634, 617, 204, 628} \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} \sqrt [3]{a} b^{5/3}}-\frac{x^2}{3 b \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 288
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^4}{\left (a+b x^3\right )^2} \, dx &=-\frac{x^2}{3 b \left (a+b x^3\right )}+\frac{2 \int \frac{x}{a+b x^3} \, dx}{3 b}\\ &=-\frac{x^2}{3 b \left (a+b x^3\right )}-\frac{2 \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 \sqrt [3]{a} b^{4/3}}+\frac{2 \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}{9 \sqrt [3]{a} b^{4/3}}\\ &=-\frac{x^2}{3 b \left (a+b x^3\right )}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{5/3}}+\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}{9 \sqrt [3]{a} b^{5/3}}+\frac{\int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 b^{4/3}}\\ &=-\frac{x^2}{3 b \left (a+b x^3\right )}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{5/3}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 \sqrt [3]{a} b^{5/3}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 \sqrt [3]{a} b^{5/3}}\\ &=-\frac{x^2}{3 b \left (a+b x^3\right )}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} \sqrt [3]{a} b^{5/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{5/3}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 \sqrt [3]{a} b^{5/3}}\\ \end{align*}
Mathematica [A] time = 0.0729075, size = 119, normalized size = 0.88 \[ \frac{\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{a}}-\frac{3 b^{2/3} x^2}{a+b x^3}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{a}}}{9 b^{5/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 108, normalized size = 0.8 \begin{align*} -{\frac{{x}^{2}}{3\,b \left ( b{x}^{3}+a \right ) }}-{\frac{2}{9\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{1}{9\,{b}^{2}}\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{2\,\sqrt{3}}{9\,{b}^{2}}\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}}}}}} \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 = 1.80404, size = 930, normalized size = 6.84 \begin{align*} \left [-\frac{3 \, a b^{2} x^{2} - 3 \, \sqrt{\frac{1}{3}}{\left (a b^{2} x^{3} + a^{2} b\right )} \sqrt{\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} \log \left (\frac{2 \, b^{2} x^{3} - a b + 3 \, \sqrt{\frac{1}{3}}{\left (a b x + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} x^{2} + \left (-a b^{2}\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac{2}{3}} x}{b x^{3} + a}\right ) -{\left (b x^{3} + a\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac{1}{3}} b x + \left (-a b^{2}\right )^{\frac{2}{3}}\right ) + 2 \,{\left (b x^{3} + a\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac{1}{3}}\right )}{9 \,{\left (a b^{4} x^{3} + a^{2} b^{3}\right )}}, -\frac{3 \, a b^{2} x^{2} - 6 \, \sqrt{\frac{1}{3}}{\left (a b^{2} x^{3} + a^{2} b\right )} \sqrt{-\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, b x + \left (-a b^{2}\right )^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}}}{b}\right ) -{\left (b x^{3} + a\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac{1}{3}} b x + \left (-a b^{2}\right )^{\frac{2}{3}}\right ) + 2 \,{\left (b x^{3} + a\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac{1}{3}}\right )}{9 \,{\left (a b^{4} x^{3} + a^{2} b^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.537006, size = 44, normalized size = 0.32 \begin{align*} - \frac{x^{2}}{3 a b + 3 b^{2} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a b^{5} + 8, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a b^{3}}{4} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14293, size = 178, normalized size = 1.31 \begin{align*} -\frac{x^{2}}{3 \,{\left (b x^{3} + a\right )} b} - \frac{2 \, \left (-\frac{a}{b}\right )^{\frac{2}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b} - \frac{2 \, \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 )}{9 \, a b^{3}} + \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 )}{9 \, a b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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