Optimal. Leaf size=142 \[ -\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{18 a^{4/3} b^{2/3}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{36 a^{4/3} b^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{6 \sqrt{3} a^{4/3} b^{2/3}}+\frac{x^4}{6 a \left (a+b x^6\right )} \]
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Rubi [A] time = 0.10935, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {275, 290, 292, 31, 634, 617, 204, 628} \[ -\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{18 a^{4/3} b^{2/3}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{36 a^{4/3} b^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{6 \sqrt{3} a^{4/3} b^{2/3}}+\frac{x^4}{6 a \left (a+b x^6\right )} \]
Antiderivative was successfully verified.
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Rule 275
Rule 290
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+b x^6\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{\left (a+b x^3\right )^2} \, dx,x,x^2\right )\\ &=\frac{x^4}{6 a \left (a+b x^6\right )}+\frac{\operatorname{Subst}\left (\int \frac{x}{a+b x^3} \, dx,x,x^2\right )}{6 a}\\ &=\frac{x^4}{6 a \left (a+b x^6\right )}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,x^2\right )}{18 a^{4/3} \sqrt [3]{b}}+\frac{\operatorname{Subst}\left (\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,x,x^2\right )}{18 a^{4/3} \sqrt [3]{b}}\\ &=\frac{x^4}{6 a \left (a+b x^6\right )}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{18 a^{4/3} b^{2/3}}+\frac{\operatorname{Subst}\left (\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,x,x^2\right )}{36 a^{4/3} b^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,x^2\right )}{12 a \sqrt [3]{b}}\\ &=\frac{x^4}{6 a \left (a+b x^6\right )}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{18 a^{4/3} b^{2/3}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{36 a^{4/3} b^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x^2}{\sqrt [3]{a}}\right )}{6 a^{4/3} b^{2/3}}\\ &=\frac{x^4}{6 a \left (a+b x^6\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{6 \sqrt{3} a^{4/3} b^{2/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{18 a^{4/3} b^{2/3}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{36 a^{4/3} b^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.105244, size = 195, normalized size = 1.37 \[ \frac{-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{b^{2/3}}+\frac{\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{b^{2/3}}+\frac{\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{b^{2/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{b^{2/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{b^{2/3}}+\frac{6 \sqrt [3]{a} x^4}{a+b x^6}}{36 a^{4/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 123, normalized size = 0.9 \begin{align*}{\frac{{x}^{4}}{6\,a \left ( b{x}^{6}+a \right ) }}-{\frac{1}{18\,ab}\ln \left ({x}^{2}+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{1}{36\,ab}\ln \left ({x}^{4}-\sqrt [3]{{\frac{a}{b}}}{x}^{2}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{\sqrt{3}}{18\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{{x}^{2}{\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 = 1.49905, size = 954, normalized size = 6.72 \begin{align*} \left [\frac{6 \, a b^{2} x^{4} + 3 \, \sqrt{\frac{1}{3}}{\left (a b^{2} x^{6} + a^{2} b\right )} \sqrt{\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} \log \left (\frac{2 \, b^{2} x^{6} - 3 \, \left (-a b^{2}\right )^{\frac{2}{3}} x^{2} - a b + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, \left (-a b^{2}\right )^{\frac{2}{3}} x^{4} + a b x^{2} + \left (-a b^{2}\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}}}{b x^{6} + a}\right ) +{\left (b x^{6} + a\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b^{2} x^{4} + \left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} + \left (-a b^{2}\right )^{\frac{2}{3}}\right ) - 2 \,{\left (b x^{6} + a\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b x^{2} - \left (-a b^{2}\right )^{\frac{1}{3}}\right )}{36 \,{\left (a^{2} b^{3} x^{6} + a^{3} b^{2}\right )}}, \frac{6 \, a b^{2} x^{4} + 6 \, \sqrt{\frac{1}{3}}{\left (a b^{2} x^{6} + 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^{2} + \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^{6} + a\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b^{2} x^{4} + \left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} + \left (-a b^{2}\right )^{\frac{2}{3}}\right ) - 2 \,{\left (b x^{6} + a\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b x^{2} - \left (-a b^{2}\right )^{\frac{1}{3}}\right )}{36 \,{\left (a^{2} b^{3} x^{6} + a^{3} b^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.931117, size = 46, normalized size = 0.32 \begin{align*} \frac{x^{4}}{6 a^{2} + 6 a b x^{6}} + \operatorname{RootSum}{\left (5832 t^{3} a^{4} b^{2} + 1, \left ( t \mapsto t \log{\left (324 t^{2} a^{3} b + x^{2} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1917, size = 182, normalized size = 1.28 \begin{align*} \frac{x^{4}}{6 \,{\left (b x^{6} + a\right )} a} - \frac{\left (-\frac{a}{b}\right )^{\frac{2}{3}} \log \left ({\left | x^{2} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{18 \, a^{2}} - \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{2} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{18 \, a^{2} b^{2}} + \frac{\left (-a b^{2}\right )^{\frac{2}{3}} \log \left (x^{4} + x^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{36 \, a^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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