Optimal. Leaf size=134 \[ -\frac{\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{9 \sqrt [3]{a} b^{5/3}}+\frac{2 \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{3 \sqrt{3} \sqrt [3]{a} b^{5/3}}+\frac{x}{3 b \left (a x^3+b\right )} \]
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Rubi [A] time = 0.0625091, antiderivative size = 134, 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 = {263, 199, 200, 31, 634, 617, 204, 628} \[ -\frac{\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{9 \sqrt [3]{a} b^{5/3}}+\frac{2 \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{3 \sqrt{3} \sqrt [3]{a} b^{5/3}}+\frac{x}{3 b \left (a x^3+b\right )} \]
Antiderivative was successfully verified.
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Rule 263
Rule 199
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^3}\right )^2 x^6} \, dx &=\int \frac{1}{\left (b+a x^3\right )^2} \, dx\\ &=\frac{x}{3 b \left (b+a x^3\right )}+\frac{2 \int \frac{1}{b+a x^3} \, dx}{3 b}\\ &=\frac{x}{3 b \left (b+a x^3\right )}+\frac{2 \int \frac{1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{9 b^{5/3}}+\frac{2 \int \frac{2 \sqrt [3]{b}-\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{9 b^{5/3}}\\ &=\frac{x}{3 b \left (b+a x^3\right )}+\frac{2 \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac{\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}{9 \sqrt [3]{a} b^{5/3}}+\frac{\int \frac{1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{3 b^{4/3}}\\ &=\frac{x}{3 b \left (b+a x^3\right )}+\frac{2 \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac{\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{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]{a} x}{\sqrt [3]{b}}\right )}{3 \sqrt [3]{a} b^{5/3}}\\ &=\frac{x}{3 b \left (b+a x^3\right )}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{3 \sqrt{3} \sqrt [3]{a} b^{5/3}}+\frac{2 \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac{\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{9 \sqrt [3]{a} b^{5/3}}\\ \end{align*}
Mathematica [A] time = 0.0577116, size = 118, normalized size = 0.88 \[ \frac{-\frac{\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{\sqrt [3]{a}}+\frac{3 b^{2/3} x}{a x^3+b}+\frac{2 \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{a}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{\sqrt [3]{a}}}{9 b^{5/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 115, normalized size = 0.9 \begin{align*}{\frac{x}{3\,b \left ( a{x}^{3}+b \right ) }}+{\frac{2}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{9\,ab}\ln \left ({x}^{2}-\sqrt [3]{{\frac{b}{a}}}x+ \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,\sqrt{3}}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}} \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 = 1.56179, size = 927, normalized size = 6.92 \begin{align*} \left [\frac{3 \, a b^{2} x + 3 \, \sqrt{\frac{1}{3}}{\left (a^{2} b x^{3} + a b^{2}\right )} \sqrt{-\frac{\left (a b^{2}\right )^{\frac{1}{3}}}{a}} \log \left (\frac{2 \, a b x^{3} - 3 \, \left (a b^{2}\right )^{\frac{1}{3}} b x - b^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a b x^{2} + \left (a b^{2}\right )^{\frac{2}{3}} x - \left (a b^{2}\right )^{\frac{1}{3}} b\right )} \sqrt{-\frac{\left (a b^{2}\right )^{\frac{1}{3}}}{a}}}{a x^{3} + b}\right ) -{\left (a x^{3} + b\right )} \left (a b^{2}\right )^{\frac{2}{3}} \log \left (a b x^{2} - \left (a b^{2}\right )^{\frac{2}{3}} x + \left (a b^{2}\right )^{\frac{1}{3}} b\right ) + 2 \,{\left (a x^{3} + b\right )} \left (a b^{2}\right )^{\frac{2}{3}} \log \left (a b x + \left (a b^{2}\right )^{\frac{2}{3}}\right )}{9 \,{\left (a^{2} b^{3} x^{3} + a b^{4}\right )}}, \frac{3 \, a b^{2} x + 6 \, \sqrt{\frac{1}{3}}{\left (a^{2} b x^{3} + a b^{2}\right )} \sqrt{\frac{\left (a b^{2}\right )^{\frac{1}{3}}}{a}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, \left (a b^{2}\right )^{\frac{2}{3}} x - \left (a b^{2}\right )^{\frac{1}{3}} b\right )} \sqrt{\frac{\left (a b^{2}\right )^{\frac{1}{3}}}{a}}}{b^{2}}\right ) -{\left (a x^{3} + b\right )} \left (a b^{2}\right )^{\frac{2}{3}} \log \left (a b x^{2} - \left (a b^{2}\right )^{\frac{2}{3}} x + \left (a b^{2}\right )^{\frac{1}{3}} b\right ) + 2 \,{\left (a x^{3} + b\right )} \left (a b^{2}\right )^{\frac{2}{3}} \log \left (a b x + \left (a b^{2}\right )^{\frac{2}{3}}\right )}{9 \,{\left (a^{2} b^{3} x^{3} + a b^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.573865, size = 39, normalized size = 0.29 \begin{align*} \frac{x}{3 a b x^{3} + 3 b^{2}} + \operatorname{RootSum}{\left (729 t^{3} a b^{5} - 8, \left ( t \mapsto t \log{\left (\frac{9 t b^{2}}{2} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.71404, size = 171, normalized size = 1.28 \begin{align*} -\frac{2 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{9 \, b^{2}} + \frac{x}{3 \,{\left (a x^{3} + b\right )} b} + \frac{2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{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 b^{2}} + \frac{\left (-a^{2} b\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{9 \, a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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