Optimal. Leaf size=124 \[ \frac{a^{2/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 b^{5/3}}-\frac{a^{2/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 b^{5/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} b^{5/3}}-\frac{1}{2 b x^2} \]
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Rubi [A] time = 0.0647602, antiderivative size = 124, 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, 325, 200, 31, 634, 617, 204, 628} \[ \frac{a^{2/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 b^{5/3}}-\frac{a^{2/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 b^{5/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} b^{5/3}}-\frac{1}{2 b x^2} \]
Antiderivative was successfully verified.
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Rule 263
Rule 325
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 ) x^6} \, dx &=\int \frac{1}{x^3 \left (b+a x^3\right )} \, dx\\ &=-\frac{1}{2 b x^2}-\frac{a \int \frac{1}{b+a x^3} \, dx}{b}\\ &=-\frac{1}{2 b x^2}-\frac{a \int \frac{1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{3 b^{5/3}}-\frac{a \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}{3 b^{5/3}}\\ &=-\frac{1}{2 b x^2}-\frac{a^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 b^{5/3}}+\frac{a^{2/3} \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}{6 b^{5/3}}-\frac{a \int \frac{1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 b^{4/3}}\\ &=-\frac{1}{2 b x^2}-\frac{a^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 b^{5/3}}+\frac{a^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 b^{5/3}}-\frac{a^{2/3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{b^{5/3}}\\ &=-\frac{1}{2 b x^2}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} b^{5/3}}-\frac{a^{2/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 b^{5/3}}+\frac{a^{2/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 b^{5/3}}\\ \end{align*}
Mathematica [A] time = 0.0202203, size = 119, normalized size = 0.96 \[ \frac{a^{2/3} x^2 \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )-2 a^{2/3} x^2 \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )+2 \sqrt{3} a^{2/3} x^2 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )-3 b^{2/3}}{6 b^{5/3} x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 99, normalized size = 0.8 \begin{align*} -{\frac{1}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}+{\frac{1}{6\,b}\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{\sqrt{3}}{3\,b}\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}}}}-{\frac{1}{2\,b{x}^{2}}} \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 = 1.4547, size = 335, normalized size = 2.7 \begin{align*} \frac{2 \, \sqrt{3} x^{2} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} - \sqrt{3} a}{3 \, a}\right ) - x^{2} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a^{2} x^{2} + a b x \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} + b^{2} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right ) + 2 \, x^{2} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x - b \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) - 3}{6 \, b x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.413136, size = 32, normalized size = 0.26 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} b^{5} + a^{2}, \left ( t \mapsto t \log{\left (- \frac{3 t b^{2}}{a} + x \right )} \right )\right )} - \frac{1}{2 b x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2122, size = 155, normalized size = 1.25 \begin{align*} \frac{a \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{3 \, b^{2}} - \frac{\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 )}{3 \, 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 )}{6 \, b^{2}} - \frac{1}{2 \, b x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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