Optimal. Leaf size=122 \[ -\frac{\sqrt [3]{a} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 b^{4/3}}+\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 b^{4/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} b^{4/3}}-\frac{1}{b x} \]
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Rubi [A] time = 0.0646309, antiderivative size = 122, 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, 292, 31, 634, 617, 204, 628} \[ -\frac{\sqrt [3]{a} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 b^{4/3}}+\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 b^{4/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} b^{4/3}}-\frac{1}{b x} \]
Antiderivative was successfully verified.
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Rule 263
Rule 325
Rule 292
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^5} \, dx &=\int \frac{1}{x^2 \left (b+a x^3\right )} \, dx\\ &=-\frac{1}{b x}-\frac{a \int \frac{x}{b+a x^3} \, dx}{b}\\ &=-\frac{1}{b x}+\frac{a^{2/3} \int \frac{1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{3 b^{4/3}}-\frac{a^{2/3} \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}{3 b^{4/3}}\\ &=-\frac{1}{b x}+\frac{\sqrt [3]{a} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 b^{4/3}}-\frac{\sqrt [3]{a} \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^{4/3}}-\frac{a^{2/3} \int \frac{1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 b}\\ &=-\frac{1}{b x}+\frac{\sqrt [3]{a} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 b^{4/3}}-\frac{\sqrt [3]{a} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 b^{4/3}}-\frac{\sqrt [3]{a} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{b^{4/3}}\\ &=-\frac{1}{b x}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} b^{4/3}}+\frac{\sqrt [3]{a} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 b^{4/3}}-\frac{\sqrt [3]{a} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 b^{4/3}}\\ \end{align*}
Mathematica [A] time = 0.0185458, size = 114, normalized size = 0.93 \[ \frac{-\sqrt [3]{a} x \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )+2 \sqrt [3]{a} x \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )+2 \sqrt{3} \sqrt [3]{a} x \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )-6 \sqrt [3]{b}}{6 b^{4/3} x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 99, normalized size = 0.8 \begin{align*}{\frac{1}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-{\frac{1}{6\,b}\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{\sqrt{3}}{3\,b}\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}}}}}}-{\frac{1}{bx}} \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 = 1.4481, size = 262, normalized size = 2.15 \begin{align*} -\frac{2 \, \sqrt{3} x \left (\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2}{3} \, \sqrt{3} x \left (\frac{a}{b}\right )^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right ) + x \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (a x^{2} - b x \left (\frac{a}{b}\right )^{\frac{2}{3}} + b \left (\frac{a}{b}\right )^{\frac{1}{3}}\right ) - 2 \, x \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (a x + b \left (\frac{a}{b}\right )^{\frac{2}{3}}\right ) + 6}{6 \, b x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.451219, size = 29, normalized size = 0.24 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} b^{4} - a, \left ( t \mapsto t \log{\left (\frac{9 t^{2} b^{3}}{a} + x \right )} \right )\right )} - \frac{1}{b x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18712, size = 163, normalized size = 1.34 \begin{align*} \frac{a \left (-\frac{b}{a}\right )^{\frac{2}{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{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 )}{3 \, a b^{2}} - \frac{\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 )}{6 \, a b^{2}} - \frac{1}{b x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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