Optimal. Leaf size=115 \[ \frac{\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{2/3}}-\frac{\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} b^{2/3}} \]
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Rubi [A] time = 0.110498, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{2/3}}-\frac{\log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} b^{2/3}} \]
Antiderivative was successfully verified.
[In] Int[x/(a - b*x^3),x]
[Out]
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Rubi in Sympy [A] time = 24.7164, size = 109, normalized size = 0.95 \[ - \frac{\log{\left (\sqrt [3]{a} - \sqrt [3]{b} x \right )}}{3 \sqrt [3]{a} b^{\frac{2}{3}}} + \frac{\log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 \sqrt [3]{a} b^{\frac{2}{3}}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 \sqrt [3]{a} b^{\frac{2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(-b*x**3+a),x)
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Mathematica [A] time = 0.0263807, size = 89, normalized size = 0.77 \[ \frac{\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )}{6 \sqrt [3]{a} b^{2/3}} \]
Antiderivative was successfully verified.
[In] Integrate[x/(a - b*x^3),x]
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Maple [A] time = 0.003, size = 92, normalized size = 0.8 \[ -{\frac{1}{3\,b}\ln \left ( x-\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{1}{6\,b}\ln \left ({x}^{2}+x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(-b*x^3+a),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-x/(b*x^3 - a),x, algorithm="maxima")
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Fricas [A] time = 0.222023, size = 136, normalized size = 1.18 \[ -\frac{\sqrt{3}{\left (\sqrt{3} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b - \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 2 \, \sqrt{3} \log \left (-a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 6 \, \arctan \left (\frac{\sqrt{3} a b + 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right )\right )}}{18 \, \left (-a b^{2}\right )^{\frac{1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-x/(b*x^3 - a),x, algorithm="fricas")
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Sympy [A] time = 0.336863, size = 26, normalized size = 0.23 \[ - \operatorname{RootSum}{\left (27 t^{3} a b^{2} - 1, \left ( t \mapsto t \log{\left (- 9 t^{2} a b + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(-b*x**3+a),x)
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GIAC/XCAS [A] time = 0.252477, size = 140, normalized size = 1.22 \[ -\frac{\left (\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a} - \frac{\sqrt{3} \left (a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{2}} + \frac{\left (a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (\frac{a}{b}\right )^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-x/(b*x^3 - a),x, algorithm="giac")
[Out]