Optimal. Leaf size=136 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{4/3} b^{2/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{4/3} b^{2/3}}+\frac{x^2}{3 a \left (a+b x^3\right )} \]
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Rubi [A] time = 0.138034, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{4/3} b^{2/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{4/3} b^{2/3}}+\frac{x^2}{3 a \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In] Int[x/(a + b*x^3)^2,x]
[Out]
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Rubi in Sympy [A] time = 29.9145, size = 122, normalized size = 0.9 \[ \frac{x^{2}}{3 a \left (a + b x^{3}\right )} - \frac{\log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{4}{3}} b^{\frac{2}{3}}} + \frac{\log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{4}{3}} b^{\frac{2}{3}}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{4}{3}} b^{\frac{2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(b*x**3+a)**2,x)
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Mathematica [A] time = 0.12746, size = 119, normalized size = 0.88 \[ \frac{\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{2/3}}+\frac{6 \sqrt [3]{a} x^2}{a+b x^3}}{18 a^{4/3}} \]
Antiderivative was successfully verified.
[In] Integrate[x/(a + b*x^3)^2,x]
[Out]
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Maple [A] time = 0.006, size = 117, normalized size = 0.9 \[{\frac{{x}^{2}}{3\,a \left ( b{x}^{3}+a \right ) }}-{\frac{1}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{1}{18\,ab}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{\sqrt{3}}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(b*x^3+a)^2,x)
[Out]
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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)^2,x, algorithm="maxima")
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Fricas [A] time = 0.235544, size = 201, normalized size = 1.48 \[ \frac{\sqrt{3}{\left (6 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} x^{2} - \sqrt{3}{\left (b x^{3} + a\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 2 \, \sqrt{3}{\left (b x^{3} + a\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 6 \,{\left (b x^{3} + a\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right )\right )}}{54 \,{\left (a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x^3 + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.55576, size = 44, normalized size = 0.32 \[ \frac{x^{2}}{3 a^{2} + 3 a b x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a^{4} b^{2} + 1, \left ( t \mapsto t \log{\left (81 t^{2} a^{3} b + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x**3+a)**2,x)
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GIAC/XCAS [A] time = 0.219194, size = 174, normalized size = 1.28 \[ \frac{x^{2}}{3 \,{\left (b x^{3} + a\right )} a} - \frac{\left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{2}} - \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{2} b^{2}} + \frac{\left (-a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x^3 + a)^2,x, algorithm="giac")
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