Optimal. Leaf size=202 \[ -\frac{\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]
[Out]
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Rubi [A] time = 0.19144, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{\left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]
Antiderivative was successfully verified.
[In] Int[x/Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{\left (a + b x^{3}\right )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/((b*x**3+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0478474, size = 109, normalized size = 0.54 \[ \frac{\left (a+b x^3\right ) \left (\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )\right )}{6 \sqrt [3]{a} b^{2/3} \sqrt{\left (a+b x^3\right )^2}} \]
Antiderivative was successfully verified.
[In] Integrate[x/Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]
[Out]
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Maple [A] time = 0.008, size = 97, normalized size = 0.5 \[ -{\frac{b{x}^{3}+a}{6\,b} \left ( 2\,\sqrt{3}\arctan \left ( 1/3\,{\sqrt{3} \left ( -2\,x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ) +2\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) -\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \right ){\frac{1}{\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/((b*x^3+a)^2)^(1/2),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/sqrt((b*x^3 + a)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.282005, size = 134, normalized size = 0.66 \[ -\frac{\sqrt{3}{\left (\sqrt{3} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 2 \, \sqrt{3} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 6 \, \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right )\right )}}{18 \, \left (-a b^{2}\right )^{\frac{1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt((b*x^3 + a)^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.431876, size = 24, normalized size = 0.12 \[ \operatorname{RootSum}{\left (27 t^{3} a b^{2} + 1, \left ( t \mapsto t \log{\left (9 t^{2} a b + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x**3+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.28277, size = 184, normalized size = 0.91 \[ -\frac{\left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right ){\rm sign}\left (b x^{3} + a\right )}{3 \, a} - \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right ){\rm sign}\left (b x^{3} + a\right )}{3 \, a b^{2}} + \frac{\left (-a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ){\rm sign}\left (b x^{3} + a\right )}{6 \, a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt((b*x^3 + a)^2),x, algorithm="giac")
[Out]