Optimal. Leaf size=24 \[ -\frac{2}{3} x \sqrt{\frac{a}{x^2}} \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]
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Rubi [A] time = 0.0255874, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{2}{3} x \sqrt{\frac{a}{x^2}} \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a/x^2]/Sqrt[1 + x^3],x]
[Out]
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Rubi in Sympy [A] time = 8.47861, size = 24, normalized size = 1. \[ - \frac{2 x \sqrt{\frac{a}{x^{2}}} \operatorname{atanh}{\left (\sqrt{x^{3} + 1} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a/x**2)**(1/2)/(x**3+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.018103, size = 24, normalized size = 1. \[ -\frac{2}{3} x \sqrt{\frac{a}{x^2}} \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a/x^2]/Sqrt[1 + x^3],x]
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Maple [A] time = 0.007, size = 19, normalized size = 0.8 \[ -{\frac{2\,x}{3}{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) \sqrt{{\frac{a}{{x}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a/x^2)^(1/2)/(x^3+1)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{a}{x^{2}}}}{\sqrt{x^{3} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a/x^2)/sqrt(x^3 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.285416, size = 1, normalized size = 0.04 \[ \left [\frac{1}{3} \, x \sqrt{\frac{a}{x^{2}}} \log \left (\frac{x^{3} - 2 \, \sqrt{x^{3} + 1} + 2}{x^{3}}\right ), \frac{2}{3} \, \sqrt{-a} \arctan \left (\frac{\sqrt{-a} x \sqrt{\frac{a}{x^{2}}}}{\sqrt{x^{3} + 1} a}\right )\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a/x^2)/sqrt(x^3 + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{a}{x^{2}}}}{\sqrt{\left (x + 1\right ) \left (x^{2} - x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a/x**2)**(1/2)/(x**3+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.263721, size = 42, normalized size = 1.75 \[ -\frac{1}{3} \, \sqrt{a}{\left ({\rm ln}\left (\sqrt{x^{3} + 1} + 1\right ) -{\rm ln}\left ({\left | \sqrt{x^{3} + 1} - 1 \right |}\right )\right )}{\rm sign}\left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a/x^2)/sqrt(x^3 + 1),x, algorithm="giac")
[Out]