Optimal. Leaf size=44 \[ \frac{\sqrt{a x^3} \tanh ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}-\frac{\sqrt{a x^3} \tan ^{-1}\left (\sqrt{x}\right )}{x^{3/2}} \]
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Rubi [A] time = 0.0327554, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{\sqrt{a x^3} \tanh ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}-\frac{\sqrt{a x^3} \tan ^{-1}\left (\sqrt{x}\right )}{x^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a*x^3]/(x - x^3),x]
[Out]
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Rubi in Sympy [A] time = 18.3921, size = 39, normalized size = 0.89 \[ - \frac{\sqrt{a x^{3}} \operatorname{atan}{\left (\sqrt{x} \right )}}{x^{\frac{3}{2}}} + \frac{\sqrt{a x^{3}} \operatorname{atanh}{\left (\sqrt{x} \right )}}{x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*x**3)**(1/2)/(-x**3+x),x)
[Out]
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Mathematica [A] time = 0.0327653, size = 47, normalized size = 1.07 \[ -\frac{\sqrt{a x^3} \left (\log \left (1-\sqrt{x}\right )-\log \left (\sqrt{x}+1\right )+2 \tan ^{-1}\left (\sqrt{x}\right )\right )}{2 x^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a*x^3]/(x - x^3),x]
[Out]
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Maple [A] time = 0.019, size = 43, normalized size = 1. \[{\frac{1}{x}\sqrt{a{x}^{3}}\sqrt{a} \left ({\it Artanh} \left ({1\sqrt{ax}{\frac{1}{\sqrt{a}}}} \right ) -\arctan \left ({1\sqrt{ax}{\frac{1}{\sqrt{a}}}} \right ) \right ){\frac{1}{\sqrt{ax}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*x^3)^(1/2)/(-x^3+x),x)
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Maxima [A] time = 0.79082, size = 43, normalized size = 0.98 \[ -\sqrt{a} \arctan \left (\sqrt{x}\right ) + \frac{1}{2} \, \sqrt{a} \log \left (\sqrt{x} + 1\right ) - \frac{1}{2} \, \sqrt{a} \log \left (\sqrt{x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(a*x^3)/(x^3 - x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.288693, size = 1, normalized size = 0.02 \[ \left [-\sqrt{a} \arctan \left (\frac{\sqrt{a x^{3}}}{\sqrt{a} x}\right ) + \frac{1}{2} \, \sqrt{a} \log \left (\frac{a x^{2} + a x + 2 \, \sqrt{a x^{3}} \sqrt{a}}{x^{2} - x}\right ), \sqrt{-a} \arctan \left (\frac{\sqrt{a x^{3}}}{\sqrt{-a} x}\right ) + \frac{1}{2} \, \sqrt{-a} \log \left (\frac{a x^{2} - a x - 2 \, \sqrt{a x^{3}} \sqrt{-a}}{x^{2} + x}\right )\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(a*x^3)/(x^3 - x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{\sqrt{a x^{3}}}{x^{3} - x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x**3)**(1/2)/(-x**3+x),x)
[Out]
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GIAC/XCAS [A] time = 0.265323, size = 51, normalized size = 1.16 \[ -{\left (\frac{a \arctan \left (\frac{\sqrt{a x}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \sqrt{a} \arctan \left (\frac{\sqrt{a x}}{\sqrt{a}}\right )\right )}{\rm sign}\left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(a*x^3)/(x^3 - x),x, algorithm="giac")
[Out]