3.215 \(\int \frac{\sqrt{a x^3}}{x-x^3} \, dx\)

Optimal. Leaf size=44 \[ \frac{\sqrt{a x^3} \tanh ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}-\frac{\sqrt{a x^3} \tan ^{-1}\left (\sqrt{x}\right )}{x^{3/2}} \]

[Out]

-((Sqrt[a*x^3]*ArcTan[Sqrt[x]])/x^(3/2)) + (Sqrt[a*x^3]*ArcTanh[Sqrt[x]])/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]

-((Sqrt[a*x^3]*ArcTan[Sqrt[x]])/x^(3/2)) + (Sqrt[a*x^3]*ArcTanh[Sqrt[x]])/x^(3/2
)

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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]

-sqrt(a*x**3)*atan(sqrt(x))/x**(3/2) + sqrt(a*x**3)*atanh(sqrt(x))/x**(3/2)

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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]

-(Sqrt[a*x^3]*(2*ArcTan[Sqrt[x]] + Log[1 - Sqrt[x]] - Log[1 + Sqrt[x]]))/(2*x^(3
/2))

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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)

[Out]

(a*x^3)^(1/2)*a^(1/2)*(arctanh((a*x)^(1/2)/a^(1/2))-arctan((a*x)^(1/2)/a^(1/2)))
/x/(a*x)^(1/2)

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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]

-sqrt(a)*arctan(sqrt(x)) + 1/2*sqrt(a)*log(sqrt(x) + 1) - 1/2*sqrt(a)*log(sqrt(x
) - 1)

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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]

[-sqrt(a)*arctan(sqrt(a*x^3)/(sqrt(a)*x)) + 1/2*sqrt(a)*log((a*x^2 + a*x + 2*sqr
t(a*x^3)*sqrt(a))/(x^2 - x)), sqrt(-a)*arctan(sqrt(a*x^3)/(sqrt(-a)*x)) + 1/2*sq
rt(-a)*log((a*x^2 - a*x - 2*sqrt(a*x^3)*sqrt(-a))/(x^2 + x))]

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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]

-Integral(sqrt(a*x**3)/(x**3 - x), x)

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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]

-(a*arctan(sqrt(a*x)/sqrt(-a))/sqrt(-a) + sqrt(a)*arctan(sqrt(a*x)/sqrt(a)))*sig
n(x)