3.211 \(\int \frac{\sqrt{a x^6}}{x-x^5} \, dx\)

Optimal. Leaf size=37 \[ \frac{\sqrt{a x^6} \tanh ^{-1}(x)}{2 x^3}-\frac{\sqrt{a x^6} \tan ^{-1}(x)}{2 x^3} \]

[Out]

-(Sqrt[a*x^6]*ArcTan[x])/(2*x^3) + (Sqrt[a*x^6]*ArcTanh[x])/(2*x^3)

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Rubi [A]  time = 0.0247574, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{\sqrt{a x^6} \tanh ^{-1}(x)}{2 x^3}-\frac{\sqrt{a x^6} \tan ^{-1}(x)}{2 x^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*x^6]/(x - x^5),x]

[Out]

-(Sqrt[a*x^6]*ArcTan[x])/(2*x^3) + (Sqrt[a*x^6]*ArcTanh[x])/(2*x^3)

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Rubi in Sympy [A]  time = 14.9667, size = 32, normalized size = 0.86 \[ - \frac{\sqrt{a x^{6}} \operatorname{atan}{\left (x \right )}}{2 x^{3}} + \frac{\sqrt{a x^{6}} \operatorname{atanh}{\left (x \right )}}{2 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*x**6)**(1/2)/(-x**5+x),x)

[Out]

-sqrt(a*x**6)*atan(x)/(2*x**3) + sqrt(a*x**6)*atanh(x)/(2*x**3)

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Mathematica [A]  time = 0.00652605, size = 33, normalized size = 0.89 \[ -\frac{\sqrt{a x^6} \left (\log (1-x)-\log (x+1)+2 \tan ^{-1}(x)\right )}{4 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a*x^6]/(x - x^5),x]

[Out]

-(Sqrt[a*x^6]*(2*ArcTan[x] + Log[1 - x] - Log[1 + x]))/(4*x^3)

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Maple [A]  time = 0.01, size = 28, normalized size = 0.8 \[ -{\frac{\ln \left ( -1+x \right ) -\ln \left ( 1+x \right ) +2\,\arctan \left ( x \right ) }{4\,{x}^{3}}\sqrt{a{x}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x^6)^(1/2)/(-x^5+x),x)

[Out]

-1/4*(a*x^6)^(1/2)*(ln(-1+x)-ln(1+x)+2*arctan(x))/x^3

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Maxima [A]  time = 0.769744, size = 35, normalized size = 0.95 \[ -\frac{1}{2} \, \sqrt{a} \arctan \left (x\right ) + \frac{1}{4} \, \sqrt{a} \log \left (x + 1\right ) - \frac{1}{4} \, \sqrt{a} \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(a*x^6)/(x^5 - x),x, algorithm="maxima")

[Out]

-1/2*sqrt(a)*arctan(x) + 1/4*sqrt(a)*log(x + 1) - 1/4*sqrt(a)*log(x - 1)

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Fricas [A]  time = 0.298899, size = 39, normalized size = 1.05 \[ -\frac{\sqrt{a x^{6}}{\left (2 \, \arctan \left (x\right ) - \log \left (\frac{x + 1}{x - 1}\right )\right )}}{4 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(a*x^6)/(x^5 - x),x, algorithm="fricas")

[Out]

-1/4*sqrt(a*x^6)*(2*arctan(x) - log((x + 1)/(x - 1)))/x^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{\sqrt{a x^{6}}}{x^{5} - x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x**6)**(1/2)/(-x**5+x),x)

[Out]

-Integral(sqrt(a*x**6)/(x**5 - x), x)

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GIAC/XCAS [A]  time = 0.259908, size = 39, normalized size = 1.05 \[ -\frac{1}{4} \,{\left (2 \, \arctan \left (x\right ){\rm sign}\left (x\right ) -{\rm ln}\left ({\left | x + 1 \right |}\right ){\rm sign}\left (x\right ) +{\rm ln}\left ({\left | x - 1 \right |}\right ){\rm sign}\left (x\right )\right )} \sqrt{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(a*x^6)/(x^5 - x),x, algorithm="giac")

[Out]

-1/4*(2*arctan(x)*sign(x) - ln(abs(x + 1))*sign(x) + ln(abs(x - 1))*sign(x))*sqr
t(a)