3.849 \(\int \frac{1}{\sqrt{1-x} \sqrt{x} \sqrt{1+x}} \, dx\)

Optimal. Leaf size=10 \[ 2 F\left (\left .\sin ^{-1}\left (\sqrt{x}\right )\right |-1\right ) \]

[Out]

2*EllipticF[ArcSin[Sqrt[x]], -1]

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Rubi [A]  time = 0.0271326, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ 2 F\left (\left .\sin ^{-1}\left (\sqrt{x}\right )\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[1 - x]*Sqrt[x]*Sqrt[1 + x]),x]

[Out]

2*EllipticF[ArcSin[Sqrt[x]], -1]

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Rubi in Sympy [A]  time = 2.99924, size = 10, normalized size = 1. \[ 2 F\left (\operatorname{asin}{\left (\sqrt{x} \right )}\middle | -1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*elliptic_f(asin(sqrt(x)), -1)

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Mathematica [C]  time = 0.115731, size = 66, normalized size = 6.6 \[ \frac{2 i \sqrt{\frac{1}{x-1}+1} \sqrt{\frac{2}{x-1}+1} (x-1)^{3/2} F\left (\left .i \sinh ^{-1}\left (\frac{1}{\sqrt{x-1}}\right )\right |2\right )}{\sqrt{-(x-1) x} \sqrt{x+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[1 - x]*Sqrt[x]*Sqrt[1 + x]),x]

[Out]

((2*I)*Sqrt[1 + (-1 + x)^(-1)]*Sqrt[1 + 2/(-1 + x)]*(-1 + x)^(3/2)*EllipticF[I*A
rcSinh[1/Sqrt[-1 + x]], 2])/(Sqrt[-((-1 + x)*x)]*Sqrt[1 + x])

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Maple [B]  time = 0.045, size = 24, normalized size = 2.4 \[{\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( \sqrt{1+x},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-x)^(1/2)/x^(1/2)/(1+x)^(1/2),x)

[Out]

2^(1/2)*(-x)^(1/2)*EllipticF((1+x)^(1/2),1/2*2^(1/2))/x^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x + 1} \sqrt{x} \sqrt{-x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x + 1)*sqrt(x)*sqrt(-x + 1)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x + 1)*sqrt(x)*sqrt(-x + 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x + 1} \sqrt{x} \sqrt{-x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x + 1)*sqrt(x)*sqrt(-x + 1)),x, algorithm="fricas")

[Out]

integral(1/(sqrt(x + 1)*sqrt(x)*sqrt(-x + 1)), x)

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Sympy [A]  time = 18.8399, size = 66, normalized size = 6.6 \[ \frac{i{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{1}{2}, 1, 1 & \frac{3}{4}, \frac{3}{4}, \frac{5}{4} \\\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4} & 0 \end{matrix} \middle |{\frac{1}{x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{i{G_{6, 6}^{3, 5}\left (\begin{matrix} - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4} & 1 \\0, \frac{1}{2}, 0 & - \frac{1}{4}, \frac{1}{4}, \frac{1}{4} \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

I*meijerg(((1/2, 1, 1), (3/4, 3/4, 5/4)), ((1/4, 1/2, 3/4, 1, 5/4), (0,)), x**(-
2))/(4*pi**(3/2)) - I*meijerg(((-1/4, 0, 1/4, 1/2, 3/4), (1,)), ((0, 1/2, 0), (-
1/4, 1/4, 1/4)), exp_polar(-2*I*pi)/x**2)/(4*pi**(3/2))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x + 1} \sqrt{x} \sqrt{-x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x + 1)*sqrt(x)*sqrt(-x + 1)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x + 1)*sqrt(x)*sqrt(-x + 1)), x)