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

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

[Out]

2*EllipticF[ArcSin[Sqrt[x]], -1]

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Rubi [A]  time = 0.0037423, 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, Rules used = {115} \[ 2 F\left (\left .\sin ^{-1}\left (\sqrt{x}\right )\right |-1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*EllipticF[ArcSin[Sqrt[x]], -1]

Rule 115

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d), 2]*E
llipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/(b*Sqrt[e]), x] /; FreeQ[{b, c, d, e, f}, x]
&& GtQ[c, 0] && GtQ[e, 0] && (GtQ[-(b/d), 0] || LtQ[-(b/f), 0])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{1-x} \sqrt{x} \sqrt{1+x}} \, dx &=2 F\left (\left .\sin ^{-1}\left (\sqrt{x}\right )\right |-1\right )\\ \end{align*}

Mathematica [C]  time = 0.0139205, size = 44, normalized size = 4.4 \[ \frac{2 x \sqrt{1-x^2} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};x^2\right )}{\sqrt{-(x-1) x} \sqrt{x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*Sqrt[1 - x^2]*Hypergeometric2F1[1/4, 1/2, 5/4, x^2])/(Sqrt[-((-1 + x)*x)]*Sqrt[1 + x])

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Maple [B]  time = 0.021, size = 24, normalized size = 2.4 \begin{align*}{\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( \sqrt{1+x},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{x}}}} \end{align*}

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. \begin{align*} \int \frac{1}{\sqrt{x + 1} \sqrt{x} \sqrt{-x + 1}}\,{d x} \end{align*}

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, 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. \begin{align*}{\rm integral}\left (-\frac{\sqrt{x + 1} \sqrt{x} \sqrt{-x + 1}}{x^{3} - x}, x\right ) \end{align*}

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, algorithm="fricas")

[Out]

integral(-sqrt(x + 1)*sqrt(x)*sqrt(-x + 1)/(x^3 - x), x)

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Sympy [B]  time = 6.253, size = 66, normalized size = 6.6 \begin{align*} \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}}} \end{align*}

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

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, algorithm="giac")

[Out]

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