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

Optimal. Leaf size=24 \[ -\frac{2 \sqrt{-x} E\left (\left .\sin ^{-1}\left (\sqrt{-x}\right )\right |-1\right )}{\sqrt{x}} \]

[Out]

(-2*Sqrt[-x]*EllipticE[ArcSin[Sqrt[-x]], -1])/Sqrt[x]

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Rubi [A]  time = 0.0080433, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {111, 110} \[ -\frac{2 \sqrt{-x} E\left (\left .\sin ^{-1}\left (\sqrt{-x}\right )\right |-1\right )}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[-x]*EllipticE[ArcSin[Sqrt[-x]], -1])/Sqrt[x]

Rule 111

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[Sqrt[-(b*x)]/Sqrt[b*
x], Int[Sqrt[e + f*x]/(Sqrt[-(b*x)]*Sqrt[c + d*x]), x], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] &
& GtQ[c, 0] && GtQ[e, 0] && LtQ[-(b/d), 0]

Rule 110

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

Rubi steps

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

Mathematica [C]  time = 0.0196932, size = 40, normalized size = 1.67 \[ -\frac{2}{3} \sqrt{x} \left (x \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};x^2\right )-3 \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};x^2\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*Sqrt[x]*(-3*Hypergeometric2F1[1/4, 1/2, 5/4, x^2] + x*Hypergeometric2F1[1/2, 3/4, 7/4, x^2]))/3

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Maple [A]  time = 0.009, size = 25, normalized size = 1. \begin{align*} 2\,{\frac{\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( \sqrt{1+x},1/2\,\sqrt{2} \right ) }{\sqrt{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-x)^(1/2)/x^(1/2)/(1+x)^(1/2),x)

[Out]

2*2^(1/2)*(-x)^(1/2)*EllipticE((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{\sqrt{-x + 1}}{\sqrt{x + 1} \sqrt{x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(1/2)/x^(1/2)/(1+x)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-x + 1)/(sqrt(x + 1)*sqrt(x)), 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^{2} + x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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^2 + x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)**(1/2)/x**(1/2)/(1+x)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^(1/2)/x^(1/2)/(1+x)^(1/2),x, algorithm="giac")

[Out]

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