3.9.69 \(\int \frac {\sqrt {1-x}}{\sqrt {x} \sqrt {1+x}} \, dx\) [869]

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

[Out]

-2*EllipticE((-x)^(1/2),I)*(-x)^(1/2)/x^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 = {112, 111} \begin {gather*} -\frac {2 \sqrt {-x} E\left (\left .\text {ArcSin}\left (\sqrt {-x}\right )\right |-1\right )}{\sqrt {x}} \end {gather*}

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] :> Simp[2*(Sqrt[e]/b)*Rt[-b/
d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*e))], 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 112

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]

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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.50, size = 40, normalized size = 1.67 \begin {gather*} -\frac {2}{3} \sqrt {x} \left (-3 \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};x^2\right )+x \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[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 = 0.08, size = 25, normalized size = 1.04

method	result	size
default	\(\frac {2 \sqrt {2}\, \sqrt {-x}\, \EllipticE \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x}}\)	\(25\)
elliptic	\(\frac {\sqrt {-x \left (x^{2}-1\right )}\, \left (\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticF \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-x^{3}+x}}-\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \left (-2 \EllipticE \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )+\EllipticF \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {-x^{3}+x}}\right )}{\sqrt {1-x}\, \sqrt {x}\, \sqrt {1+x}}\)	\(119\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-x)^(1/2)/x^(1/2)/(1+x)^(1/2),x,method=_RETURNVERBOSE)

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

0

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1 - x}}{\sqrt {x} \sqrt {x + 1}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\sqrt {1-x}}{\sqrt {x}\,\sqrt {x+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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