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*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 = {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 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]

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]

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]  time = 0.02, 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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fricas [F]  time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x + 1} \sqrt {x} \sqrt {-x + 1}}{x^{2} + x}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-x + 1}}{\sqrt {x + 1} \sqrt {x}}\,{d x} \]

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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maple [A]  time = 0.01, size = 25, normalized size = 1.04 \[ \frac {2 \sqrt {2}\, \sqrt {-x}\, \EllipticE \left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*2^(1/2)*(-x)^(1/2)*EllipticE((x+1)^(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 \[ \int \frac {\sqrt {-x + 1}}{\sqrt {x + 1} \sqrt {x}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sqrt {1-x}}{\sqrt {x}\,\sqrt {x+1}} \,d x \]

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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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {1 - x}}{\sqrt {x} \sqrt {x + 1}}\, dx \]

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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