3.433 \(\int \frac{\sqrt{1-x}}{\sqrt{-x-x^2}} \, dx\)

Optimal. Leaf size=12 \[ -2 E\left (\left .\sin ^{-1}\left (\sqrt{-x}\right )\right |-1\right ) \]

[Out]

-2*EllipticE[ArcSin[Sqrt[-x]], -1]

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Rubi [A]  time = 0.0511323, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ -2 E\left (\left .\sin ^{-1}\left (\sqrt{-x}\right )\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - x]/Sqrt[-x - x^2],x]

[Out]

-2*EllipticE[ArcSin[Sqrt[-x]], -1]

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Rubi in Sympy [A]  time = 14.2598, size = 14, normalized size = 1.17 \[ - 2 E\left (\operatorname{asin}{\left (\sqrt{- x} \right )}\middle | -1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*elliptic_e(asin(sqrt(-x)), -1)

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Mathematica [B]  time = 0.0154677, size = 60, normalized size = 5. \[ -\frac{2 \sqrt{2} \sqrt{x} \sqrt{x+1} \left (F\left (\sin ^{-1}\left (\sqrt{1-x}\right )|\frac{1}{2}\right )-E\left (\sin ^{-1}\left (\sqrt{1-x}\right )|\frac{1}{2}\right )\right )}{\sqrt{-x (x+1)}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - x]/Sqrt[-x - x^2],x]

[Out]

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

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Maple [B]  time = 0.013, size = 38, normalized size = 3.2 \[ -2\,{\frac{{\it EllipticE} \left ( \sqrt{1+x},1/2\,\sqrt{2} \right ) \sqrt{-x}\sqrt{-x \left ( 1+x \right ) }\sqrt{2}}{\sqrt{1+x}x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-x + 1}}{\sqrt{-x^{2} - x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x + 1)/sqrt(-x^2 - x),x, algorithm="maxima")

[Out]

integrate(sqrt(-x + 1)/sqrt(-x^2 - x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x + 1)/sqrt(-x^2 - x),x, algorithm="fricas")

[Out]

integral(sqrt(-x + 1)/sqrt(-x^2 - x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- x + 1}}{\sqrt{- x \left (x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-x + 1}}{\sqrt{-x^{2} - x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x + 1)/sqrt(-x^2 - x),x, algorithm="giac")

[Out]

integrate(sqrt(-x + 1)/sqrt(-x^2 - x), x)