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

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

[Out]

2*EllipticE[ArcSin[Sqrt[x]], -1]

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Rubi [A]  time = 0.0453259, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ 2 E\left (\left .\sin ^{-1}\left (\sqrt{x}\right )\right |-1\right ) \]

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 = 5.54435, size = 10, normalized size = 1. \[ 2 E\left (\operatorname{asin}{\left (\sqrt{x} \right )}\middle | -1\right ) \]

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 [C]  time = 0.125739, size = 104, normalized size = 10.4 \[ \frac{2 \sqrt{\frac{x-1}{x+1}} \sqrt{\frac{x+1}{x-1}} \left (\sqrt{x-1} \sqrt{\frac{x+1}{x-1}} x+\frac{i \sqrt{2} x E\left (i \sinh ^{-1}\left (\frac{\sqrt{2}}{\sqrt{x-1}}\right )|\frac{1}{2}\right )}{\sqrt{\frac{x}{x-1}}}\right )}{\sqrt{-(x-1) x}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*Sqrt[(-1 + x)/(1 + x)]*Sqrt[(1 + x)/(-1 + x)]*(Sqrt[-1 + x]*x*Sqrt[(1 + x)/(-
1 + x)] + (I*Sqrt[2]*x*EllipticE[I*ArcSinh[Sqrt[2]/Sqrt[-1 + x]], 1/2])/Sqrt[x/(
-1 + x)]))/Sqrt[-((-1 + x)*x)]

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Maple [B]  time = 0.015, size = 56, normalized size = 5.6 \[ -2\,{\frac{ \left ({\it EllipticF} \left ( \sqrt{1+x},1/2\,\sqrt{2} \right ) -{\it EllipticE} \left ( \sqrt{1+x},1/2\,\sqrt{2} \right ) \right ) \sqrt{-x}\sqrt{2-2\,x}\sqrt{-x \left ( -1+x \right ) }}{x \left ( -1+x \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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