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

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

[Out]

2*EllipticF[ArcSin[Sqrt[x]], -1]

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Rubi [A]  time = 0.0434691, 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 \[ 2 F\left (\left .\sin ^{-1}\left (\sqrt{x}\right )\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[1 + x]*Sqrt[x - x^2]),x]

[Out]

2*EllipticF[ArcSin[Sqrt[x]], -1]

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Rubi in Sympy [A]  time = 5.93002, size = 10, normalized size = 1. \[ 2 F\left (\operatorname{asin}{\left (\sqrt{x} \right )}\middle | -1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*elliptic_f(asin(sqrt(x)), -1)

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Mathematica [C]  time = 0.0197596, size = 66, normalized size = 6.6 \[ \frac{2 i \sqrt{\frac{1}{x-1}+1} \sqrt{\frac{2}{x-1}+1} (x-1)^{3/2} F\left (\left .i \sinh ^{-1}\left (\frac{1}{\sqrt{x-1}}\right )\right |2\right )}{\sqrt{-(x-1) x} \sqrt{x+1}} \]

Antiderivative was successfully verified.

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

[Out]

((2*I)*Sqrt[1 + (-1 + x)^(-1)]*Sqrt[1 + 2/(-1 + x)]*(-1 + x)^(3/2)*EllipticF[I*A
rcSinh[1/Sqrt[-1 + x]], 2])/(Sqrt[-((-1 + x)*x)]*Sqrt[1 + x])

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Maple [B]  time = 0.02, size = 43, normalized size = 4.3 \[{\frac{1}{ \left ( 1-x \right ) x}{\it EllipticF} \left ( \sqrt{1+x},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-x}\sqrt{2-2\,x}\sqrt{-x \left ( -1+x \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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