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