3.647 \(\int \frac{\sqrt{x}}{\sqrt{1-x}} \, dx\)

Optimal. Leaf size=27 \[ -\sqrt{1-x} \sqrt{x}-\frac{1}{2} \sin ^{-1}(1-2 x) \]

[Out]

-(Sqrt[1 - x]*Sqrt[x]) - ArcSin[1 - 2*x]/2

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Rubi [A]  time = 0.0200479, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\sqrt{1-x} \sqrt{x}-\frac{1}{2} \sin ^{-1}(1-2 x) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/Sqrt[1 - x],x]

[Out]

-(Sqrt[1 - x]*Sqrt[x]) - ArcSin[1 - 2*x]/2

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Rubi in Sympy [A]  time = 3.1598, size = 19, normalized size = 0.7 \[ - \sqrt{x} \sqrt{- x + 1} + \frac{\operatorname{asin}{\left (2 x - 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(x)*sqrt(-x + 1) + asin(2*x - 1)/2

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Mathematica [A]  time = 0.0153409, size = 19, normalized size = 0.7 \[ \sin ^{-1}\left (\sqrt{x}\right )-\sqrt{-(x-1) x} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[-((-1 + x)*x)] + ArcSin[Sqrt[x]]

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Maple [A]  time = 0.026, size = 41, normalized size = 1.5 \[ -\sqrt{1-x}\sqrt{x}+{\frac{\arcsin \left ( -1+2\,x \right ) }{2}\sqrt{x \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(1-x)^(1/2),x)

[Out]

-(1-x)^(1/2)*x^(1/2)+1/2*(x*(1-x))^(1/2)/x^(1/2)/(1-x)^(1/2)*arcsin(-1+2*x)

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Maxima [A]  time = 1.49528, size = 50, normalized size = 1.85 \[ \frac{\sqrt{-x + 1}}{\sqrt{x}{\left (\frac{x - 1}{x} - 1\right )}} - \arctan \left (\frac{\sqrt{-x + 1}}{\sqrt{x}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-x + 1)/(sqrt(x)*((x - 1)/x - 1)) - arctan(sqrt(-x + 1)/sqrt(x))

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Fricas [A]  time = 0.2257, size = 36, normalized size = 1.33 \[ -\sqrt{x} \sqrt{-x + 1} - \arctan \left (\frac{\sqrt{-x + 1}}{\sqrt{x}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(x)*sqrt(-x + 1) - arctan(sqrt(-x + 1)/sqrt(x))

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Sympy [A]  time = 5.8027, size = 54, normalized size = 2. \[ \begin{cases} - i \sqrt{x} \sqrt{x - 1} - i \operatorname{acosh}{\left (\sqrt{x} \right )} & \text{for}\: \left |{x}\right | > 1 \\\frac{x^{\frac{3}{2}}}{\sqrt{- x + 1}} - \frac{\sqrt{x}}{\sqrt{- x + 1}} + \operatorname{asin}{\left (\sqrt{x} \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-I*sqrt(x)*sqrt(x - 1) - I*acosh(sqrt(x)), Abs(x) > 1), (x**(3/2)/sqr
t(-x + 1) - sqrt(x)/sqrt(-x + 1) + asin(sqrt(x)), True))

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GIAC/XCAS [A]  time = 0.208235, size = 23, normalized size = 0.85 \[ -\sqrt{x} \sqrt{-x + 1} + \arcsin \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(x)*sqrt(-x + 1) + arcsin(sqrt(x))