3.838 \(\int \sqrt{(4-x) x} \, dx\)

Optimal. Leaf size=33 \[ -\frac{1}{2} \sqrt{4 x-x^2} (2-x)-2 \sin ^{-1}\left (1-\frac{x}{2}\right ) \]

[Out]

-((2 - x)*Sqrt[4*x - x^2])/2 - 2*ArcSin[1 - x/2]

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Rubi [A]  time = 0.0270427, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{1}{2} \sqrt{4 x-x^2} (2-x)-2 \sin ^{-1}\left (1-\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[(4 - x)*x],x]

[Out]

-((2 - x)*Sqrt[4*x - x^2])/2 - 2*ArcSin[1 - x/2]

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Rubi in Sympy [A]  time = 0.953432, size = 24, normalized size = 0.73 \[ - \frac{\left (- 2 x + 4\right ) \sqrt{- x^{2} + 4 x}}{4} + 2 \operatorname{asin}{\left (\frac{x}{2} - 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((4-x)*x)**(1/2),x)

[Out]

-(-2*x + 4)*sqrt(-x**2 + 4*x)/4 + 2*asin(x/2 - 1)

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Mathematica [A]  time = 0.0427216, size = 45, normalized size = 1.36 \[ \frac{1}{2} \sqrt{-(x-4) x} \left (x-\frac{8 \log \left (\sqrt{x-4}+\sqrt{x}\right )}{\sqrt{x-4} \sqrt{x}}-2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[(4 - x)*x],x]

[Out]

(Sqrt[-((-4 + x)*x)]*(-2 + x - (8*Log[Sqrt[-4 + x] + Sqrt[x]])/(Sqrt[-4 + x]*Sqr
t[x])))/2

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Maple [A]  time = 0.01, size = 28, normalized size = 0.9 \[ -{\frac{-2\,x+4}{4}\sqrt{-{x}^{2}+4\,x}}+2\,\arcsin \left ( x/2-1 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((4-x)*x)^(1/2),x)

[Out]

-1/4*(-2*x+4)*(-x^2+4*x)^(1/2)+2*arcsin(1/2*x-1)

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Maxima [A]  time = 0.752189, size = 49, normalized size = 1.48 \[ \frac{1}{2} \, \sqrt{-x^{2} + 4 \, x} x - \sqrt{-x^{2} + 4 \, x} - 2 \, \arcsin \left (-\frac{1}{2} \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-(x - 4)*x),x, algorithm="maxima")

[Out]

1/2*sqrt(-x^2 + 4*x)*x - sqrt(-x^2 + 4*x) - 2*arcsin(-1/2*x + 1)

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Fricas [A]  time = 0.265156, size = 47, normalized size = 1.42 \[ \frac{1}{2} \, \sqrt{-x^{2} + 4 \, x}{\left (x - 2\right )} - 4 \, \arctan \left (\frac{\sqrt{-x^{2} + 4 \, x}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-(x - 4)*x),x, algorithm="fricas")

[Out]

1/2*sqrt(-x^2 + 4*x)*(x - 2) - 4*arctan(sqrt(-x^2 + 4*x)/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x \left (- x + 4\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((4-x)*x)**(1/2),x)

[Out]

Integral(sqrt(x*(-x + 4)), x)

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GIAC/XCAS [A]  time = 0.260836, size = 34, normalized size = 1.03 \[ \frac{1}{2} \, \sqrt{-x^{2} + 4 \, x}{\left (x - 2\right )} + 2 \, \arcsin \left (\frac{1}{2} \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-(x - 4)*x),x, algorithm="giac")

[Out]

1/2*sqrt(-x^2 + 4*x)*(x - 2) + 2*arcsin(1/2*x - 1)