3.39 \(\int x \sqrt{2 x-x^2} \, dx\)

Optimal. Leaf size=50 \[ -\frac{1}{3} \left (2 x-x^2\right )^{3/2}-\frac{1}{2} (1-x) \sqrt{2 x-x^2}-\frac{1}{2} \sin ^{-1}(1-x) \]

[Out]

-((1 - x)*Sqrt[2*x - x^2])/2 - (2*x - x^2)^(3/2)/3 - ArcSin[1 - x]/2

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Rubi [A]  time = 0.0394325, antiderivative size = 50, 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 \[ -\frac{1}{3} \left (2 x-x^2\right )^{3/2}-\frac{1}{2} (1-x) \sqrt{2 x-x^2}-\frac{1}{2} \sin ^{-1}(1-x) \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[2*x - x^2],x]

[Out]

-((1 - x)*Sqrt[2*x - x^2])/2 - (2*x - x^2)^(3/2)/3 - ArcSin[1 - x]/2

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Rubi in Sympy [A]  time = 3.85398, size = 34, normalized size = 0.68 \[ - \frac{\left (- 2 x + 2\right ) \sqrt{- x^{2} + 2 x}}{4} - \frac{\left (- x^{2} + 2 x\right )^{\frac{3}{2}}}{3} + \frac{\operatorname{asin}{\left (x - 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 2)*sqrt(-x**2 + 2*x)/4 - (-x**2 + 2*x)**(3/2)/3 + asin(x - 1)/2

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Mathematica [A]  time = 0.0492451, size = 52, normalized size = 1.04 \[ \frac{1}{6} \sqrt{-(x-2) x} \left (2 x^2-x-\frac{6 \log \left (\sqrt{x-2}+\sqrt{x}\right )}{\sqrt{x-2} \sqrt{x}}-3\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[2*x - x^2],x]

[Out]

(Sqrt[-((-2 + x)*x)]*(-3 - x + 2*x^2 - (6*Log[Sqrt[-2 + x] + Sqrt[x]])/(Sqrt[-2
+ x]*Sqrt[x])))/6

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Maple [A]  time = 0.005, size = 39, normalized size = 0.8 \[ -{\frac{1}{3} \left ( -{x}^{2}+2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2-2\,x}{4}\sqrt{-{x}^{2}+2\,x}}+{\frac{\arcsin \left ( -1+x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(-x^2+2*x)^(1/2),x)

[Out]

-1/3*(-x^2+2*x)^(3/2)-1/4*(2-2*x)*(-x^2+2*x)^(1/2)+1/2*arcsin(-1+x)

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Maxima [A]  time = 0.810706, size = 66, normalized size = 1.32 \[ -\frac{1}{3} \,{\left (-x^{2} + 2 \, x\right )}^{\frac{3}{2}} + \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} x - \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} - \frac{1}{2} \, \arcsin \left (-x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(-x^2 + 2*x)^(3/2) + 1/2*sqrt(-x^2 + 2*x)*x - 1/2*sqrt(-x^2 + 2*x) - 1/2*ar
csin(-x + 1)

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Fricas [A]  time = 0.21873, size = 57, normalized size = 1.14 \[ \frac{1}{6} \,{\left (2 \, x^{2} - x - 3\right )} \sqrt{-x^{2} + 2 \, x} - \arctan \left (\frac{\sqrt{-x^{2} + 2 \, x}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*x^2 - x - 3)*sqrt(-x^2 + 2*x) - arctan(sqrt(-x^2 + 2*x)/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x \sqrt{- x \left (x - 2\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*sqrt(-x*(x - 2)), x)

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GIAC/XCAS [A]  time = 0.214063, size = 39, normalized size = 0.78 \[ \frac{1}{6} \,{\left ({\left (2 \, x - 1\right )} x - 3\right )} \sqrt{-x^{2} + 2 \, x} + \frac{1}{2} \, \arcsin \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*((2*x - 1)*x - 3)*sqrt(-x^2 + 2*x) + 1/2*arcsin(x - 1)