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

Optimal. Leaf size=46 \[ -\frac{1}{2} \sqrt{2 x-x^2} x-\frac{3}{2} \sqrt{2 x-x^2}-\frac{3}{2} \sin ^{-1}(1-x) \]

[Out]

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

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Rubi [A]  time = 0.0533722, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{1}{2} \sqrt{2 x-x^2} x-\frac{3}{2} \sqrt{2 x-x^2}-\frac{3}{2} \sin ^{-1}(1-x) \]

Antiderivative was successfully verified.

[In]  Int[x^2/Sqrt[2*x - x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 5.51362, size = 34, normalized size = 0.74 \[ - \frac{x \sqrt{- x^{2} + 2 x}}{2} - \frac{3 \sqrt{- x^{2} + 2 x}}{2} + \frac{3 \operatorname{asin}{\left (x - 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0310124, size = 51, normalized size = 1.11 \[ \frac{x \left (x^2+x-6\right )+6 \sqrt{x-2} \sqrt{x} \log \left (\sqrt{x-2}+\sqrt{x}\right )}{2 \sqrt{-(x-2) x}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/Sqrt[2*x - x^2],x]

[Out]

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

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Maple [A]  time = 0.006, size = 35, normalized size = 0.8 \[{\frac{3\,\arcsin \left ( -1+x \right ) }{2}}-{\frac{3}{2}\sqrt{-{x}^{2}+2\,x}}-{\frac{x}{2}\sqrt{-{x}^{2}+2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-x^2+2*x)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.784022, size = 49, normalized size = 1.07 \[ -\frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} x - \frac{3}{2} \, \sqrt{-x^{2} + 2 \, x} - \frac{3}{2} \, \arcsin \left (-x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.232911, size = 47, normalized size = 1.02 \[ -\frac{1}{2} \, \sqrt{-x^{2} + 2 \, x}{\left (x + 3\right )} - 3 \, \arctan \left (\frac{\sqrt{-x^{2} + 2 \, x}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2/sqrt(-x*(x - 2)), x)

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GIAC/XCAS [A]  time = 0.213876, size = 31, normalized size = 0.67 \[ -\frac{1}{2} \, \sqrt{-x^{2} + 2 \, x}{\left (x + 3\right )} + \frac{3}{2} \, \arcsin \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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