3.10 \(\int \sqrt{6 x-x^2} \, dx\)

Optimal. Leaf size=35 \[ -\frac{1}{2} \sqrt{6 x-x^2} (3-x)-\frac{9}{2} \sin ^{-1}\left (1-\frac{x}{3}\right ) \]

[Out]

-((3 - x)*Sqrt[6*x - x^2])/2 - (9*ArcSin[1 - x/3])/2

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Rubi [A]  time = 0.0244905, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{1}{2} \sqrt{6 x-x^2} (3-x)-\frac{9}{2} \sin ^{-1}\left (1-\frac{x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[6*x - x^2],x]

[Out]

-((3 - x)*Sqrt[6*x - x^2])/2 - (9*ArcSin[1 - x/3])/2

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Rubi in Sympy [A]  time = 1.68944, size = 26, normalized size = 0.74 \[ - \frac{\left (- 2 x + 6\right ) \sqrt{- x^{2} + 6 x}}{4} + \frac{9 \operatorname{asin}{\left (\frac{x}{3} - 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**2+6*x)**(1/2),x)

[Out]

-(-2*x + 6)*sqrt(-x**2 + 6*x)/4 + 9*asin(x/3 - 1)/2

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Mathematica [A]  time = 0.0427161, size = 45, normalized size = 1.29 \[ \frac{1}{2} \sqrt{-(x-6) x} \left (x-\frac{18 \log \left (\sqrt{x-6}+\sqrt{x}\right )}{\sqrt{x-6} \sqrt{x}}-3\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[6*x - x^2],x]

[Out]

(Sqrt[-((-6 + x)*x)]*(-3 + x - (18*Log[Sqrt[-6 + x] + Sqrt[x]])/(Sqrt[-6 + x]*Sq
rt[x])))/2

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Maple [A]  time = 0.005, size = 28, normalized size = 0.8 \[ -{\frac{-2\,x+6}{4}\sqrt{-{x}^{2}+6\,x}}+{\frac{9}{2}\arcsin \left ( -1+{\frac{x}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+6*x)^(1/2),x)

[Out]

-1/4*(-2*x+6)*(-x^2+6*x)^(1/2)+9/2*arcsin(-1+1/3*x)

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Maxima [A]  time = 0.798836, size = 49, normalized size = 1.4 \[ \frac{1}{2} \, \sqrt{-x^{2} + 6 \, x} x - \frac{3}{2} \, \sqrt{-x^{2} + 6 \, x} - \frac{9}{2} \, \arcsin \left (-\frac{1}{3} \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + 6*x),x, algorithm="maxima")

[Out]

1/2*sqrt(-x^2 + 6*x)*x - 3/2*sqrt(-x^2 + 6*x) - 9/2*arcsin(-1/3*x + 1)

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Fricas [A]  time = 0.213235, size = 47, normalized size = 1.34 \[ \frac{1}{2} \, \sqrt{-x^{2} + 6 \, x}{\left (x - 3\right )} - 9 \, \arctan \left (\frac{\sqrt{-x^{2} + 6 \, x}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + 6*x),x, algorithm="fricas")

[Out]

1/2*sqrt(-x^2 + 6*x)*(x - 3) - 9*arctan(sqrt(-x^2 + 6*x)/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- x^{2} + 6 x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**2+6*x)**(1/2),x)

[Out]

Integral(sqrt(-x**2 + 6*x), x)

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GIAC/XCAS [A]  time = 0.208856, size = 34, normalized size = 0.97 \[ \frac{1}{2} \, \sqrt{-x^{2} + 6 \, x}{\left (x - 3\right )} + \frac{9}{2} \, \arcsin \left (\frac{1}{3} \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + 6*x),x, algorithm="giac")

[Out]

1/2*sqrt(-x^2 + 6*x)*(x - 3) + 9/2*arcsin(1/3*x - 1)