3.2417 \(\int x \sqrt{8+2 x-x^2} \, dx\)

Optimal. Leaf size=56 \[ -\frac{1}{3} \left (-x^2+2 x+8\right )^{3/2}-\frac{1}{2} (1-x) \sqrt{-x^2+2 x+8}-\frac{9}{2} \sin ^{-1}\left (\frac{1-x}{3}\right ) \]

[Out]

-((1 - x)*Sqrt[8 + 2*x - x^2])/2 - (8 + 2*x - x^2)^(3/2)/3 - (9*ArcSin[(1 - x)/3
])/2

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Rubi [A]  time = 0.0468366, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{1}{3} \left (-x^2+2 x+8\right )^{3/2}-\frac{1}{2} (1-x) \sqrt{-x^2+2 x+8}-\frac{9}{2} \sin ^{-1}\left (\frac{1-x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[8 + 2*x - x^2],x]

[Out]

-((1 - x)*Sqrt[8 + 2*x - x^2])/2 - (8 + 2*x - x^2)^(3/2)/3 - (9*ArcSin[(1 - x)/3
])/2

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Rubi in Sympy [A]  time = 4.66661, size = 56, normalized size = 1. \[ - \frac{\left (- 2 x + 2\right ) \sqrt{- x^{2} + 2 x + 8}}{4} - \frac{\left (- x^{2} + 2 x + 8\right )^{\frac{3}{2}}}{3} - \frac{9 \operatorname{atan}{\left (\frac{- 2 x + 2}{2 \sqrt{- x^{2} + 2 x + 8}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(-x**2+2*x+8)**(1/2),x)

[Out]

-(-2*x + 2)*sqrt(-x**2 + 2*x + 8)/4 - (-x**2 + 2*x + 8)**(3/2)/3 - 9*atan((-2*x
+ 2)/(2*sqrt(-x**2 + 2*x + 8)))/2

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Mathematica [A]  time = 0.0315145, size = 43, normalized size = 0.77 \[ \frac{1}{6} \sqrt{-x^2+2 x+8} \left (2 x^2-x-19\right )-\frac{9}{2} \sin ^{-1}\left (\frac{1-x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[8 + 2*x - x^2],x]

[Out]

(Sqrt[8 + 2*x - x^2]*(-19 - x + 2*x^2))/6 - (9*ArcSin[(1 - x)/3])/2

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Maple [A]  time = 0.007, size = 43, normalized size = 0.8 \[ -{\frac{1}{3} \left ( -{x}^{2}+2\,x+8 \right ) ^{{\frac{3}{2}}}}-{\frac{2-2\,x}{4}\sqrt{-{x}^{2}+2\,x+8}}+{\frac{9}{2}\arcsin \left ( -{\frac{1}{3}}+{\frac{x}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(-x^2+2*x+8)^(1/2),x)

[Out]

-1/3*(-x^2+2*x+8)^(3/2)-1/4*(2-2*x)*(-x^2+2*x+8)^(1/2)+9/2*arcsin(-1/3+1/3*x)

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Maxima [A]  time = 0.746649, size = 70, normalized size = 1.25 \[ -\frac{1}{3} \,{\left (-x^{2} + 2 \, x + 8\right )}^{\frac{3}{2}} + \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x + 8} x - \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x + 8} - \frac{9}{2} \, \arcsin \left (-\frac{1}{3} \, x + \frac{1}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + 2*x + 8)*x,x, algorithm="maxima")

[Out]

-1/3*(-x^2 + 2*x + 8)^(3/2) + 1/2*sqrt(-x^2 + 2*x + 8)*x - 1/2*sqrt(-x^2 + 2*x +
8) - 9/2*arcsin(-1/3*x + 1/3)

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Fricas [A]  time = 0.224976, size = 59, normalized size = 1.05 \[ \frac{1}{6} \,{\left (2 \, x^{2} - x - 19\right )} \sqrt{-x^{2} + 2 \, x + 8} + \frac{9}{2} \, \arctan \left (\frac{x - 1}{\sqrt{-x^{2} + 2 \, x + 8}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + 2*x + 8)*x,x, algorithm="fricas")

[Out]

1/6*(2*x^2 - x - 19)*sqrt(-x^2 + 2*x + 8) + 9/2*arctan((x - 1)/sqrt(-x^2 + 2*x +
8))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x \sqrt{- \left (x - 4\right ) \left (x + 2\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(-x**2+2*x+8)**(1/2),x)

[Out]

Integral(x*sqrt(-(x - 4)*(x + 2)), x)

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GIAC/XCAS [A]  time = 0.208042, size = 43, normalized size = 0.77 \[ \frac{1}{6} \,{\left ({\left (2 \, x - 1\right )} x - 19\right )} \sqrt{-x^{2} + 2 \, x + 8} + \frac{9}{2} \, \arcsin \left (\frac{1}{3} \, x - \frac{1}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + 2*x + 8)*x,x, algorithm="giac")

[Out]

1/6*((2*x - 1)*x - 19)*sqrt(-x^2 + 2*x + 8) + 9/2*arcsin(1/3*x - 1/3)