3.2416 \(\int x \sqrt{3-2 x-x^2} \, dx\)

Optimal. Leaf size=52 \[ -\frac{1}{3} \left (-x^2-2 x+3\right )^{3/2}-\frac{1}{2} (x+1) \sqrt{-x^2-2 x+3}+2 \sin ^{-1}\left (\frac{1}{2} (-x-1)\right ) \]

[Out]

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

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Rubi [A]  time = 0.0458968, antiderivative size = 52, 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 \[ -\frac{1}{3} \left (-x^2-2 x+3\right )^{3/2}-\frac{1}{2} (x+1) \sqrt{-x^2-2 x+3}+2 \sin ^{-1}\left (\frac{1}{2} (-x-1)\right ) \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[3 - 2*x - x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 4.85216, size = 58, normalized size = 1.12 \[ - \frac{\left (2 x + 2\right ) \sqrt{- x^{2} - 2 x + 3}}{4} - \frac{\left (- x^{2} - 2 x + 3\right )^{\frac{3}{2}}}{3} - 2 \operatorname{atan}{\left (- \frac{- 2 x - 2}{2 \sqrt{- x^{2} - 2 x + 3}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(-x**2-2*x+3)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0327067, size = 37, normalized size = 0.71 \[ \frac{1}{6} \sqrt{-x^2-2 x+3} \left (2 x^2+x-9\right )-2 \sin ^{-1}\left (\frac{x+1}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[3 - 2*x - x^2],x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(-x^2-2*x+3)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.75913, size = 70, normalized size = 1.35 \[ -\frac{1}{3} \,{\left (-x^{2} - 2 \, x + 3\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{-x^{2} - 2 \, x + 3} x - \frac{1}{2} \, \sqrt{-x^{2} - 2 \, x + 3} + 2 \, \arcsin \left (-\frac{1}{2} \, x - \frac{1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 - 2*x + 3)*x,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.224655, size = 57, normalized size = 1.1 \[ \frac{1}{6} \,{\left (2 \, x^{2} + x - 9\right )} \sqrt{-x^{2} - 2 \, x + 3} - 2 \, \arctan \left (\frac{x + 1}{\sqrt{-x^{2} - 2 \, x + 3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 - 2*x + 3)*x,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x \sqrt{- \left (x - 1\right ) \left (x + 3\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(-x**2-2*x+3)**(1/2),x)

[Out]

Integral(x*sqrt(-(x - 1)*(x + 3)), x)

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GIAC/XCAS [A]  time = 0.208364, size = 43, normalized size = 0.83 \[ \frac{1}{6} \,{\left ({\left (2 \, x + 1\right )} x - 9\right )} \sqrt{-x^{2} - 2 \, x + 3} - 2 \, \arcsin \left (\frac{1}{2} \, x + \frac{1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 - 2*x + 3)*x,x, algorithm="giac")

[Out]

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