3.353 \(\int \sqrt{5-4 x-x^2} \, dx\)

Optimal. Leaf size=36 \[ \frac{1}{2} (x+2) \sqrt{-x^2-4 x+5}-\frac{9}{2} \sin ^{-1}\left (\frac{1}{3} (-x-2)\right ) \]

[Out]

((2 + x)*Sqrt[5 - 4*x - x^2])/2 - (9*ArcSin[(-2 - x)/3])/2

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Rubi [A]  time = 0.0223694, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{1}{2} (x+2) \sqrt{-x^2-4 x+5}-\frac{9}{2} \sin ^{-1}\left (\frac{1}{3} (-x-2)\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[5 - 4*x - x^2],x]

[Out]

((2 + x)*Sqrt[5 - 4*x - x^2])/2 - (9*ArcSin[(-2 - x)/3])/2

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Rubi in Sympy [A]  time = 0.918535, size = 44, normalized size = 1.22 \[ \frac{\left (2 x + 4\right ) \sqrt{- x^{2} - 4 x + 5}}{4} + \frac{9 \operatorname{atan}{\left (- \frac{- 2 x - 4}{2 \sqrt{- x^{2} - 4 x + 5}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**2-4*x+5)**(1/2),x)

[Out]

(2*x + 4)*sqrt(-x**2 - 4*x + 5)/4 + 9*atan(-(-2*x - 4)/(2*sqrt(-x**2 - 4*x + 5))
)/2

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Mathematica [A]  time = 0.0275601, size = 33, normalized size = 0.92 \[ \frac{1}{2} \left (\sqrt{-x^2-4 x+5} (x+2)+9 \sin ^{-1}\left (\frac{x+2}{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[5 - 4*x - x^2],x]

[Out]

((2 + x)*Sqrt[5 - 4*x - x^2] + 9*ArcSin[(2 + x)/3])/2

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Maple [A]  time = 0.004, size = 29, normalized size = 0.8 \[ -{\frac{-2\,x-4}{4}\sqrt{-{x}^{2}-4\,x+5}}+{\frac{9}{2}\arcsin \left ({\frac{2}{3}}+{\frac{x}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2-4*x+5)^(1/2),x)

[Out]

-1/4*(-2*x-4)*(-x^2-4*x+5)^(1/2)+9/2*arcsin(2/3+1/3*x)

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Maxima [A]  time = 1.51817, size = 49, normalized size = 1.36 \[ \frac{1}{2} \, \sqrt{-x^{2} - 4 \, x + 5} x + \sqrt{-x^{2} - 4 \, x + 5} - \frac{9}{2} \, \arcsin \left (-\frac{1}{3} \, x - \frac{2}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 - 4*x + 5),x, algorithm="maxima")

[Out]

1/2*sqrt(-x^2 - 4*x + 5)*x + sqrt(-x^2 - 4*x + 5) - 9/2*arcsin(-1/3*x - 2/3)

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Fricas [A]  time = 0.204392, size = 50, normalized size = 1.39 \[ \frac{1}{2} \, \sqrt{-x^{2} - 4 \, x + 5}{\left (x + 2\right )} + \frac{9}{2} \, \arctan \left (\frac{x + 2}{\sqrt{-x^{2} - 4 \, x + 5}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 - 4*x + 5),x, algorithm="fricas")

[Out]

1/2*sqrt(-x^2 - 4*x + 5)*(x + 2) + 9/2*arctan((x + 2)/sqrt(-x^2 - 4*x + 5))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- x^{2} - 4 x + 5}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**2-4*x+5)**(1/2),x)

[Out]

Integral(sqrt(-x**2 - 4*x + 5), x)

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GIAC/XCAS [A]  time = 0.205528, size = 35, normalized size = 0.97 \[ \frac{1}{2} \, \sqrt{-x^{2} - 4 \, x + 5}{\left (x + 2\right )} + \frac{9}{2} \, \arcsin \left (\frac{1}{3} \, x + \frac{2}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 - 4*x + 5),x, algorithm="giac")

[Out]

1/2*sqrt(-x^2 - 4*x + 5)*(x + 2) + 9/2*arcsin(1/3*x + 2/3)