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

Optimal. Leaf size=27 \[ \sin ^{-1}\left (\frac{1}{2} (-x-1)\right )-\sqrt{-x^2-2 x+3} \]

[Out]

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

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Rubi [A]  time = 0.0264479, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \sin ^{-1}\left (\frac{1}{2} (-x-1)\right )-\sqrt{-x^2-2 x+3} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[3 - 2*x - x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 1.91482, size = 36, normalized size = 1.33 \[ - \sqrt{- x^{2} - 2 x + 3} - \operatorname{atan}{\left (- \frac{- 2 x - 2}{2 \sqrt{- x^{2} - 2 x + 3}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0166762, size = 27, normalized size = 1. \[ \sin ^{-1}\left (\frac{1}{2} (-x-1)\right )-\sqrt{-x^2-2 x+3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 24, normalized size = 0.9 \[ -\arcsin \left ({\frac{1}{2}}+{\frac{x}{2}} \right ) -\sqrt{-{x}^{2}-2\,x+3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-x^2-2*x+3)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.54533, size = 28, normalized size = 1.04 \[ -\sqrt{-x^{2} - 2 \, x + 3} + \arcsin \left (-\frac{1}{2} \, x - \frac{1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.216014, size = 46, normalized size = 1.7 \[ -\sqrt{-x^{2} - 2 \, x + 3} - \arctan \left (\frac{x + 1}{\sqrt{-x^{2} - 2 \, x + 3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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 = 0.206193, size = 31, normalized size = 1.15 \[ -\sqrt{-x^{2} - 2 \, x + 3} - \arcsin \left (\frac{1}{2} \, x + \frac{1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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