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 = 0.0104876, 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, Rules used = {640, 619, 216} \[ \sin ^{-1}\left (\frac{1}{2} (-x-1)\right )-\sqrt{-x^2-2 x+3} \]

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]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{x}{\sqrt{3-2 x-x^2}} \, dx &=-\sqrt{3-2 x-x^2}-\int \frac{1}{\sqrt{3-2 x-x^2}} \, dx\\ &=-\sqrt{3-2 x-x^2}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{16}}} \, dx,x,-2-2 x\right )\\ &=-\sqrt{3-2 x-x^2}+\sin ^{-1}\left (\frac{1}{2} (-1-x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0070841, size = 27, normalized size = 1. \[ \sin ^{-1}\left (\frac{1}{4} (-2 x-2)\right )-\sqrt{-x^2-2 x+3} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[3 - 2*x - x^2],x]

[Out]

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

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Maple [A]  time = 0.005, size = 24, normalized size = 0.9 \begin{align*} -\arcsin \left ({\frac{1}{2}}+{\frac{x}{2}} \right ) -\sqrt{-{x}^{2}-2\,x+3} \end{align*}

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 = 1.40455, size = 28, normalized size = 1.04 \begin{align*} -\sqrt{-x^{2} - 2 \, x + 3} + \arcsin \left (-\frac{1}{2} \, x - \frac{1}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^2-2*x+3)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.02817, size = 105, normalized size = 3.89 \begin{align*} -\sqrt{-x^{2} - 2 \, x + 3} + \arctan \left (\frac{\sqrt{-x^{2} - 2 \, x + 3}{\left (x + 1\right )}}{x^{2} + 2 \, x - 3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^2-2*x+3)^(1/2),x, algorithm="fricas")

[Out]

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

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

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 [A]  time = 1.06826, size = 31, normalized size = 1.15 \begin{align*} -\sqrt{-x^{2} - 2 \, x + 3} - \arcsin \left (\frac{1}{2} \, x + \frac{1}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^2-2*x+3)^(1/2),x, algorithm="giac")

[Out]

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