### 3.2427 $$\int x \sqrt{3-2 x-x^2} \, dx$$

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 = 0.0137335, 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, Rules used = {640, 612, 619, 216} $-\frac{1}{3} \left (-x^2-2 x+3\right )^{3/2}-\frac{1}{2} (x+1) \sqrt{-x^2-2 x+3}+2 \sin ^{-1}\left (\frac{1}{2} (-x-1)\right )$

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]

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 612

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

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

Mathematica [A]  time = 0.0174071, 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 = 0.042, size = 43, normalized size = 0.8 \begin{align*} -{\frac{1}{3} \left ( -{x}^{2}-2\,x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{-2\,x-2}{4}\sqrt{-{x}^{2}-2\,x+3}}-2\,\arcsin \left ( 1/2+x/2 \right ) \end{align*}

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

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Maxima [A]  time = 1.49706, size = 70, normalized size = 1.35 \begin{align*} -\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 ) \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]

-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 = 2.08042, size = 134, normalized size = 2.58 \begin{align*} \frac{1}{6} \,{\left (2 \, x^{2} + x - 9\right )} \sqrt{-x^{2} - 2 \, x + 3} + 2 \, \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]

1/6*(2*x^2 + x - 9)*sqrt(-x^2 - 2*x + 3) + 2*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 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 = 1.09121, size = 43, normalized size = 0.83 \begin{align*} \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 ) \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]

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