\(\int x \sqrt {3-2 x-x^2} \, dx\) [2427]

Optimal result

Integrand size = 16, antiderivative size = 52 \[ \int x \sqrt {3-2 x-x^2} \, dx=-\frac {1}{2} (1+x) \sqrt {3-2 x-x^2}-\frac {1}{3} \left (3-2 x-x^2\right )^{3/2}+2 \arcsin \left (\frac {1}{2} (-1-x)\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {654, 626, 633, 222} \[ \int x \sqrt {3-2 x-x^2} \, dx=2 \arcsin \left (\frac {1}{2} (-x-1)\right )-\frac {1}{3} \left (-x^2-2 x+3\right )^{3/2}-\frac {1}{2} (x+1) \sqrt {-x^2-2 x+3} \]

[In]

Int[x*Sqrt[3 - 2*x - x^2],x]

[Out]

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

Rule 222

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]

Rule 626

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 633

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 654

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]

Rubi steps \begin{align*} \text {integral}& = -\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} \text {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] (verified)

Time = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.96 \[ \int x \sqrt {3-2 x-x^2} \, dx=\frac {1}{6} \sqrt {3-2 x-x^2} \left (-9+x+2 x^2\right )+4 \arctan \left (\frac {\sqrt {3-2 x-x^2}}{3+x}\right ) \]

[In]

Integrate[x*Sqrt[3 - 2*x - x^2],x]

[Out]

(Sqrt[3 - 2*x - x^2]*(-9 + x + 2*x^2))/6 + 4*ArcTan[Sqrt[3 - 2*x - x^2]/(3 + x)]

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.77

method	result	size
risch	\(-\frac {\left (2 x^{2}+x -9\right ) \left (x^{2}+2 x -3\right )}{6 \sqrt {-x^{2}-2 x +3}}-2 \arcsin \left (\frac {1}{2}+\frac {x}{2}\right )\)	\(40\)
default	\(-\frac {\left (-x^{2}-2 x +3\right )^{\frac {3}{2}}}{3}+\frac {\left (-2-2 x \right ) \sqrt {-x^{2}-2 x +3}}{4}-2 \arcsin \left (\frac {1}{2}+\frac {x}{2}\right )\)	\(43\)
trager	\(\left (\frac {1}{3} x^{2}+\frac {1}{6} x -\frac {3}{2}\right ) \sqrt {-x^{2}-2 x +3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +\sqrt {-x^{2}-2 x +3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )\)	\(61\)

[In]

int(x*(-x^2-2*x+3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00 \[ \int x \sqrt {3-2 x-x^2} \, dx=\frac {1}{6} \, {\left (2 \, x^{2} + x - 9\right )} \sqrt {-x^{2} - 2 \, x + 3} + 2 \, \arctan \left (\frac {\sqrt {-x^{2} - 2 \, x + 3} {\left (x + 1\right )}}{x^{2} + 2 \, x - 3}\right ) \]

[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))

Sympy [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.62 \[ \int x \sqrt {3-2 x-x^2} \, dx=\sqrt {- x^{2} - 2 x + 3} \left (\frac {x^{2}}{3} + \frac {x}{6} - \frac {3}{2}\right ) - 2 \operatorname {asin}{\left (\frac {x}{2} + \frac {1}{2} \right )} \]

[In]

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

[Out]

sqrt(-x**2 - 2*x + 3)*(x**2/3 + x/6 - 3/2) - 2*asin(x/2 + 1/2)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00 \[ \int x \sqrt {3-2 x-x^2} \, dx=-\frac {1}{3} \, {\left (-x^{2} - 2 \, x + 3\right )}^{\frac {3}{2}} - \frac {1}{2} \, \sqrt {-x^{2} - 2 \, x + 3} x - \frac {1}{2} \, \sqrt {-x^{2} - 2 \, x + 3} + 2 \, \arcsin \left (-\frac {1}{2} \, x - \frac {1}{2}\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.62 \[ \int x \sqrt {3-2 x-x^2} \, dx=\frac {1}{6} \, {\left ({\left (2 \, x + 1\right )} x - 9\right )} \sqrt {-x^{2} - 2 \, x + 3} - 2 \, \arcsin \left (\frac {1}{2} \, x + \frac {1}{2}\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.90 \[ \int x \sqrt {3-2 x-x^2} \, dx=\frac {\sqrt {-x^2-2\,x+3}\,\left (8\,x^2+4\,x-36\right )}{24}+\ln \left (x+1-\sqrt {-x^2-2\,x+3}\,1{}\mathrm {i}\right )\,2{}\mathrm {i} \]

[In]

int(x*(3 - x^2 - 2*x)^(1/2),x)

[Out]

log(x - (3 - x^2 - 2*x)^(1/2)*1i + 1)*2i + ((3 - x^2 - 2*x)^(1/2)*(4*x + 8*x^2 - 36))/24