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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {612, 619, 216} \[ \frac {1}{2} (x+2) \sqrt {-x^2-4 x+5}-\frac {9}{2} \sin ^{-1}\left (\frac {1}{3} (-x-2)\right ) \]

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

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]

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]

Rubi steps

\begin {align*} \int \sqrt {5-4 x-x^2} \, dx &=\frac {1}{2} (2+x) \sqrt {5-4 x-x^2}+\frac {9}{2} \int \frac {1}{\sqrt {5-4 x-x^2}} \, dx\\ &=\frac {1}{2} (2+x) \sqrt {5-4 x-x^2}-\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{36}}} \, dx,x,-4-2 x\right )\\ &=\frac {1}{2} (2+x) \sqrt {5-4 x-x^2}-\frac {9}{2} \sin ^{-1}\left (\frac {1}{3} (-2-x)\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, 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

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 47, normalized size = 1.31 \[ \frac {1}{2} \, \sqrt {-x^{2} - 4 \, x + 5} {\left (x + 2\right )} - \frac {9}{2} \, \arctan \left (\frac {\sqrt {-x^{2} - 4 \, x + 5} {\left (x + 2\right )}}{x^{2} + 4 \, x - 5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2-4*x+5)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 1.09, size = 26, normalized size = 0.72 \[ \frac {1}{2} \, \sqrt {-x^{2} - 4 \, x + 5} {\left (x + 2\right )} + \frac {9}{2} \, \arcsin \left (\frac {1}{3} \, x + \frac {2}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2-4*x+5)^(1/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.00, size = 29, normalized size = 0.81 \[ \frac {9 \arcsin \left (\frac {x}{3}+\frac {2}{3}\right )}{2}-\frac {\left (-2 x -4\right ) \sqrt {-x^{2}-4 x +5}}{4} \]

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(1/3*x+2/3)

________________________________________________________________________________________

maxima [A]  time = 1.12, size = 36, normalized size = 1.00 \[ \frac {1}{2} \, \sqrt {-x^{2} - 4 \, x + 5} x + \sqrt {-x^{2} - 4 \, x + 5} - \frac {9}{2} \, \arcsin \left (-\frac {1}{3} \, x - \frac {2}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2-4*x+5)^(1/2),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)

________________________________________________________________________________________

mupad [B]  time = 0.18, size = 27, normalized size = 0.75 \[ \frac {9\,\mathrm {asin}\left (\frac {x}{3}+\frac {2}{3}\right )}{2}+\left (\frac {x}{2}+1\right )\,\sqrt {-x^2-4\,x+5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5 - x^2 - 4*x)^(1/2),x)

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- x^{2} - 4 x + 5}\, dx \]

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)

________________________________________________________________________________________