\(\int \sqrt {5-4 x-x^2} \, dx\) [353]

Optimal result

Integrand size = 14, antiderivative size = 36 \[ \int \sqrt {5-4 x-x^2} \, dx=\frac {1}{2} (2+x) \sqrt {5-4 x-x^2}-\frac {9}{2} \arcsin \left (\frac {1}{3} (-2-x)\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {626, 633, 222} \[ \int \sqrt {5-4 x-x^2} \, dx=\frac {1}{2} (x+2) \sqrt {-x^2-4 x+5}-\frac {9}{2} \arcsin \left (\frac {1}{3} (-x-2)\right ) \]

[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 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]

Rubi steps \begin{align*} \text {integral}& = \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} \text {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} \arcsin \left (\frac {1}{3} (-2-x)\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.25 \[ \int \sqrt {5-4 x-x^2} \, dx=\frac {1}{2} (2+x) \sqrt {5-4 x-x^2}-9 \arctan \left (\frac {\sqrt {5-4 x-x^2}}{5+x}\right ) \]

[In]

Integrate[Sqrt[5 - 4*x - x^2],x]

[Out]

((2 + x)*Sqrt[5 - 4*x - x^2])/2 - 9*ArcTan[Sqrt[5 - 4*x - x^2]/(5 + x)]

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81

method	result	size
default	\(-\frac {\left (-2 x -4\right ) \sqrt {-x^{2}-4 x +5}}{4}+\frac {9 \arcsin \left (\frac {2}{3}+\frac {x}{3}\right )}{2}\)	\(29\)
risch	\(-\frac {\left (2+x \right ) \left (x^{2}+4 x -5\right )}{2 \sqrt {-x^{2}-4 x +5}}+\frac {9 \arcsin \left (\frac {2}{3}+\frac {x}{3}\right )}{2}\)	\(35\)
trager	\(\left (1+\frac {x}{2}\right ) \sqrt {-x^{2}-4 x +5}+\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +\sqrt {-x^{2}-4 x +5}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{2}\)	\(59\)

[In]

int((-x^2-4*x+5)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.31 \[ \int \sqrt {5-4 x-x^2} \, dx=\frac {1}{2} \, \sqrt {-x^{2} - 4 \, x + 5} {\left (x + 2\right )} - \frac {9}{2} \, \arctan \left (\frac {\sqrt {-x^{2} - 4 \, x + 5} {\left (x + 2\right )}}{x^{2} + 4 \, x - 5}\right ) \]

[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))

Sympy [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int \sqrt {5-4 x-x^2} \, dx=\left (\frac {x}{2} + 1\right ) \sqrt {- x^{2} - 4 x + 5} + \frac {9 \operatorname {asin}{\left (\frac {x}{3} + \frac {2}{3} \right )}}{2} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \sqrt {5-4 x-x^2} \, dx=\frac {1}{2} \, \sqrt {-x^{2} - 4 \, x + 5} x + \sqrt {-x^{2} - 4 \, x + 5} - \frac {9}{2} \, \arcsin \left (-\frac {1}{3} \, x - \frac {2}{3}\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \sqrt {5-4 x-x^2} \, dx=\frac {1}{2} \, \sqrt {-x^{2} - 4 \, x + 5} {\left (x + 2\right )} + \frac {9}{2} \, \arcsin \left (\frac {1}{3} \, x + \frac {2}{3}\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int \sqrt {5-4 x-x^2} \, dx=\frac {9\,\mathrm {asin}\left (\frac {x}{3}+\frac {2}{3}\right )}{2}+\left (\frac {x}{2}+1\right )\,\sqrt {-x^2-4\,x+5} \]

[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)