\(\int \sqrt {3-4 x-4 x^2} \, dx\) [104]

Optimal result

Integrand size = 14, antiderivative size = 30 \[ \int \sqrt {3-4 x-4 x^2} \, dx=\frac {1}{4} (1+2 x) \sqrt {3-4 x-4 x^2}+\arcsin \left (\frac {1}{2}+x\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, 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} \[ \int \sqrt {3-4 x-4 x^2} \, dx=\arcsin \left (x+\frac {1}{2}\right )+\frac {1}{4} \sqrt {-4 x^2-4 x+3} (2 x+1) \]

[In]

Int[Sqrt[3 - 4*x - 4*x^2],x]

[Out]

((1 + 2*x)*Sqrt[3 - 4*x - 4*x^2])/4 + ArcSin[1/2 + x]

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}{4} (1+2 x) \sqrt {3-4 x-4 x^2}+2 \int \frac {1}{\sqrt {3-4 x-4 x^2}} \, dx \\ & = \frac {1}{4} (1+2 x) \sqrt {3-4 x-4 x^2}-\frac {1}{8} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{64}}} \, dx,x,-4-8 x\right ) \\ & = \frac {1}{4} (1+2 x) \sqrt {3-4 x-4 x^2}+\sin ^{-1}\left (\frac {1}{2}+x\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63 \[ \int \sqrt {3-4 x-4 x^2} \, dx=\frac {1}{4} (1+2 x) \sqrt {3-4 x-4 x^2}-2 \arctan \left (\frac {\sqrt {3-4 x-4 x^2}}{3+2 x}\right ) \]

[In]

Integrate[Sqrt[3 - 4*x - 4*x^2],x]

[Out]

((1 + 2*x)*Sqrt[3 - 4*x - 4*x^2])/4 - 2*ArcTan[Sqrt[3 - 4*x - 4*x^2]/(3 + 2*x)]

Maple [A] (verified)

Time = 2.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83

method	result	size
default	\(-\frac {\left (-8 x -4\right ) \sqrt {-4 x^{2}-4 x +3}}{16}+\arcsin \left (x +\frac {1}{2}\right )\)	\(25\)
risch	\(-\frac {\left (4 x^{2}+4 x -3\right ) \left (1+2 x \right )}{4 \sqrt {-4 x^{2}-4 x +3}}+\arcsin \left (x +\frac {1}{2}\right )\)	\(35\)
trager	\(\left (\frac {1}{4}+\frac {x}{2}\right ) \sqrt {-4 x^{2}-4 x +3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {-4 x^{2}-4 x +3}\right )\)	\(58\)

[In]

int((-4*x^2-4*x+3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/16*(-8*x-4)*(-4*x^2-4*x+3)^(1/2)+arcsin(x+1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).

Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.77 \[ \int \sqrt {3-4 x-4 x^2} \, dx=\frac {1}{4} \, \sqrt {-4 \, x^{2} - 4 \, x + 3} {\left (2 \, x + 1\right )} - \arctan \left (\frac {\sqrt {-4 \, x^{2} - 4 \, x + 3} {\left (2 \, x + 1\right )}}{4 \, x^{2} + 4 \, x - 3}\right ) \]

[In]

integrate((-4*x^2-4*x+3)^(1/2),x, algorithm="fricas")

[Out]

1/4*sqrt(-4*x^2 - 4*x + 3)*(2*x + 1) - arctan(sqrt(-4*x^2 - 4*x + 3)*(2*x + 1)/(4*x^2 + 4*x - 3))

Sympy [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \sqrt {3-4 x-4 x^2} \, dx=\left (\frac {x}{2} + \frac {1}{4}\right ) \sqrt {- 4 x^{2} - 4 x + 3} + \operatorname {asin}{\left (x + \frac {1}{2} \right )} \]

[In]

integrate((-4*x**2-4*x+3)**(1/2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27 \[ \int \sqrt {3-4 x-4 x^2} \, dx=\frac {1}{2} \, \sqrt {-4 \, x^{2} - 4 \, x + 3} x + \frac {1}{4} \, \sqrt {-4 \, x^{2} - 4 \, x + 3} - \arcsin \left (-x - \frac {1}{2}\right ) \]

[In]

integrate((-4*x^2-4*x+3)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \sqrt {3-4 x-4 x^2} \, dx=\frac {1}{4} \, \sqrt {-4 \, x^{2} - 4 \, x + 3} {\left (2 \, x + 1\right )} + \arcsin \left (x + \frac {1}{2}\right ) \]

[In]

integrate((-4*x^2-4*x+3)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.77 \[ \int \sqrt {3-4 x-4 x^2} \, dx=\mathrm {asin}\left (x+\frac {1}{2}\right )+\left (\frac {x}{2}+\frac {1}{4}\right )\,\sqrt {-4\,x^2-4\,x+3} \]

[In]

int((3 - 4*x^2 - 4*x)^(1/2),x)

[Out]

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