3.104 \(\int \sqrt{3-4 x-4 x^2} \, dx\)

Optimal. Leaf size=30 \[ \frac{1}{4} \sqrt{-4 x^2-4 x+3} (2 x+1)+\sin ^{-1}\left (x+\frac{1}{2}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0101856, antiderivative size = 30, normalized size of antiderivative = 1., 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} \[ \frac{1}{4} \sqrt{-4 x^2-4 x+3} (2 x+1)+\sin ^{-1}\left (x+\frac{1}{2}\right ) \]

Antiderivative was successfully verified.

[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 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 \sqrt{3-4 x-4 x^2} \, dx &=\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} \operatorname{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]  time = 0.0129401, size = 30, normalized size = 1. \[ \frac{1}{4} \sqrt{-4 x^2-4 x+3} (2 x+1)+\sin ^{-1}\left (x+\frac{1}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[3 - 4*x - 4*x^2],x]

[Out]

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

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Maple [A]  time = 0.048, size = 25, normalized size = 0.8 \begin{align*} -{\frac{-8\,x-4}{16}\sqrt{-4\,{x}^{2}-4\,x+3}}+\arcsin \left ( x+{\frac{1}{2}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.70079, size = 51, normalized size = 1.7 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 1.90776, size = 134, normalized size = 4.47 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- 4 x^{2} - 4 x + 3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-4*x**2 - 4*x + 3), x)

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Giac [A]  time = 1.26053, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{4} \, \sqrt{-4 \, x^{2} - 4 \, x + 3}{\left (2 \, x + 1\right )} + \arcsin \left (x + \frac{1}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)