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]

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

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Rubi [A]  time = 0.01, antiderivative size = 30, 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} \[ \frac {1}{4} \sqrt {-4 x^2-4 x+3} (2 x+1)+\sin ^{-1}\left (x+\frac {1}{2}\right ) \]

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 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 {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*}

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Mathematica [A]  time = 0.01, size = 30, normalized size = 1.00 \[ \frac {1}{4} \sqrt {-4 x^2-4 x+3} (2 x+1)+\sin ^{-1}\left (x+\frac {1}{2}\right ) \]

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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fricas [B]  time = 0.78, size = 53, normalized size = 1.77 \[ \frac {1}{4} \, \sqrt {-4 \, x^{2} - 4 \, x + 3} {\left (2 \, x + 1\right )} - \arctan \left (\frac {\sqrt {-4 \, x^{2} - 4 \, x + 3} {\left (2 \, x + 1\right )}}{4 \, x^{2} + 4 \, x - 3}\right ) \]

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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giac [A]  time = 1.60, size = 24, normalized size = 0.80 \[ \frac {1}{4} \, \sqrt {-4 \, x^{2} - 4 \, x + 3} {\left (2 \, x + 1\right )} + \arcsin \left (x + \frac {1}{2}\right ) \]

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)

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maple [A]  time = 0.04, size = 25, normalized size = 0.83 \[ \arcsin \left (x +\frac {1}{2}\right )-\frac {\left (-8 x -4\right ) \sqrt {-4 x^{2}-4 x +3}}{16} \]

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 = 3.00, size = 38, normalized size = 1.27 \[ \frac {1}{2} \, \sqrt {-4 \, x^{2} - 4 \, x + 3} x + \frac {1}{4} \, \sqrt {-4 \, x^{2} - 4 \, x + 3} - \arcsin \left (-x - \frac {1}{2}\right ) \]

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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mupad [B]  time = 0.05, size = 23, normalized size = 0.77 \[ \mathrm {asin}\left (x+\frac {1}{2}\right )+\left (\frac {x}{2}+\frac {1}{4}\right )\,\sqrt {-4\,x^2-4\,x+3} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- 4 x^{2} - 4 x + 3}\, dx \]

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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