3.72 \(\int \sqrt{-4-12 x-9 x^2} \, dx\)

Optimal. Leaf size=23 \[ \frac{1}{6} (3 x+2) \sqrt{-9 x^2-12 x-4} \]

[Out]

((2 + 3*x)*Sqrt[-4 - 12*x - 9*x^2])/6

________________________________________________________________________________________

Rubi [A]  time = 0.0025759, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {609} \[ \frac{1}{6} (3 x+2) \sqrt{-9 x^2-12 x-4} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-4 - 12*x - 9*x^2],x]

[Out]

((2 + 3*x)*Sqrt[-4 - 12*x - 9*x^2])/6

Rule 609

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] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && NeQ[p, -2^(-1)]

Rubi steps

\begin{align*} \int \sqrt{-4-12 x-9 x^2} \, dx &=\frac{1}{6} (2+3 x) \sqrt{-4-12 x-9 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0057645, size = 27, normalized size = 1.17 \[ \frac{x \sqrt{-(3 x+2)^2} (3 x+4)}{6 x+4} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-4 - 12*x - 9*x^2],x]

[Out]

(x*Sqrt[-(2 + 3*x)^2]*(4 + 3*x))/(4 + 6*x)

________________________________________________________________________________________

Maple [A]  time = 0.069, size = 27, normalized size = 1.2 \begin{align*}{\frac{x \left ( 3\,x+4 \right ) }{4+6\,x}\sqrt{- \left ( 2+3\,x \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.73642, size = 41, normalized size = 1.78 \begin{align*} \frac{1}{2} \, \sqrt{-9 \, x^{2} - 12 \, x - 4} x + \frac{1}{3} \, \sqrt{-9 \, x^{2} - 12 \, x - 4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-9*x^2 - 12*x - 4)*x + 1/3*sqrt(-9*x^2 - 12*x - 4)

________________________________________________________________________________________

Fricas [C]  time = 2.03856, size = 26, normalized size = 1.13 \begin{align*} \frac{3}{2} i \, x^{2} + 2 i \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*I*x^2 + 2*I*x

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-(3*x + 2)**2), x)

________________________________________________________________________________________

Giac [C]  time = 1.23801, size = 35, normalized size = 1.52 \begin{align*} -\frac{1}{2} i \,{\left (3 \, x^{2} + 4 \, x\right )} \mathrm{sgn}\left (-3 \, x - 2\right ) - \frac{2}{3} i \, \mathrm{sgn}\left (-3 \, x - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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