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]

1/6*(2+3*x)*(-(2+3*x)^2)^(1/2)

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Rubi [A]  time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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)

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fricas [C]  time = 0.85, size = 9, normalized size = 0.39 \[ \frac {3}{2} i \, x^{2} + 2 i \, x \]

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

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giac [C]  time = 0.41, size = 26, normalized size = 1.13 \[ -\frac {1}{2} i \, {\left (3 \, x^{2} + 4 \, x\right )} \mathrm {sgn}\left (-3 \, x - 2\right ) - \frac {2}{3} i \, \mathrm {sgn}\left (-3 \, x - 2\right ) \]

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)

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maple [A]  time = 0.04, size = 27, normalized size = 1.17 \[ \frac {\left (3 x +4\right ) \sqrt {-\left (3 x +2\right )^{2}}\, x}{6 x +4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.96, size = 30, normalized size = 1.30 \[ \frac {1}{2} \, \sqrt {-9 \, x^{2} - 12 \, x - 4} x + \frac {1}{3} \, \sqrt {-9 \, x^{2} - 12 \, x - 4} \]

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)

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mupad [B]  time = 0.07, size = 18, normalized size = 0.78 \[ \frac {\left (3\,x+2\right )\,\sqrt {-{\left (3\,x+2\right )}^2}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-(3*x + 2)^2)^(1/2),x)

[Out]

((3*x + 2)*(-(3*x + 2)^2)^(1/2))/6

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

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)

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