\(\int \sqrt {-4-12 x-9 x^2} \, dx\) [72]

Optimal result

Integrand size = 14, antiderivative size = 23 \[ \int \sqrt {-4-12 x-9 x^2} \, dx=\frac {1}{6} (2+3 x) \sqrt {-4-12 x-9 x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {623} \[ \int \sqrt {-4-12 x-9 x^2} \, dx=\frac {1}{6} (3 x+2) \sqrt {-9 x^2-12 x-4} \]

[In]

Int[Sqrt[-4 - 12*x - 9*x^2],x]

[Out]

((2 + 3*x)*Sqrt[-4 - 12*x - 9*x^2])/6

Rule 623

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*} \text {integral}& = \frac {1}{6} (2+3 x) \sqrt {-4-12 x-9 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \sqrt {-4-12 x-9 x^2} \, dx=\frac {x \sqrt {-(2+3 x)^2} (4+3 x)}{4+6 x} \]

[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] (verified)

Time = 2.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17

method	result	size
gosper	\(\frac {x \left (4+3 x \right ) \sqrt {-\left (2+3 x \right )^{2}}}{4+6 x}\)	\(27\)
default	\(\frac {x \left (4+3 x \right ) \sqrt {-\left (2+3 x \right )^{2}}}{4+6 x}\)	\(27\)
risch	\(\frac {2 \sqrt {-\left (2+3 x \right )^{2}}\, x}{2+3 x}+\frac {3 \sqrt {-\left (2+3 x \right )^{2}}\, x^{2}}{2 \left (2+3 x \right )}\)	\(46\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.39 \[ \int \sqrt {-4-12 x-9 x^2} \, dx=\frac {3}{2} i \, x^{2} + 2 i \, x \]

[In]

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

[Out]

3/2*I*x^2 + 2*I*x

Sympy [A] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \sqrt {-4-12 x-9 x^2} \, dx=\left (\frac {x}{2} + \frac {1}{3}\right ) \sqrt {- 9 x^{2} - 12 x - 4} \]

[In]

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

[Out]

(x/2 + 1/3)*sqrt(-9*x**2 - 12*x - 4)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \sqrt {-4-12 x-9 x^2} \, dx=\frac {1}{2} \, \sqrt {-9 \, x^{2} - 12 \, x - 4} x + \frac {1}{3} \, \sqrt {-9 \, x^{2} - 12 \, x - 4} \]

[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)

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \sqrt {-4-12 x-9 x^2} \, dx=-\frac {1}{2} i \, {\left (3 \, x^{2} + 4 \, x\right )} \mathrm {sgn}\left (-3 \, x - 2\right ) - \frac {2}{3} i \, \mathrm {sgn}\left (-3 \, x - 2\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \sqrt {-4-12 x-9 x^2} \, dx=\frac {\left (3\,x+2\right )\,\sqrt {-{\left (3\,x+2\right )}^2}}{6} \]

[In]

int((-(3*x + 2)^2)^(1/2),x)

[Out]

((3*x + 2)*(-(3*x + 2)^2)^(1/2))/6