\(\int \sqrt {4+12 x+9 x^2} \, dx\) [65]

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 = 25, normalized size of antiderivative = 1.09 \[ \int \sqrt {4+12 x+9 x^2} \, dx=\frac {x \sqrt {(2+3 x)^2} (4+3 x)}{4+6 x} \]

[In]

Integrate[Sqrt[4 + 12*x + 9*x^2],x]

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 2.

Time = 2.07 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.70

method	result	size
default	\(\frac {\operatorname {csgn}\left (2+3 x \right ) \left (2+3 x \right )^{2}}{6}\)	\(16\)
gosper	\(\frac {x \left (4+3 x \right ) \sqrt {\left (2+3 x \right )^{2}}}{4+6 x}\)	\(25\)
risch	\(\frac {3 \sqrt {\left (2+3 x \right )^{2}}\, x^{2}}{2 \left (2+3 x \right )}+\frac {2 \sqrt {\left (2+3 x \right )^{2}}\, x}{2+3 x}\)	\(42\)

[In]

int((9*x^2+12*x+4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.45 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.39 \[ \int \sqrt {4+12 x+9 x^2} \, dx=\frac {3}{2} \, x^{2} + 2 \, x \]

[In]

integrate((9*x^2+12*x+4)^(1/2),x, algorithm="fricas")

[Out]

3/2*x^2 + 2*x

Sympy [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \sqrt {4+12 x+9 x^2} \, dx=\left (\frac {x}{2} + \frac {1}{3}\right ) \sqrt {9 x^{2} + 12 x + 4} \]

[In]

integrate((9*x**2+12*x+4)**(1/2),x)

[Out]

(x/2 + 1/3)*sqrt(9*x**2 + 12*x + 4)

Maxima [A] (verification not implemented)

none

Time = 0.27 (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((9*x^2+12*x+4)^(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 [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \sqrt {4+12 x+9 x^2} \, dx=\frac {1}{2} \, {\left (3 \, x^{2} + 4 \, x\right )} \mathrm {sgn}\left (3 \, x + 2\right ) + \frac {2}{3} \, \mathrm {sgn}\left (3 \, x + 2\right ) \]

[In]

integrate((9*x^2+12*x+4)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \sqrt {4+12 x+9 x^2} \, dx=\frac {\left (3\,x+2\right )\,\sqrt {9\,x^2+12\,x+4}}{6} \]

[In]

int((12*x + 9*x^2 + 4)^(1/2),x)

[Out]

((3*x + 2)*(12*x + 9*x^2 + 4)^(1/2))/6