\(\int x \sqrt {9+12 x+4 x^2} \, dx\) [210]

Optimal result

Integrand size = 16, antiderivative size = 42 \[ \int x \sqrt {9+12 x+4 x^2} \, dx=-\frac {3}{8} (3+2 x) \sqrt {9+12 x+4 x^2}+\frac {1}{12} \left (9+12 x+4 x^2\right )^{3/2} \]

[Out]

1/12*(4*x^2+12*x+9)^(3/2)-3/8*(3+2*x)*((3+2*x)^2)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {654, 623} \[ \int x \sqrt {9+12 x+4 x^2} \, dx=\frac {1}{12} \left (4 x^2+12 x+9\right )^{3/2}-\frac {3}{8} (2 x+3) \sqrt {4 x^2+12 x+9} \]

[In]

Int[x*Sqrt[9 + 12*x + 4*x^2],x]

[Out]

(-3*(3 + 2*x)*Sqrt[9 + 12*x + 4*x^2])/8 + (9 + 12*x + 4*x^2)^(3/2)/12

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)]

Rule 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{12} \left (9+12 x+4 x^2\right )^{3/2}-\frac {3}{2} \int \sqrt {9+12 x+4 x^2} \, dx \\ & = -\frac {3}{8} (3+2 x) \sqrt {9+12 x+4 x^2}+\frac {1}{12} \left (9+12 x+4 x^2\right )^{3/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71 \[ \int x \sqrt {9+12 x+4 x^2} \, dx=\frac {x^2 \sqrt {(3+2 x)^2} (9+4 x)}{6 (3+2 x)} \]

[In]

Integrate[x*Sqrt[9 + 12*x + 4*x^2],x]

[Out]

(x^2*Sqrt[(3 + 2*x)^2]*(9 + 4*x))/(6*(3 + 2*x))

Maple [C] (warning: unable to verify)

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

Time = 2.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.50

method	result	size
default	\(\frac {\operatorname {csgn}\left (2 x +3\right ) \left (2 x +3\right )^{2} \left (4 x -3\right )}{24}\)	\(21\)
gosper	\(\frac {x^{2} \left (9+4 x \right ) \sqrt {\left (2 x +3\right )^{2}}}{12 x +18}\)	\(27\)
risch	\(\frac {2 \sqrt {\left (2 x +3\right )^{2}}\, x^{3}}{3 \left (2 x +3\right )}+\frac {3 \sqrt {\left (2 x +3\right )^{2}}\, x^{2}}{2 \left (2 x +3\right )}\)	\(44\)

[In]

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

[Out]

1/24*csgn(2*x+3)*(2*x+3)^2*(4*x-3)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.26 \[ \int x \sqrt {9+12 x+4 x^2} \, dx=\frac {2}{3} \, x^{3} + \frac {3}{2} \, x^{2} \]

[In]

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

[Out]

2/3*x^3 + 3/2*x^2

Sympy [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.57 \[ \int x \sqrt {9+12 x+4 x^2} \, dx=\left (\frac {x^{2}}{3} + \frac {x}{4} - \frac {3}{8}\right ) \sqrt {4 x^{2} + 12 x + 9} \]

[In]

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

[Out]

(x**2/3 + x/4 - 3/8)*sqrt(4*x**2 + 12*x + 9)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.05 \[ \int x \sqrt {9+12 x+4 x^2} \, dx=\frac {1}{12} \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac {3}{2}} - \frac {3}{4} \, \sqrt {4 \, x^{2} + 12 \, x + 9} x - \frac {9}{8} \, \sqrt {4 \, x^{2} + 12 \, x + 9} \]

[In]

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

[Out]

1/12*(4*x^2 + 12*x + 9)^(3/2) - 3/4*sqrt(4*x^2 + 12*x + 9)*x - 9/8*sqrt(4*x^2 + 12*x + 9)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.74 \[ \int x \sqrt {9+12 x+4 x^2} \, dx=\frac {2}{3} \, x^{3} \mathrm {sgn}\left (2 \, x + 3\right ) + \frac {3}{2} \, x^{2} \mathrm {sgn}\left (2 \, x + 3\right ) - \frac {9}{8} \, \mathrm {sgn}\left (2 \, x + 3\right ) \]

[In]

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

[Out]

2/3*x^3*sgn(2*x + 3) + 3/2*x^2*sgn(2*x + 3) - 9/8*sgn(2*x + 3)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.55 \[ \int x \sqrt {9+12 x+4 x^2} \, dx=\left (\frac {x^2}{3}+\frac {x}{4}-\frac {3}{8}\right )\,\sqrt {4\,x^2+12\,x+9} \]

[In]

int(x*(12*x + 4*x^2 + 9)^(1/2),x)

[Out]

(x/4 + x^2/3 - 3/8)*(12*x + 4*x^2 + 9)^(1/2)