\(\int \sqrt {4-12 x+9 x^2} \, dx\) [68]

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

[In]

Int[Sqrt[4 - 12*x + 9*x^2],x]

[Out]

-1/6*((2 - 3*x)*Sqrt[4 - 12*x + 9*x^2])

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 \[ \int \sqrt {4-12 x+9 x^2} \, dx=\frac {\sqrt {(2-3 x)^2} x (-4+3 x)}{-4+6 x} \]

[In]

Integrate[Sqrt[4 - 12*x + 9*x^2],x]

[Out]

(Sqrt[(2 - 3*x)^2]*x*(-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 = 0.17 (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	\(\frac {x \left (3 x -4\right ) \sqrt {\left (-2+3 x \right )^{2}}}{-4+6 x}\)	\(25\)
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(((-2+3*x)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

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

[In]

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

[Out]

3/2*x^2 - 2*x

Sympy [A] (verification not implemented)

Time = 0.42 (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(((-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 = 0.27 (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(((-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 [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(((-2+3*x)^2)^(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 = 0.11 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.57 \[ \int \sqrt {4-12 x+9 x^2} \, dx=\frac {\left |3\,x-2\right |\,\left (3\,x-2\right )}{6} \]

[In]

int(((3*x - 2)^2)^(1/2),x)

[Out]

(abs(3*x - 2)*(3*x - 2))/6