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

Optimal. Leaf size=23 \[ -\frac{1}{6} (2-3 x) \sqrt{-9 x^2+12 x-4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0026217, antiderivative size = 23, normalized size of antiderivative = 1., 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} \[ -\frac{1}{6} (2-3 x) \sqrt{-9 x^2+12 x-4} \]

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*}

Mathematica [A]  time = 0.0076881, size = 27, normalized size = 1.17 \[ \frac{\sqrt{-(2-3 x)^2} x (3 x-4)}{6 x-4} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-4 + 12*x - 9*x^2],x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.066, size = 27, normalized size = 1.2 \begin{align*}{\frac{x \left ( 3\,x-4 \right ) }{-4+6\,x}\sqrt{- \left ( -2+3\,x \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.69619, size = 41, normalized size = 1.78 \begin{align*} \frac{1}{2} \, \sqrt{-9 \, x^{2} + 12 \, x - 4} x - \frac{1}{3} \, \sqrt{-9 \, x^{2} + 12 \, x - 4} \end{align*}

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)

________________________________________________________________________________________

Fricas [C]  time = 2.04588, size = 26, normalized size = 1.13 \begin{align*} \frac{3}{2} i \, x^{2} - 2 i \, x \end{align*}

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- \left (3 x - 2\right )^{2}}\, dx \end{align*}

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)

________________________________________________________________________________________

Giac [C]  time = 1.33137, size = 35, normalized size = 1.52 \begin{align*} -\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 ) \end{align*}

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)