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

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

[Out]

((2 + 3*x)*Sqrt[4 + 12*x + 9*x^2])/6

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Rubi [A]  time = 0.0031282, 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} \[ \frac{1}{6} (3 x+2) \sqrt{9 x^2+12 x+4} \]

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 = 0.0054391, size = 25, normalized size = 1.09 \[ \frac{x \sqrt{(3 x+2)^2} (3 x+4)}{6 x+4} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[4 + 12*x + 9*x^2],x]

[Out]

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

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Maple [A]  time = 0.069, size = 25, normalized size = 1.1 \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((9*x^2+12*x+4)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.76622, 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((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)

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Fricas [A]  time = 2.13546, size = 20, normalized size = 0.87 \begin{align*} \frac{3}{2} \, x^{2} + 2 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*x^2 + 2*x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{9 x^{2} + 12 x + 4}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(9*x**2 + 12*x + 4), x)

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Giac [A]  time = 1.24368, size = 35, normalized size = 1.52 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)