### 3.210 $$\int x \sqrt{9+12 x+4 x^2} \, dx$$

Optimal. Leaf size=42 $\frac{1}{12} \left (4 x^2+12 x+9\right )^{3/2}-\frac{3}{8} (2 x+3) \sqrt{4 x^2+12 x+9}$

[Out]

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

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Rubi [A]  time = 0.0079994, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.125, Rules used = {640, 609} $\frac{1}{12} \left (4 x^2+12 x+9\right )^{3/2}-\frac{3}{8} (2 x+3) \sqrt{4 x^2+12 x+9}$

Antiderivative was successfully veriﬁed.

[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 640

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]

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 x \sqrt{9+12 x+4 x^2} \, dx &=\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]  time = 0.0064007, size = 30, normalized size = 0.71 $\frac{x^2 \sqrt{(2 x+3)^2} (4 x+9)}{6 (2 x+3)}$

Antiderivative was successfully veriﬁed.

[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))

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Maple [A]  time = 0.062, size = 27, normalized size = 0.6 \begin{align*}{\frac{{x}^{2} \left ( 4\,x+9 \right ) }{18+12\,x}\sqrt{ \left ( 3+2\,x \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.70717, size = 59, normalized size = 1.4 \begin{align*} \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} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.53914, size = 26, normalized size = 0.62 \begin{align*} \frac{2}{3} \, x^{3} + \frac{3}{2} \, x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x^3 + 3/2*x^2

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*sqrt((2*x + 3)**2), x)

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Giac [A]  time = 1.31284, size = 42, normalized size = 1. \begin{align*} \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 ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)