### 3.212 $$\int \frac{x}{(9+12 x+4 x^2)^{3/2}} \, dx$$

Optimal. Leaf size=44 $\frac{3}{8 (2 x+3) \sqrt{4 x^2+12 x+9}}-\frac{1}{4 \sqrt{4 x^2+12 x+9}}$

[Out]

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

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Rubi [A]  time = 0.0086371, antiderivative size = 44, 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, 607} $\frac{3}{8 (2 x+3) \sqrt{4 x^2+12 x+9}}-\frac{1}{4 \sqrt{4 x^2+12 x+9}}$

Antiderivative was successfully veriﬁed.

[In]

Int[x/(9 + 12*x + 4*x^2)^(3/2),x]

[Out]

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

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 607

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(2*(a + b*x + c*x^2)^(p + 1))/((2*p + 1)*(b + 2
*c*x)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rubi steps

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

Mathematica [A]  time = 0.0081203, size = 27, normalized size = 0.61 $\frac{-4 x-3}{8 (2 x+3) \sqrt{(2 x+3)^2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x/(9 + 12*x + 4*x^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.08, size = 22, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 3+2\,x \right ) \left ( 4\,x+3 \right ) }{8} \left ( \left ( 3+2\,x \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x/(4*x^2+12*x+9)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.71029, size = 32, normalized size = 0.73 \begin{align*} -\frac{1}{4 \, \sqrt{4 \, x^{2} + 12 \, x + 9}} + \frac{3}{8 \,{\left (2 \, x + 3\right )}^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(4*x^2+12*x+9)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.63125, size = 47, normalized size = 1.07 \begin{align*} -\frac{4 \, x + 3}{8 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(4*x^2+12*x+9)^(3/2),x, algorithm="fricas")

[Out]

-1/8*(4*x + 3)/(4*x^2 + 12*x + 9)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(4*x**2+12*x+9)**(3/2),x)

[Out]

Integral(x/((2*x + 3)**2)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(4*x^2+12*x+9)^(3/2),x, algorithm="giac")

[Out]

sage0*x