3.904 \(\int \frac{3+2 x}{(13+12 x+4 x^2)^2} \, dx\)

Optimal. Leaf size=16 \[ -\frac{1}{4 \left (4 x^2+12 x+13\right )} \]

[Out]

-1/(4*(13 + 12*x + 4*x^2))

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Rubi [A]  time = 0.0036779, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {629} \[ -\frac{1}{4 \left (4 x^2+12 x+13\right )} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 2*x)/(13 + 12*x + 4*x^2)^2,x]

[Out]

-1/(4*(13 + 12*x + 4*x^2))

Rule 629

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

Rubi steps

\begin{align*} \int \frac{3+2 x}{\left (13+12 x+4 x^2\right )^2} \, dx &=-\frac{1}{4 \left (13+12 x+4 x^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.005657, size = 16, normalized size = 1. \[ -\frac{1}{4 \left (4 x^2+12 x+13\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 2*x)/(13 + 12*x + 4*x^2)^2,x]

[Out]

-1/(4*(13 + 12*x + 4*x^2))

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Maple [A]  time = 0.002, size = 15, normalized size = 0.9 \begin{align*} -{\frac{1}{16\,{x}^{2}+48\,x+52}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+2*x)/(4*x^2+12*x+13)^2,x)

[Out]

-1/4/(4*x^2+12*x+13)

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Maxima [A]  time = 1.01458, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{4 \,{\left (4 \, x^{2} + 12 \, x + 13\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+2*x)/(4*x^2+12*x+13)^2,x, algorithm="maxima")

[Out]

-1/4/(4*x^2 + 12*x + 13)

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Fricas [A]  time = 1.48829, size = 35, normalized size = 2.19 \begin{align*} -\frac{1}{4 \,{\left (4 \, x^{2} + 12 \, x + 13\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+2*x)/(4*x^2+12*x+13)^2,x, algorithm="fricas")

[Out]

-1/4/(4*x^2 + 12*x + 13)

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Sympy [A]  time = 0.191294, size = 12, normalized size = 0.75 \begin{align*} - \frac{1}{16 x^{2} + 48 x + 52} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+2*x)/(4*x**2+12*x+13)**2,x)

[Out]

-1/(16*x**2 + 48*x + 52)

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Giac [A]  time = 1.3026, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{4 \,{\left (4 \, x^{2} + 12 \, x + 13\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+2*x)/(4*x^2+12*x+13)^2,x, algorithm="giac")

[Out]

-1/4/(4*x^2 + 12*x + 13)