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

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

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {1}{4 \left (4 x^2+12 x+13\right )} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} -\frac {1}{4 \left (4 x^2+12 x+13\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*1/(13 + 12*x + 4*x^2)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3+2 x}{\left (13+12 x+4 x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.44, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{4 \, {\left (4 \, x^{2} + 12 \, x + 13\right )}} \end {gather*}

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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giac [A]  time = 0.15, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{4 \, {\left (4 \, x^{2} + 12 \, x + 13\right )}} \end {gather*}

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)

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maple [A]  time = 0.05, size = 15, normalized size = 0.94 \begin {gather*} -\frac {1}{4 \left (4 x^{2}+12 x +13\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.53, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{4 \, {\left (4 \, x^{2} + 12 \, x + 13\right )}} \end {gather*}

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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mupad [B]  time = 1.16, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{16\,x^2+48\,x+52} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(48*x + 16*x^2 + 52)

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sympy [A]  time = 0.11, size = 12, normalized size = 0.75 \begin {gather*} - \frac {1}{16 x^{2} + 48 x + 52} \end {gather*}

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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