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

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 = 0.00844147, 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 \[ -\frac{1}{4 \left (4 x^2+12 x+13\right )} \]

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))

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Rubi in Sympy [A]  time = 3.05497, size = 12, normalized size = 0.75 \[ - \frac{1}{4 \left (4 x^{2} + 12 x + 13\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+2*x)/(4*x**2+12*x+13)**2,x)

[Out]

-1/(4*(4*x**2 + 12*x + 13))

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Mathematica [A]  time = 0.0104798, 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 \[ -{\frac{1}{16\,{x}^{2}+48\,x+52}} \]

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 = 0.682622, size = 19, normalized size = 1.19 \[ -\frac{1}{4 \,{\left (4 \, x^{2} + 12 \, x + 13\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + 3)/(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 = 0.276629, size = 19, normalized size = 1.19 \[ -\frac{1}{4 \,{\left (4 \, x^{2} + 12 \, x + 13\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + 3)/(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 = 0.225544, size = 12, normalized size = 0.75 \[ - \frac{1}{16 x^{2} + 48 x + 52} \]

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/XCAS [A]  time = 0.269334, size = 19, normalized size = 1.19 \[ -\frac{1}{4 \,{\left (4 \, x^{2} + 12 \, x + 13\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/(4*x^2 + 12*x + 13)