3.1530 \(\int \frac{d+e x}{\left (9+12 x+4 x^2\right )^2} \, dx\)

Optimal. Leaf size=31 \[ -\frac{2 d-3 e}{12 (2 x+3)^3}-\frac{e}{8 (2 x+3)^2} \]

[Out]

-(2*d - 3*e)/(12*(3 + 2*x)^3) - e/(8*(3 + 2*x)^2)

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Rubi [A]  time = 0.0443564, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{2 d-3 e}{12 (2 x+3)^3}-\frac{e}{8 (2 x+3)^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)/(9 + 12*x + 4*x^2)^2,x]

[Out]

-(2*d - 3*e)/(12*(3 + 2*x)^3) - e/(8*(3 + 2*x)^2)

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Rubi in Sympy [A]  time = 9.70317, size = 24, normalized size = 0.77 \[ - \frac{e}{8 \left (2 x + 3\right )^{2}} - \frac{\frac{d}{6} - \frac{e}{4}}{\left (2 x + 3\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)/(4*x**2+12*x+9)**2,x)

[Out]

-e/(8*(2*x + 3)**2) - (d/6 - e/4)/(2*x + 3)**3

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Mathematica [A]  time = 0.0131996, size = 22, normalized size = 0.71 \[ -\frac{4 d+6 e x+3 e}{24 (2 x+3)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)/(9 + 12*x + 4*x^2)^2,x]

[Out]

-(4*d + 3*e + 6*e*x)/(24*(3 + 2*x)^3)

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Maple [A]  time = 0.008, size = 28, normalized size = 0.9 \[ -{\frac{e}{8\, \left ( 2\,x+3 \right ) ^{2}}}-{\frac{1}{3\, \left ( 2\,x+3 \right ) ^{3}} \left ({\frac{d}{2}}-{\frac{3\,e}{4}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(4*x^2+12*x+9)^2,x)

[Out]

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

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Maxima [A]  time = 0.691078, size = 41, normalized size = 1.32 \[ -\frac{6 \, e x + 4 \, d + 3 \, e}{24 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(4*x^2 + 12*x + 9)^2,x, algorithm="maxima")

[Out]

-1/24*(6*e*x + 4*d + 3*e)/(8*x^3 + 36*x^2 + 54*x + 27)

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Fricas [A]  time = 0.19663, size = 41, normalized size = 1.32 \[ -\frac{6 \, e x + 4 \, d + 3 \, e}{24 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(4*x^2 + 12*x + 9)^2,x, algorithm="fricas")

[Out]

-1/24*(6*e*x + 4*d + 3*e)/(8*x^3 + 36*x^2 + 54*x + 27)

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Sympy [A]  time = 1.4651, size = 27, normalized size = 0.87 \[ - \frac{4 d + 6 e x + 3 e}{192 x^{3} + 864 x^{2} + 1296 x + 648} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)/(4*x**2+12*x+9)**2,x)

[Out]

-(4*d + 6*e*x + 3*e)/(192*x**3 + 864*x**2 + 1296*x + 648)

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GIAC/XCAS [A]  time = 0.207897, size = 30, normalized size = 0.97 \[ -\frac{6 \, x e + 4 \, d + 3 \, e}{24 \,{\left (2 \, x + 3\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(4*x^2 + 12*x + 9)^2,x, algorithm="giac")

[Out]

-1/24*(6*x*e + 4*d + 3*e)/(2*x + 3)^3