3.1531 \(\int \frac{d+e x}{\left (9+12 x+4 x^2\right )^3} \, dx\)

Optimal. Leaf size=31 \[ -\frac{2 d-3 e}{20 (2 x+3)^5}-\frac{e}{16 (2 x+3)^4} \]

[Out]

-(2*d - 3*e)/(20*(3 + 2*x)^5) - e/(16*(3 + 2*x)^4)

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Rubi [A]  time = 0.046083, 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 \[ -\frac{2 d-3 e}{20 (2 x+3)^5}-\frac{e}{16 (2 x+3)^4} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)/(9 + 12*x + 4*x^2)^3,x]

[Out]

-(2*d - 3*e)/(20*(3 + 2*x)^5) - e/(16*(3 + 2*x)^4)

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Rubi in Sympy [A]  time = 9.85082, size = 26, normalized size = 0.84 \[ - \frac{e}{16 \left (2 x + 3\right )^{4}} - \frac{\frac{d}{10} - \frac{3 e}{20}}{\left (2 x + 3\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)/(4*x**2+12*x+9)**3,x)

[Out]

-e/(16*(2*x + 3)**4) - (d/10 - 3*e/20)/(2*x + 3)**5

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Mathematica [A]  time = 0.0166647, size = 22, normalized size = 0.71 \[ -\frac{8 d+e (10 x+3)}{80 (2 x+3)^5} \]

Antiderivative was successfully verified.

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

[Out]

-(8*d + e*(3 + 10*x))/(80*(3 + 2*x)^5)

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Maple [A]  time = 0.008, size = 28, normalized size = 0.9 \[ -{\frac{e}{16\, \left ( 2\,x+3 \right ) ^{4}}}-{\frac{1}{5\, \left ( 2\,x+3 \right ) ^{5}} \left ({\frac{d}{2}}-{\frac{3\,e}{4}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(4*x^2+12*x+9)^3,x)

[Out]

-1/16*e/(2*x+3)^4-1/5*(1/2*d-3/4*e)/(2*x+3)^5

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Maxima [A]  time = 0.683122, size = 54, normalized size = 1.74 \[ -\frac{10 \, e x + 8 \, d + 3 \, e}{80 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(4*x^2 + 12*x + 9)^3,x, algorithm="maxima")

[Out]

-1/80*(10*e*x + 8*d + 3*e)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)

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Fricas [A]  time = 0.193985, size = 54, normalized size = 1.74 \[ -\frac{10 \, e x + 8 \, d + 3 \, e}{80 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(4*x^2 + 12*x + 9)^3,x, algorithm="fricas")

[Out]

-1/80*(10*e*x + 8*d + 3*e)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)

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Sympy [A]  time = 1.75718, size = 37, normalized size = 1.19 \[ - \frac{8 d + 10 e x + 3 e}{2560 x^{5} + 19200 x^{4} + 57600 x^{3} + 86400 x^{2} + 64800 x + 19440} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)/(4*x**2+12*x+9)**3,x)

[Out]

-(8*d + 10*e*x + 3*e)/(2560*x**5 + 19200*x**4 + 57600*x**3 + 86400*x**2 + 64800*
x + 19440)

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GIAC/XCAS [A]  time = 0.207665, size = 30, normalized size = 0.97 \[ -\frac{10 \, x e + 8 \, d + 3 \, e}{80 \,{\left (2 \, x + 3\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(4*x^2 + 12*x + 9)^3,x, algorithm="giac")

[Out]

-1/80*(10*x*e + 8*d + 3*e)/(2*x + 3)^5