3.1542 \(\int \frac{d+e x}{(9+12 x+4 x^2)^3} \, dx\)

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 = 0.0165174, 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, Rules used = {27, 43} \[ -\frac{2 d-3 e}{20 (2 x+3)^5}-\frac{e}{16 (2 x+3)^4} \]

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)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{d+e x}{\left (9+12 x+4 x^2\right )^3} \, dx &=\int \frac{d+e x}{(3+2 x)^6} \, dx\\ &=\int \left (\frac{2 d-3 e}{2 (3+2 x)^6}+\frac{e}{2 (3+2 x)^5}\right ) \, dx\\ &=-\frac{2 d-3 e}{20 (3+2 x)^5}-\frac{e}{16 (3+2 x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0074101, 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 = 0.045, size = 28, normalized size = 0.9 \begin{align*} -{\frac{e}{16\, \left ( 3+2\,x \right ) ^{4}}}-{\frac{1}{5\, \left ( 3+2\,x \right ) ^{5}} \left ({\frac{d}{2}}-{\frac{3\,e}{4}} \right ) } \end{align*}

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

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Maxima [A]  time = 1.0152, size = 54, normalized size = 1.74 \begin{align*} -\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 )}} \end{align*}

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 = 1.67236, size = 111, normalized size = 3.58 \begin{align*} -\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 )}} \end{align*}

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 = 0.496349, size = 37, normalized size = 1.19 \begin{align*} - \frac{8 d + 10 e x + 3 e}{2560 x^{5} + 19200 x^{4} + 57600 x^{3} + 86400 x^{2} + 64800 x + 19440} \end{align*}

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 [A]  time = 1.13009, size = 30, normalized size = 0.97 \begin{align*} -\frac{10 \, x e + 8 \, d + 3 \, e}{80 \,{\left (2 \, x + 3\right )}^{5}} \end{align*}

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