### 3.1541 $$\int \frac{d+e x}{(9+12 x+4 x^2)^2} \, dx$$

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 = 0.0162679, 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} $-\frac{2 d-3 e}{12 (2 x+3)^3}-\frac{e}{8 (2 x+3)^2}$

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)

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 )^2} \, dx &=\int \frac{d+e x}{(3+2 x)^4} \, dx\\ &=\int \left (\frac{2 d-3 e}{2 (3+2 x)^4}+\frac{e}{2 (3+2 x)^3}\right ) \, dx\\ &=-\frac{2 d-3 e}{12 (3+2 x)^3}-\frac{e}{8 (3+2 x)^2}\\ \end{align*}

Mathematica [A]  time = 0.0077466, 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 = 0.044, size = 28, normalized size = 0.9 \begin{align*} -{\frac{1}{3\, \left ( 3+2\,x \right ) ^{3}} \left ({\frac{d}{2}}-{\frac{3\,e}{4}} \right ) }-{\frac{e}{8\, \left ( 3+2\,x \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/(4*x^2+12*x+9)^2,x)

[Out]

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

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Maxima [A]  time = 1.15613, size = 41, normalized size = 1.32 \begin{align*} -\frac{6 \, e x + 4 \, d + 3 \, e}{24 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \end{align*}

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 = 1.39814, size = 76, normalized size = 2.45 \begin{align*} -\frac{6 \, e x + 4 \, d + 3 \, e}{24 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \end{align*}

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 = 0.374043, size = 27, normalized size = 0.87 \begin{align*} - \frac{4 d + 6 e x + 3 e}{192 x^{3} + 864 x^{2} + 1296 x + 648} \end{align*}

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 [A]  time = 1.19541, size = 30, normalized size = 0.97 \begin{align*} -\frac{6 \, x e + 4 \, d + 3 \, e}{24 \,{\left (2 \, x + 3\right )}^{3}} \end{align*}

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