\(\int \frac {d+e x}{(9+12 x+4 x^2)^2} \, dx\) [1541]

Optimal result

Integrand size = 18, antiderivative size = 31 \[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^2} \, dx=-\frac {2 d-3 e}{12 (3+2 x)^3}-\frac {e}{8 (3+2 x)^2} \]

[Out]

1/12*(-2*d+3*e)/(3+2*x)^3-1/8*e/(3+2*x)^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {27, 45} \[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^2} \, dx=-\frac {2 d-3 e}{12 (2 x+3)^3}-\frac {e}{8 (2 x+3)^2} \]

[In]

Int[(d + e*x)/(9 + 12*x + 4*x^2)^2,x]

[Out]

-1/12*(2*d - 3*e)/(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 45

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*} \text {integral}& = \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] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^2} \, dx=-\frac {4 d+3 e+6 e x}{24 (3+2 x)^3} \]

[In]

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

[Out]

-1/24*(4*d + 3*e + 6*e*x)/(3 + 2*x)^3

Maple [A] (verified)

Time = 2.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65

method	result	size
norman	\(\frac {-\frac {1}{4} e x -\frac {1}{8} e -\frac {1}{6} d}{\left (2 x +3\right )^{3}}\)	\(20\)
risch	\(\frac {-\frac {1}{4} e x -\frac {1}{8} e -\frac {1}{6} d}{\left (2 x +3\right )^{3}}\)	\(21\)
default	\(-\frac {e}{8 \left (2 x +3\right )^{2}}-\frac {\frac {d}{2}-\frac {3 e}{4}}{3 \left (2 x +3\right )^{3}}\)	\(28\)
gosper	\(-\frac {6 e x +4 d +3 e}{24 \left (2 x +3\right ) \left (4 x^{2}+12 x +9\right )}\)	\(33\)
parallelrisch	\(\frac {-6 e x -4 d -3 e}{24 \left (4 x^{2}+12 x +9\right ) \left (2 x +3\right )}\)	\(33\)
meijerg	\(\frac {e \,x^{2} \left (3+\frac {2 x}{3}\right )}{486 \left (1+\frac {2 x}{3}\right )^{3}}+\frac {d x \left (\frac {4}{9} x^{2}+2 x +3\right )}{243 \left (1+\frac {2 x}{3}\right )^{3}}\)	\(41\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^2} \, dx=-\frac {6 \, e x + 4 \, d + 3 \, e}{24 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^2} \, dx=\frac {- 4 d - 6 e x - 3 e}{192 x^{3} + 864 x^{2} + 1296 x + 648} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^2} \, dx=-\frac {6 \, e x + 4 \, d + 3 \, e}{24 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^2} \, dx=-\frac {6 \, e x + 4 \, d + 3 \, e}{24 \, {\left (2 \, x + 3\right )}^{3}} \]

[In]

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

[Out]

-1/24*(6*e*x + 4*d + 3*e)/(2*x + 3)^3

Mupad [B] (verification not implemented)

Time = 9.91 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {d+e x}{\left (9+12 x+4 x^2\right )^2} \, dx=-\frac {4\,d+3\,e+6\,e\,x}{24\,{\left (2\,x+3\right )}^3} \]

[In]

int((d + e*x)/(12*x + 4*x^2 + 9)^2,x)

[Out]

-(4*d + 3*e + 6*e*x)/(24*(2*x + 3)^3)