\(\int (e x)^m (A+B x^2) (c+d x^2)^2 \, dx\) [11]

Optimal result

Integrand size = 22, antiderivative size = 91 \[ \int (e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\frac {A c^2 (e x)^{1+m}}{e (1+m)}+\frac {c (B c+2 A d) (e x)^{3+m}}{e^3 (3+m)}+\frac {d (2 B c+A d) (e x)^{5+m}}{e^5 (5+m)}+\frac {B d^2 (e x)^{7+m}}{e^7 (7+m)} \] Output:

A*c^2*(e*x)^(1+m)/e/(1+m)+c*(2*A*d+B*c)*(e*x)^(3+m)/e^3/(3+m)+d*(A*d+2*B*c 
)*(e*x)^(5+m)/e^5/(5+m)+B*d^2*(e*x)^(7+m)/e^7/(7+m)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.74 \[ \int (e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=x (e x)^m \left (\frac {A c^2}{1+m}+\frac {c (B c+2 A d) x^2}{3+m}+\frac {d (2 B c+A d) x^4}{5+m}+\frac {B d^2 x^6}{7+m}\right ) \] Input:

Integrate[(e*x)^m*(A + B*x^2)*(c + d*x^2)^2,x]
 

Output:

x*(e*x)^m*((A*c^2)/(1 + m) + (c*(B*c + 2*A*d)*x^2)/(3 + m) + (d*(2*B*c + A 
*d)*x^4)/(5 + m) + (B*d^2*x^6)/(7 + m))
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {355, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \left (A+B x^2\right ) \left (c+d x^2\right )^2 (e x)^m \, dx\)

\(\Big \downarrow \) 355

\(\displaystyle \int \left (\frac {d (e x)^{m+4} (A d+2 B c)}{e^4}+\frac {c (e x)^{m+2} (2 A d+B c)}{e^2}+A c^2 (e x)^m+\frac {B d^2 (e x)^{m+6}}{e^6}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d (e x)^{m+5} (A d+2 B c)}{e^5 (m+5)}+\frac {c (e x)^{m+3} (2 A d+B c)}{e^3 (m+3)}+\frac {A c^2 (e x)^{m+1}}{e (m+1)}+\frac {B d^2 (e x)^{m+7}}{e^7 (m+7)}\)

Input:

Int[(e*x)^m*(A + B*x^2)*(c + d*x^2)^2,x]
 

Output:

(A*c^2*(e*x)^(1 + m))/(e*(1 + m)) + (c*(B*c + 2*A*d)*(e*x)^(3 + m))/(e^3*( 
3 + m)) + (d*(2*B*c + A*d)*(e*x)^(5 + m))/(e^5*(5 + m)) + (B*d^2*(e*x)^(7 
+ m))/(e^7*(7 + m))
 

Defintions of rubi rules used

rule 355

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q 
_.), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(a + b*x^2)^p*(c + d*x^2)^q, 
x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] & 
& IGtQ[q, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.99

method	result	size
norman	\(\frac {A \,c^{2} x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}+\frac {B \,d^{2} x^{7} {\mathrm e}^{m \ln \left (e x \right )}}{7+m}+\frac {c \left (2 A d +B c \right ) x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}+\frac {d \left (A d +2 B c \right ) x^{5} {\mathrm e}^{m \ln \left (e x \right )}}{5+m}\)	\(90\)
gosper	\(\frac {x \left (B \,d^{2} m^{3} x^{6}+9 B \,d^{2} m^{2} x^{6}+A \,d^{2} m^{3} x^{4}+2 B c d \,m^{3} x^{4}+23 m \,x^{6} B \,d^{2}+11 A \,d^{2} m^{2} x^{4}+22 B c d \,m^{2} x^{4}+15 B \,d^{2} x^{6}+2 A c d \,m^{3} x^{2}+31 A \,d^{2} x^{4} m +B \,c^{2} m^{3} x^{2}+62 B c d \,x^{4} m +26 A c d \,m^{2} x^{2}+21 A \,d^{2} x^{4}+13 B \,c^{2} m^{2} x^{2}+42 B c d \,x^{4}+A \,c^{2} m^{3}+94 A c d \,x^{2} m +47 B \,c^{2} x^{2} m +15 A \,c^{2} m^{2}+70 A c d \,x^{2}+35 B \,c^{2} x^{2}+71 A \,c^{2} m +105 A \,c^{2}\right ) \left (e x \right )^{m}}{\left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\)	\(263\)
risch	\(\frac {x \left (B \,d^{2} m^{3} x^{6}+9 B \,d^{2} m^{2} x^{6}+A \,d^{2} m^{3} x^{4}+2 B c d \,m^{3} x^{4}+23 m \,x^{6} B \,d^{2}+11 A \,d^{2} m^{2} x^{4}+22 B c d \,m^{2} x^{4}+15 B \,d^{2} x^{6}+2 A c d \,m^{3} x^{2}+31 A \,d^{2} x^{4} m +B \,c^{2} m^{3} x^{2}+62 B c d \,x^{4} m +26 A c d \,m^{2} x^{2}+21 A \,d^{2} x^{4}+13 B \,c^{2} m^{2} x^{2}+42 B c d \,x^{4}+A \,c^{2} m^{3}+94 A c d \,x^{2} m +47 B \,c^{2} x^{2} m +15 A \,c^{2} m^{2}+70 A c d \,x^{2}+35 B \,c^{2} x^{2}+71 A \,c^{2} m +105 A \,c^{2}\right ) \left (e x \right )^{m}}{\left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\)	\(263\)
orering	\(\frac {x \left (B \,d^{2} m^{3} x^{6}+9 B \,d^{2} m^{2} x^{6}+A \,d^{2} m^{3} x^{4}+2 B c d \,m^{3} x^{4}+23 m \,x^{6} B \,d^{2}+11 A \,d^{2} m^{2} x^{4}+22 B c d \,m^{2} x^{4}+15 B \,d^{2} x^{6}+2 A c d \,m^{3} x^{2}+31 A \,d^{2} x^{4} m +B \,c^{2} m^{3} x^{2}+62 B c d \,x^{4} m +26 A c d \,m^{2} x^{2}+21 A \,d^{2} x^{4}+13 B \,c^{2} m^{2} x^{2}+42 B c d \,x^{4}+A \,c^{2} m^{3}+94 A c d \,x^{2} m +47 B \,c^{2} x^{2} m +15 A \,c^{2} m^{2}+70 A c d \,x^{2}+35 B \,c^{2} x^{2}+71 A \,c^{2} m +105 A \,c^{2}\right ) \left (e x \right )^{m}}{\left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\)	\(263\)
parallelrisch	\(\frac {62 B \,x^{5} \left (e x \right )^{m} c d m +26 A \,x^{3} \left (e x \right )^{m} c d \,m^{2}+94 A \,x^{3} \left (e x \right )^{m} c d m +22 B \,x^{5} \left (e x \right )^{m} c d \,m^{2}+2 A \,x^{3} \left (e x \right )^{m} c d \,m^{3}+11 A \,x^{5} \left (e x \right )^{m} d^{2} m^{2}+31 A \,x^{5} \left (e x \right )^{m} d^{2} m +B \,x^{3} \left (e x \right )^{m} c^{2} m^{3}+42 B \,x^{5} \left (e x \right )^{m} c d +13 B \,x^{3} \left (e x \right )^{m} c^{2} m^{2}+A x \left (e x \right )^{m} c^{2} m^{3}+47 B \,x^{3} \left (e x \right )^{m} c^{2} m +70 A \,x^{3} \left (e x \right )^{m} c d +2 B \,x^{5} \left (e x \right )^{m} c d \,m^{3}+B \,x^{7} \left (e x \right )^{m} d^{2} m^{3}+9 B \,x^{7} \left (e x \right )^{m} d^{2} m^{2}+A \,x^{5} \left (e x \right )^{m} d^{2} m^{3}+23 B \,x^{7} \left (e x \right )^{m} d^{2} m +15 A x \left (e x \right )^{m} c^{2} m^{2}+71 A x \left (e x \right )^{m} c^{2} m +15 B \,x^{7} \left (e x \right )^{m} d^{2}+21 A \,x^{5} \left (e x \right )^{m} d^{2}+35 B \,x^{3} \left (e x \right )^{m} c^{2}+105 A x \left (e x \right )^{m} c^{2}}{\left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\)	\(381\)

Input:

int((e*x)^m*(B*x^2+A)*(d*x^2+c)^2,x,method=_RETURNVERBOSE)
 

Output:

A*c^2/(1+m)*x*exp(m*ln(e*x))+B*d^2/(7+m)*x^7*exp(m*ln(e*x))+c*(2*A*d+B*c)/ 
(3+m)*x^3*exp(m*ln(e*x))+d*(A*d+2*B*c)/(5+m)*x^5*exp(m*ln(e*x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (91) = 182\).

Time = 0.08 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.38 \[ \int (e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\frac {{\left ({\left (B d^{2} m^{3} + 9 \, B d^{2} m^{2} + 23 \, B d^{2} m + 15 \, B d^{2}\right )} x^{7} + {\left ({\left (2 \, B c d + A d^{2}\right )} m^{3} + 42 \, B c d + 21 \, A d^{2} + 11 \, {\left (2 \, B c d + A d^{2}\right )} m^{2} + 31 \, {\left (2 \, B c d + A d^{2}\right )} m\right )} x^{5} + {\left ({\left (B c^{2} + 2 \, A c d\right )} m^{3} + 35 \, B c^{2} + 70 \, A c d + 13 \, {\left (B c^{2} + 2 \, A c d\right )} m^{2} + 47 \, {\left (B c^{2} + 2 \, A c d\right )} m\right )} x^{3} + {\left (A c^{2} m^{3} + 15 \, A c^{2} m^{2} + 71 \, A c^{2} m + 105 \, A c^{2}\right )} x\right )} \left (e x\right )^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \] Input:

integrate((e*x)^m*(B*x^2+A)*(d*x^2+c)^2,x, algorithm="fricas")
 

Output:

((B*d^2*m^3 + 9*B*d^2*m^2 + 23*B*d^2*m + 15*B*d^2)*x^7 + ((2*B*c*d + A*d^2 
)*m^3 + 42*B*c*d + 21*A*d^2 + 11*(2*B*c*d + A*d^2)*m^2 + 31*(2*B*c*d + A*d 
^2)*m)*x^5 + ((B*c^2 + 2*A*c*d)*m^3 + 35*B*c^2 + 70*A*c*d + 13*(B*c^2 + 2* 
A*c*d)*m^2 + 47*(B*c^2 + 2*A*c*d)*m)*x^3 + (A*c^2*m^3 + 15*A*c^2*m^2 + 71* 
A*c^2*m + 105*A*c^2)*x)*(e*x)^m/(m^4 + 16*m^3 + 86*m^2 + 176*m + 105)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1096 vs. \(2 (82) = 164\).

Time = 0.57 (sec) , antiderivative size = 1096, normalized size of antiderivative = 12.04 \[ \int (e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\text {Too large to display} \] Input:

integrate((e*x)**m*(B*x**2+A)*(d*x**2+c)**2,x)
 

Output:

Piecewise(((-A*c**2/(6*x**6) - A*c*d/(2*x**4) - A*d**2/(2*x**2) - B*c**2/( 
4*x**4) - B*c*d/x**2 + B*d**2*log(x))/e**7, Eq(m, -7)), ((-A*c**2/(4*x**4) 
 - A*c*d/x**2 + A*d**2*log(x) - B*c**2/(2*x**2) + 2*B*c*d*log(x) + B*d**2* 
x**2/2)/e**5, Eq(m, -5)), ((-A*c**2/(2*x**2) + 2*A*c*d*log(x) + A*d**2*x** 
2/2 + B*c**2*log(x) + B*c*d*x**2 + B*d**2*x**4/4)/e**3, Eq(m, -3)), ((A*c* 
*2*log(x) + A*c*d*x**2 + A*d**2*x**4/4 + B*c**2*x**2/2 + B*c*d*x**4/2 + B* 
d**2*x**6/6)/e, Eq(m, -1)), (A*c**2*m**3*x*(e*x)**m/(m**4 + 16*m**3 + 86*m 
**2 + 176*m + 105) + 15*A*c**2*m**2*x*(e*x)**m/(m**4 + 16*m**3 + 86*m**2 + 
 176*m + 105) + 71*A*c**2*m*x*(e*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 
 105) + 105*A*c**2*x*(e*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 2 
*A*c*d*m**3*x**3*(e*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 26*A* 
c*d*m**2*x**3*(e*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 94*A*c*d 
*m*x**3*(e*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 70*A*c*d*x**3* 
(e*x)**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + A*d**2*m**3*x**5*(e*x) 
**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 11*A*d**2*m**2*x**5*(e*x)** 
m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 31*A*d**2*m*x**5*(e*x)**m/(m* 
*4 + 16*m**3 + 86*m**2 + 176*m + 105) + 21*A*d**2*x**5*(e*x)**m/(m**4 + 16 
*m**3 + 86*m**2 + 176*m + 105) + B*c**2*m**3*x**3*(e*x)**m/(m**4 + 16*m**3 
 + 86*m**2 + 176*m + 105) + 13*B*c**2*m**2*x**3*(e*x)**m/(m**4 + 16*m**3 + 
 86*m**2 + 176*m + 105) + 47*B*c**2*m*x**3*(e*x)**m/(m**4 + 16*m**3 + 8...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.27 \[ \int (e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\frac {B d^{2} e^{m} x^{7} x^{m}}{m + 7} + \frac {2 \, B c d e^{m} x^{5} x^{m}}{m + 5} + \frac {A d^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {B c^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {2 \, A c d e^{m} x^{3} x^{m}}{m + 3} + \frac {\left (e x\right )^{m + 1} A c^{2}}{e {\left (m + 1\right )}} \] Input:

integrate((e*x)^m*(B*x^2+A)*(d*x^2+c)^2,x, algorithm="maxima")
 

Output:

B*d^2*e^m*x^7*x^m/(m + 7) + 2*B*c*d*e^m*x^5*x^m/(m + 5) + A*d^2*e^m*x^5*x^ 
m/(m + 5) + B*c^2*e^m*x^3*x^m/(m + 3) + 2*A*c*d*e^m*x^3*x^m/(m + 3) + (e*x 
)^(m + 1)*A*c^2/(e*(m + 1))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (91) = 182\).

Time = 0.13 (sec) , antiderivative size = 380, normalized size of antiderivative = 4.18 \[ \int (e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\frac {\left (e x\right )^{m} B d^{2} m^{3} x^{7} + 9 \, \left (e x\right )^{m} B d^{2} m^{2} x^{7} + 2 \, \left (e x\right )^{m} B c d m^{3} x^{5} + \left (e x\right )^{m} A d^{2} m^{3} x^{5} + 23 \, \left (e x\right )^{m} B d^{2} m x^{7} + 22 \, \left (e x\right )^{m} B c d m^{2} x^{5} + 11 \, \left (e x\right )^{m} A d^{2} m^{2} x^{5} + 15 \, \left (e x\right )^{m} B d^{2} x^{7} + \left (e x\right )^{m} B c^{2} m^{3} x^{3} + 2 \, \left (e x\right )^{m} A c d m^{3} x^{3} + 62 \, \left (e x\right )^{m} B c d m x^{5} + 31 \, \left (e x\right )^{m} A d^{2} m x^{5} + 13 \, \left (e x\right )^{m} B c^{2} m^{2} x^{3} + 26 \, \left (e x\right )^{m} A c d m^{2} x^{3} + 42 \, \left (e x\right )^{m} B c d x^{5} + 21 \, \left (e x\right )^{m} A d^{2} x^{5} + \left (e x\right )^{m} A c^{2} m^{3} x + 47 \, \left (e x\right )^{m} B c^{2} m x^{3} + 94 \, \left (e x\right )^{m} A c d m x^{3} + 15 \, \left (e x\right )^{m} A c^{2} m^{2} x + 35 \, \left (e x\right )^{m} B c^{2} x^{3} + 70 \, \left (e x\right )^{m} A c d x^{3} + 71 \, \left (e x\right )^{m} A c^{2} m x + 105 \, \left (e x\right )^{m} A c^{2} x}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \] Input:

integrate((e*x)^m*(B*x^2+A)*(d*x^2+c)^2,x, algorithm="giac")
 

Output:

((e*x)^m*B*d^2*m^3*x^7 + 9*(e*x)^m*B*d^2*m^2*x^7 + 2*(e*x)^m*B*c*d*m^3*x^5 
 + (e*x)^m*A*d^2*m^3*x^5 + 23*(e*x)^m*B*d^2*m*x^7 + 22*(e*x)^m*B*c*d*m^2*x 
^5 + 11*(e*x)^m*A*d^2*m^2*x^5 + 15*(e*x)^m*B*d^2*x^7 + (e*x)^m*B*c^2*m^3*x 
^3 + 2*(e*x)^m*A*c*d*m^3*x^3 + 62*(e*x)^m*B*c*d*m*x^5 + 31*(e*x)^m*A*d^2*m 
*x^5 + 13*(e*x)^m*B*c^2*m^2*x^3 + 26*(e*x)^m*A*c*d*m^2*x^3 + 42*(e*x)^m*B* 
c*d*x^5 + 21*(e*x)^m*A*d^2*x^5 + (e*x)^m*A*c^2*m^3*x + 47*(e*x)^m*B*c^2*m* 
x^3 + 94*(e*x)^m*A*c*d*m*x^3 + 15*(e*x)^m*A*c^2*m^2*x + 35*(e*x)^m*B*c^2*x 
^3 + 70*(e*x)^m*A*c*d*x^3 + 71*(e*x)^m*A*c^2*m*x + 105*(e*x)^m*A*c^2*x)/(m 
^4 + 16*m^3 + 86*m^2 + 176*m + 105)
 

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.97 \[ \int (e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx={\left (e\,x\right )}^m\,\left (\frac {B\,d^2\,x^7\,\left (m^3+9\,m^2+23\,m+15\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {A\,c^2\,x\,\left (m^3+15\,m^2+71\,m+105\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {c\,x^3\,\left (2\,A\,d+B\,c\right )\,\left (m^3+13\,m^2+47\,m+35\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {d\,x^5\,\left (A\,d+2\,B\,c\right )\,\left (m^3+11\,m^2+31\,m+21\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}\right ) \] Input:

int((A + B*x^2)*(e*x)^m*(c + d*x^2)^2,x)
 

Output:

(e*x)^m*((B*d^2*x^7*(23*m + 9*m^2 + m^3 + 15))/(176*m + 86*m^2 + 16*m^3 + 
m^4 + 105) + (A*c^2*x*(71*m + 15*m^2 + m^3 + 105))/(176*m + 86*m^2 + 16*m^ 
3 + m^4 + 105) + (c*x^3*(2*A*d + B*c)*(47*m + 13*m^2 + m^3 + 35))/(176*m + 
 86*m^2 + 16*m^3 + m^4 + 105) + (d*x^5*(A*d + 2*B*c)*(31*m + 11*m^2 + m^3 
+ 21))/(176*m + 86*m^2 + 16*m^3 + m^4 + 105))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 263, normalized size of antiderivative = 2.89 \[ \int (e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\frac {x^{m} e^{m} x \left (b \,d^{2} m^{3} x^{6}+9 b \,d^{2} m^{2} x^{6}+a \,d^{2} m^{3} x^{4}+2 b c d \,m^{3} x^{4}+23 b \,d^{2} m \,x^{6}+11 a \,d^{2} m^{2} x^{4}+22 b c d \,m^{2} x^{4}+15 b \,d^{2} x^{6}+2 a c d \,m^{3} x^{2}+31 a \,d^{2} m \,x^{4}+b \,c^{2} m^{3} x^{2}+62 b c d m \,x^{4}+26 a c d \,m^{2} x^{2}+21 a \,d^{2} x^{4}+13 b \,c^{2} m^{2} x^{2}+42 b c d \,x^{4}+a \,c^{2} m^{3}+94 a c d m \,x^{2}+47 b \,c^{2} m \,x^{2}+15 a \,c^{2} m^{2}+70 a c d \,x^{2}+35 b \,c^{2} x^{2}+71 a \,c^{2} m +105 a \,c^{2}\right )}{m^{4}+16 m^{3}+86 m^{2}+176 m +105} \] Input:

int((e*x)^m*(B*x^2+A)*(d*x^2+c)^2,x)
 

Output:

(x**m*e**m*x*(a*c**2*m**3 + 15*a*c**2*m**2 + 71*a*c**2*m + 105*a*c**2 + 2* 
a*c*d*m**3*x**2 + 26*a*c*d*m**2*x**2 + 94*a*c*d*m*x**2 + 70*a*c*d*x**2 + a 
*d**2*m**3*x**4 + 11*a*d**2*m**2*x**4 + 31*a*d**2*m*x**4 + 21*a*d**2*x**4 
+ b*c**2*m**3*x**2 + 13*b*c**2*m**2*x**2 + 47*b*c**2*m*x**2 + 35*b*c**2*x* 
*2 + 2*b*c*d*m**3*x**4 + 22*b*c*d*m**2*x**4 + 62*b*c*d*m*x**4 + 42*b*c*d*x 
**4 + b*d**2*m**3*x**6 + 9*b*d**2*m**2*x**6 + 23*b*d**2*m*x**6 + 15*b*d**2 
*x**6))/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105)