3.18 \(\int (e x)^m (A+B x^n) (c+d x^n)^3 \, dx\)
Optimal. Leaf size=137 \[ \frac{c^2 x^{n+1} (e x)^m (3 A d+B c)}{m+n+1}+\frac{d^2 x^{3 n+1} (e x)^m (A d+3 B c)}{m+3 n+1}+\frac{3 c d x^{2 n+1} (e x)^m (A d+B c)}{m+2 n+1}+\frac{A c^3 (e x)^{m+1}}{e (m+1)}+\frac{B d^3 x^{4 n+1} (e x)^m}{m+4 n+1} \]
[Out]
(c^2*(B*c + 3*A*d)*x^(1 + n)*(e*x)^m)/(1 + m + n) + (3*c*d*(B*c + A*d)*x^(1 + 2*n)*(e*x)^m)/(1 + m + 2*n) + (d
^2*(3*B*c + A*d)*x^(1 + 3*n)*(e*x)^m)/(1 + m + 3*n) + (B*d^3*x^(1 + 4*n)*(e*x)^m)/(1 + m + 4*n) + (A*c^3*(e*x)
^(1 + m))/(e*(1 + m))
________________________________________________________________________________________
Rubi [A] time = 0.110753, antiderivative size = 137, normalized size of antiderivative = 1.,
number of steps used = 10, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used
= {448, 20, 30} \[ \frac{c^2 x^{n+1} (e x)^m (3 A d+B c)}{m+n+1}+\frac{d^2 x^{3 n+1} (e x)^m (A d+3 B c)}{m+3 n+1}+\frac{3 c d x^{2 n+1} (e x)^m (A d+B c)}{m+2 n+1}+\frac{A c^3 (e x)^{m+1}}{e (m+1)}+\frac{B d^3 x^{4 n+1} (e x)^m}{m+4 n+1} \]
Antiderivative was successfully verified.
[In]
Int[(e*x)^m*(A + B*x^n)*(c + d*x^n)^3,x]
[Out]
(c^2*(B*c + 3*A*d)*x^(1 + n)*(e*x)^m)/(1 + m + n) + (3*c*d*(B*c + A*d)*x^(1 + 2*n)*(e*x)^m)/(1 + m + 2*n) + (d
^2*(3*B*c + A*d)*x^(1 + 3*n)*(e*x)^m)/(1 + m + 3*n) + (B*d^3*x^(1 + 4*n)*(e*x)^m)/(1 + m + 4*n) + (A*c^3*(e*x)
^(1 + m))/(e*(1 + m))
Rule 448
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]
Rule 20
Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] && !IntegerQ[m] && !IntegerQ[n]
&& !IntegerQ[m + n]
Rule 30
Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
Rubi steps
\begin{align*} \int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^3 \, dx &=\int \left (A c^3 (e x)^m+c^2 (B c+3 A d) x^n (e x)^m+3 c d (B c+A d) x^{2 n} (e x)^m+d^2 (3 B c+A d) x^{3 n} (e x)^m+B d^3 x^{4 n} (e x)^m\right ) \, dx\\ &=\frac{A c^3 (e x)^{1+m}}{e (1+m)}+\left (B d^3\right ) \int x^{4 n} (e x)^m \, dx+(3 c d (B c+A d)) \int x^{2 n} (e x)^m \, dx+\left (d^2 (3 B c+A d)\right ) \int x^{3 n} (e x)^m \, dx+\left (c^2 (B c+3 A d)\right ) \int x^n (e x)^m \, dx\\ &=\frac{A c^3 (e x)^{1+m}}{e (1+m)}+\left (B d^3 x^{-m} (e x)^m\right ) \int x^{m+4 n} \, dx+\left (3 c d (B c+A d) x^{-m} (e x)^m\right ) \int x^{m+2 n} \, dx+\left (d^2 (3 B c+A d) x^{-m} (e x)^m\right ) \int x^{m+3 n} \, dx+\left (c^2 (B c+3 A d) x^{-m} (e x)^m\right ) \int x^{m+n} \, dx\\ &=\frac{c^2 (B c+3 A d) x^{1+n} (e x)^m}{1+m+n}+\frac{3 c d (B c+A d) x^{1+2 n} (e x)^m}{1+m+2 n}+\frac{d^2 (3 B c+A d) x^{1+3 n} (e x)^m}{1+m+3 n}+\frac{B d^3 x^{1+4 n} (e x)^m}{1+m+4 n}+\frac{A c^3 (e x)^{1+m}}{e (1+m)}\\ \end{align*}
Mathematica [A] time = 0.14954, size = 106, normalized size = 0.77 \[ x (e x)^m \left (\frac{c^2 x^n (3 A d+B c)}{m+n+1}+\frac{d^2 x^{3 n} (A d+3 B c)}{m+3 n+1}+\frac{3 c d x^{2 n} (A d+B c)}{m+2 n+1}+\frac{A c^3}{m+1}+\frac{B d^3 x^{4 n}}{m+4 n+1}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[(e*x)^m*(A + B*x^n)*(c + d*x^n)^3,x]
[Out]
x*(e*x)^m*((A*c^3)/(1 + m) + (c^2*(B*c + 3*A*d)*x^n)/(1 + m + n) + (3*c*d*(B*c + A*d)*x^(2*n))/(1 + m + 2*n) +
(d^2*(3*B*c + A*d)*x^(3*n))/(1 + m + 3*n) + (B*d^3*x^(4*n))/(1 + m + 4*n))
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Maple [C] time = 0.067, size = 1609, normalized size = 11.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x)^m*(A+B*x^n)*(c+d*x^n)^3,x)
[Out]
x*(42*B*c*d^2*n^2*(x^n)^3+12*A*c^2*d*m^3*x^n+72*A*c^2*d*n^3*x^n+18*A*c*d^2*m^2*(x^n)^2+7*A*d^3*m^3*n*(x^n)^3+1
4*A*d^3*m^2*n^2*(x^n)^3+22*B*d^3*m*n^2*(x^n)^4+21*B*c*d^2*m^3*n*(x^n)^3+42*B*c*d^2*m^2*n^2*(x^n)^3+24*B*c*d^2*
m*n^3*(x^n)^3+24*A*c*d^2*m^3*n*(x^n)^2+57*A*c*d^2*m^2*n^2*(x^n)^2+A*c^3+12*A*c*d^2*(x^n)^2*m+24*A*c*d^2*(x^n)^
2*n+27*B*c^3*m*n*x^n+12*B*c^2*d*(x^n)^2*m+10*A*c^3*m^3*n+35*A*c^3*m^2*n^2+50*A*c^3*m*n^3+30*A*c^3*m^2*n+70*A*c
^3*m*n^2+30*A*c^3*m*n+8*A*d^3*m*n^3*(x^n)^3+3*B*c*d^2*m^4*(x^n)^3+18*B*d^3*m^2*n*(x^n)^4+6*B*d^3*m^3*n*(x^n)^4
+11*B*d^3*m^2*n^2*(x^n)^4+6*B*d^3*m*n^3*(x^n)^4+18*B*d^3*m*n*(x^n)^4+57*A*c*d^2*n^2*(x^n)^2+12*B*c*d^2*m^3*(x^
n)^3+24*B*c*d^2*n^3*(x^n)^3+3*A*c*d^2*m^4*(x^n)^2+21*A*d^3*m^2*n*(x^n)^3+28*A*d^3*m*n^2*(x^n)^3+3*B*c^2*d*m^4*
(x^n)^2+4*A*c^3*m+10*A*c^3*n+3*A*c^2*d*m^4*x^n+12*A*c*d^2*m^3*(x^n)^2+36*A*c*d^2*n^3*(x^n)^2+21*A*d^3*m*n*(x^n
)^3+(x^n)^4*B*d^3+x^n*B*c^3+(x^n)^3*A*d^3+A*c^3*m^4+24*A*c^3*n^4+4*A*c^3*m^3+50*A*c^3*n^3+6*A*c^3*m^2+35*A*c^3
*n^2+24*B*c^3*m*n^3*x^n+12*B*c*d^2*(x^n)^3*m+21*B*c*d^2*(x^n)^3*n+18*A*c^2*d*m^2*x^n+78*A*c^2*d*n^2*x^n+12*B*c
^2*d*m^3*(x^n)^2+27*B*c^3*m^2*n*x^n+52*B*c^3*m*n^2*x^n+18*B*c^2*d*m^2*(x^n)^2+57*B*c^2*d*n^2*(x^n)^2+4*A*d^3*(
x^n)^3*m+7*A*d^3*(x^n)^3*n+4*B*c^3*m^3*x^n+24*B*c^3*n^3*x^n+6*B*c^3*m^2*x^n+9*B*c^3*m^3*n*x^n+26*B*c^3*m^2*n^2
*x^n+4*B*d^3*m^3*(x^n)^4+6*B*d^3*n^3*(x^n)^4+4*A*d^3*m^3*(x^n)^3+8*A*d^3*n^3*(x^n)^3+6*B*d^3*m^2*(x^n)^4+B*c^3
*m^4*x^n+4*m*B*d^3*(x^n)^4+6*B*d^3*(x^n)^4*n+B*d^3*m^4*(x^n)^4+A*d^3*m^4*(x^n)^3+24*B*c^2*d*(x^n)^2*n+12*A*c^2
*d*x^n*m+27*A*c^2*d*x^n*n+72*A*c^2*d*m*n^3*x^n+72*A*c*d^2*m^2*n*(x^n)^2+114*A*c*d^2*m*n^2*(x^n)^2+72*B*c^2*d*m
^2*n*(x^n)^2+114*B*c^2*d*m*n^2*(x^n)^2+81*A*c^2*d*m*n*x^n+63*B*c*d^2*m*n*(x^n)^3+81*A*c^2*d*m^2*n*x^n+26*B*c^3
*n^2*x^n+3*B*c*d^2*(x^n)^3+3*A*c*d^2*(x^n)^2+4*B*c^3*x^n*m+9*B*c^3*x^n*n+3*B*c^2*d*(x^n)^2+3*A*c^2*d*x^n+11*B*
d^3*n^2*(x^n)^4+6*A*d^3*m^2*(x^n)^3+14*A*d^3*n^2*(x^n)^3+36*B*c^2*d*n^3*(x^n)^2+18*B*c*d^2*m^2*(x^n)^3+72*B*c^
2*d*m*n*(x^n)^2+36*A*c*d^2*m*n^3*(x^n)^2+24*B*c^2*d*m^3*n*(x^n)^2+57*B*c^2*d*m^2*n^2*(x^n)^2+36*B*c^2*d*m*n^3*
(x^n)^2+63*B*c*d^2*m^2*n*(x^n)^3+84*B*c*d^2*m*n^2*(x^n)^3+27*A*c^2*d*m^3*n*x^n+78*A*c^2*d*m^2*n^2*x^n+156*A*c^
2*d*m*n^2*x^n+72*A*c*d^2*m*n*(x^n)^2)/(1+m)/(m+n+1)/(1+m+2*n)/(1+m+3*n)/(1+m+4*n)*exp(1/2*m*(-I*Pi*csgn(I*e*x)
^3+I*Pi*csgn(I*e*x)^2*csgn(I*e)+I*Pi*csgn(I*e*x)^2*csgn(I*x)-I*Pi*csgn(I*e*x)*csgn(I*e)*csgn(I*x)+2*ln(e)+2*ln
(x)))
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(A+B*x^n)*(c+d*x^n)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [B] time = 1.16382, size = 2437, normalized size = 17.79 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(A+B*x^n)*(c+d*x^n)^3,x, algorithm="fricas")
[Out]
((B*d^3*m^4 + 4*B*d^3*m^3 + 6*B*d^3*m^2 + 4*B*d^3*m + B*d^3 + 6*(B*d^3*m + B*d^3)*n^3 + 11*(B*d^3*m^2 + 2*B*d^
3*m + B*d^3)*n^2 + 6*(B*d^3*m^3 + 3*B*d^3*m^2 + 3*B*d^3*m + B*d^3)*n)*x*x^(4*n)*e^(m*log(e) + m*log(x)) + ((3*
B*c*d^2 + A*d^3)*m^4 + 3*B*c*d^2 + A*d^3 + 4*(3*B*c*d^2 + A*d^3)*m^3 + 8*(3*B*c*d^2 + A*d^3 + (3*B*c*d^2 + A*d
^3)*m)*n^3 + 6*(3*B*c*d^2 + A*d^3)*m^2 + 14*(3*B*c*d^2 + A*d^3 + (3*B*c*d^2 + A*d^3)*m^2 + 2*(3*B*c*d^2 + A*d^
3)*m)*n^2 + 4*(3*B*c*d^2 + A*d^3)*m + 7*(3*B*c*d^2 + A*d^3 + (3*B*c*d^2 + A*d^3)*m^3 + 3*(3*B*c*d^2 + A*d^3)*m
^2 + 3*(3*B*c*d^2 + A*d^3)*m)*n)*x*x^(3*n)*e^(m*log(e) + m*log(x)) + 3*((B*c^2*d + A*c*d^2)*m^4 + B*c^2*d + A*
c*d^2 + 4*(B*c^2*d + A*c*d^2)*m^3 + 12*(B*c^2*d + A*c*d^2 + (B*c^2*d + A*c*d^2)*m)*n^3 + 6*(B*c^2*d + A*c*d^2)
*m^2 + 19*(B*c^2*d + A*c*d^2 + (B*c^2*d + A*c*d^2)*m^2 + 2*(B*c^2*d + A*c*d^2)*m)*n^2 + 4*(B*c^2*d + A*c*d^2)*
m + 8*(B*c^2*d + A*c*d^2 + (B*c^2*d + A*c*d^2)*m^3 + 3*(B*c^2*d + A*c*d^2)*m^2 + 3*(B*c^2*d + A*c*d^2)*m)*n)*x
*x^(2*n)*e^(m*log(e) + m*log(x)) + ((B*c^3 + 3*A*c^2*d)*m^4 + B*c^3 + 3*A*c^2*d + 4*(B*c^3 + 3*A*c^2*d)*m^3 +
24*(B*c^3 + 3*A*c^2*d + (B*c^3 + 3*A*c^2*d)*m)*n^3 + 6*(B*c^3 + 3*A*c^2*d)*m^2 + 26*(B*c^3 + 3*A*c^2*d + (B*c^
3 + 3*A*c^2*d)*m^2 + 2*(B*c^3 + 3*A*c^2*d)*m)*n^2 + 4*(B*c^3 + 3*A*c^2*d)*m + 9*(B*c^3 + 3*A*c^2*d + (B*c^3 +
3*A*c^2*d)*m^3 + 3*(B*c^3 + 3*A*c^2*d)*m^2 + 3*(B*c^3 + 3*A*c^2*d)*m)*n)*x*x^n*e^(m*log(e) + m*log(x)) + (A*c^
3*m^4 + 24*A*c^3*n^4 + 4*A*c^3*m^3 + 6*A*c^3*m^2 + 4*A*c^3*m + A*c^3 + 50*(A*c^3*m + A*c^3)*n^3 + 35*(A*c^3*m^
2 + 2*A*c^3*m + A*c^3)*n^2 + 10*(A*c^3*m^3 + 3*A*c^3*m^2 + 3*A*c^3*m + A*c^3)*n)*x*e^(m*log(e) + m*log(x)))/(m
^5 + 24*(m + 1)*n^4 + 5*m^4 + 50*(m^2 + 2*m + 1)*n^3 + 10*m^3 + 35*(m^3 + 3*m^2 + 3*m + 1)*n^2 + 10*m^2 + 10*(
m^4 + 4*m^3 + 6*m^2 + 4*m + 1)*n + 5*m + 1)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)**m*(A+B*x**n)*(c+d*x**n)**3,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.15775, size = 3075, normalized size = 22.45 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(A+B*x^n)*(c+d*x^n)^3,x, algorithm="giac")
[Out]
(B*d^3*m^4*x*x^m*x^(4*n)*e^m + 6*B*d^3*m^3*n*x*x^m*x^(4*n)*e^m + 11*B*d^3*m^2*n^2*x*x^m*x^(4*n)*e^m + 6*B*d^3*
m*n^3*x*x^m*x^(4*n)*e^m + 3*B*c*d^2*m^4*x*x^m*x^(3*n)*e^m + A*d^3*m^4*x*x^m*x^(3*n)*e^m + 21*B*c*d^2*m^3*n*x*x
^m*x^(3*n)*e^m + 7*A*d^3*m^3*n*x*x^m*x^(3*n)*e^m + 42*B*c*d^2*m^2*n^2*x*x^m*x^(3*n)*e^m + 14*A*d^3*m^2*n^2*x*x
^m*x^(3*n)*e^m + 24*B*c*d^2*m*n^3*x*x^m*x^(3*n)*e^m + 8*A*d^3*m*n^3*x*x^m*x^(3*n)*e^m + 3*B*c^2*d*m^4*x*x^m*x^
(2*n)*e^m + 3*A*c*d^2*m^4*x*x^m*x^(2*n)*e^m + 24*B*c^2*d*m^3*n*x*x^m*x^(2*n)*e^m + 24*A*c*d^2*m^3*n*x*x^m*x^(2
*n)*e^m + 57*B*c^2*d*m^2*n^2*x*x^m*x^(2*n)*e^m + 57*A*c*d^2*m^2*n^2*x*x^m*x^(2*n)*e^m + 36*B*c^2*d*m*n^3*x*x^m
*x^(2*n)*e^m + 36*A*c*d^2*m*n^3*x*x^m*x^(2*n)*e^m + B*c^3*m^4*x*x^m*x^n*e^m + 3*A*c^2*d*m^4*x*x^m*x^n*e^m + 9*
B*c^3*m^3*n*x*x^m*x^n*e^m + 27*A*c^2*d*m^3*n*x*x^m*x^n*e^m + 26*B*c^3*m^2*n^2*x*x^m*x^n*e^m + 78*A*c^2*d*m^2*n
^2*x*x^m*x^n*e^m + 24*B*c^3*m*n^3*x*x^m*x^n*e^m + 72*A*c^2*d*m*n^3*x*x^m*x^n*e^m + A*c^3*m^4*x*x^m*e^m + 10*A*
c^3*m^3*n*x*x^m*e^m + 35*A*c^3*m^2*n^2*x*x^m*e^m + 50*A*c^3*m*n^3*x*x^m*e^m + 24*A*c^3*n^4*x*x^m*e^m + 4*B*d^3
*m^3*x*x^m*x^(4*n)*e^m + 18*B*d^3*m^2*n*x*x^m*x^(4*n)*e^m + 22*B*d^3*m*n^2*x*x^m*x^(4*n)*e^m + 6*B*d^3*n^3*x*x
^m*x^(4*n)*e^m + 12*B*c*d^2*m^3*x*x^m*x^(3*n)*e^m + 4*A*d^3*m^3*x*x^m*x^(3*n)*e^m + 63*B*c*d^2*m^2*n*x*x^m*x^(
3*n)*e^m + 21*A*d^3*m^2*n*x*x^m*x^(3*n)*e^m + 84*B*c*d^2*m*n^2*x*x^m*x^(3*n)*e^m + 28*A*d^3*m*n^2*x*x^m*x^(3*n
)*e^m + 24*B*c*d^2*n^3*x*x^m*x^(3*n)*e^m + 8*A*d^3*n^3*x*x^m*x^(3*n)*e^m + 12*B*c^2*d*m^3*x*x^m*x^(2*n)*e^m +
12*A*c*d^2*m^3*x*x^m*x^(2*n)*e^m + 72*B*c^2*d*m^2*n*x*x^m*x^(2*n)*e^m + 72*A*c*d^2*m^2*n*x*x^m*x^(2*n)*e^m + 1
14*B*c^2*d*m*n^2*x*x^m*x^(2*n)*e^m + 114*A*c*d^2*m*n^2*x*x^m*x^(2*n)*e^m + 36*B*c^2*d*n^3*x*x^m*x^(2*n)*e^m +
36*A*c*d^2*n^3*x*x^m*x^(2*n)*e^m + 4*B*c^3*m^3*x*x^m*x^n*e^m + 12*A*c^2*d*m^3*x*x^m*x^n*e^m + 27*B*c^3*m^2*n*x
*x^m*x^n*e^m + 81*A*c^2*d*m^2*n*x*x^m*x^n*e^m + 52*B*c^3*m*n^2*x*x^m*x^n*e^m + 156*A*c^2*d*m*n^2*x*x^m*x^n*e^m
+ 24*B*c^3*n^3*x*x^m*x^n*e^m + 72*A*c^2*d*n^3*x*x^m*x^n*e^m + 4*A*c^3*m^3*x*x^m*e^m + 30*A*c^3*m^2*n*x*x^m*e^
m + 70*A*c^3*m*n^2*x*x^m*e^m + 50*A*c^3*n^3*x*x^m*e^m + 6*B*d^3*m^2*x*x^m*x^(4*n)*e^m + 18*B*d^3*m*n*x*x^m*x^(
4*n)*e^m + 11*B*d^3*n^2*x*x^m*x^(4*n)*e^m + 18*B*c*d^2*m^2*x*x^m*x^(3*n)*e^m + 6*A*d^3*m^2*x*x^m*x^(3*n)*e^m +
63*B*c*d^2*m*n*x*x^m*x^(3*n)*e^m + 21*A*d^3*m*n*x*x^m*x^(3*n)*e^m + 42*B*c*d^2*n^2*x*x^m*x^(3*n)*e^m + 14*A*d
^3*n^2*x*x^m*x^(3*n)*e^m + 18*B*c^2*d*m^2*x*x^m*x^(2*n)*e^m + 18*A*c*d^2*m^2*x*x^m*x^(2*n)*e^m + 72*B*c^2*d*m*
n*x*x^m*x^(2*n)*e^m + 72*A*c*d^2*m*n*x*x^m*x^(2*n)*e^m + 57*B*c^2*d*n^2*x*x^m*x^(2*n)*e^m + 57*A*c*d^2*n^2*x*x
^m*x^(2*n)*e^m + 6*B*c^3*m^2*x*x^m*x^n*e^m + 18*A*c^2*d*m^2*x*x^m*x^n*e^m + 27*B*c^3*m*n*x*x^m*x^n*e^m + 81*A*
c^2*d*m*n*x*x^m*x^n*e^m + 26*B*c^3*n^2*x*x^m*x^n*e^m + 78*A*c^2*d*n^2*x*x^m*x^n*e^m + 6*A*c^3*m^2*x*x^m*e^m +
30*A*c^3*m*n*x*x^m*e^m + 35*A*c^3*n^2*x*x^m*e^m + 4*B*d^3*m*x*x^m*x^(4*n)*e^m + 6*B*d^3*n*x*x^m*x^(4*n)*e^m +
12*B*c*d^2*m*x*x^m*x^(3*n)*e^m + 4*A*d^3*m*x*x^m*x^(3*n)*e^m + 21*B*c*d^2*n*x*x^m*x^(3*n)*e^m + 7*A*d^3*n*x*x^
m*x^(3*n)*e^m + 12*B*c^2*d*m*x*x^m*x^(2*n)*e^m + 12*A*c*d^2*m*x*x^m*x^(2*n)*e^m + 24*B*c^2*d*n*x*x^m*x^(2*n)*e
^m + 24*A*c*d^2*n*x*x^m*x^(2*n)*e^m + 4*B*c^3*m*x*x^m*x^n*e^m + 12*A*c^2*d*m*x*x^m*x^n*e^m + 9*B*c^3*n*x*x^m*x
^n*e^m + 27*A*c^2*d*n*x*x^m*x^n*e^m + 4*A*c^3*m*x*x^m*e^m + 10*A*c^3*n*x*x^m*e^m + B*d^3*x*x^m*x^(4*n)*e^m + 3
*B*c*d^2*x*x^m*x^(3*n)*e^m + A*d^3*x*x^m*x^(3*n)*e^m + 3*B*c^2*d*x*x^m*x^(2*n)*e^m + 3*A*c*d^2*x*x^m*x^(2*n)*e
^m + B*c^3*x*x^m*x^n*e^m + 3*A*c^2*d*x*x^m*x^n*e^m + A*c^3*x*x^m*e^m)/(m^5 + 10*m^4*n + 35*m^3*n^2 + 50*m^2*n^
3 + 24*m*n^4 + 5*m^4 + 40*m^3*n + 105*m^2*n^2 + 100*m*n^3 + 24*n^4 + 10*m^3 + 60*m^2*n + 105*m*n^2 + 50*n^3 +
10*m^2 + 40*m*n + 35*n^2 + 5*m + 10*n + 1)