Integrand size = 22, antiderivative size = 137 \[ \int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^3 \, dx=\frac {c^2 (B c+3 A d) x^{1+n} (e x)^m}{1+m+n}+\frac {3 c d (B c+A d) x^{1+2 n} (e x)^m}{1+m+2 n}+\frac {d^2 (3 B c+A d) x^{1+3 n} (e x)^m}{1+m+3 n}+\frac {B d^3 x^{1+4 n} (e x)^m}{1+m+4 n}+\frac {A c^3 (e x)^{1+m}}{e (1+m)} \]
c^2*(3*A*d+B*c)*x^(1+n)*(e*x)^m/(1+m+n)+3*c*d*(A*d+B*c)*x^(1+2*n)*(e*x)^m/ (1+m+2*n)+d^2*(A*d+3*B*c)*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)
Time = 0.22 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.77 \[ \int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^3 \, dx=x (e x)^m \left (\frac {A c^3}{1+m}+\frac {c^2 (B c+3 A d) x^n}{1+m+n}+\frac {3 c d (B c+A d) x^{2 n}}{1+m+2 n}+\frac {d^2 (3 B c+A d) x^{3 n}}{1+m+3 n}+\frac {B d^3 x^{4 n}}{1+m+4 n}\right ) \]
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))
Time = 0.30 (sec) , antiderivative size = 137, 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 = {950, 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.
\(\displaystyle \int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^3 \, dx\) |
\(\Big \downarrow \) 950 |
\(\displaystyle \int \left (c^2 x^n (e x)^m (3 A d+B c)+d^2 x^{3 n} (e x)^m (A d+3 B c)+3 c d x^{2 n} (e x)^m (A d+B c)+A c^3 (e x)^m+B d^3 x^{4 n} (e x)^m\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {c^2 x^{n+1} (e x)^m (3 A d+B c)}{m+n+1}+\frac {d^2 x^{3 n+1} (e x)^m (A d+3 B c)}{m+3 n+1}+\frac {3 c d x^{2 n+1} (e x)^m (A d+B c)}{m+2 n+1}+\frac {A c^3 (e x)^{m+1}}{e (m+1)}+\frac {B d^3 x^{4 n+1} (e x)^m}{m+4 n+1}\) |
(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))
3.1.18.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(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] && IGt Q[p, 0] && IGtQ[q, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.43 (sec) , antiderivative size = 1576, normalized size of antiderivative = 11.50
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1576\) |
parallelrisch | \(\text {Expression too large to display}\) | \(2207\) |
x*(B*d^3*m^4*(x^n)^4+63*B*c*d^2*m*n*(x^n)^3+114*A*c*d^2*m*n^2*(x^n)^2+57*B *c^2*d*m^2*n^2*(x^n)^2+72*B*c^2*d*m^2*n*(x^n)^2+24*A*c*d^2*m^3*n*(x^n)^2+1 2*A*c*d^2*m^3*(x^n)^2+81*A*c^2*d*m^2*n*x^n+57*A*c*d^2*m^2*n^2*(x^n)^2+36*A *c*d^2*m*n^3*(x^n)^2+9*B*c^3*x^n*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+24*B*c*d^2*m*n^3*(x^n)^3+57*A*c*d^2*n^2*(x^n)^2+10*A* c^3*m^3*n+12*A*c^2*d*m^3*x^n+A*c^3+72*B*c^2*d*m*n*(x^n)^2+81*A*c^2*d*m*n*x ^n+36*B*c^2*d*m*n^3*(x^n)^2+156*A*c^2*d*m*n^2*x^n+78*A*c^2*d*m^2*n^2*x^n+4 *B*c^3*x^n*m+35*A*c^3*n^2+35*A*c^3*m^2*n^2+24*B*c^2*d*(x^n)^2*n+42*B*c*d^2 *n^2*(x^n)^3+3*(x^n)^2*A*c*d^2+4*A*c^3*m+10*A*c^3*n+6*A*d^3*m^2*(x^n)^3+14 *A*d^3*n^2*(x^n)^3+B*c^3*m^4*x^n+21*B*c*d^2*m^3*n*(x^n)^3+50*A*c^3*n^3+(x^ n)^3*A*d^3+3*A*c^2*d*x^n+6*B*d^3*(x^n)^4*n+12*B*c^2*d*m^3*(x^n)^2+36*B*c^2 *d*n^3*(x^n)^2+A*d^3*m^4*(x^n)^3+50*A*c^3*m*n^3+84*B*c*d^2*m*n^2*(x^n)^3+3 0*A*c^3*m^2*n+70*A*c^3*m*n^2+30*A*c^3*m*n+6*B*c^3*m^2*x^n+A*c^3*m^4+3*B*c* d^2*(x^n)^3+72*A*c*d^2*m*n*(x^n)^2+72*A*c*d^2*m^2*n*(x^n)^2+3*B*c*d^2*m^4* (x^n)^3+36*A*c*d^2*n^3*(x^n)^2+18*B*c*d^2*m^2*(x^n)^3+42*B*c*d^2*m^2*n^2*( x^n)^3+24*B*c^2*d*m^3*n*(x^n)^2+114*B*c^2*d*m*n^2*(x^n)^2+24*A*c^3*n^4+4*A *c^3*m^3+21*A*d^3*m*n*(x^n)^3+63*B*c*d^2*m^2*n*(x^n)^3+6*A*c^3*m^2+9*B*c^3 *m^3*n*x^n+27*A*c^2*d*m^3*n*x^n+72*A*c^2*d*m*n^3*x^n+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+B*d^3*(x^n)^4+26*B*c^3*n^2*x^n+22*B* d^3*m*n^2*(x^n)^4+3*A*c*d^2*m^4*(x^n)^2+x^n*B*c^3+4*B*c^3*m^3*x^n+72*A*...
Leaf count of result is larger than twice the leaf count of optimal. 1104 vs. \(2 (137) = 274\).
Time = 0.31 (sec) , antiderivative size = 1104, normalized size of antiderivative = 8.06 \[ \int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^3 \, dx=\text {Too large to display} \]
((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*lo g(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^...
Leaf count of result is larger than twice the leaf count of optimal. 16781 vs. \(2 (128) = 256\).
Time = 5.49 (sec) , antiderivative size = 16781, normalized size of antiderivative = 122.49 \[ \int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^3 \, dx=\text {Too large to display} \]
Piecewise(((A + B)*(c + d)**3*log(x)/e, Eq(m, -1) & Eq(n, 0)), ((A*c**3*lo g(x) + 3*A*c**2*d*x**n/n + 3*A*c*d**2*x**(2*n)/(2*n) + A*d**3*x**(3*n)/(3* n) + B*c**3*x**n/n + 3*B*c**2*d*x**(2*n)/(2*n) + B*c*d**2*x**(3*n)/n + B*d **3*x**(4*n)/(4*n))/e, Eq(m, -1)), (A*c**3*Piecewise((0**(-4*n - 1)*x, Eq( e, 0)), (Piecewise((-1/(4*n*(e*x)**(4*n)), Ne(n, 0)), (log(e*x), True))/e, True)) + 3*A*c**2*d*Piecewise((-x*x**n*(e*x)**(-4*n - 1)/(3*n), Ne(n, 0)) , (x*x**n*(e*x)**(-4*n - 1)*log(x), True)) + 3*A*c*d**2*Piecewise((-x*x**( 2*n)*(e*x)**(-4*n - 1)/(2*n), Ne(n, 0)), (x*x**(2*n)*(e*x)**(-4*n - 1)*log (x), True)) + A*d**3*Piecewise((-x*x**(3*n)*(e*x)**(-4*n - 1)/n, Ne(n, 0)) , (x*x**(3*n)*(e*x)**(-4*n - 1)*log(x), True)) + B*c**3*Piecewise((-x*x**n *(e*x)**(-4*n - 1)/(3*n), Ne(n, 0)), (x*x**n*(e*x)**(-4*n - 1)*log(x), Tru e)) + 3*B*c**2*d*Piecewise((-x*x**(2*n)*(e*x)**(-4*n - 1)/(2*n), Ne(n, 0)) , (x*x**(2*n)*(e*x)**(-4*n - 1)*log(x), True)) + 3*B*c*d**2*Piecewise((-x* x**(3*n)*(e*x)**(-4*n - 1)/n, Ne(n, 0)), (x*x**(3*n)*(e*x)**(-4*n - 1)*log (x), True)) + B*d**3*x*x**(4*n)*(e*x)**(-4*n - 1)*log(x), Eq(m, -4*n - 1)) , (A*c**3*Piecewise((0**(-3*n - 1)*x, Eq(e, 0)), (Piecewise((-1/(3*n*(e*x) **(3*n)), Ne(n, 0)), (log(e*x), True))/e, True)) + 3*A*c**2*d*Piecewise((- x*x**n*(e*x)**(-3*n - 1)/(2*n), Ne(n, 0)), (x*x**n*(e*x)**(-3*n - 1)*log(x ), True)) + 3*A*c*d**2*Piecewise((-x*x**(2*n)*(e*x)**(-3*n - 1)/n, Ne(n, 0 )), (x*x**(2*n)*(e*x)**(-3*n - 1)*log(x), True)) + A*d**3*x*x**(3*n)*(e...
Time = 0.21 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.60 \[ \int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^3 \, dx=\frac {B d^{3} e^{m} x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right )\right )}}{m + 4 \, n + 1} + \frac {3 \, B c d^{2} e^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 1} + \frac {A d^{3} e^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 1} + \frac {3 \, B c^{2} d e^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {3 \, A c d^{2} e^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {B c^{3} e^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {3 \, A c^{2} d e^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {\left (e x\right )^{m + 1} A c^{3}}{e {\left (m + 1\right )}} \]
B*d^3*e^m*x*e^(m*log(x) + 4*n*log(x))/(m + 4*n + 1) + 3*B*c*d^2*e^m*x*e^(m *log(x) + 3*n*log(x))/(m + 3*n + 1) + A*d^3*e^m*x*e^(m*log(x) + 3*n*log(x) )/(m + 3*n + 1) + 3*B*c^2*d*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + 3*A*c*d^2*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + B*c^3*e^m*x*e^ (m*log(x) + n*log(x))/(m + n + 1) + 3*A*c^2*d*e^m*x*e^(m*log(x) + n*log(x) )/(m + n + 1) + (e*x)^(m + 1)*A*c^3/(e*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 7893 vs. \(2 (137) = 274\).
Time = 0.32 (sec) , antiderivative size = 7893, normalized size of antiderivative = 57.61 \[ \int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^3 \, dx=\text {Too large to display} \]
(B*d^3*m^4*x*x^(4*n)*e^(m*log(e) + m*log(x)) + 6*B*d^3*m^3*n*x*x^(4*n)*e^( m*log(e) + m*log(x)) + 11*B*d^3*m^2*n^2*x*x^(4*n)*e^(m*log(e) + m*log(x)) + 6*B*d^3*m*n^3*x*x^(4*n)*e^(m*log(e) + m*log(x)) + 3*B*c*d^2*m^4*x*x^(3*n )*e^(m*log(e) + m*log(x)) + A*d^3*m^4*x*x^(3*n)*e^(m*log(e) + m*log(x)) + B*d^3*m^4*x*x^(3*n)*e^(m*log(e) + m*log(x)) + 21*B*c*d^2*m^3*n*x*x^(3*n)*e ^(m*log(e) + m*log(x)) + 7*A*d^3*m^3*n*x*x^(3*n)*e^(m*log(e) + m*log(x)) + 6*B*d^3*m^3*n*x*x^(3*n)*e^(m*log(e) + m*log(x)) + 42*B*c*d^2*m^2*n^2*x*x^ (3*n)*e^(m*log(e) + m*log(x)) + 14*A*d^3*m^2*n^2*x*x^(3*n)*e^(m*log(e) + m *log(x)) + 11*B*d^3*m^2*n^2*x*x^(3*n)*e^(m*log(e) + m*log(x)) + 24*B*c*d^2 *m*n^3*x*x^(3*n)*e^(m*log(e) + m*log(x)) + 8*A*d^3*m*n^3*x*x^(3*n)*e^(m*lo g(e) + m*log(x)) + 6*B*d^3*m*n^3*x*x^(3*n)*e^(m*log(e) + m*log(x)) + 3*B*c ^2*d*m^4*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 3*A*c*d^2*m^4*x*x^(2*n)*e^(m* log(e) + m*log(x)) + 3*B*c*d^2*m^4*x*x^(2*n)*e^(m*log(e) + m*log(x)) + A*d ^3*m^4*x*x^(2*n)*e^(m*log(e) + m*log(x)) + B*d^3*m^4*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 24*B*c^2*d*m^3*n*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 24*A*c *d^2*m^3*n*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 21*B*c*d^2*m^3*n*x*x^(2*n)* e^(m*log(e) + m*log(x)) + 7*A*d^3*m^3*n*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 6*B*d^3*m^3*n*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 57*B*c^2*d*m^2*n^2*x*x ^(2*n)*e^(m*log(e) + m*log(x)) + 57*A*c*d^2*m^2*n^2*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 42*B*c*d^2*m^2*n^2*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 14...
Time = 9.29 (sec) , antiderivative size = 563, normalized size of antiderivative = 4.11 \[ \int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^3 \, dx=\frac {A\,c^3\,x\,{\left (e\,x\right )}^m}{m+1}+\frac {d^2\,x\,x^{3\,n}\,{\left (e\,x\right )}^m\,\left (A\,d+3\,B\,c\right )\,\left (m^3+7\,m^2\,n+3\,m^2+14\,m\,n^2+14\,m\,n+3\,m+8\,n^3+14\,n^2+7\,n+1\right )}{m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1}+\frac {c^2\,x\,x^n\,{\left (e\,x\right )}^m\,\left (3\,A\,d+B\,c\right )\,\left (m^3+9\,m^2\,n+3\,m^2+26\,m\,n^2+18\,m\,n+3\,m+24\,n^3+26\,n^2+9\,n+1\right )}{m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1}+\frac {B\,d^3\,x\,x^{4\,n}\,{\left (e\,x\right )}^m\,\left (m^3+6\,m^2\,n+3\,m^2+11\,m\,n^2+12\,m\,n+3\,m+6\,n^3+11\,n^2+6\,n+1\right )}{m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1}+\frac {3\,c\,d\,x\,x^{2\,n}\,{\left (e\,x\right )}^m\,\left (A\,d+B\,c\right )\,\left (m^3+8\,m^2\,n+3\,m^2+19\,m\,n^2+16\,m\,n+3\,m+12\,n^3+19\,n^2+8\,n+1\right )}{m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1} \]
(A*c^3*x*(e*x)^m)/(m + 1) + (d^2*x*x^(3*n)*(e*x)^m*(A*d + 3*B*c)*(3*m + 7* n + 14*m*n + 14*m*n^2 + 7*m^2*n + 3*m^2 + m^3 + 14*n^2 + 8*n^3 + 1))/(4*m + 10*n + 30*m*n + 70*m*n^2 + 30*m^2*n + 50*m*n^3 + 10*m^3*n + 6*m^2 + 4*m^ 3 + m^4 + 35*n^2 + 50*n^3 + 24*n^4 + 35*m^2*n^2 + 1) + (c^2*x*x^n*(e*x)^m* (3*A*d + B*c)*(3*m + 9*n + 18*m*n + 26*m*n^2 + 9*m^2*n + 3*m^2 + m^3 + 26* n^2 + 24*n^3 + 1))/(4*m + 10*n + 30*m*n + 70*m*n^2 + 30*m^2*n + 50*m*n^3 + 10*m^3*n + 6*m^2 + 4*m^3 + m^4 + 35*n^2 + 50*n^3 + 24*n^4 + 35*m^2*n^2 + 1) + (B*d^3*x*x^(4*n)*(e*x)^m*(3*m + 6*n + 12*m*n + 11*m*n^2 + 6*m^2*n + 3 *m^2 + m^3 + 11*n^2 + 6*n^3 + 1))/(4*m + 10*n + 30*m*n + 70*m*n^2 + 30*m^2 *n + 50*m*n^3 + 10*m^3*n + 6*m^2 + 4*m^3 + m^4 + 35*n^2 + 50*n^3 + 24*n^4 + 35*m^2*n^2 + 1) + (3*c*d*x*x^(2*n)*(e*x)^m*(A*d + B*c)*(3*m + 8*n + 16*m *n + 19*m*n^2 + 8*m^2*n + 3*m^2 + m^3 + 19*n^2 + 12*n^3 + 1))/(4*m + 10*n + 30*m*n + 70*m*n^2 + 30*m^2*n + 50*m*n^3 + 10*m^3*n + 6*m^2 + 4*m^3 + m^4 + 35*n^2 + 50*n^3 + 24*n^4 + 35*m^2*n^2 + 1)