Integrand size = 31, antiderivative size = 310 \[ \int (e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right ) \left (c+d x^n\right )^3 \, dx=\frac {a c^2 (2 A b c+a B c+3 a A d) x^{1+n} (e x)^m}{1+m+n}+\frac {c \left (a B c (2 b c+3 a d)+A \left (b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) x^{1+2 n} (e x)^m}{1+m+2 n}+\frac {\left (6 a b c d (B c+A d)+a^2 d^2 (3 B c+A d)+b^2 c^2 (B c+3 A d)\right ) x^{1+3 n} (e x)^m}{1+m+3 n}+\frac {d \left (a^2 B d^2+3 b^2 c (B c+A d)+2 a b d (3 B c+A d)\right ) x^{1+4 n} (e x)^m}{1+m+4 n}+\frac {b d^2 (3 b B c+A b d+2 a B d) x^{1+5 n} (e x)^m}{1+m+5 n}+\frac {b^2 B d^3 x^{1+6 n} (e x)^m}{1+m+6 n}+\frac {a^2 A c^3 (e x)^{1+m}}{e (1+m)} \]
a*c^2*(3*A*a*d+2*A*b*c+B*a*c)*x^(1+n)*(e*x)^m/(1+m+n)+c*(a*B*c*(3*a*d+2*b* c)+A*(3*a^2*d^2+6*a*b*c*d+b^2*c^2))*x^(1+2*n)*(e*x)^m/(1+m+2*n)+(6*a*b*c*d *(A*d+B*c)+a^2*d^2*(A*d+3*B*c)+b^2*c^2*(3*A*d+B*c))*x^(1+3*n)*(e*x)^m/(1+m +3*n)+d*(a^2*B*d^2+3*b^2*c*(A*d+B*c)+2*a*b*d*(A*d+3*B*c))*x^(1+4*n)*(e*x)^ m/(1+m+4*n)+b*d^2*(A*b*d+2*B*a*d+3*B*b*c)*x^(1+5*n)*(e*x)^m/(1+m+5*n)+b^2* B*d^3*x^(1+6*n)*(e*x)^m/(1+m+6*n)+a^2*A*c^3*(e*x)^(1+m)/e/(1+m)
Time = 1.64 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.85 \[ \int (e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right ) \left (c+d x^n\right )^3 \, dx=x (e x)^m \left (\frac {a^2 A c^3}{1+m}+\frac {a c^2 (2 A b c+a B c+3 a A d) x^n}{1+m+n}+\frac {c \left (a B c (2 b c+3 a d)+A \left (b^2 c^2+6 a b c d+3 a^2 d^2\right )\right ) x^{2 n}}{1+m+2 n}+\frac {\left (6 a b c d (B c+A d)+a^2 d^2 (3 B c+A d)+b^2 c^2 (B c+3 A d)\right ) x^{3 n}}{1+m+3 n}+\frac {d \left (a^2 B d^2+3 b^2 c (B c+A d)+2 a b d (3 B c+A d)\right ) x^{4 n}}{1+m+4 n}+\frac {b d^2 (3 b B c+A b d+2 a B d) x^{5 n}}{1+m+5 n}+\frac {b^2 B d^3 x^{6 n}}{1+m+6 n}\right ) \]
x*(e*x)^m*((a^2*A*c^3)/(1 + m) + (a*c^2*(2*A*b*c + a*B*c + 3*a*A*d)*x^n)/( 1 + m + n) + (c*(a*B*c*(2*b*c + 3*a*d) + A*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^ 2))*x^(2*n))/(1 + m + 2*n) + ((6*a*b*c*d*(B*c + A*d) + a^2*d^2*(3*B*c + A* d) + b^2*c^2*(B*c + 3*A*d))*x^(3*n))/(1 + m + 3*n) + (d*(a^2*B*d^2 + 3*b^2 *c*(B*c + A*d) + 2*a*b*d*(3*B*c + A*d))*x^(4*n))/(1 + m + 4*n) + (b*d^2*(3 *b*B*c + A*b*d + 2*a*B*d)*x^(5*n))/(1 + m + 5*n) + (b^2*B*d^3*x^(6*n))/(1 + m + 6*n))
Time = 0.61 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1040, 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.
| \(\displaystyle \int (e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right ) \left (c+d x^n\right )^3 \, dx\) |
| \(\Big \downarrow \) 1040 |
| \(\displaystyle \int \left (c x^{2 n} (e x)^m \left (A \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a B c (3 a d+2 b c)\right )+x^{3 n} (e x)^m \left (a^2 d^2 (A d+3 B c)+6 a b c d (A d+B c)+b^2 c^2 (3 A d+B c)\right )+d x^{4 n} (e x)^m \left (a^2 B d^2+2 a b d (A d+3 B c)+3 b^2 c (A d+B c)\right )+a^2 A c^3 (e x)^m+a c^2 x^n (e x)^m (3 a A d+a B c+2 A b c)+b d^2 x^{5 n} (e x)^m (2 a B d+A b d+3 b B c)+b^2 B d^3 x^{6 n} (e x)^m\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {c x^{2 n+1} (e x)^m \left (A \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a B c (3 a d+2 b c)\right )}{m+2 n+1}+\frac {x^{3 n+1} (e x)^m \left (a^2 d^2 (A d+3 B c)+6 a b c d (A d+B c)+b^2 c^2 (3 A d+B c)\right )}{m+3 n+1}+\frac {d x^{4 n+1} (e x)^m \left (a^2 B d^2+2 a b d (A d+3 B c)+3 b^2 c (A d+B c)\right )}{m+4 n+1}+\frac {a^2 A c^3 (e x)^{m+1}}{e (m+1)}+\frac {a c^2 x^{n+1} (e x)^m (3 a A d+a B c+2 A b c)}{m+n+1}+\frac {b d^2 x^{5 n+1} (e x)^m (2 a B d+A b d+3 b B c)}{m+5 n+1}+\frac {b^2 B d^3 x^{6 n+1} (e x)^m}{m+6 n+1}\) |
(a*c^2*(2*A*b*c + a*B*c + 3*a*A*d)*x^(1 + n)*(e*x)^m)/(1 + m + n) + (c*(a* B*c*(2*b*c + 3*a*d) + A*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2))*x^(1 + 2*n)*(e* x)^m)/(1 + m + 2*n) + ((6*a*b*c*d*(B*c + A*d) + a^2*d^2*(3*B*c + A*d) + b^ 2*c^2*(B*c + 3*A*d))*x^(1 + 3*n)*(e*x)^m)/(1 + m + 3*n) + (d*(a^2*B*d^2 + 3*b^2*c*(B*c + A*d) + 2*a*b*d*(3*B*c + A*d))*x^(1 + 4*n)*(e*x)^m)/(1 + m + 4*n) + (b*d^2*(3*b*B*c + A*b*d + 2*a*B*d)*x^(1 + 5*n)*(e*x)^m)/(1 + m + 5 *n) + (b^2*B*d^3*x^(1 + 6*n)*(e*x)^m)/(1 + m + 6*n) + (a^2*A*c^3*(e*x)^(1 + m))/(e*(1 + m))
3.1.16.3.1 Defintions of rubi rules used
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x_Symbol] :> Int[ExpandIntegrand[ (g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a, b, c , d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 5.10 (sec) , antiderivative size = 11356, normalized size of antiderivative = 36.63
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(11356\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(15203\) |
Leaf count of result is larger than twice the leaf count of optimal. 6557 vs. \(2 (310) = 620\).
Time = 0.48 (sec) , antiderivative size = 6557, normalized size of antiderivative = 21.15 \[ \int (e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right ) \left (c+d x^n\right )^3 \, dx=\text {Too large to display} \]
Leaf count of result is larger than twice the leaf count of optimal. 168099 vs. \(2 (311) = 622\).
Time = 23.29 (sec) , antiderivative size = 168099, normalized size of antiderivative = 542.25 \[ \int (e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right ) \left (c+d x^n\right )^3 \, dx=\text {Too large to display} \]
Piecewise(((A + B)*(a + b)**2*(c + d)**3*log(x)/e, Eq(m, -1) & Eq(n, 0)), ((A*a**2*c**3*log(x) + 3*A*a**2*c**2*d*x**n/n + 3*A*a**2*c*d**2*x**(2*n)/( 2*n) + A*a**2*d**3*x**(3*n)/(3*n) + 2*A*a*b*c**3*x**n/n + 3*A*a*b*c**2*d*x **(2*n)/n + 2*A*a*b*c*d**2*x**(3*n)/n + A*a*b*d**3*x**(4*n)/(2*n) + A*b**2 *c**3*x**(2*n)/(2*n) + A*b**2*c**2*d*x**(3*n)/n + 3*A*b**2*c*d**2*x**(4*n) /(4*n) + A*b**2*d**3*x**(5*n)/(5*n) + B*a**2*c**3*x**n/n + 3*B*a**2*c**2*d *x**(2*n)/(2*n) + B*a**2*c*d**2*x**(3*n)/n + B*a**2*d**3*x**(4*n)/(4*n) + B*a*b*c**3*x**(2*n)/n + 2*B*a*b*c**2*d*x**(3*n)/n + 3*B*a*b*c*d**2*x**(4*n )/(2*n) + 2*B*a*b*d**3*x**(5*n)/(5*n) + B*b**2*c**3*x**(3*n)/(3*n) + 3*B*b **2*c**2*d*x**(4*n)/(4*n) + 3*B*b**2*c*d**2*x**(5*n)/(5*n) + B*b**2*d**3*x **(6*n)/(6*n))/e, Eq(m, -1)), (A*a**2*c**3*Piecewise((0**(-6*n - 1)*x, Eq( e, 0)), (Piecewise((-1/(6*n*(e*x)**(6*n)), Ne(n, 0)), (log(e*x), True))/e, True)) + 3*A*a**2*c**2*d*Piecewise((-x*x**n*(e*x)**(-6*n - 1)/(5*n), Ne(n , 0)), (x*x**n*(e*x)**(-6*n - 1)*log(x), True)) + 3*A*a**2*c*d**2*Piecewis e((-x*x**(2*n)*(e*x)**(-6*n - 1)/(4*n), Ne(n, 0)), (x*x**(2*n)*(e*x)**(-6* n - 1)*log(x), True)) + A*a**2*d**3*Piecewise((-x*x**(3*n)*(e*x)**(-6*n - 1)/(3*n), Ne(n, 0)), (x*x**(3*n)*(e*x)**(-6*n - 1)*log(x), True)) + 2*A*a* b*c**3*Piecewise((-x*x**n*(e*x)**(-6*n - 1)/(5*n), Ne(n, 0)), (x*x**n*(e*x )**(-6*n - 1)*log(x), True)) + 6*A*a*b*c**2*d*Piecewise((-x*x**(2*n)*(e*x) **(-6*n - 1)/(4*n), Ne(n, 0)), (x*x**(2*n)*(e*x)**(-6*n - 1)*log(x), Tr...
Leaf count of result is larger than twice the leaf count of optimal. 748 vs. \(2 (310) = 620\).
Time = 0.29 (sec) , antiderivative size = 748, normalized size of antiderivative = 2.41 \[ \int (e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right ) \left (c+d x^n\right )^3 \, dx=\frac {B b^{2} d^{3} e^{m} x e^{\left (m \log \left (x\right ) + 6 \, n \log \left (x\right )\right )}}{m + 6 \, n + 1} + \frac {3 \, B b^{2} c d^{2} e^{m} x e^{\left (m \log \left (x\right ) + 5 \, n \log \left (x\right )\right )}}{m + 5 \, n + 1} + \frac {2 \, B a b d^{3} e^{m} x e^{\left (m \log \left (x\right ) + 5 \, n \log \left (x\right )\right )}}{m + 5 \, n + 1} + \frac {A b^{2} d^{3} e^{m} x e^{\left (m \log \left (x\right ) + 5 \, n \log \left (x\right )\right )}}{m + 5 \, n + 1} + \frac {3 \, B b^{2} c^{2} d e^{m} x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right )\right )}}{m + 4 \, n + 1} + \frac {6 \, B a b c d^{2} e^{m} x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right )\right )}}{m + 4 \, n + 1} + \frac {3 \, A b^{2} c d^{2} e^{m} x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right )\right )}}{m + 4 \, n + 1} + \frac {B a^{2} d^{3} e^{m} x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right )\right )}}{m + 4 \, n + 1} + \frac {2 \, A a 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 {B b^{2} c^{3} e^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 1} + \frac {6 \, B a b c^{2} d e^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 1} + \frac {3 \, A b^{2} c^{2} d e^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 1} + \frac {3 \, B a^{2} 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 {6 \, A a 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 a^{2} d^{3} e^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 1} + \frac {2 \, B a b c^{3} e^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {A b^{2} c^{3} e^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {3 \, B a^{2} 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 {6 \, A a 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 a^{2} 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 a^{2} c^{3} e^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {2 \, A a 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 a^{2} 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 a^{2} c^{3}}{e {\left (m + 1\right )}} \]
B*b^2*d^3*e^m*x*e^(m*log(x) + 6*n*log(x))/(m + 6*n + 1) + 3*B*b^2*c*d^2*e^ m*x*e^(m*log(x) + 5*n*log(x))/(m + 5*n + 1) + 2*B*a*b*d^3*e^m*x*e^(m*log(x ) + 5*n*log(x))/(m + 5*n + 1) + A*b^2*d^3*e^m*x*e^(m*log(x) + 5*n*log(x))/ (m + 5*n + 1) + 3*B*b^2*c^2*d*e^m*x*e^(m*log(x) + 4*n*log(x))/(m + 4*n + 1 ) + 6*B*a*b*c*d^2*e^m*x*e^(m*log(x) + 4*n*log(x))/(m + 4*n + 1) + 3*A*b^2* c*d^2*e^m*x*e^(m*log(x) + 4*n*log(x))/(m + 4*n + 1) + B*a^2*d^3*e^m*x*e^(m *log(x) + 4*n*log(x))/(m + 4*n + 1) + 2*A*a*b*d^3*e^m*x*e^(m*log(x) + 4*n* log(x))/(m + 4*n + 1) + B*b^2*c^3*e^m*x*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 1) + 6*B*a*b*c^2*d*e^m*x*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 1) + 3*A* b^2*c^2*d*e^m*x*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 1) + 3*B*a^2*c*d^2*e^ m*x*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 1) + 6*A*a*b*c*d^2*e^m*x*e^(m*log (x) + 3*n*log(x))/(m + 3*n + 1) + A*a^2*d^3*e^m*x*e^(m*log(x) + 3*n*log(x) )/(m + 3*n + 1) + 2*B*a*b*c^3*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1 ) + A*b^2*c^3*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + 3*B*a^2*c^2* d*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + 6*A*a*b*c^2*d*e^m*x*e^(m *log(x) + 2*n*log(x))/(m + 2*n + 1) + 3*A*a^2*c*d^2*e^m*x*e^(m*log(x) + 2* n*log(x))/(m + 2*n + 1) + B*a^2*c^3*e^m*x*e^(m*log(x) + n*log(x))/(m + n + 1) + 2*A*a*b*c^3*e^m*x*e^(m*log(x) + n*log(x))/(m + n + 1) + 3*A*a^2*c^2* d*e^m*x*e^(m*log(x) + n*log(x))/(m + n + 1) + (e*x)^(m + 1)*A*a^2*c^3/(e*( m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 70422 vs. \(2 (310) = 620\).
Time = 0.85 (sec) , antiderivative size = 70422, normalized size of antiderivative = 227.17 \[ \int (e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right ) \left (c+d x^n\right )^3 \, dx=\text {Too large to display} \]
(B*b^2*d^3*m^6*x*x^(6*n)*e^(m*log(e) + m*log(x)) + 15*B*b^2*d^3*m^5*n*x*x^ (6*n)*e^(m*log(e) + m*log(x)) + 85*B*b^2*d^3*m^4*n^2*x*x^(6*n)*e^(m*log(e) + m*log(x)) + 225*B*b^2*d^3*m^3*n^3*x*x^(6*n)*e^(m*log(e) + m*log(x)) + 2 74*B*b^2*d^3*m^2*n^4*x*x^(6*n)*e^(m*log(e) + m*log(x)) + 120*B*b^2*d^3*m*n ^5*x*x^(6*n)*e^(m*log(e) + m*log(x)) + 3*B*b^2*c*d^2*m^6*x*x^(5*n)*e^(m*lo g(e) + m*log(x)) + 2*B*a*b*d^3*m^6*x*x^(5*n)*e^(m*log(e) + m*log(x)) + A*b ^2*d^3*m^6*x*x^(5*n)*e^(m*log(e) + m*log(x)) + B*b^2*d^3*m^6*x*x^(5*n)*e^( m*log(e) + m*log(x)) + 48*B*b^2*c*d^2*m^5*n*x*x^(5*n)*e^(m*log(e) + m*log( x)) + 32*B*a*b*d^3*m^5*n*x*x^(5*n)*e^(m*log(e) + m*log(x)) + 16*A*b^2*d^3* m^5*n*x*x^(5*n)*e^(m*log(e) + m*log(x)) + 15*B*b^2*d^3*m^5*n*x*x^(5*n)*e^( m*log(e) + m*log(x)) + 285*B*b^2*c*d^2*m^4*n^2*x*x^(5*n)*e^(m*log(e) + m*l og(x)) + 190*B*a*b*d^3*m^4*n^2*x*x^(5*n)*e^(m*log(e) + m*log(x)) + 95*A*b^ 2*d^3*m^4*n^2*x*x^(5*n)*e^(m*log(e) + m*log(x)) + 85*B*b^2*d^3*m^4*n^2*x*x ^(5*n)*e^(m*log(e) + m*log(x)) + 780*B*b^2*c*d^2*m^3*n^3*x*x^(5*n)*e^(m*lo g(e) + m*log(x)) + 520*B*a*b*d^3*m^3*n^3*x*x^(5*n)*e^(m*log(e) + m*log(x)) + 260*A*b^2*d^3*m^3*n^3*x*x^(5*n)*e^(m*log(e) + m*log(x)) + 225*B*b^2*d^3 *m^3*n^3*x*x^(5*n)*e^(m*log(e) + m*log(x)) + 972*B*b^2*c*d^2*m^2*n^4*x*x^( 5*n)*e^(m*log(e) + m*log(x)) + 648*B*a*b*d^3*m^2*n^4*x*x^(5*n)*e^(m*log(e) + m*log(x)) + 324*A*b^2*d^3*m^2*n^4*x*x^(5*n)*e^(m*log(e) + m*log(x)) + 2 74*B*b^2*d^3*m^2*n^4*x*x^(5*n)*e^(m*log(e) + m*log(x)) + 432*B*b^2*c*d^...
Time = 10.84 (sec) , antiderivative size = 1882, normalized size of antiderivative = 6.07 \[ \int (e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right ) \left (c+d x^n\right )^3 \, dx=\text {Too large to display} \]
(x*x^(3*n)*(e*x)^m*(A*a^2*d^3 + B*b^2*c^3 + 3*A*b^2*c^2*d + 3*B*a^2*c*d^2 + 6*A*a*b*c*d^2 + 6*B*a*b*c^2*d)*(5*m + 18*n + 72*m*n + 363*m*n^2 + 108*m^ 2*n + 744*m*n^3 + 72*m^3*n + 508*m*n^4 + 18*m^4*n + 10*m^2 + 10*m^3 + 5*m^ 4 + m^5 + 121*n^2 + 372*n^3 + 508*n^4 + 240*n^5 + 363*m^2*n^2 + 372*m^2*n^ 3 + 121*m^3*n^2 + 1))/(6*m + 21*n + 105*m*n + 700*m*n^2 + 210*m^2*n + 2205 *m*n^3 + 210*m^3*n + 3248*m*n^4 + 105*m^4*n + 1764*m*n^5 + 21*m^5*n + 15*m ^2 + 20*m^3 + 15*m^4 + 6*m^5 + m^6 + 175*n^2 + 735*n^3 + 1624*n^4 + 1764*n ^5 + 720*n^6 + 1050*m^2*n^2 + 2205*m^2*n^3 + 700*m^3*n^2 + 1624*m^2*n^4 + 735*m^3*n^3 + 175*m^4*n^2 + 1) + (A*a^2*c^3*x*(e*x)^m)/(m + 1) + (c*x*x^(2 *n)*(e*x)^m*(3*A*a^2*d^2 + A*b^2*c^2 + 2*B*a*b*c^2 + 3*B*a^2*c*d + 6*A*a*b *c*d)*(5*m + 19*n + 76*m*n + 411*m*n^2 + 114*m^2*n + 922*m*n^3 + 76*m^3*n + 702*m*n^4 + 19*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 137*n^2 + 461*n^3 + 702*n^4 + 360*n^5 + 411*m^2*n^2 + 461*m^2*n^3 + 137*m^3*n^2 + 1))/(6*m + 21*n + 105*m*n + 700*m*n^2 + 210*m^2*n + 2205*m*n^3 + 210*m^3*n + 3248*m *n^4 + 105*m^4*n + 1764*m*n^5 + 21*m^5*n + 15*m^2 + 20*m^3 + 15*m^4 + 6*m^ 5 + m^6 + 175*n^2 + 735*n^3 + 1624*n^4 + 1764*n^5 + 720*n^6 + 1050*m^2*n^2 + 2205*m^2*n^3 + 700*m^3*n^2 + 1624*m^2*n^4 + 735*m^3*n^3 + 175*m^4*n^2 + 1) + (d*x*x^(4*n)*(e*x)^m*(B*a^2*d^2 + 3*B*b^2*c^2 + 2*A*a*b*d^2 + 3*A*b^ 2*c*d + 6*B*a*b*c*d)*(5*m + 17*n + 68*m*n + 321*m*n^2 + 102*m^2*n + 614*m* n^3 + 68*m^3*n + 396*m*n^4 + 17*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 +...