Integrand size = 29, antiderivative size = 160 \[ \int (e x)^m \left (a+b x^n\right ) \left (A+B x^n\right ) \left (c+d x^n\right )^2 \, dx=\frac {c (A b c+a B c+2 a A d) x^{1+n} (e x)^m}{1+m+n}+\frac {(a d (2 B c+A d)+b c (B c+2 A d)) x^{1+2 n} (e x)^m}{1+m+2 n}+\frac {d (2 b B c+A b d+a B d) x^{1+3 n} (e x)^m}{1+m+3 n}+\frac {b B d^2 x^{1+4 n} (e x)^m}{1+m+4 n}+\frac {a A c^2 (e x)^{1+m}}{e (1+m)} \]
c*(2*A*a*d+A*b*c+B*a*c)*x^(1+n)*(e*x)^m/(1+m+n)+(a*d*(A*d+2*B*c)+b*c*(2*A* d+B*c))*x^(1+2*n)*(e*x)^m/(1+m+2*n)+d*(A*b*d+B*a*d+2*B*b*c)*x^(1+3*n)*(e*x )^m/(1+m+3*n)+b*B*d^2*x^(1+4*n)*(e*x)^m/(1+m+4*n)+a*A*c^2*(e*x)^(1+m)/e/(1 +m)
Time = 0.42 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.81 \[ \int (e x)^m \left (a+b x^n\right ) \left (A+B x^n\right ) \left (c+d x^n\right )^2 \, dx=x (e x)^m \left (\frac {a A c^2}{1+m}+\frac {c (A b c+a B c+2 a A d) x^n}{1+m+n}+\frac {(a d (2 B c+A d)+b c (B c+2 A d)) x^{2 n}}{1+m+2 n}+\frac {d (2 b B c+A b d+a B d) x^{3 n}}{1+m+3 n}+\frac {b B d^2 x^{4 n}}{1+m+4 n}\right ) \]
x*(e*x)^m*((a*A*c^2)/(1 + m) + (c*(A*b*c + a*B*c + 2*a*A*d)*x^n)/(1 + m + n) + ((a*d*(2*B*c + A*d) + b*c*(B*c + 2*A*d))*x^(2*n))/(1 + m + 2*n) + (d* (2*b*B*c + A*b*d + a*B*d)*x^(3*n))/(1 + m + 3*n) + (b*B*d^2*x^(4*n))/(1 + m + 4*n))
Time = 0.35 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, 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.
\(\displaystyle \int (e x)^m \left (a+b x^n\right ) \left (A+B x^n\right ) \left (c+d x^n\right )^2 \, dx\) |
\(\Big \downarrow \) 1040 |
\(\displaystyle \int \left (x^{2 n} (e x)^m (a d (A d+2 B c)+b c (2 A d+B c))+d x^{3 n} (e x)^m (a B d+A b d+2 b B c)+c x^n (e x)^m (2 a A d+a B c+A b c)+a A c^2 (e x)^m+b B d^2 x^{4 n} (e x)^m\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {c x^{n+1} (e x)^m (2 a A d+a B c+A b c)}{m+n+1}+\frac {x^{2 n+1} (e x)^m (a d (A d+2 B c)+b c (2 A d+B c))}{m+2 n+1}+\frac {d x^{3 n+1} (e x)^m (a B d+A b d+2 b B c)}{m+3 n+1}+\frac {a A c^2 (e x)^{m+1}}{e (m+1)}+\frac {b B d^2 x^{4 n+1} (e x)^m}{m+4 n+1}\) |
(c*(A*b*c + a*B*c + 2*a*A*d)*x^(1 + n)*(e*x)^m)/(1 + m + n) + ((a*d*(2*B*c + A*d) + b*c*(B*c + 2*A*d))*x^(1 + 2*n)*(e*x)^m)/(1 + m + 2*n) + (d*(2*b* B*c + A*b*d + a*B*d)*x^(1 + 3*n)*(e*x)^m)/(1 + m + 3*n) + (b*B*d^2*x^(1 + 4*n)*(e*x)^m)/(1 + m + 4*n) + (a*A*c^2*(e*x)^(1 + m))/(e*(1 + m))
3.1.10.3.1 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 = 2.41 (sec) , antiderivative size = 2377, normalized size of antiderivative = 14.86
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2377\) |
parallelrisch | \(\text {Expression too large to display}\) | \(3344\) |
x*(54*A*a*c*d*m*n*x^n+B*a*d^2*(x^n)^3+18*B*b*d^2*m*n*(x^n)^4+16*B*b*c*d*m* n^3*(x^n)^3+28*B*b*c*d*n^2*(x^n)^3+8*A*a*c*d*m^3*x^n+2*B*b*c*d*m^4*(x^n)^3 +7*B*a*d^2*m^3*n*(x^n)^3+8*B*b*c*d*m^3*(x^n)^3+14*B*a*d^2*m^2*n^2*(x^n)^3+ 52*A*a*c*d*n^2*x^n+27*B*a*c^2*m^2*n*x^n+52*A*b*c^2*m*n^2*x^n+28*B*a*d^2*m* n^2*(x^n)^3+24*A*a*d^2*m*n*(x^n)^2+27*A*b*c^2*m^2*n*x^n+2*A*b*c*d*m^4*(x^n )^2+14*B*b*c*d*m^3*n*(x^n)^3+38*B*a*c*d*m^2*n^2*(x^n)^2+8*B*a*c*d*(x^n)^2* m+27*B*a*c^2*m*n*x^n+8*A*b*c*d*(x^n)^2*m+16*A*b*c*d*(x^n)^2*n+B*b*c^2*(x^n )^2+54*A*a*c*d*m^2*n*x^n+A*a*c^2+24*B*a*c*d*m*n^3*(x^n)^2+b*B*d^2*(x^n)^4+ 24*A*b*c*d*m*n^3*(x^n)^2+18*A*a*c*d*m^3*n*x^n+4*A*a*c^2*m+10*A*a*c^2*n+42* B*b*c*d*m^2*n*(x^n)^3+52*B*a*c^2*m*n^2*x^n+52*A*a*c*d*m^2*n^2*x^n+19*B*b*c ^2*m^2*n^2*(x^n)^2+6*B*b*d^2*m^3*n*(x^n)^4+11*B*b*d^2*m^2*n^2*(x^n)^4+A*b* d^2*(x^n)^3+24*B*b*c^2*m^2*n*(x^n)^2+38*B*b*c^2*m*n^2*(x^n)^2+A*a*d^2*(x^n )^2+8*B*b*c^2*m^3*n*(x^n)^2+A*b*c^2*x^n+12*B*b*c^2*n^3*(x^n)^2+16*B*a*c*d* (x^n)^2*n+27*A*b*c^2*m*n*x^n+22*B*b*d^2*m*n^2*(x^n)^4+6*B*b*d^2*m*n^3*(x^n )^4+16*A*b*c*d*m^3*n*(x^n)^2+56*B*b*c*d*m*n^2*(x^n)^3+x^n*c^2*B*a+12*A*b*c *d*m^2*(x^n)^2+24*A*b*c^2*m*n^3*x^n+8*A*b*c*d*m^3*(x^n)^2+A*a*c^2*m^4+76*B *a*c*d*m*n^2*(x^n)^2+42*B*b*c*d*m*n*(x^n)^3+48*A*a*c*d*m*n^3*x^n+48*A*b*c* d*m^2*n*(x^n)^2+8*B*a*c*d*m^3*(x^n)^2+24*B*a*c*d*n^3*(x^n)^2+21*B*a*d^2*m* n*(x^n)^3+4*A*a*c^2*m^3+50*A*a*c^2*n^3+6*A*a*c^2*m^2+35*A*a*c^2*n^2+24*A*a *c^2*n^4+7*B*a*d^2*(x^n)^3*n+6*B*b*c^2*m^2*(x^n)^2+19*B*b*c^2*n^2*(x^n)...
Leaf count of result is larger than twice the leaf count of optimal. 1426 vs. \(2 (160) = 320\).
Time = 0.34 (sec) , antiderivative size = 1426, normalized size of antiderivative = 8.91 \[ \int (e x)^m \left (a+b x^n\right ) \left (A+B x^n\right ) \left (c+d x^n\right )^2 \, dx=\text {Too large to display} \]
((B*b*d^2*m^4 + 4*B*b*d^2*m^3 + 6*B*b*d^2*m^2 + 4*B*b*d^2*m + B*b*d^2 + 6* (B*b*d^2*m + B*b*d^2)*n^3 + 11*(B*b*d^2*m^2 + 2*B*b*d^2*m + B*b*d^2)*n^2 + 6*(B*b*d^2*m^3 + 3*B*b*d^2*m^2 + 3*B*b*d^2*m + B*b*d^2)*n)*x*x^(4*n)*e^(m *log(e) + m*log(x)) + ((2*B*b*c*d + (B*a + A*b)*d^2)*m^4 + 2*B*b*c*d + 4*( 2*B*b*c*d + (B*a + A*b)*d^2)*m^3 + 8*(2*B*b*c*d + (B*a + A*b)*d^2 + (2*B*b *c*d + (B*a + A*b)*d^2)*m)*n^3 + (B*a + A*b)*d^2 + 6*(2*B*b*c*d + (B*a + A *b)*d^2)*m^2 + 14*(2*B*b*c*d + (B*a + A*b)*d^2 + (2*B*b*c*d + (B*a + A*b)* d^2)*m^2 + 2*(2*B*b*c*d + (B*a + A*b)*d^2)*m)*n^2 + 4*(2*B*b*c*d + (B*a + A*b)*d^2)*m + 7*(2*B*b*c*d + (2*B*b*c*d + (B*a + A*b)*d^2)*m^3 + (B*a + A* b)*d^2 + 3*(2*B*b*c*d + (B*a + A*b)*d^2)*m^2 + 3*(2*B*b*c*d + (B*a + A*b)* d^2)*m)*n)*x*x^(3*n)*e^(m*log(e) + m*log(x)) + ((B*b*c^2 + A*a*d^2 + 2*(B* a + A*b)*c*d)*m^4 + B*b*c^2 + A*a*d^2 + 4*(B*b*c^2 + A*a*d^2 + 2*(B*a + A* b)*c*d)*m^3 + 12*(B*b*c^2 + A*a*d^2 + 2*(B*a + A*b)*c*d + (B*b*c^2 + A*a*d ^2 + 2*(B*a + A*b)*c*d)*m)*n^3 + 2*(B*a + A*b)*c*d + 6*(B*b*c^2 + A*a*d^2 + 2*(B*a + A*b)*c*d)*m^2 + 19*(B*b*c^2 + A*a*d^2 + 2*(B*a + A*b)*c*d + (B* b*c^2 + A*a*d^2 + 2*(B*a + A*b)*c*d)*m^2 + 2*(B*b*c^2 + A*a*d^2 + 2*(B*a + A*b)*c*d)*m)*n^2 + 4*(B*b*c^2 + A*a*d^2 + 2*(B*a + A*b)*c*d)*m + 8*(B*b*c ^2 + A*a*d^2 + (B*b*c^2 + A*a*d^2 + 2*(B*a + A*b)*c*d)*m^3 + 2*(B*a + A*b) *c*d + 3*(B*b*c^2 + A*a*d^2 + 2*(B*a + A*b)*c*d)*m^2 + 3*(B*b*c^2 + A*a*d^ 2 + 2*(B*a + A*b)*c*d)*m)*n)*x*x^(2*n)*e^(m*log(e) + m*log(x)) + ((2*A*...
Leaf count of result is larger than twice the leaf count of optimal. 25315 vs. \(2 (156) = 312\).
Time = 7.30 (sec) , antiderivative size = 25315, normalized size of antiderivative = 158.22 \[ \int (e x)^m \left (a+b x^n\right ) \left (A+B x^n\right ) \left (c+d x^n\right )^2 \, dx=\text {Too large to display} \]
Piecewise(((A + B)*(a + b)*(c + d)**2*log(x)/e, Eq(m, -1) & Eq(n, 0)), ((A *a*c**2*log(x) + 2*A*a*c*d*x**n/n + A*a*d**2*x**(2*n)/(2*n) + A*b*c**2*x** n/n + A*b*c*d*x**(2*n)/n + A*b*d**2*x**(3*n)/(3*n) + B*a*c**2*x**n/n + B*a *c*d*x**(2*n)/n + B*a*d**2*x**(3*n)/(3*n) + B*b*c**2*x**(2*n)/(2*n) + 2*B* b*c*d*x**(3*n)/(3*n) + B*b*d**2*x**(4*n)/(4*n))/e, Eq(m, -1)), (A*a*c**2*P iecewise((0**(-4*n - 1)*x, Eq(e, 0)), (Piecewise((-1/(4*n*(e*x)**(4*n)), N e(n, 0)), (log(e*x), True))/e, True)) + 2*A*a*c*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)) + A*a*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*b*c**2*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)) + 2*A*b*c*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)) + A*b*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*a*c**2*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)) + 2*B*a*c*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), Tru e)) + B*a*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*b*c**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(...
Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (160) = 320\).
Time = 0.24 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.08 \[ \int (e x)^m \left (a+b x^n\right ) \left (A+B x^n\right ) \left (c+d x^n\right )^2 \, dx=\frac {B b d^{2} e^{m} x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right )\right )}}{m + 4 \, n + 1} + \frac {2 \, B b c d e^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 1} + \frac {B a d^{2} e^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 1} + \frac {A b d^{2} e^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 1} + \frac {B b c^{2} e^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {2 \, B a c d e^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {2 \, A b c d e^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {A a d^{2} e^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {B a c^{2} e^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {A b c^{2} e^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {2 \, A a c d e^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {\left (e x\right )^{m + 1} A a c^{2}}{e {\left (m + 1\right )}} \]
B*b*d^2*e^m*x*e^(m*log(x) + 4*n*log(x))/(m + 4*n + 1) + 2*B*b*c*d*e^m*x*e^ (m*log(x) + 3*n*log(x))/(m + 3*n + 1) + B*a*d^2*e^m*x*e^(m*log(x) + 3*n*lo g(x))/(m + 3*n + 1) + A*b*d^2*e^m*x*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 1 ) + B*b*c^2*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + 2*B*a*c*d*e^m* x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + 2*A*b*c*d*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + A*a*d^2*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2 *n + 1) + B*a*c^2*e^m*x*e^(m*log(x) + n*log(x))/(m + n + 1) + A*b*c^2*e^m* x*e^(m*log(x) + n*log(x))/(m + n + 1) + 2*A*a*c*d*e^m*x*e^(m*log(x) + n*lo g(x))/(m + n + 1) + (e*x)^(m + 1)*A*a*c^2/(e*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 11834 vs. \(2 (160) = 320\).
Time = 0.39 (sec) , antiderivative size = 11834, normalized size of antiderivative = 73.96 \[ \int (e x)^m \left (a+b x^n\right ) \left (A+B x^n\right ) \left (c+d x^n\right )^2 \, dx=\text {Too large to display} \]
(B*b*d^2*m^4*x*x^(4*n)*e^(m*log(e) + m*log(x)) + 6*B*b*d^2*m^3*n*x*x^(4*n) *e^(m*log(e) + m*log(x)) + 11*B*b*d^2*m^2*n^2*x*x^(4*n)*e^(m*log(e) + m*lo g(x)) + 6*B*b*d^2*m*n^3*x*x^(4*n)*e^(m*log(e) + m*log(x)) + 2*B*b*c*d*m^4* x*x^(3*n)*e^(m*log(e) + m*log(x)) + B*a*d^2*m^4*x*x^(3*n)*e^(m*log(e) + m* log(x)) + A*b*d^2*m^4*x*x^(3*n)*e^(m*log(e) + m*log(x)) + B*b*d^2*m^4*x*x^ (3*n)*e^(m*log(e) + m*log(x)) + 14*B*b*c*d*m^3*n*x*x^(3*n)*e^(m*log(e) + m *log(x)) + 7*B*a*d^2*m^3*n*x*x^(3*n)*e^(m*log(e) + m*log(x)) + 7*A*b*d^2*m ^3*n*x*x^(3*n)*e^(m*log(e) + m*log(x)) + 6*B*b*d^2*m^3*n*x*x^(3*n)*e^(m*lo g(e) + m*log(x)) + 28*B*b*c*d*m^2*n^2*x*x^(3*n)*e^(m*log(e) + m*log(x)) + 14*B*a*d^2*m^2*n^2*x*x^(3*n)*e^(m*log(e) + m*log(x)) + 14*A*b*d^2*m^2*n^2* x*x^(3*n)*e^(m*log(e) + m*log(x)) + 11*B*b*d^2*m^2*n^2*x*x^(3*n)*e^(m*log( e) + m*log(x)) + 16*B*b*c*d*m*n^3*x*x^(3*n)*e^(m*log(e) + m*log(x)) + 8*B* a*d^2*m*n^3*x*x^(3*n)*e^(m*log(e) + m*log(x)) + 8*A*b*d^2*m*n^3*x*x^(3*n)* e^(m*log(e) + m*log(x)) + 6*B*b*d^2*m*n^3*x*x^(3*n)*e^(m*log(e) + m*log(x) ) + B*b*c^2*m^4*x*x^(2*n)*e^(m*log(e) + m*log(x)) + 2*B*a*c*d*m^4*x*x^(2*n )*e^(m*log(e) + m*log(x)) + 2*A*b*c*d*m^4*x*x^(2*n)*e^(m*log(e) + m*log(x) ) + 2*B*b*c*d*m^4*x*x^(2*n)*e^(m*log(e) + m*log(x)) + A*a*d^2*m^4*x*x^(2*n )*e^(m*log(e) + m*log(x)) + B*a*d^2*m^4*x*x^(2*n)*e^(m*log(e) + m*log(x)) + A*b*d^2*m^4*x*x^(2*n)*e^(m*log(e) + m*log(x)) + B*b*d^2*m^4*x*x^(2*n)*e^ (m*log(e) + m*log(x)) + 8*B*b*c^2*m^3*n*x*x^(2*n)*e^(m*log(e) + m*log(x...
Time = 9.47 (sec) , antiderivative size = 588, normalized size of antiderivative = 3.68 \[ \int (e x)^m \left (a+b x^n\right ) \left (A+B x^n\right ) \left (c+d x^n\right )^2 \, dx=\frac {x\,x^{2\,n}\,{\left (e\,x\right )}^m\,\left (A\,a\,d^2+B\,b\,c^2+2\,A\,b\,c\,d+2\,B\,a\,c\,d\right )\,\left (m^3+8\,m^2\,n+3\,m^2+19\,m\,n^2+16\,m\,n+3\,m+12\,n^3+19\,n^2+8\,n+1\right )}{m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1}+\frac {A\,a\,c^2\,x\,{\left (e\,x\right )}^m}{m+1}+\frac {c\,x\,x^n\,{\left (e\,x\right )}^m\,\left (2\,A\,a\,d+A\,b\,c+B\,a\,c\right )\,\left (m^3+9\,m^2\,n+3\,m^2+26\,m\,n^2+18\,m\,n+3\,m+24\,n^3+26\,n^2+9\,n+1\right )}{m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1}+\frac {d\,x\,x^{3\,n}\,{\left (e\,x\right )}^m\,\left (A\,b\,d+B\,a\,d+2\,B\,b\,c\right )\,\left (m^3+7\,m^2\,n+3\,m^2+14\,m\,n^2+14\,m\,n+3\,m+8\,n^3+14\,n^2+7\,n+1\right )}{m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1}+\frac {B\,b\,d^2\,x\,x^{4\,n}\,{\left (e\,x\right )}^m\,\left (m^3+6\,m^2\,n+3\,m^2+11\,m\,n^2+12\,m\,n+3\,m+6\,n^3+11\,n^2+6\,n+1\right )}{m^4+10\,m^3\,n+4\,m^3+35\,m^2\,n^2+30\,m^2\,n+6\,m^2+50\,m\,n^3+70\,m\,n^2+30\,m\,n+4\,m+24\,n^4+50\,n^3+35\,n^2+10\,n+1} \]
(x*x^(2*n)*(e*x)^m*(A*a*d^2 + B*b*c^2 + 2*A*b*c*d + 2*B*a*c*d)*(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) + (A*a*c^2*x*(e*x)^m)/ (m + 1) + (c*x*x^n*(e*x)^m*(2*A*a*d + A*b*c + B*a*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 + 3 5*n^2 + 50*n^3 + 24*n^4 + 35*m^2*n^2 + 1) + (d*x*x^(3*n)*(e*x)^m*(A*b*d + B*a*d + 2*B*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) + (B*b*d^2*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)