Integrand size = 29, antiderivative size = 210 \[ \int (e x)^m \left (a+b x^n\right )^3 \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\frac {a^2 (3 A b c+a B c+a A d) x^{1+n} (e x)^m}{1+m+n}+\frac {a (3 A b (b c+a d)+a B (3 b c+a d)) x^{1+2 n} (e x)^m}{1+m+2 n}+\frac {b (3 a B (b c+a d)+A b (b c+3 a d)) x^{1+3 n} (e x)^m}{1+m+3 n}+\frac {b^2 (b B c+A b d+3 a B d) x^{1+4 n} (e x)^m}{1+m+4 n}+\frac {b^3 B d x^{1+5 n} (e x)^m}{1+m+5 n}+\frac {a^3 A c (e x)^{1+m}}{e (1+m)} \]
a^2*(A*a*d+3*A*b*c+B*a*c)*x^(1+n)*(e*x)^m/(1+m+n)+a*(3*A*b*(a*d+b*c)+a*B*( a*d+3*b*c))*x^(1+2*n)*(e*x)^m/(1+m+2*n)+b*(3*a*B*(a*d+b*c)+A*b*(3*a*d+b*c) )*x^(1+3*n)*(e*x)^m/(1+m+3*n)+b^2*(A*b*d+3*B*a*d+B*b*c)*x^(1+4*n)*(e*x)^m/ (1+m+4*n)+b^3*B*d*x^(1+5*n)*(e*x)^m/(1+m+5*n)+a^3*A*c*(e*x)^(1+m)/e/(1+m)
Time = 0.92 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.82 \[ \int (e x)^m \left (a+b x^n\right )^3 \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=x (e x)^m \left (\frac {a^3 A c}{1+m}+\frac {a^2 (3 A b c+a B c+a A d) x^n}{1+m+n}+\frac {a (3 A b (b c+a d)+a B (3 b c+a d)) x^{2 n}}{1+m+2 n}+\frac {b (3 a B (b c+a d)+A b (b c+3 a d)) x^{3 n}}{1+m+3 n}+\frac {b^2 (b B c+A b d+3 a B d) x^{4 n}}{1+m+4 n}+\frac {b^3 B d x^{5 n}}{1+m+5 n}\right ) \]
x*(e*x)^m*((a^3*A*c)/(1 + m) + (a^2*(3*A*b*c + a*B*c + a*A*d)*x^n)/(1 + m + n) + (a*(3*A*b*(b*c + a*d) + a*B*(3*b*c + a*d))*x^(2*n))/(1 + m + 2*n) + (b*(3*a*B*(b*c + a*d) + A*b*(b*c + 3*a*d))*x^(3*n))/(1 + m + 3*n) + (b^2* (b*B*c + A*b*d + 3*a*B*d)*x^(4*n))/(1 + m + 4*n) + (b^3*B*d*x^(5*n))/(1 + m + 5*n))
Time = 0.43 (sec) , antiderivative size = 210, 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.
| \(\displaystyle \int (e x)^m \left (a+b x^n\right )^3 \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx\) |
| \(\Big \downarrow \) 1040 |
| \(\displaystyle \int \left (a^3 A c (e x)^m+a^2 x^n (e x)^m (a A d+a B c+3 A b c)+b^2 x^{4 n} (e x)^m (3 a B d+A b d+b B c)+a x^{2 n} (e x)^m (3 A b (a d+b c)+a B (a d+3 b c))+b x^{3 n} (e x)^m (A b (3 a d+b c)+3 a B (a d+b c))+b^3 B d x^{5 n} (e x)^m\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 A c (e x)^{m+1}}{e (m+1)}+\frac {a^2 x^{n+1} (e x)^m (a A d+a B c+3 A b c)}{m+n+1}+\frac {b^2 x^{4 n+1} (e x)^m (3 a B d+A b d+b B c)}{m+4 n+1}+\frac {a x^{2 n+1} (e x)^m (3 A b (a d+b c)+a B (a d+3 b c))}{m+2 n+1}+\frac {b x^{3 n+1} (e x)^m (A b (3 a d+b c)+3 a B (a d+b c))}{m+3 n+1}+\frac {b^3 B d x^{5 n+1} (e x)^m}{m+5 n+1}\) |
(a^2*(3*A*b*c + a*B*c + a*A*d)*x^(1 + n)*(e*x)^m)/(1 + m + n) + (a*(3*A*b* (b*c + a*d) + a*B*(3*b*c + a*d))*x^(1 + 2*n)*(e*x)^m)/(1 + m + 2*n) + (b*( 3*a*B*(b*c + a*d) + A*b*(b*c + 3*a*d))*x^(1 + 3*n)*(e*x)^m)/(1 + m + 3*n) + (b^2*(b*B*c + A*b*d + 3*a*B*d)*x^(1 + 4*n)*(e*x)^m)/(1 + m + 4*n) + (b^3 *B*d*x^(1 + 5*n)*(e*x)^m)/(1 + m + 5*n) + (a^3*A*c*(e*x)^(1 + m))/(e*(1 + m))
3.1.1.3.1 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 = 7.53 (sec) , antiderivative size = 4939, normalized size of antiderivative = 23.52
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(4939\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(6818\) |
x*(147*A*b^3*c*m*n^2*(x^n)^3+3*(x^n)^2*d*a^2*b*A+90*B*a*b^2*d*n^4*(x^n)^4+ 44*B*b^3*c*m^3*n*(x^n)^4+120*B*a*b^2*c*n^4*(x^n)^3+180*B*a^2*b*c*n^4*(x^n) ^2+84*B*a^3*c*m^2*n*x^n+144*B*a*b^2*c*m*n*(x^n)^3+132*B*a*b^2*d*m^3*n*(x^n )^4+122*A*b^3*d*m*n^3*(x^n)^4+132*B*a*b^2*d*m*n*(x^n)^4+61*A*b^3*d*m^2*n^3 *(x^n)^4+366*B*a*b^2*d*m*n^3*(x^n)^4+39*A*a^2*b*d*m^4*n*(x^n)^2+30*B*a*b^2 *d*m^3*(x^n)^4+177*A*a*b^2*c*n^2*(x^n)^2+14*A*a^3*d*m^4*n*x^n+15*B*a*b^2*c *m^4*(x^n)^3+216*B*a*b^2*c*m^2*n*(x^n)^3+44*A*b^3*d*m^3*n*(x^n)^4+177*B*a^ 2*b*c*m^3*n^2*(x^n)^2+234*B*a^2*b*c*m^2*n*(x^n)^2+531*B*a^2*b*c*m*n^2*(x^n )^2+144*B*a^2*b*d*m*n*(x^n)^3+441*A*a*b^2*d*m^2*n^2*(x^n)^3+234*A*a*b^2*c* m^2*n*(x^n)^2+147*B*a^2*b*d*m^3*n^2*(x^n)^3+234*B*a^2*b*d*m^2*n^3*(x^n)^3+ 308*B*a^3*c*m*n^3*x^n+24*B*b^3*d*m*n^4*(x^n)^5+30*A*b^3*d*m*n^4*(x^n)^4+53 1*A*a*b^2*c*m^2*n^2*(x^n)^2+30*A*a^2*b*c*m^2*x^n+213*A*a^2*b*c*n^2*x^n+213 *B*a^3*c*m^2*n^2*x^n+15*A*a*b^2*c*(x^n)^2*m+213*A*a^3*d*m*n^2*x^n+30*A*a^2 *b*c*m^3*x^n+56*A*a^3*d*m^3*n*x^n+15*B*a^2*b*c*(x^n)^2*m+15*A*a*b^2*d*(x^n )^3*m+36*A*a*b^2*d*(x^n)^3*n+30*A*a^2*b*d*m^3*(x^n)^2+59*B*a^3*d*m^3*n^2*( x^n)^2+107*B*a^3*d*m^2*n^3*(x^n)^2+642*A*a*b^2*c*m*n^3*(x^n)^2+441*B*a*b^2 *c*m*n^2*(x^n)^3+44*B*b^3*c*m*n*(x^n)^4+30*B*a*b^2*c*m^3*(x^n)^3+180*A*a^2 *b*d*m*n^4*(x^n)^2+39*A*a*b^2*c*m^4*n*(x^n)^2+123*B*b^3*c*m*n^2*(x^n)^4+3* B*a*b^2*c*m^5*(x^n)^3+15*B*a*b^2*d*m^4*(x^n)^4+12*A*b^3*c*m^4*n*(x^n)^3+12 3*A*b^3*d*m^2*n^2*(x^n)^4+120*B*a^2*b*d*n^4*(x^n)^3+180*B*a^2*b*c*m*n^4...
Leaf count of result is larger than twice the leaf count of optimal. 3073 vs. \(2 (210) = 420\).
Time = 0.37 (sec) , antiderivative size = 3073, normalized size of antiderivative = 14.63 \[ \int (e x)^m \left (a+b x^n\right )^3 \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\text {Too large to display} \]
((B*b^3*d*m^5 + 5*B*b^3*d*m^4 + 10*B*b^3*d*m^3 + 10*B*b^3*d*m^2 + 5*B*b^3* d*m + B*b^3*d + 24*(B*b^3*d*m + B*b^3*d)*n^4 + 50*(B*b^3*d*m^2 + 2*B*b^3*d *m + B*b^3*d)*n^3 + 35*(B*b^3*d*m^3 + 3*B*b^3*d*m^2 + 3*B*b^3*d*m + B*b^3* d)*n^2 + 10*(B*b^3*d*m^4 + 4*B*b^3*d*m^3 + 6*B*b^3*d*m^2 + 4*B*b^3*d*m + B *b^3*d)*n)*x*x^(5*n)*e^(m*log(e) + m*log(x)) + ((B*b^3*c + (3*B*a*b^2 + A* b^3)*d)*m^5 + B*b^3*c + 5*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^4 + 30*(B*b^ 3*c + (3*B*a*b^2 + A*b^3)*d + (B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m)*n^4 + 1 0*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^3 + 61*(B*b^3*c + (B*b^3*c + (3*B*a* b^2 + A*b^3)*d)*m^2 + (3*B*a*b^2 + A*b^3)*d + 2*(B*b^3*c + (3*B*a*b^2 + A* b^3)*d)*m)*n^3 + 10*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^2 + 41*(B*b^3*c + (B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^3 + 3*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d )*m^2 + (3*B*a*b^2 + A*b^3)*d + 3*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m)*n^2 + (3*B*a*b^2 + A*b^3)*d + 5*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m + 11*(B*b ^3*c + (B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^4 + 4*(B*b^3*c + (3*B*a*b^2 + A *b^3)*d)*m^3 + 6*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m^2 + (3*B*a*b^2 + A*b^ 3)*d + 4*(B*b^3*c + (3*B*a*b^2 + A*b^3)*d)*m)*n)*x*x^(4*n)*e^(m*log(e) + m *log(x)) + (((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m^5 + 5*((3* B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m^4 + 40*((3*B*a*b^2 + A*b^3 )*c + 3*(B*a^2*b + A*a*b^2)*d + ((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a* b^2)*d)*m)*n^4 + 10*((3*B*a*b^2 + A*b^3)*c + 3*(B*a^2*b + A*a*b^2)*d)*m...
Leaf count of result is larger than twice the leaf count of optimal. 64068 vs. \(2 (206) = 412\).
Time = 13.38 (sec) , antiderivative size = 64068, normalized size of antiderivative = 305.09 \[ \int (e x)^m \left (a+b x^n\right )^3 \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\text {Too large to display} \]
Piecewise(((A + B)*(a + b)**3*(c + d)*log(x)/e, Eq(m, -1) & Eq(n, 0)), ((A *a**3*c*log(x) + A*a**3*d*x**n/n + 3*A*a**2*b*c*x**n/n + 3*A*a**2*b*d*x**( 2*n)/(2*n) + 3*A*a*b**2*c*x**(2*n)/(2*n) + A*a*b**2*d*x**(3*n)/n + A*b**3* c*x**(3*n)/(3*n) + A*b**3*d*x**(4*n)/(4*n) + B*a**3*c*x**n/n + B*a**3*d*x* *(2*n)/(2*n) + 3*B*a**2*b*c*x**(2*n)/(2*n) + B*a**2*b*d*x**(3*n)/n + B*a*b **2*c*x**(3*n)/n + 3*B*a*b**2*d*x**(4*n)/(4*n) + B*b**3*c*x**(4*n)/(4*n) + B*b**3*d*x**(5*n)/(5*n))/e, Eq(m, -1)), (A*a**3*c*Piecewise((0**(-5*n - 1 )*x, Eq(e, 0)), (Piecewise((-1/(5*n*(e*x)**(5*n)), Ne(n, 0)), (log(e*x), T rue))/e, True)) + A*a**3*d*Piecewise((-x*x**n*(e*x)**(-5*n - 1)/(4*n), Ne( n, 0)), (x*x**n*(e*x)**(-5*n - 1)*log(x), True)) + 3*A*a**2*b*c*Piecewise( (-x*x**n*(e*x)**(-5*n - 1)/(4*n), Ne(n, 0)), (x*x**n*(e*x)**(-5*n - 1)*log (x), True)) + 3*A*a**2*b*d*Piecewise((-x*x**(2*n)*(e*x)**(-5*n - 1)/(3*n), Ne(n, 0)), (x*x**(2*n)*(e*x)**(-5*n - 1)*log(x), True)) + 3*A*a*b**2*c*Pi ecewise((-x*x**(2*n)*(e*x)**(-5*n - 1)/(3*n), Ne(n, 0)), (x*x**(2*n)*(e*x) **(-5*n - 1)*log(x), True)) + 3*A*a*b**2*d*Piecewise((-x*x**(3*n)*(e*x)**( -5*n - 1)/(2*n), Ne(n, 0)), (x*x**(3*n)*(e*x)**(-5*n - 1)*log(x), True)) + A*b**3*c*Piecewise((-x*x**(3*n)*(e*x)**(-5*n - 1)/(2*n), Ne(n, 0)), (x*x* *(3*n)*(e*x)**(-5*n - 1)*log(x), True)) + A*b**3*d*Piecewise((-x*x**(4*n)* (e*x)**(-5*n - 1)/n, Ne(n, 0)), (x*x**(4*n)*(e*x)**(-5*n - 1)*log(x), True )) + B*a**3*c*Piecewise((-x*x**n*(e*x)**(-5*n - 1)/(4*n), Ne(n, 0)), (x...
Leaf count of result is larger than twice the leaf count of optimal. 464 vs. \(2 (210) = 420\).
Time = 0.25 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.21 \[ \int (e x)^m \left (a+b x^n\right )^3 \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\frac {B b^{3} d e^{m} x e^{\left (m \log \left (x\right ) + 5 \, n \log \left (x\right )\right )}}{m + 5 \, n + 1} + \frac {B b^{3} c e^{m} x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right )\right )}}{m + 4 \, n + 1} + \frac {3 \, B a b^{2} d e^{m} x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right )\right )}}{m + 4 \, n + 1} + \frac {A b^{3} d e^{m} x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right )\right )}}{m + 4 \, n + 1} + \frac {3 \, B a b^{2} c e^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 1} + \frac {A b^{3} c e^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 1} + \frac {3 \, B a^{2} b d e^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 1} + \frac {3 \, A a b^{2} d e^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 1} + \frac {3 \, B a^{2} b c e^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {3 \, A a b^{2} c e^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {B a^{3} d e^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {3 \, A a^{2} b d e^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {B a^{3} c e^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {3 \, A a^{2} b c e^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {A a^{3} d e^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {\left (e x\right )^{m + 1} A a^{3} c}{e {\left (m + 1\right )}} \]
B*b^3*d*e^m*x*e^(m*log(x) + 5*n*log(x))/(m + 5*n + 1) + B*b^3*c*e^m*x*e^(m *log(x) + 4*n*log(x))/(m + 4*n + 1) + 3*B*a*b^2*d*e^m*x*e^(m*log(x) + 4*n* log(x))/(m + 4*n + 1) + A*b^3*d*e^m*x*e^(m*log(x) + 4*n*log(x))/(m + 4*n + 1) + 3*B*a*b^2*c*e^m*x*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 1) + A*b^3*c* e^m*x*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 1) + 3*B*a^2*b*d*e^m*x*e^(m*log (x) + 3*n*log(x))/(m + 3*n + 1) + 3*A*a*b^2*d*e^m*x*e^(m*log(x) + 3*n*log( x))/(m + 3*n + 1) + 3*B*a^2*b*c*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + 3*A*a*b^2*c*e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + B*a^3*d* e^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + 3*A*a^2*b*d*e^m*x*e^(m*log (x) + 2*n*log(x))/(m + 2*n + 1) + B*a^3*c*e^m*x*e^(m*log(x) + n*log(x))/(m + n + 1) + 3*A*a^2*b*c*e^m*x*e^(m*log(x) + n*log(x))/(m + n + 1) + A*a^3* d*e^m*x*e^(m*log(x) + n*log(x))/(m + n + 1) + (e*x)^(m + 1)*A*a^3*c/(e*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 27992 vs. \(2 (210) = 420\).
Time = 0.49 (sec) , antiderivative size = 27992, normalized size of antiderivative = 133.30 \[ \int (e x)^m \left (a+b x^n\right )^3 \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\text {Too large to display} \]
(B*b^3*d*m^5*x*x^(5*n)*e^(m*log(e) + m*log(x)) + 10*B*b^3*d*m^4*n*x*x^(5*n )*e^(m*log(e) + m*log(x)) + 35*B*b^3*d*m^3*n^2*x*x^(5*n)*e^(m*log(e) + m*l og(x)) + 50*B*b^3*d*m^2*n^3*x*x^(5*n)*e^(m*log(e) + m*log(x)) + 24*B*b^3*d *m*n^4*x*x^(5*n)*e^(m*log(e) + m*log(x)) + B*b^3*c*m^5*x*x^(4*n)*e^(m*log( e) + m*log(x)) + 3*B*a*b^2*d*m^5*x*x^(4*n)*e^(m*log(e) + m*log(x)) + A*b^3 *d*m^5*x*x^(4*n)*e^(m*log(e) + m*log(x)) + B*b^3*d*m^5*x*x^(4*n)*e^(m*log( e) + m*log(x)) + 11*B*b^3*c*m^4*n*x*x^(4*n)*e^(m*log(e) + m*log(x)) + 33*B *a*b^2*d*m^4*n*x*x^(4*n)*e^(m*log(e) + m*log(x)) + 11*A*b^3*d*m^4*n*x*x^(4 *n)*e^(m*log(e) + m*log(x)) + 10*B*b^3*d*m^4*n*x*x^(4*n)*e^(m*log(e) + m*l og(x)) + 41*B*b^3*c*m^3*n^2*x*x^(4*n)*e^(m*log(e) + m*log(x)) + 123*B*a*b^ 2*d*m^3*n^2*x*x^(4*n)*e^(m*log(e) + m*log(x)) + 41*A*b^3*d*m^3*n^2*x*x^(4* n)*e^(m*log(e) + m*log(x)) + 35*B*b^3*d*m^3*n^2*x*x^(4*n)*e^(m*log(e) + m* log(x)) + 61*B*b^3*c*m^2*n^3*x*x^(4*n)*e^(m*log(e) + m*log(x)) + 183*B*a*b ^2*d*m^2*n^3*x*x^(4*n)*e^(m*log(e) + m*log(x)) + 61*A*b^3*d*m^2*n^3*x*x^(4 *n)*e^(m*log(e) + m*log(x)) + 50*B*b^3*d*m^2*n^3*x*x^(4*n)*e^(m*log(e) + m *log(x)) + 30*B*b^3*c*m*n^4*x*x^(4*n)*e^(m*log(e) + m*log(x)) + 90*B*a*b^2 *d*m*n^4*x*x^(4*n)*e^(m*log(e) + m*log(x)) + 30*A*b^3*d*m*n^4*x*x^(4*n)*e^ (m*log(e) + m*log(x)) + 24*B*b^3*d*m*n^4*x*x^(4*n)*e^(m*log(e) + m*log(x)) + 3*B*a*b^2*c*m^5*x*x^(3*n)*e^(m*log(e) + m*log(x)) + A*b^3*c*m^5*x*x^(3* n)*e^(m*log(e) + m*log(x)) + B*b^3*c*m^5*x*x^(3*n)*e^(m*log(e) + m*log(...
Time = 9.90 (sec) , antiderivative size = 1089, normalized size of antiderivative = 5.19 \[ \int (e x)^m \left (a+b x^n\right )^3 \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx=\frac {A\,a^3\,c\,x\,{\left (e\,x\right )}^m}{m+1}+\frac {b^2\,x\,x^{4\,n}\,{\left (e\,x\right )}^m\,\left (A\,b\,d+3\,B\,a\,d+B\,b\,c\right )\,\left (m^4+11\,m^3\,n+4\,m^3+41\,m^2\,n^2+33\,m^2\,n+6\,m^2+61\,m\,n^3+82\,m\,n^2+33\,m\,n+4\,m+30\,n^4+61\,n^3+41\,n^2+11\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1}+\frac {a\,x\,x^{2\,n}\,{\left (e\,x\right )}^m\,\left (3\,A\,b^2\,c+B\,a^2\,d+3\,A\,a\,b\,d+3\,B\,a\,b\,c\right )\,\left (m^4+13\,m^3\,n+4\,m^3+59\,m^2\,n^2+39\,m^2\,n+6\,m^2+107\,m\,n^3+118\,m\,n^2+39\,m\,n+4\,m+60\,n^4+107\,n^3+59\,n^2+13\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1}+\frac {b\,x\,x^{3\,n}\,{\left (e\,x\right )}^m\,\left (A\,b^2\,c+3\,B\,a^2\,d+3\,A\,a\,b\,d+3\,B\,a\,b\,c\right )\,\left (m^4+12\,m^3\,n+4\,m^3+49\,m^2\,n^2+36\,m^2\,n+6\,m^2+78\,m\,n^3+98\,m\,n^2+36\,m\,n+4\,m+40\,n^4+78\,n^3+49\,n^2+12\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1}+\frac {a^2\,x\,x^n\,{\left (e\,x\right )}^m\,\left (A\,a\,d+3\,A\,b\,c+B\,a\,c\right )\,\left (m^4+14\,m^3\,n+4\,m^3+71\,m^2\,n^2+42\,m^2\,n+6\,m^2+154\,m\,n^3+142\,m\,n^2+42\,m\,n+4\,m+120\,n^4+154\,n^3+71\,n^2+14\,n+1\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1}+\frac {B\,b^3\,d\,x\,x^{5\,n}\,{\left (e\,x\right )}^m\,\left (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\right )}{m^5+15\,m^4\,n+5\,m^4+85\,m^3\,n^2+60\,m^3\,n+10\,m^3+225\,m^2\,n^3+255\,m^2\,n^2+90\,m^2\,n+10\,m^2+274\,m\,n^4+450\,m\,n^3+255\,m\,n^2+60\,m\,n+5\,m+120\,n^5+274\,n^4+225\,n^3+85\,n^2+15\,n+1} \]
(A*a^3*c*x*(e*x)^m)/(m + 1) + (b^2*x*x^(4*n)*(e*x)^m*(A*b*d + 3*B*a*d + B* b*c)*(4*m + 11*n + 33*m*n + 82*m*n^2 + 33*m^2*n + 61*m*n^3 + 11*m^3*n + 6* m^2 + 4*m^3 + m^4 + 41*n^2 + 61*n^3 + 30*n^4 + 41*m^2*n^2 + 1))/(5*m + 15* n + 60*m*n + 255*m*n^2 + 90*m^2*n + 450*m*n^3 + 60*m^3*n + 274*m*n^4 + 15* m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 85*n^2 + 225*n^3 + 274*n^4 + 120*n ^5 + 255*m^2*n^2 + 225*m^2*n^3 + 85*m^3*n^2 + 1) + (a*x*x^(2*n)*(e*x)^m*(3 *A*b^2*c + B*a^2*d + 3*A*a*b*d + 3*B*a*b*c)*(4*m + 13*n + 39*m*n + 118*m*n ^2 + 39*m^2*n + 107*m*n^3 + 13*m^3*n + 6*m^2 + 4*m^3 + m^4 + 59*n^2 + 107* n^3 + 60*n^4 + 59*m^2*n^2 + 1))/(5*m + 15*n + 60*m*n + 255*m*n^2 + 90*m^2* n + 450*m*n^3 + 60*m^3*n + 274*m*n^4 + 15*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 85*n^2 + 225*n^3 + 274*n^4 + 120*n^5 + 255*m^2*n^2 + 225*m^2*n^3 + 85*m^3*n^2 + 1) + (b*x*x^(3*n)*(e*x)^m*(A*b^2*c + 3*B*a^2*d + 3*A*a*b*d + 3*B*a*b*c)*(4*m + 12*n + 36*m*n + 98*m*n^2 + 36*m^2*n + 78*m*n^3 + 12*m^3 *n + 6*m^2 + 4*m^3 + m^4 + 49*n^2 + 78*n^3 + 40*n^4 + 49*m^2*n^2 + 1))/(5* m + 15*n + 60*m*n + 255*m*n^2 + 90*m^2*n + 450*m*n^3 + 60*m^3*n + 274*m*n^ 4 + 15*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 85*n^2 + 225*n^3 + 274*n^4 + 120*n^5 + 255*m^2*n^2 + 225*m^2*n^3 + 85*m^3*n^2 + 1) + (a^2*x*x^n*(e*x) ^m*(A*a*d + 3*A*b*c + B*a*c)*(4*m + 14*n + 42*m*n + 142*m*n^2 + 42*m^2*n + 154*m*n^3 + 14*m^3*n + 6*m^2 + 4*m^3 + m^4 + 71*n^2 + 154*n^3 + 120*n^4 + 71*m^2*n^2 + 1))/(5*m + 15*n + 60*m*n + 255*m*n^2 + 90*m^2*n + 450*m*n...