Integrand size = 25, antiderivative size = 555 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=-\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}{5 e^8}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right ) (d+e x)^6}{6 e^8}-\frac {3 \left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) (d+e x)^7}{7 e^8}-\frac {\left (A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right ) (d+e x)^8}{8 e^8}-\frac {\left (B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )-3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) (d+e x)^9}{9 e^8}-\frac {3 c \left (A c e (2 c d-b e)-B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right ) (d+e x)^{10}}{10 e^8}-\frac {c^2 (7 B c d-3 b B e-A c e) (d+e x)^{11}}{11 e^8}+\frac {B c^3 (d+e x)^{12}}{12 e^8} \]
-1/5*(-A*e+B*d)*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^5/e^8-1/6*(a*e^2-b*d*e+c*d^2 )^2*(3*A*e*(-b*e+2*c*d)-B*(7*c*d^2-e*(-a*e+4*b*d)))*(e*x+d)^6/e^8-3/7*(a*e ^2-b*d*e+c*d^2)*(B*(7*c^2*d^3-c*d*e*(-3*a*e+8*b*d)+b*e^2*(-a*e+2*b*d))-A*e *(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d)))*(e*x+d)^7/e^8-1/8*(A*e*(-b*e+2*c*d) *(10*c^2*d^2+b^2*e^2-2*c*e*(-3*a*e+5*b*d))-B*(35*c^3*d^4-b^2*e^3*(-3*a*e+4 *b*d)-30*c^2*d^2*e*(-a*e+2*b*d)+3*c*e^2*(a^2*e^2-8*a*b*d*e+10*b^2*d^2)))*( e*x+d)^8/e^8-1/9*(B*(35*c^3*d^3-b^3*e^3+3*b*c*e^2*(-2*a*e+5*b*d)-15*c^2*d* e*(-a*e+3*b*d))-3*A*c*e*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d)))*(e*x+d)^9/e^ 8-3/10*c*(A*c*e*(-b*e+2*c*d)-B*(7*c^2*d^2+b^2*e^2-c*e*(-a*e+6*b*d)))*(e*x+ d)^10/e^8-1/11*c^2*(-A*c*e-3*B*b*e+7*B*c*d)*(e*x+d)^11/e^8+1/12*B*c^3*(e*x +d)^12/e^8
Time = 0.34 (sec) , antiderivative size = 957, normalized size of antiderivative = 1.72 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=a^3 A d^4 x+\frac {1}{2} a^2 d^3 (3 A b d+a B d+4 a A e) x^2+\frac {1}{3} a d^2 \left (a B d (3 b d+4 a e)+3 A \left (b^2 d^2+4 a b d e+a \left (c d^2+2 a e^2\right )\right )\right ) x^3+\frac {1}{4} d \left (3 a B d \left (b^2 d^2+4 a b d e+a \left (c d^2+2 a e^2\right )\right )+A \left (b^3 d^3+12 a b^2 d^2 e+4 a^2 e \left (3 c d^2+a e^2\right )+6 a b d \left (c d^2+3 a e^2\right )\right )\right ) x^4+\frac {1}{5} \left (b^3 d^3 (B d+4 A e)+3 b^2 d^2 \left (A c d^2+4 a B d e+6 a A e^2\right )+6 a b d \left (B c d^3+4 A c d^2 e+3 a B d e^2+2 a A e^3\right )+a \left (4 a B d e \left (3 c d^2+a e^2\right )+A \left (3 c^2 d^4+18 a c d^2 e^2+a^2 e^4\right )\right )\right ) x^5+\frac {1}{6} \left (2 b^3 d^2 e (2 B d+3 A e)+12 a A c d e \left (c d^2+a e^2\right )+12 a b B d e \left (2 c d^2+a e^2\right )+3 b^2 d \left (B c d^3+4 A c d^2 e+6 a B d e^2+4 a A e^3\right )+3 A b \left (c^2 d^4+12 a c d^2 e^2+a^2 e^4\right )+a B \left (3 c^2 d^4+18 a c d^2 e^2+a^2 e^4\right )\right ) x^6+\frac {1}{7} \left (2 b^3 d e^2 (3 B d+2 A e)+12 a B c d e \left (c d^2+a e^2\right )+12 A b c d e \left (c d^2+2 a e^2\right )+3 b^2 e \left (4 B c d^3+6 A c d^2 e+4 a B d e^2+a A e^3\right )+3 b B \left (c^2 d^4+12 a c d^2 e^2+a^2 e^4\right )+A c \left (c^2 d^4+18 a c d^2 e^2+3 a^2 e^4\right )\right ) x^7+\frac {1}{8} \left (A e \left (4 c^3 d^3+b^3 e^3+6 b c e^2 (2 b d+a e)+6 c^2 d e (3 b d+2 a e)\right )+B \left (c^3 d^4+6 c^2 d^2 e (2 b d+3 a e)+b^2 e^3 (4 b d+3 a e)+3 c e^2 \left (6 b^2 d^2+8 a b d e+a^2 e^2\right )\right )\right ) x^8+\frac {1}{9} e \left (3 A c e \left (2 c^2 d^2+b^2 e^2+c e (4 b d+a e)\right )+B \left (4 c^3 d^3+b^3 e^3+6 b c e^2 (2 b d+a e)+6 c^2 d e (3 b d+2 a e)\right )\right ) x^9+\frac {1}{10} c e^2 \left (A c e (4 c d+3 b e)+3 B \left (2 c^2 d^2+b^2 e^2+c e (4 b d+a e)\right )\right ) x^{10}+\frac {1}{11} c^2 e^3 (4 B c d+3 b B e+A c e) x^{11}+\frac {1}{12} B c^3 e^4 x^{12} \]
a^3*A*d^4*x + (a^2*d^3*(3*A*b*d + a*B*d + 4*a*A*e)*x^2)/2 + (a*d^2*(a*B*d* (3*b*d + 4*a*e) + 3*A*(b^2*d^2 + 4*a*b*d*e + a*(c*d^2 + 2*a*e^2)))*x^3)/3 + (d*(3*a*B*d*(b^2*d^2 + 4*a*b*d*e + a*(c*d^2 + 2*a*e^2)) + A*(b^3*d^3 + 1 2*a*b^2*d^2*e + 4*a^2*e*(3*c*d^2 + a*e^2) + 6*a*b*d*(c*d^2 + 3*a*e^2)))*x^ 4)/4 + ((b^3*d^3*(B*d + 4*A*e) + 3*b^2*d^2*(A*c*d^2 + 4*a*B*d*e + 6*a*A*e^ 2) + 6*a*b*d*(B*c*d^3 + 4*A*c*d^2*e + 3*a*B*d*e^2 + 2*a*A*e^3) + a*(4*a*B* d*e*(3*c*d^2 + a*e^2) + A*(3*c^2*d^4 + 18*a*c*d^2*e^2 + a^2*e^4)))*x^5)/5 + ((2*b^3*d^2*e*(2*B*d + 3*A*e) + 12*a*A*c*d*e*(c*d^2 + a*e^2) + 12*a*b*B* d*e*(2*c*d^2 + a*e^2) + 3*b^2*d*(B*c*d^3 + 4*A*c*d^2*e + 6*a*B*d*e^2 + 4*a *A*e^3) + 3*A*b*(c^2*d^4 + 12*a*c*d^2*e^2 + a^2*e^4) + a*B*(3*c^2*d^4 + 18 *a*c*d^2*e^2 + a^2*e^4))*x^6)/6 + ((2*b^3*d*e^2*(3*B*d + 2*A*e) + 12*a*B*c *d*e*(c*d^2 + a*e^2) + 12*A*b*c*d*e*(c*d^2 + 2*a*e^2) + 3*b^2*e*(4*B*c*d^3 + 6*A*c*d^2*e + 4*a*B*d*e^2 + a*A*e^3) + 3*b*B*(c^2*d^4 + 12*a*c*d^2*e^2 + a^2*e^4) + A*c*(c^2*d^4 + 18*a*c*d^2*e^2 + 3*a^2*e^4))*x^7)/7 + ((A*e*(4 *c^3*d^3 + b^3*e^3 + 6*b*c*e^2*(2*b*d + a*e) + 6*c^2*d*e*(3*b*d + 2*a*e)) + B*(c^3*d^4 + 6*c^2*d^2*e*(2*b*d + 3*a*e) + b^2*e^3*(4*b*d + 3*a*e) + 3*c *e^2*(6*b^2*d^2 + 8*a*b*d*e + a^2*e^2)))*x^8)/8 + (e*(3*A*c*e*(2*c^2*d^2 + b^2*e^2 + c*e*(4*b*d + a*e)) + B*(4*c^3*d^3 + b^3*e^3 + 6*b*c*e^2*(2*b*d + a*e) + 6*c^2*d*e*(3*b*d + 2*a*e)))*x^9)/9 + (c*e^2*(A*c*e*(4*c*d + 3*b*e ) + 3*B*(2*c^2*d^2 + b^2*e^2 + c*e*(4*b*d + a*e)))*x^10)/10 + (c^2*e^3*...
Time = 1.52 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1195, 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.
| \(\displaystyle \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {(d+e x)^7 \left (B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )-A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )\right )}{e^7}+\frac {3 c (d+e x)^9 \left (B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )-A c e (2 c d-b e)\right )}{e^7}+\frac {3 (d+e x)^6 \left (a e^2-b d e+c d^2\right ) \left (A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )\right )}{e^7}+\frac {(d+e x)^8 \left (3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )\right )}{e^7}+\frac {(d+e x)^5 \left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{e^7}+\frac {(d+e x)^4 (A e-B d) \left (a e^2-b d e+c d^2\right )^3}{e^7}+\frac {c^2 (d+e x)^{10} (A c e+3 b B e-7 B c d)}{e^7}+\frac {B c^3 (d+e x)^{11}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(d+e x)^8 \left (A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )\right )}{8 e^8}-\frac {3 c (d+e x)^{10} \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{10 e^8}-\frac {3 (d+e x)^7 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{7 e^8}-\frac {(d+e x)^9 \left (B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{9 e^8}+\frac {(d+e x)^6 \left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{6 e^8}-\frac {(d+e x)^5 (B d-A e) \left (a e^2-b d e+c d^2\right )^3}{5 e^8}-\frac {c^2 (d+e x)^{11} (-A c e-3 b B e+7 B c d)}{11 e^8}+\frac {B c^3 (d+e x)^{12}}{12 e^8}\) |
-1/5*((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^5)/e^8 + ((c*d^2 - b *d*e + a*e^2)^2*(7*B*c*d^2 - B*e*(4*b*d - a*e) - 3*A*e*(2*c*d - b*e))*(d + e*x)^6)/(6*e^8) - (3*(c*d^2 - b*d*e + a*e^2)*(B*(7*c^2*d^3 - c*d*e*(8*b*d - 3*a*e) + b*e^2*(2*b*d - a*e)) - A*e*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))*(d + e*x)^7)/(7*e^8) - ((A*e*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e)) - B*(35*c^3*d^4 - b^2*e^3*(4*b*d - 3*a*e) - 30*c^2 *d^2*e*(2*b*d - a*e) + 3*c*e^2*(10*b^2*d^2 - 8*a*b*d*e + a^2*e^2)))*(d + e *x)^8)/(8*e^8) - ((B*(35*c^3*d^3 - b^3*e^3 + 3*b*c*e^2*(5*b*d - 2*a*e) - 1 5*c^2*d*e*(3*b*d - a*e)) - 3*A*c*e*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e )))*(d + e*x)^9)/(9*e^8) - (3*c*(A*c*e*(2*c*d - b*e) - B*(7*c^2*d^2 + b^2* e^2 - c*e*(6*b*d - a*e)))*(d + e*x)^10)/(10*e^8) - (c^2*(7*B*c*d - 3*b*B*e - A*c*e)*(d + e*x)^11)/(11*e^8) + (B*c^3*(d + e*x)^12)/(12*e^8)
3.24.34.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 0.38 (sec) , antiderivative size = 1041, normalized size of antiderivative = 1.88
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1041\) |
| norman | \(\text {Expression too large to display}\) | \(1110\) |
| gosper | \(\text {Expression too large to display}\) | \(1354\) |
| risch | \(\text {Expression too large to display}\) | \(1354\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1354\) |
1/12*B*c^3*e^4*x^12+1/11*((A*e^4+4*B*d*e^3)*c^3+3*B*e^4*b*c^2)*x^11+1/10*( (4*A*d*e^3+6*B*d^2*e^2)*c^3+3*(A*e^4+4*B*d*e^3)*b*c^2+B*e^4*(a*c^2+2*b^2*c +c*(2*a*c+b^2)))*x^10+1/9*((6*A*d^2*e^2+4*B*d^3*e)*c^3+3*(4*A*d*e^3+6*B*d^ 2*e^2)*b*c^2+(A*e^4+4*B*d*e^3)*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+B*e^4*(4*a*b* c+b*(2*a*c+b^2)))*x^9+1/8*((4*A*d^3*e+B*d^4)*c^3+3*(6*A*d^2*e^2+4*B*d^3*e) *b*c^2+(4*A*d*e^3+6*B*d^2*e^2)*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+(A*e^4+4*B*d* e^3)*(4*a*b*c+b*(2*a*c+b^2))+B*e^4*(a*(2*a*c+b^2)+2*b^2*a+c*a^2))*x^8+1/7* (A*d^4*c^3+3*(4*A*d^3*e+B*d^4)*b*c^2+(6*A*d^2*e^2+4*B*d^3*e)*(a*c^2+2*b^2* c+c*(2*a*c+b^2))+(4*A*d*e^3+6*B*d^2*e^2)*(4*a*b*c+b*(2*a*c+b^2))+(A*e^4+4* B*d*e^3)*(a*(2*a*c+b^2)+2*b^2*a+c*a^2)+3*B*e^4*b*a^2)*x^7+1/6*(3*A*d^4*b*c ^2+(4*A*d^3*e+B*d^4)*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+(6*A*d^2*e^2+4*B*d^3*e) *(4*a*b*c+b*(2*a*c+b^2))+(4*A*d*e^3+6*B*d^2*e^2)*(a*(2*a*c+b^2)+2*b^2*a+c* a^2)+3*(A*e^4+4*B*d*e^3)*b*a^2+B*e^4*a^3)*x^6+1/5*(A*d^4*(a*c^2+2*b^2*c+c* (2*a*c+b^2))+(4*A*d^3*e+B*d^4)*(4*a*b*c+b*(2*a*c+b^2))+(6*A*d^2*e^2+4*B*d^ 3*e)*(a*(2*a*c+b^2)+2*b^2*a+c*a^2)+3*(4*A*d*e^3+6*B*d^2*e^2)*b*a^2+(A*e^4+ 4*B*d*e^3)*a^3)*x^5+1/4*(A*d^4*(4*a*b*c+b*(2*a*c+b^2))+(4*A*d^3*e+B*d^4)*( a*(2*a*c+b^2)+2*b^2*a+c*a^2)+3*(6*A*d^2*e^2+4*B*d^3*e)*b*a^2+(4*A*d*e^3+6* B*d^2*e^2)*a^3)*x^4+1/3*(A*d^4*(a*(2*a*c+b^2)+2*b^2*a+c*a^2)+3*(4*A*d^3*e+ B*d^4)*b*a^2+(6*A*d^2*e^2+4*B*d^3*e)*a^3)*x^3+1/2*(3*A*a^2*b*d^4+(4*A*d^3* e+B*d^4)*a^3)*x^2+d^4*A*a^3*x
Time = 0.31 (sec) , antiderivative size = 919, normalized size of antiderivative = 1.66 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{12} \, B c^{3} e^{4} x^{12} + \frac {1}{11} \, {\left (4 \, B c^{3} d e^{3} + {\left (3 \, B b c^{2} + A c^{3}\right )} e^{4}\right )} x^{11} + \frac {1}{10} \, {\left (6 \, B c^{3} d^{2} e^{2} + 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{3} + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} e^{4}\right )} x^{10} + \frac {1}{9} \, {\left (4 \, B c^{3} d^{3} e + 6 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{2} + 12 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d e^{3} + {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e^{4}\right )} x^{9} + A a^{3} d^{4} x + \frac {1}{8} \, {\left (B c^{3} d^{4} + 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e + 18 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} e^{2} + 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d e^{3} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} + 12 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e + 6 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} e^{2} + 4 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d e^{3} + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} + 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{3} e + 6 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{2} e^{2} + 12 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d e^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (A a^{3} e^{4} + {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{4} + 4 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{3} e + 18 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2} e^{2} + 4 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, A a^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{4} + 12 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{3} e + 6 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{3} d^{2} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{4} + 4 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{3} d^{3} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4}\right )} x^{2} \]
1/12*B*c^3*e^4*x^12 + 1/11*(4*B*c^3*d*e^3 + (3*B*b*c^2 + A*c^3)*e^4)*x^11 + 1/10*(6*B*c^3*d^2*e^2 + 4*(3*B*b*c^2 + A*c^3)*d*e^3 + 3*(B*b^2*c + (B*a + A*b)*c^2)*e^4)*x^10 + 1/9*(4*B*c^3*d^3*e + 6*(3*B*b*c^2 + A*c^3)*d^2*e^2 + 12*(B*b^2*c + (B*a + A*b)*c^2)*d*e^3 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^4)*x^9 + A*a^3*d^4*x + 1/8*(B*c^3*d^4 + 4*(3*B*b*c^2 + A*c^3 )*d^3*e + 18*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^2 + 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d*e^3 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c) *e^4)*x^8 + 1/7*((3*B*b*c^2 + A*c^3)*d^4 + 12*(B*b^2*c + (B*a + A*b)*c^2)* d^3*e + 6*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^2 + 4*(3*B*a*b ^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^3 + 3*(B*a^2*b + A*a*b^2 + A*a^2*c )*e^4)*x^7 + 1/6*(3*(B*b^2*c + (B*a + A*b)*c^2)*d^4 + 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*e + 6*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a* b)*c)*d^2*e^2 + 12*(B*a^2*b + A*a*b^2 + A*a^2*c)*d*e^3 + (B*a^3 + 3*A*a^2* b)*e^4)*x^6 + 1/5*(A*a^3*e^4 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c) *d^4 + 4*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^3*e + 18*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^2 + 4*(B*a^3 + 3*A*a^2*b)*d*e^3)*x^5 + 1/4*(4*A* a^3*d*e^3 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^4 + 12*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^3*e + 6*(B*a^3 + 3*A*a^2*b)*d^2*e^2)*x^4 + 1/3*(6*A *a^3*d^2*e^2 + 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^4 + 4*(B*a^3 + 3*A*a^2*b) *d^3*e)*x^3 + 1/2*(4*A*a^3*d^3*e + (B*a^3 + 3*A*a^2*b)*d^4)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 1401 vs. \(2 (542) = 1084\).
Time = 0.09 (sec) , antiderivative size = 1401, normalized size of antiderivative = 2.52 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
A*a**3*d**4*x + B*c**3*e**4*x**12/12 + x**11*(A*c**3*e**4/11 + 3*B*b*c**2* e**4/11 + 4*B*c**3*d*e**3/11) + x**10*(3*A*b*c**2*e**4/10 + 2*A*c**3*d*e** 3/5 + 3*B*a*c**2*e**4/10 + 3*B*b**2*c*e**4/10 + 6*B*b*c**2*d*e**3/5 + 3*B* c**3*d**2*e**2/5) + x**9*(A*a*c**2*e**4/3 + A*b**2*c*e**4/3 + 4*A*b*c**2*d *e**3/3 + 2*A*c**3*d**2*e**2/3 + 2*B*a*b*c*e**4/3 + 4*B*a*c**2*d*e**3/3 + B*b**3*e**4/9 + 4*B*b**2*c*d*e**3/3 + 2*B*b*c**2*d**2*e**2 + 4*B*c**3*d**3 *e/9) + x**8*(3*A*a*b*c*e**4/4 + 3*A*a*c**2*d*e**3/2 + A*b**3*e**4/8 + 3*A *b**2*c*d*e**3/2 + 9*A*b*c**2*d**2*e**2/4 + A*c**3*d**3*e/2 + 3*B*a**2*c*e **4/8 + 3*B*a*b**2*e**4/8 + 3*B*a*b*c*d*e**3 + 9*B*a*c**2*d**2*e**2/4 + B* b**3*d*e**3/2 + 9*B*b**2*c*d**2*e**2/4 + 3*B*b*c**2*d**3*e/2 + B*c**3*d**4 /8) + x**7*(3*A*a**2*c*e**4/7 + 3*A*a*b**2*e**4/7 + 24*A*a*b*c*d*e**3/7 + 18*A*a*c**2*d**2*e**2/7 + 4*A*b**3*d*e**3/7 + 18*A*b**2*c*d**2*e**2/7 + 12 *A*b*c**2*d**3*e/7 + A*c**3*d**4/7 + 3*B*a**2*b*e**4/7 + 12*B*a**2*c*d*e** 3/7 + 12*B*a*b**2*d*e**3/7 + 36*B*a*b*c*d**2*e**2/7 + 12*B*a*c**2*d**3*e/7 + 6*B*b**3*d**2*e**2/7 + 12*B*b**2*c*d**3*e/7 + 3*B*b*c**2*d**4/7) + x**6 *(A*a**2*b*e**4/2 + 2*A*a**2*c*d*e**3 + 2*A*a*b**2*d*e**3 + 6*A*a*b*c*d**2 *e**2 + 2*A*a*c**2*d**3*e + A*b**3*d**2*e**2 + 2*A*b**2*c*d**3*e + A*b*c** 2*d**4/2 + B*a**3*e**4/6 + 2*B*a**2*b*d*e**3 + 3*B*a**2*c*d**2*e**2 + 3*B* a*b**2*d**2*e**2 + 4*B*a*b*c*d**3*e + B*a*c**2*d**4/2 + 2*B*b**3*d**3*e/3 + B*b**2*c*d**4/2) + x**5*(A*a**3*e**4/5 + 12*A*a**2*b*d*e**3/5 + 18*A*...
Time = 0.20 (sec) , antiderivative size = 919, normalized size of antiderivative = 1.66 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{12} \, B c^{3} e^{4} x^{12} + \frac {1}{11} \, {\left (4 \, B c^{3} d e^{3} + {\left (3 \, B b c^{2} + A c^{3}\right )} e^{4}\right )} x^{11} + \frac {1}{10} \, {\left (6 \, B c^{3} d^{2} e^{2} + 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{3} + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} e^{4}\right )} x^{10} + \frac {1}{9} \, {\left (4 \, B c^{3} d^{3} e + 6 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{2} + 12 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d e^{3} + {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e^{4}\right )} x^{9} + A a^{3} d^{4} x + \frac {1}{8} \, {\left (B c^{3} d^{4} + 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e + 18 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} e^{2} + 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d e^{3} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} + 12 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e + 6 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} e^{2} + 4 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d e^{3} + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} + 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{3} e + 6 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{2} e^{2} + 12 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d e^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (A a^{3} e^{4} + {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{4} + 4 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{3} e + 18 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2} e^{2} + 4 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, A a^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{4} + 12 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{3} e + 6 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{3} d^{2} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{4} + 4 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{3} d^{3} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4}\right )} x^{2} \]
1/12*B*c^3*e^4*x^12 + 1/11*(4*B*c^3*d*e^3 + (3*B*b*c^2 + A*c^3)*e^4)*x^11 + 1/10*(6*B*c^3*d^2*e^2 + 4*(3*B*b*c^2 + A*c^3)*d*e^3 + 3*(B*b^2*c + (B*a + A*b)*c^2)*e^4)*x^10 + 1/9*(4*B*c^3*d^3*e + 6*(3*B*b*c^2 + A*c^3)*d^2*e^2 + 12*(B*b^2*c + (B*a + A*b)*c^2)*d*e^3 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^4)*x^9 + A*a^3*d^4*x + 1/8*(B*c^3*d^4 + 4*(3*B*b*c^2 + A*c^3 )*d^3*e + 18*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^2 + 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d*e^3 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c) *e^4)*x^8 + 1/7*((3*B*b*c^2 + A*c^3)*d^4 + 12*(B*b^2*c + (B*a + A*b)*c^2)* d^3*e + 6*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^2 + 4*(3*B*a*b ^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^3 + 3*(B*a^2*b + A*a*b^2 + A*a^2*c )*e^4)*x^7 + 1/6*(3*(B*b^2*c + (B*a + A*b)*c^2)*d^4 + 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*e + 6*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a* b)*c)*d^2*e^2 + 12*(B*a^2*b + A*a*b^2 + A*a^2*c)*d*e^3 + (B*a^3 + 3*A*a^2* b)*e^4)*x^6 + 1/5*(A*a^3*e^4 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c) *d^4 + 4*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^3*e + 18*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^2 + 4*(B*a^3 + 3*A*a^2*b)*d*e^3)*x^5 + 1/4*(4*A* a^3*d*e^3 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^4 + 12*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^3*e + 6*(B*a^3 + 3*A*a^2*b)*d^2*e^2)*x^4 + 1/3*(6*A *a^3*d^2*e^2 + 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^4 + 4*(B*a^3 + 3*A*a^2*b) *d^3*e)*x^3 + 1/2*(4*A*a^3*d^3*e + (B*a^3 + 3*A*a^2*b)*d^4)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 1353 vs. \(2 (540) = 1080\).
Time = 0.31 (sec) , antiderivative size = 1353, normalized size of antiderivative = 2.44 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
1/12*B*c^3*e^4*x^12 + 4/11*B*c^3*d*e^3*x^11 + 3/11*B*b*c^2*e^4*x^11 + 1/11 *A*c^3*e^4*x^11 + 3/5*B*c^3*d^2*e^2*x^10 + 6/5*B*b*c^2*d*e^3*x^10 + 2/5*A* c^3*d*e^3*x^10 + 3/10*B*b^2*c*e^4*x^10 + 3/10*B*a*c^2*e^4*x^10 + 3/10*A*b* c^2*e^4*x^10 + 4/9*B*c^3*d^3*e*x^9 + 2*B*b*c^2*d^2*e^2*x^9 + 2/3*A*c^3*d^2 *e^2*x^9 + 4/3*B*b^2*c*d*e^3*x^9 + 4/3*B*a*c^2*d*e^3*x^9 + 4/3*A*b*c^2*d*e ^3*x^9 + 1/9*B*b^3*e^4*x^9 + 2/3*B*a*b*c*e^4*x^9 + 1/3*A*b^2*c*e^4*x^9 + 1 /3*A*a*c^2*e^4*x^9 + 1/8*B*c^3*d^4*x^8 + 3/2*B*b*c^2*d^3*e*x^8 + 1/2*A*c^3 *d^3*e*x^8 + 9/4*B*b^2*c*d^2*e^2*x^8 + 9/4*B*a*c^2*d^2*e^2*x^8 + 9/4*A*b*c ^2*d^2*e^2*x^8 + 1/2*B*b^3*d*e^3*x^8 + 3*B*a*b*c*d*e^3*x^8 + 3/2*A*b^2*c*d *e^3*x^8 + 3/2*A*a*c^2*d*e^3*x^8 + 3/8*B*a*b^2*e^4*x^8 + 1/8*A*b^3*e^4*x^8 + 3/8*B*a^2*c*e^4*x^8 + 3/4*A*a*b*c*e^4*x^8 + 3/7*B*b*c^2*d^4*x^7 + 1/7*A *c^3*d^4*x^7 + 12/7*B*b^2*c*d^3*e*x^7 + 12/7*B*a*c^2*d^3*e*x^7 + 12/7*A*b* c^2*d^3*e*x^7 + 6/7*B*b^3*d^2*e^2*x^7 + 36/7*B*a*b*c*d^2*e^2*x^7 + 18/7*A* b^2*c*d^2*e^2*x^7 + 18/7*A*a*c^2*d^2*e^2*x^7 + 12/7*B*a*b^2*d*e^3*x^7 + 4/ 7*A*b^3*d*e^3*x^7 + 12/7*B*a^2*c*d*e^3*x^7 + 24/7*A*a*b*c*d*e^3*x^7 + 3/7* B*a^2*b*e^4*x^7 + 3/7*A*a*b^2*e^4*x^7 + 3/7*A*a^2*c*e^4*x^7 + 1/2*B*b^2*c* d^4*x^6 + 1/2*B*a*c^2*d^4*x^6 + 1/2*A*b*c^2*d^4*x^6 + 2/3*B*b^3*d^3*e*x^6 + 4*B*a*b*c*d^3*e*x^6 + 2*A*b^2*c*d^3*e*x^6 + 2*A*a*c^2*d^3*e*x^6 + 3*B*a* b^2*d^2*e^2*x^6 + A*b^3*d^2*e^2*x^6 + 3*B*a^2*c*d^2*e^2*x^6 + 6*A*a*b*c*d^ 2*e^2*x^6 + 2*B*a^2*b*d*e^3*x^6 + 2*A*a*b^2*d*e^3*x^6 + 2*A*a^2*c*d*e^3...
Time = 0.32 (sec) , antiderivative size = 1093, normalized size of antiderivative = 1.97 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=x^5\,\left (\frac {4\,B\,a^3\,d\,e^3}{5}+\frac {A\,a^3\,e^4}{5}+\frac {18\,B\,a^2\,b\,d^2\,e^2}{5}+\frac {12\,A\,a^2\,b\,d\,e^3}{5}+\frac {12\,B\,a^2\,c\,d^3\,e}{5}+\frac {18\,A\,a^2\,c\,d^2\,e^2}{5}+\frac {12\,B\,a\,b^2\,d^3\,e}{5}+\frac {18\,A\,a\,b^2\,d^2\,e^2}{5}+\frac {6\,B\,a\,b\,c\,d^4}{5}+\frac {24\,A\,a\,b\,c\,d^3\,e}{5}+\frac {3\,A\,a\,c^2\,d^4}{5}+\frac {B\,b^3\,d^4}{5}+\frac {4\,A\,b^3\,d^3\,e}{5}+\frac {3\,A\,b^2\,c\,d^4}{5}\right )+x^8\,\left (\frac {3\,B\,a^2\,c\,e^4}{8}+\frac {3\,B\,a\,b^2\,e^4}{8}+3\,B\,a\,b\,c\,d\,e^3+\frac {3\,A\,a\,b\,c\,e^4}{4}+\frac {9\,B\,a\,c^2\,d^2\,e^2}{4}+\frac {3\,A\,a\,c^2\,d\,e^3}{2}+\frac {B\,b^3\,d\,e^3}{2}+\frac {A\,b^3\,e^4}{8}+\frac {9\,B\,b^2\,c\,d^2\,e^2}{4}+\frac {3\,A\,b^2\,c\,d\,e^3}{2}+\frac {3\,B\,b\,c^2\,d^3\,e}{2}+\frac {9\,A\,b\,c^2\,d^2\,e^2}{4}+\frac {B\,c^3\,d^4}{8}+\frac {A\,c^3\,d^3\,e}{2}\right )+x^3\,\left (\frac {4\,B\,a^3\,d^3\,e}{3}+2\,A\,a^3\,d^2\,e^2+B\,a^2\,b\,d^4+4\,A\,a^2\,b\,d^3\,e+A\,c\,a^2\,d^4+A\,a\,b^2\,d^4\right )+x^{10}\,\left (\frac {3\,B\,b^2\,c\,e^4}{10}+\frac {6\,B\,b\,c^2\,d\,e^3}{5}+\frac {3\,A\,b\,c^2\,e^4}{10}+\frac {3\,B\,c^3\,d^2\,e^2}{5}+\frac {2\,A\,c^3\,d\,e^3}{5}+\frac {3\,B\,a\,c^2\,e^4}{10}\right )+x^6\,\left (\frac {B\,a^3\,e^4}{6}+2\,B\,a^2\,b\,d\,e^3+\frac {A\,a^2\,b\,e^4}{2}+3\,B\,a^2\,c\,d^2\,e^2+2\,A\,a^2\,c\,d\,e^3+3\,B\,a\,b^2\,d^2\,e^2+2\,A\,a\,b^2\,d\,e^3+4\,B\,a\,b\,c\,d^3\,e+6\,A\,a\,b\,c\,d^2\,e^2+\frac {B\,a\,c^2\,d^4}{2}+2\,A\,a\,c^2\,d^3\,e+\frac {2\,B\,b^3\,d^3\,e}{3}+A\,b^3\,d^2\,e^2+\frac {B\,b^2\,c\,d^4}{2}+2\,A\,b^2\,c\,d^3\,e+\frac {A\,b\,c^2\,d^4}{2}\right )+x^7\,\left (\frac {3\,B\,a^2\,b\,e^4}{7}+\frac {12\,B\,a^2\,c\,d\,e^3}{7}+\frac {3\,A\,a^2\,c\,e^4}{7}+\frac {12\,B\,a\,b^2\,d\,e^3}{7}+\frac {3\,A\,a\,b^2\,e^4}{7}+\frac {36\,B\,a\,b\,c\,d^2\,e^2}{7}+\frac {24\,A\,a\,b\,c\,d\,e^3}{7}+\frac {12\,B\,a\,c^2\,d^3\,e}{7}+\frac {18\,A\,a\,c^2\,d^2\,e^2}{7}+\frac {6\,B\,b^3\,d^2\,e^2}{7}+\frac {4\,A\,b^3\,d\,e^3}{7}+\frac {12\,B\,b^2\,c\,d^3\,e}{7}+\frac {18\,A\,b^2\,c\,d^2\,e^2}{7}+\frac {3\,B\,b\,c^2\,d^4}{7}+\frac {12\,A\,b\,c^2\,d^3\,e}{7}+\frac {A\,c^3\,d^4}{7}\right )+x^4\,\left (\frac {3\,B\,a^3\,d^2\,e^2}{2}+A\,a^3\,d\,e^3+3\,B\,a^2\,b\,d^3\,e+\frac {9\,A\,a^2\,b\,d^2\,e^2}{2}+\frac {3\,B\,c\,a^2\,d^4}{4}+3\,A\,c\,a^2\,d^3\,e+\frac {3\,B\,a\,b^2\,d^4}{4}+3\,A\,a\,b^2\,d^3\,e+\frac {3\,A\,c\,a\,b\,d^4}{2}+\frac {A\,b^3\,d^4}{4}\right )+x^9\,\left (\frac {B\,b^3\,e^4}{9}+\frac {4\,B\,b^2\,c\,d\,e^3}{3}+\frac {A\,b^2\,c\,e^4}{3}+2\,B\,b\,c^2\,d^2\,e^2+\frac {4\,A\,b\,c^2\,d\,e^3}{3}+\frac {2\,B\,a\,b\,c\,e^4}{3}+\frac {4\,B\,c^3\,d^3\,e}{9}+\frac {2\,A\,c^3\,d^2\,e^2}{3}+\frac {4\,B\,a\,c^2\,d\,e^3}{3}+\frac {A\,a\,c^2\,e^4}{3}\right )+\frac {a^2\,d^3\,x^2\,\left (4\,A\,a\,e+3\,A\,b\,d+B\,a\,d\right )}{2}+\frac {c^2\,e^3\,x^{11}\,\left (A\,c\,e+3\,B\,b\,e+4\,B\,c\,d\right )}{11}+A\,a^3\,d^4\,x+\frac {B\,c^3\,e^4\,x^{12}}{12} \]
x^5*((A*a^3*e^4)/5 + (B*b^3*d^4)/5 + (3*A*a*c^2*d^4)/5 + (3*A*b^2*c*d^4)/5 + (4*A*b^3*d^3*e)/5 + (4*B*a^3*d*e^3)/5 + (18*A*a*b^2*d^2*e^2)/5 + (18*A* a^2*c*d^2*e^2)/5 + (18*B*a^2*b*d^2*e^2)/5 + (6*B*a*b*c*d^4)/5 + (12*A*a^2* b*d*e^3)/5 + (12*B*a*b^2*d^3*e)/5 + (12*B*a^2*c*d^3*e)/5 + (24*A*a*b*c*d^3 *e)/5) + x^8*((A*b^3*e^4)/8 + (B*c^3*d^4)/8 + (3*B*a*b^2*e^4)/8 + (3*B*a^2 *c*e^4)/8 + (A*c^3*d^3*e)/2 + (B*b^3*d*e^3)/2 + (9*A*b*c^2*d^2*e^2)/4 + (9 *B*a*c^2*d^2*e^2)/4 + (9*B*b^2*c*d^2*e^2)/4 + (3*A*a*b*c*e^4)/4 + (3*A*a*c ^2*d*e^3)/2 + (3*A*b^2*c*d*e^3)/2 + (3*B*b*c^2*d^3*e)/2 + 3*B*a*b*c*d*e^3) + x^3*(A*a*b^2*d^4 + A*a^2*c*d^4 + B*a^2*b*d^4 + (4*B*a^3*d^3*e)/3 + 2*A* a^3*d^2*e^2 + 4*A*a^2*b*d^3*e) + x^10*((3*A*b*c^2*e^4)/10 + (3*B*a*c^2*e^4 )/10 + (3*B*b^2*c*e^4)/10 + (2*A*c^3*d*e^3)/5 + (3*B*c^3*d^2*e^2)/5 + (6*B *b*c^2*d*e^3)/5) + x^6*((B*a^3*e^4)/6 + (A*a^2*b*e^4)/2 + (A*b*c^2*d^4)/2 + (B*a*c^2*d^4)/2 + (B*b^2*c*d^4)/2 + (2*B*b^3*d^3*e)/3 + A*b^3*d^2*e^2 + 3*B*a*b^2*d^2*e^2 + 3*B*a^2*c*d^2*e^2 + 2*A*a*b^2*d*e^3 + 2*A*a*c^2*d^3*e + 2*A*a^2*c*d*e^3 + 2*B*a^2*b*d*e^3 + 2*A*b^2*c*d^3*e + 6*A*a*b*c*d^2*e^2 + 4*B*a*b*c*d^3*e) + x^7*((A*c^3*d^4)/7 + (3*A*a*b^2*e^4)/7 + (3*A*a^2*c*e ^4)/7 + (3*B*a^2*b*e^4)/7 + (3*B*b*c^2*d^4)/7 + (4*A*b^3*d*e^3)/7 + (6*B*b ^3*d^2*e^2)/7 + (18*A*a*c^2*d^2*e^2)/7 + (18*A*b^2*c*d^2*e^2)/7 + (12*B*a* b^2*d*e^3)/7 + (12*A*b*c^2*d^3*e)/7 + (12*B*a*c^2*d^3*e)/7 + (12*B*a^2*c*d *e^3)/7 + (12*B*b^2*c*d^3*e)/7 + (36*B*a*b*c*d^2*e^2)/7 + (24*A*a*b*c*d...