Integrand size = 20, antiderivative size = 443 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^4 \, dx=\frac {\left (c d^2-b d e+a e^2\right )^4 (d+e x)^5}{5 e^9}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^6}{3 e^9}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^7}{7 e^9}-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^8}{2 e^9}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^9}{9 e^9}-\frac {2 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{10}}{5 e^9}+\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{11}}{11 e^9}-\frac {c^3 (2 c d-b e) (d+e x)^{12}}{3 e^9}+\frac {c^4 (d+e x)^{13}}{13 e^9} \]
1/5*(a*e^2-b*d*e+c*d^2)^4*(e*x+d)^5/e^9-2/3*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^ 2)^3*(e*x+d)^6/e^9+2/7*(a*e^2-b*d*e+c*d^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*( -a*e+7*b*d))*(e*x+d)^7/e^9-1/2*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(7*c^2*d^2 +b^2*e^2-c*e*(-3*a*e+7*b*d))*(e*x+d)^8/e^9+1/9*(70*c^4*d^4+b^4*e^4-4*b^2*c *e^3*(-3*a*e+5*b*d)-20*c^3*d^2*e*(-3*a*e+7*b*d)+6*c^2*e^2*(a^2*e^2-10*a*b* d*e+15*b^2*d^2))*(e*x+d)^9/e^9-2/5*c*(-b*e+2*c*d)*(7*c^2*d^2+b^2*e^2-c*e*( -3*a*e+7*b*d))*(e*x+d)^10/e^9+2/11*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7 *b*d))*(e*x+d)^11/e^9-1/3*c^3*(-b*e+2*c*d)*(e*x+d)^12/e^9+1/13*c^4*(e*x+d) ^13/e^9
Time = 0.16 (sec) , antiderivative size = 766, normalized size of antiderivative = 1.73 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^4 \, dx=a^4 d^4 x+2 a^3 d^3 (b d+a e) x^2+\frac {2}{3} a^2 d^2 \left (3 b^2 d^2+8 a b d e+a \left (2 c d^2+3 a e^2\right )\right ) x^3+a d \left (b^3 d^3+6 a b^2 d^2 e+a^2 e \left (4 c d^2+a e^2\right )+3 a b d \left (c d^2+2 a e^2\right )\right ) x^4+\frac {1}{5} \left (b^4 d^4+16 a b^3 d^3 e+16 a^2 b d e \left (3 c d^2+a e^2\right )+12 a b^2 d^2 \left (c d^2+3 a e^2\right )+a^2 \left (6 c^2 d^4+24 a c d^2 e^2+a^2 e^4\right )\right ) x^5+\frac {2}{3} \left (b^4 d^3 e+6 a b^2 d e \left (2 c d^2+a e^2\right )+2 a^2 c d e \left (3 c d^2+2 a e^2\right )+b^3 \left (c d^4+6 a d^2 e^2\right )+a b \left (3 c^2 d^4+18 a c d^2 e^2+a^2 e^4\right )\right ) x^6+\frac {2}{7} \left (3 b^4 d^2 e^2+24 a b c d e \left (c d^2+a e^2\right )+8 b^3 \left (c d^3 e+a d e^3\right )+2 a c \left (c^2 d^4+9 a c d^2 e^2+a^2 e^4\right )+3 b^2 \left (c^2 d^4+12 a c d^2 e^2+a^2 e^4\right )\right ) x^7+\frac {1}{2} \left (b^4 d e^3+6 b^2 c d e \left (c d^2+2 a e^2\right )+2 a c^2 d e \left (2 c d^2+3 a e^2\right )+b^3 \left (6 c d^2 e^2+a e^4\right )+b c \left (c^2 d^4+18 a c d^2 e^2+3 a^2 e^4\right )\right ) x^8+\frac {1}{9} \left (c^4 d^4+b^4 e^4+8 c^3 d^2 e (2 b d+3 a e)+4 b^2 c e^3 (4 b d+3 a e)+6 c^2 e^2 \left (6 b^2 d^2+8 a b d e+a^2 e^2\right )\right ) x^9+\frac {2}{5} c e \left (c^3 d^3+b^3 e^3+3 b c e^2 (2 b d+a e)+2 c^2 d e (3 b d+2 a e)\right ) x^{10}+\frac {2}{11} c^2 e^2 \left (3 c^2 d^2+3 b^2 e^2+2 c e (4 b d+a e)\right ) x^{11}+\frac {1}{3} c^3 e^3 (c d+b e) x^{12}+\frac {1}{13} c^4 e^4 x^{13} \]
a^4*d^4*x + 2*a^3*d^3*(b*d + a*e)*x^2 + (2*a^2*d^2*(3*b^2*d^2 + 8*a*b*d*e + a*(2*c*d^2 + 3*a*e^2))*x^3)/3 + a*d*(b^3*d^3 + 6*a*b^2*d^2*e + a^2*e*(4* c*d^2 + a*e^2) + 3*a*b*d*(c*d^2 + 2*a*e^2))*x^4 + ((b^4*d^4 + 16*a*b^3*d^3 *e + 16*a^2*b*d*e*(3*c*d^2 + a*e^2) + 12*a*b^2*d^2*(c*d^2 + 3*a*e^2) + a^2 *(6*c^2*d^4 + 24*a*c*d^2*e^2 + a^2*e^4))*x^5)/5 + (2*(b^4*d^3*e + 6*a*b^2* d*e*(2*c*d^2 + a*e^2) + 2*a^2*c*d*e*(3*c*d^2 + 2*a*e^2) + b^3*(c*d^4 + 6*a *d^2*e^2) + a*b*(3*c^2*d^4 + 18*a*c*d^2*e^2 + a^2*e^4))*x^6)/3 + (2*(3*b^4 *d^2*e^2 + 24*a*b*c*d*e*(c*d^2 + a*e^2) + 8*b^3*(c*d^3*e + a*d*e^3) + 2*a* c*(c^2*d^4 + 9*a*c*d^2*e^2 + a^2*e^4) + 3*b^2*(c^2*d^4 + 12*a*c*d^2*e^2 + a^2*e^4))*x^7)/7 + ((b^4*d*e^3 + 6*b^2*c*d*e*(c*d^2 + 2*a*e^2) + 2*a*c^2*d *e*(2*c*d^2 + 3*a*e^2) + b^3*(6*c*d^2*e^2 + a*e^4) + b*c*(c^2*d^4 + 18*a*c *d^2*e^2 + 3*a^2*e^4))*x^8)/2 + ((c^4*d^4 + b^4*e^4 + 8*c^3*d^2*e*(2*b*d + 3*a*e) + 4*b^2*c*e^3*(4*b*d + 3*a*e) + 6*c^2*e^2*(6*b^2*d^2 + 8*a*b*d*e + a^2*e^2))*x^9)/9 + (2*c*e*(c^3*d^3 + b^3*e^3 + 3*b*c*e^2*(2*b*d + a*e) + 2*c^2*d*e*(3*b*d + 2*a*e))*x^10)/5 + (2*c^2*e^2*(3*c^2*d^2 + 3*b^2*e^2 + 2 *c*e*(4*b*d + a*e))*x^11)/11 + (c^3*e^3*(c*d + b*e)*x^12)/3 + (c^4*e^4*x^1 3)/13
Time = 1.10 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1140, 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.
| \(\displaystyle \int (d+e x)^4 \left (a+b x+c x^2\right )^4 \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {(d+e x)^8 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^8}+\frac {2 c^2 (d+e x)^{10} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8}+\frac {4 c (d+e x)^9 (2 c d-b e) \left (c e (7 b d-3 a e)-b^2 e^2-7 c^2 d^2\right )}{e^8}+\frac {4 (d+e x)^7 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-3 a c e^2-b^2 e^2+7 b c d e-7 c^2 d^2\right )}{e^8}+\frac {2 (d+e x)^6 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8}+\frac {4 (d+e x)^5 (b e-2 c d) \left (a e^2-b d e+c d^2\right )^3}{e^8}+\frac {(d+e x)^4 \left (a e^2-b d e+c d^2\right )^4}{e^8}-\frac {4 c^3 (d+e x)^{11} (2 c d-b e)}{e^8}+\frac {c^4 (d+e x)^{12}}{e^8}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^9 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{9 e^9}+\frac {2 c^2 (d+e x)^{11} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{11 e^9}-\frac {2 c (d+e x)^{10} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^9}-\frac {(d+e x)^8 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 e^9}+\frac {2 (d+e x)^7 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{7 e^9}-\frac {2 (d+e x)^6 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^9}+\frac {(d+e x)^5 \left (a e^2-b d e+c d^2\right )^4}{5 e^9}-\frac {c^3 (d+e x)^{12} (2 c d-b e)}{3 e^9}+\frac {c^4 (d+e x)^{13}}{13 e^9}\) |
((c*d^2 - b*d*e + a*e^2)^4*(d + e*x)^5)/(5*e^9) - (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^6)/(3*e^9) + (2*(c*d^2 - b*d*e + a*e^2)^2*(14 *c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^7)/(7*e^9) - ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e) )*(d + e*x)^8)/(2*e^9) + ((70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a *e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(d + e*x)^9)/(9*e^9) - (2*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^10)/(5*e^9) + (2*c^2*(14*c^2*d^2 + 3*b^2*e ^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^11)/(11*e^9) - (c^3*(2*c*d - b*e)*(d + e*x)^12)/(3*e^9) + (c^4*(d + e*x)^13)/(13*e^9)
3.22.46.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Time = 3.18 (sec) , antiderivative size = 838, normalized size of antiderivative = 1.89
| method | result | size |
| norman | \(\frac {c^{4} e^{4} x^{13}}{13}+\left (\frac {1}{3} e^{4} c^{3} b +\frac {1}{3} d \,e^{3} c^{4}\right ) x^{12}+\left (\frac {4}{11} a \,c^{3} e^{4}+\frac {6}{11} b^{2} c^{2} e^{4}+\frac {16}{11} d \,e^{3} c^{3} b +\frac {6}{11} d^{2} e^{2} c^{4}\right ) x^{11}+\left (\frac {6}{5} a b \,c^{2} e^{4}+\frac {8}{5} a \,c^{3} d \,e^{3}+\frac {2}{5} b^{3} c \,e^{4}+\frac {12}{5} b^{2} c^{2} d \,e^{3}+\frac {12}{5} b \,c^{3} d^{2} e^{2}+\frac {2}{5} c^{4} d^{3} e \right ) x^{10}+\left (\frac {2}{3} c^{2} a^{2} e^{4}+\frac {4}{3} a \,b^{2} c \,e^{4}+\frac {16}{3} a b \,c^{2} d \,e^{3}+\frac {8}{3} c^{3} a \,d^{2} e^{2}+\frac {1}{9} b^{4} e^{4}+\frac {16}{9} b^{3} c d \,e^{3}+4 b^{2} c^{2} d^{2} e^{2}+\frac {16}{9} b \,c^{3} d^{3} e +\frac {1}{9} c^{4} d^{4}\right ) x^{9}+\left (\frac {3}{2} a^{2} b c \,e^{4}+3 a^{2} c^{2} d \,e^{3}+\frac {1}{2} a \,b^{3} e^{4}+6 a \,b^{2} c d \,e^{3}+9 a b \,c^{2} d^{2} e^{2}+2 a \,c^{3} d^{3} e +\frac {1}{2} b^{4} d \,e^{3}+3 b^{3} c \,d^{2} e^{2}+3 b^{2} c^{2} d^{3} e +\frac {1}{2} d^{4} c^{3} b \right ) x^{8}+\left (\frac {4}{7} a^{3} c \,e^{4}+\frac {6}{7} a^{2} b^{2} e^{4}+\frac {48}{7} a^{2} b c d \,e^{3}+\frac {36}{7} a^{2} c^{2} d^{2} e^{2}+\frac {16}{7} a \,b^{3} d \,e^{3}+\frac {72}{7} a \,b^{2} c \,d^{2} e^{2}+\frac {48}{7} a b \,c^{2} d^{3} e +\frac {4}{7} a \,c^{3} d^{4}+\frac {6}{7} b^{4} d^{2} e^{2}+\frac {16}{7} b^{3} c \,d^{3} e +\frac {6}{7} b^{2} c^{2} d^{4}\right ) x^{7}+\left (\frac {2}{3} a^{3} b \,e^{4}+\frac {8}{3} a^{3} c d \,e^{3}+4 a^{2} b^{2} d \,e^{3}+12 a^{2} b c \,d^{2} e^{2}+4 a^{2} c^{2} d^{3} e +4 a \,b^{3} d^{2} e^{2}+8 a \,b^{2} c \,d^{3} e +2 b \,c^{2} a \,d^{4}+\frac {2}{3} b^{4} d^{3} e +\frac {2}{3} b^{3} d^{4} c \right ) x^{6}+\left (\frac {1}{5} e^{4} a^{4}+\frac {16}{5} b \,e^{3} d \,a^{3}+\frac {24}{5} a^{3} c \,d^{2} e^{2}+\frac {36}{5} b^{2} e^{2} d^{2} a^{2}+\frac {48}{5} a^{2} b c \,d^{3} e +\frac {6}{5} d^{4} a^{2} c^{2}+\frac {16}{5} a \,b^{3} d^{3} e +\frac {12}{5} b^{2} c \,d^{4} a +\frac {1}{5} b^{4} d^{4}\right ) x^{5}+\left (a^{4} d \,e^{3}+6 d^{2} e^{2} a^{3} b +4 a^{3} c \,d^{3} e +6 a^{2} b^{2} d^{3} e +3 a^{2} b c \,d^{4}+a \,b^{3} d^{4}\right ) x^{4}+\left (2 d^{2} e^{2} a^{4}+\frac {16}{3} a^{3} b \,d^{3} e +\frac {4}{3} a^{3} c \,d^{4}+2 a^{2} b^{2} d^{4}\right ) x^{3}+\left (2 d^{3} e \,a^{4}+2 d^{4} a^{3} b \right ) x^{2}+a^{4} d^{4} x\) | \(838\) |
| default | \(\frac {c^{4} e^{4} x^{13}}{13}+\frac {\left (4 e^{4} c^{3} b +4 d \,e^{3} c^{4}\right ) x^{12}}{12}+\frac {\left (6 d^{2} e^{2} c^{4}+16 d \,e^{3} c^{3} b +e^{4} \left (2 \left (2 a c +b^{2}\right ) c^{2}+4 b^{2} c^{2}\right )\right ) x^{11}}{11}+\frac {\left (4 c^{4} d^{3} e +24 b \,c^{3} d^{2} e^{2}+4 d \,e^{3} \left (2 \left (2 a c +b^{2}\right ) c^{2}+4 b^{2} c^{2}\right )+e^{4} \left (4 b \,c^{2} a +4 \left (2 a c +b^{2}\right ) b c \right )\right ) x^{10}}{10}+\frac {\left (c^{4} d^{4}+16 b \,c^{3} d^{3} e +6 d^{2} e^{2} \left (2 \left (2 a c +b^{2}\right ) c^{2}+4 b^{2} c^{2}\right )+4 d \,e^{3} \left (4 b \,c^{2} a +4 \left (2 a c +b^{2}\right ) b c \right )+e^{4} \left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right )\right ) x^{9}}{9}+\frac {\left (4 d^{4} c^{3} b +4 d^{3} e \left (2 \left (2 a c +b^{2}\right ) c^{2}+4 b^{2} c^{2}\right )+6 d^{2} e^{2} \left (4 b \,c^{2} a +4 \left (2 a c +b^{2}\right ) b c \right )+4 d \,e^{3} \left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right )+e^{4} \left (4 a^{2} b c +4 a b \left (2 a c +b^{2}\right )\right )\right ) x^{8}}{8}+\frac {\left (d^{4} \left (2 \left (2 a c +b^{2}\right ) c^{2}+4 b^{2} c^{2}\right )+4 d^{3} e \left (4 b \,c^{2} a +4 \left (2 a c +b^{2}\right ) b c \right )+6 d^{2} e^{2} \left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right )+4 d \,e^{3} \left (4 a^{2} b c +4 a b \left (2 a c +b^{2}\right )\right )+e^{4} \left (2 a^{2} \left (2 a c +b^{2}\right )+4 a^{2} b^{2}\right )\right ) x^{7}}{7}+\frac {\left (d^{4} \left (4 b \,c^{2} a +4 \left (2 a c +b^{2}\right ) b c \right )+4 d^{3} e \left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right )+6 d^{2} e^{2} \left (4 a^{2} b c +4 a b \left (2 a c +b^{2}\right )\right )+4 d \,e^{3} \left (2 a^{2} \left (2 a c +b^{2}\right )+4 a^{2} b^{2}\right )+4 a^{3} b \,e^{4}\right ) x^{6}}{6}+\frac {\left (d^{4} \left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right )+4 d^{3} e \left (4 a^{2} b c +4 a b \left (2 a c +b^{2}\right )\right )+6 d^{2} e^{2} \left (2 a^{2} \left (2 a c +b^{2}\right )+4 a^{2} b^{2}\right )+16 b \,e^{3} d \,a^{3}+e^{4} a^{4}\right ) x^{5}}{5}+\frac {\left (d^{4} \left (4 a^{2} b c +4 a b \left (2 a c +b^{2}\right )\right )+4 d^{3} e \left (2 a^{2} \left (2 a c +b^{2}\right )+4 a^{2} b^{2}\right )+24 d^{2} e^{2} a^{3} b +4 a^{4} d \,e^{3}\right ) x^{4}}{4}+\frac {\left (d^{4} \left (2 a^{2} \left (2 a c +b^{2}\right )+4 a^{2} b^{2}\right )+16 a^{3} b \,d^{3} e +6 d^{2} e^{2} a^{4}\right ) x^{3}}{3}+\frac {\left (4 d^{3} e \,a^{4}+4 d^{4} a^{3} b \right ) x^{2}}{2}+a^{4} d^{4} x\) | \(949\) |
| gosper | \(\text {Expression too large to display}\) | \(1002\) |
| risch | \(\text {Expression too large to display}\) | \(1002\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1002\) |
1/13*c^4*e^4*x^13+(1/3*e^4*c^3*b+1/3*d*e^3*c^4)*x^12+(4/11*a*c^3*e^4+6/11* b^2*c^2*e^4+16/11*d*e^3*c^3*b+6/11*d^2*e^2*c^4)*x^11+(6/5*a*b*c^2*e^4+8/5* a*c^3*d*e^3+2/5*b^3*c*e^4+12/5*b^2*c^2*d*e^3+12/5*b*c^3*d^2*e^2+2/5*c^4*d^ 3*e)*x^10+(2/3*c^2*a^2*e^4+4/3*a*b^2*c*e^4+16/3*a*b*c^2*d*e^3+8/3*c^3*a*d^ 2*e^2+1/9*b^4*e^4+16/9*b^3*c*d*e^3+4*b^2*c^2*d^2*e^2+16/9*b*c^3*d^3*e+1/9* c^4*d^4)*x^9+(3/2*a^2*b*c*e^4+3*a^2*c^2*d*e^3+1/2*a*b^3*e^4+6*a*b^2*c*d*e^ 3+9*a*b*c^2*d^2*e^2+2*a*c^3*d^3*e+1/2*b^4*d*e^3+3*b^3*c*d^2*e^2+3*b^2*c^2* d^3*e+1/2*d^4*c^3*b)*x^8+(4/7*a^3*c*e^4+6/7*a^2*b^2*e^4+48/7*a^2*b*c*d*e^3 +36/7*a^2*c^2*d^2*e^2+16/7*a*b^3*d*e^3+72/7*a*b^2*c*d^2*e^2+48/7*a*b*c^2*d ^3*e+4/7*a*c^3*d^4+6/7*b^4*d^2*e^2+16/7*b^3*c*d^3*e+6/7*b^2*c^2*d^4)*x^7+( 2/3*a^3*b*e^4+8/3*a^3*c*d*e^3+4*a^2*b^2*d*e^3+12*a^2*b*c*d^2*e^2+4*a^2*c^2 *d^3*e+4*a*b^3*d^2*e^2+8*a*b^2*c*d^3*e+2*b*c^2*a*d^4+2/3*b^4*d^3*e+2/3*b^3 *d^4*c)*x^6+(1/5*e^4*a^4+16/5*b*e^3*d*a^3+24/5*a^3*c*d^2*e^2+36/5*b^2*e^2* d^2*a^2+48/5*a^2*b*c*d^3*e+6/5*d^4*a^2*c^2+16/5*a*b^3*d^3*e+12/5*b^2*c*d^4 *a+1/5*b^4*d^4)*x^5+(a^4*d*e^3+6*a^3*b*d^2*e^2+4*a^3*c*d^3*e+6*a^2*b^2*d^3 *e+3*a^2*b*c*d^4+a*b^3*d^4)*x^4+(2*d^2*e^2*a^4+16/3*a^3*b*d^3*e+4/3*a^3*c* d^4+2*a^2*b^2*d^4)*x^3+(2*a^4*d^3*e+2*a^3*b*d^4)*x^2+a^4*d^4*x
Time = 0.31 (sec) , antiderivative size = 762, normalized size of antiderivative = 1.72 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^4 \, dx=\frac {1}{13} \, c^{4} e^{4} x^{13} + \frac {1}{3} \, {\left (c^{4} d e^{3} + b c^{3} e^{4}\right )} x^{12} + \frac {2}{11} \, {\left (3 \, c^{4} d^{2} e^{2} + 8 \, b c^{3} d e^{3} + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{4}\right )} x^{11} + \frac {2}{5} \, {\left (c^{4} d^{3} e + 6 \, b c^{3} d^{2} e^{2} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{3} + {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{4}\right )} x^{10} + \frac {1}{9} \, {\left (c^{4} d^{4} + 16 \, b c^{3} d^{3} e + 12 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{2} + 16 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{4}\right )} x^{9} + a^{4} d^{4} x + \frac {1}{2} \, {\left (b c^{3} d^{4} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e + 6 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{2} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{3} + {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{4}\right )} x^{8} + \frac {2}{7} \, {\left ({\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} + 8 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{2} + 8 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{3} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{4}\right )} x^{7} + \frac {2}{3} \, {\left (a^{3} b e^{4} + {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e + 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{2} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (16 \, a^{3} b d e^{3} + a^{4} e^{4} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} + 16 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e + 12 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{2}\right )} x^{5} + {\left (6 \, a^{3} b d^{2} e^{2} + a^{4} d e^{3} + {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{4} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{3} e\right )} x^{4} + \frac {2}{3} \, {\left (8 \, a^{3} b d^{3} e + 3 \, a^{4} d^{2} e^{2} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{4}\right )} x^{3} + 2 \, {\left (a^{3} b d^{4} + a^{4} d^{3} e\right )} x^{2} \]
1/13*c^4*e^4*x^13 + 1/3*(c^4*d*e^3 + b*c^3*e^4)*x^12 + 2/11*(3*c^4*d^2*e^2 + 8*b*c^3*d*e^3 + (3*b^2*c^2 + 2*a*c^3)*e^4)*x^11 + 2/5*(c^4*d^3*e + 6*b* c^3*d^2*e^2 + 2*(3*b^2*c^2 + 2*a*c^3)*d*e^3 + (b^3*c + 3*a*b*c^2)*e^4)*x^1 0 + 1/9*(c^4*d^4 + 16*b*c^3*d^3*e + 12*(3*b^2*c^2 + 2*a*c^3)*d^2*e^2 + 16* (b^3*c + 3*a*b*c^2)*d*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^4)*x^9 + a^4* d^4*x + 1/2*(b*c^3*d^4 + 2*(3*b^2*c^2 + 2*a*c^3)*d^3*e + 6*(b^3*c + 3*a*b* c^2)*d^2*e^2 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^3 + (a*b^3 + 3*a^2*b*c)* e^4)*x^8 + 2/7*((3*b^2*c^2 + 2*a*c^3)*d^4 + 8*(b^3*c + 3*a*b*c^2)*d^3*e + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^2 + 8*(a*b^3 + 3*a^2*b*c)*d*e^3 + ( 3*a^2*b^2 + 2*a^3*c)*e^4)*x^7 + 2/3*(a^3*b*e^4 + (b^3*c + 3*a*b*c^2)*d^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e + 6*(a*b^3 + 3*a^2*b*c)*d^2*e^2 + 2* (3*a^2*b^2 + 2*a^3*c)*d*e^3)*x^6 + 1/5*(16*a^3*b*d*e^3 + a^4*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4 + 16*(a*b^3 + 3*a^2*b*c)*d^3*e + 12*(3*a^2*b^2 + 2*a^3*c)*d^2*e^2)*x^5 + (6*a^3*b*d^2*e^2 + a^4*d*e^3 + (a*b^3 + 3*a^2*b *c)*d^4 + 2*(3*a^2*b^2 + 2*a^3*c)*d^3*e)*x^4 + 2/3*(8*a^3*b*d^3*e + 3*a^4* d^2*e^2 + (3*a^2*b^2 + 2*a^3*c)*d^4)*x^3 + 2*(a^3*b*d^4 + a^4*d^3*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 998 vs. \(2 (439) = 878\).
Time = 0.07 (sec) , antiderivative size = 998, normalized size of antiderivative = 2.25 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^4 \, dx=a^{4} d^{4} x + \frac {c^{4} e^{4} x^{13}}{13} + x^{12} \left (\frac {b c^{3} e^{4}}{3} + \frac {c^{4} d e^{3}}{3}\right ) + x^{11} \cdot \left (\frac {4 a c^{3} e^{4}}{11} + \frac {6 b^{2} c^{2} e^{4}}{11} + \frac {16 b c^{3} d e^{3}}{11} + \frac {6 c^{4} d^{2} e^{2}}{11}\right ) + x^{10} \cdot \left (\frac {6 a b c^{2} e^{4}}{5} + \frac {8 a c^{3} d e^{3}}{5} + \frac {2 b^{3} c e^{4}}{5} + \frac {12 b^{2} c^{2} d e^{3}}{5} + \frac {12 b c^{3} d^{2} e^{2}}{5} + \frac {2 c^{4} d^{3} e}{5}\right ) + x^{9} \cdot \left (\frac {2 a^{2} c^{2} e^{4}}{3} + \frac {4 a b^{2} c e^{4}}{3} + \frac {16 a b c^{2} d e^{3}}{3} + \frac {8 a c^{3} d^{2} e^{2}}{3} + \frac {b^{4} e^{4}}{9} + \frac {16 b^{3} c d e^{3}}{9} + 4 b^{2} c^{2} d^{2} e^{2} + \frac {16 b c^{3} d^{3} e}{9} + \frac {c^{4} d^{4}}{9}\right ) + x^{8} \cdot \left (\frac {3 a^{2} b c e^{4}}{2} + 3 a^{2} c^{2} d e^{3} + \frac {a b^{3} e^{4}}{2} + 6 a b^{2} c d e^{3} + 9 a b c^{2} d^{2} e^{2} + 2 a c^{3} d^{3} e + \frac {b^{4} d e^{3}}{2} + 3 b^{3} c d^{2} e^{2} + 3 b^{2} c^{2} d^{3} e + \frac {b c^{3} d^{4}}{2}\right ) + x^{7} \cdot \left (\frac {4 a^{3} c e^{4}}{7} + \frac {6 a^{2} b^{2} e^{4}}{7} + \frac {48 a^{2} b c d e^{3}}{7} + \frac {36 a^{2} c^{2} d^{2} e^{2}}{7} + \frac {16 a b^{3} d e^{3}}{7} + \frac {72 a b^{2} c d^{2} e^{2}}{7} + \frac {48 a b c^{2} d^{3} e}{7} + \frac {4 a c^{3} d^{4}}{7} + \frac {6 b^{4} d^{2} e^{2}}{7} + \frac {16 b^{3} c d^{3} e}{7} + \frac {6 b^{2} c^{2} d^{4}}{7}\right ) + x^{6} \cdot \left (\frac {2 a^{3} b e^{4}}{3} + \frac {8 a^{3} c d e^{3}}{3} + 4 a^{2} b^{2} d e^{3} + 12 a^{2} b c d^{2} e^{2} + 4 a^{2} c^{2} d^{3} e + 4 a b^{3} d^{2} e^{2} + 8 a b^{2} c d^{3} e + 2 a b c^{2} d^{4} + \frac {2 b^{4} d^{3} e}{3} + \frac {2 b^{3} c d^{4}}{3}\right ) + x^{5} \left (\frac {a^{4} e^{4}}{5} + \frac {16 a^{3} b d e^{3}}{5} + \frac {24 a^{3} c d^{2} e^{2}}{5} + \frac {36 a^{2} b^{2} d^{2} e^{2}}{5} + \frac {48 a^{2} b c d^{3} e}{5} + \frac {6 a^{2} c^{2} d^{4}}{5} + \frac {16 a b^{3} d^{3} e}{5} + \frac {12 a b^{2} c d^{4}}{5} + \frac {b^{4} d^{4}}{5}\right ) + x^{4} \left (a^{4} d e^{3} + 6 a^{3} b d^{2} e^{2} + 4 a^{3} c d^{3} e + 6 a^{2} b^{2} d^{3} e + 3 a^{2} b c d^{4} + a b^{3} d^{4}\right ) + x^{3} \cdot \left (2 a^{4} d^{2} e^{2} + \frac {16 a^{3} b d^{3} e}{3} + \frac {4 a^{3} c d^{4}}{3} + 2 a^{2} b^{2} d^{4}\right ) + x^{2} \cdot \left (2 a^{4} d^{3} e + 2 a^{3} b d^{4}\right ) \]
a**4*d**4*x + c**4*e**4*x**13/13 + x**12*(b*c**3*e**4/3 + c**4*d*e**3/3) + x**11*(4*a*c**3*e**4/11 + 6*b**2*c**2*e**4/11 + 16*b*c**3*d*e**3/11 + 6*c **4*d**2*e**2/11) + x**10*(6*a*b*c**2*e**4/5 + 8*a*c**3*d*e**3/5 + 2*b**3* c*e**4/5 + 12*b**2*c**2*d*e**3/5 + 12*b*c**3*d**2*e**2/5 + 2*c**4*d**3*e/5 ) + x**9*(2*a**2*c**2*e**4/3 + 4*a*b**2*c*e**4/3 + 16*a*b*c**2*d*e**3/3 + 8*a*c**3*d**2*e**2/3 + b**4*e**4/9 + 16*b**3*c*d*e**3/9 + 4*b**2*c**2*d**2 *e**2 + 16*b*c**3*d**3*e/9 + c**4*d**4/9) + x**8*(3*a**2*b*c*e**4/2 + 3*a* *2*c**2*d*e**3 + a*b**3*e**4/2 + 6*a*b**2*c*d*e**3 + 9*a*b*c**2*d**2*e**2 + 2*a*c**3*d**3*e + b**4*d*e**3/2 + 3*b**3*c*d**2*e**2 + 3*b**2*c**2*d**3* e + b*c**3*d**4/2) + x**7*(4*a**3*c*e**4/7 + 6*a**2*b**2*e**4/7 + 48*a**2* b*c*d*e**3/7 + 36*a**2*c**2*d**2*e**2/7 + 16*a*b**3*d*e**3/7 + 72*a*b**2*c *d**2*e**2/7 + 48*a*b*c**2*d**3*e/7 + 4*a*c**3*d**4/7 + 6*b**4*d**2*e**2/7 + 16*b**3*c*d**3*e/7 + 6*b**2*c**2*d**4/7) + x**6*(2*a**3*b*e**4/3 + 8*a* *3*c*d*e**3/3 + 4*a**2*b**2*d*e**3 + 12*a**2*b*c*d**2*e**2 + 4*a**2*c**2*d **3*e + 4*a*b**3*d**2*e**2 + 8*a*b**2*c*d**3*e + 2*a*b*c**2*d**4 + 2*b**4* d**3*e/3 + 2*b**3*c*d**4/3) + x**5*(a**4*e**4/5 + 16*a**3*b*d*e**3/5 + 24* a**3*c*d**2*e**2/5 + 36*a**2*b**2*d**2*e**2/5 + 48*a**2*b*c*d**3*e/5 + 6*a **2*c**2*d**4/5 + 16*a*b**3*d**3*e/5 + 12*a*b**2*c*d**4/5 + b**4*d**4/5) + x**4*(a**4*d*e**3 + 6*a**3*b*d**2*e**2 + 4*a**3*c*d**3*e + 6*a**2*b**2*d* *3*e + 3*a**2*b*c*d**4 + a*b**3*d**4) + x**3*(2*a**4*d**2*e**2 + 16*a**...
Time = 0.21 (sec) , antiderivative size = 762, normalized size of antiderivative = 1.72 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^4 \, dx=\frac {1}{13} \, c^{4} e^{4} x^{13} + \frac {1}{3} \, {\left (c^{4} d e^{3} + b c^{3} e^{4}\right )} x^{12} + \frac {2}{11} \, {\left (3 \, c^{4} d^{2} e^{2} + 8 \, b c^{3} d e^{3} + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{4}\right )} x^{11} + \frac {2}{5} \, {\left (c^{4} d^{3} e + 6 \, b c^{3} d^{2} e^{2} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{3} + {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{4}\right )} x^{10} + \frac {1}{9} \, {\left (c^{4} d^{4} + 16 \, b c^{3} d^{3} e + 12 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{2} + 16 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{4}\right )} x^{9} + a^{4} d^{4} x + \frac {1}{2} \, {\left (b c^{3} d^{4} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e + 6 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{2} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{3} + {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{4}\right )} x^{8} + \frac {2}{7} \, {\left ({\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} + 8 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{2} + 8 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{3} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{4}\right )} x^{7} + \frac {2}{3} \, {\left (a^{3} b e^{4} + {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e + 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{2} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (16 \, a^{3} b d e^{3} + a^{4} e^{4} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} + 16 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e + 12 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{2}\right )} x^{5} + {\left (6 \, a^{3} b d^{2} e^{2} + a^{4} d e^{3} + {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{4} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{3} e\right )} x^{4} + \frac {2}{3} \, {\left (8 \, a^{3} b d^{3} e + 3 \, a^{4} d^{2} e^{2} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{4}\right )} x^{3} + 2 \, {\left (a^{3} b d^{4} + a^{4} d^{3} e\right )} x^{2} \]
1/13*c^4*e^4*x^13 + 1/3*(c^4*d*e^3 + b*c^3*e^4)*x^12 + 2/11*(3*c^4*d^2*e^2 + 8*b*c^3*d*e^3 + (3*b^2*c^2 + 2*a*c^3)*e^4)*x^11 + 2/5*(c^4*d^3*e + 6*b* c^3*d^2*e^2 + 2*(3*b^2*c^2 + 2*a*c^3)*d*e^3 + (b^3*c + 3*a*b*c^2)*e^4)*x^1 0 + 1/9*(c^4*d^4 + 16*b*c^3*d^3*e + 12*(3*b^2*c^2 + 2*a*c^3)*d^2*e^2 + 16* (b^3*c + 3*a*b*c^2)*d*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^4)*x^9 + a^4* d^4*x + 1/2*(b*c^3*d^4 + 2*(3*b^2*c^2 + 2*a*c^3)*d^3*e + 6*(b^3*c + 3*a*b* c^2)*d^2*e^2 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^3 + (a*b^3 + 3*a^2*b*c)* e^4)*x^8 + 2/7*((3*b^2*c^2 + 2*a*c^3)*d^4 + 8*(b^3*c + 3*a*b*c^2)*d^3*e + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^2 + 8*(a*b^3 + 3*a^2*b*c)*d*e^3 + ( 3*a^2*b^2 + 2*a^3*c)*e^4)*x^7 + 2/3*(a^3*b*e^4 + (b^3*c + 3*a*b*c^2)*d^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e + 6*(a*b^3 + 3*a^2*b*c)*d^2*e^2 + 2* (3*a^2*b^2 + 2*a^3*c)*d*e^3)*x^6 + 1/5*(16*a^3*b*d*e^3 + a^4*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4 + 16*(a*b^3 + 3*a^2*b*c)*d^3*e + 12*(3*a^2*b^2 + 2*a^3*c)*d^2*e^2)*x^5 + (6*a^3*b*d^2*e^2 + a^4*d*e^3 + (a*b^3 + 3*a^2*b *c)*d^4 + 2*(3*a^2*b^2 + 2*a^3*c)*d^3*e)*x^4 + 2/3*(8*a^3*b*d^3*e + 3*a^4* d^2*e^2 + (3*a^2*b^2 + 2*a^3*c)*d^4)*x^3 + 2*(a^3*b*d^4 + a^4*d^3*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 1001 vs. \(2 (425) = 850\).
Time = 0.27 (sec) , antiderivative size = 1001, normalized size of antiderivative = 2.26 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^4 \, dx=\frac {1}{13} \, c^{4} e^{4} x^{13} + \frac {1}{3} \, c^{4} d e^{3} x^{12} + \frac {1}{3} \, b c^{3} e^{4} x^{12} + \frac {6}{11} \, c^{4} d^{2} e^{2} x^{11} + \frac {16}{11} \, b c^{3} d e^{3} x^{11} + \frac {6}{11} \, b^{2} c^{2} e^{4} x^{11} + \frac {4}{11} \, a c^{3} e^{4} x^{11} + \frac {2}{5} \, c^{4} d^{3} e x^{10} + \frac {12}{5} \, b c^{3} d^{2} e^{2} x^{10} + \frac {12}{5} \, b^{2} c^{2} d e^{3} x^{10} + \frac {8}{5} \, a c^{3} d e^{3} x^{10} + \frac {2}{5} \, b^{3} c e^{4} x^{10} + \frac {6}{5} \, a b c^{2} e^{4} x^{10} + \frac {1}{9} \, c^{4} d^{4} x^{9} + \frac {16}{9} \, b c^{3} d^{3} e x^{9} + 4 \, b^{2} c^{2} d^{2} e^{2} x^{9} + \frac {8}{3} \, a c^{3} d^{2} e^{2} x^{9} + \frac {16}{9} \, b^{3} c d e^{3} x^{9} + \frac {16}{3} \, a b c^{2} d e^{3} x^{9} + \frac {1}{9} \, b^{4} e^{4} x^{9} + \frac {4}{3} \, a b^{2} c e^{4} x^{9} + \frac {2}{3} \, a^{2} c^{2} e^{4} x^{9} + \frac {1}{2} \, b c^{3} d^{4} x^{8} + 3 \, b^{2} c^{2} d^{3} e x^{8} + 2 \, a c^{3} d^{3} e x^{8} + 3 \, b^{3} c d^{2} e^{2} x^{8} + 9 \, a b c^{2} d^{2} e^{2} x^{8} + \frac {1}{2} \, b^{4} d e^{3} x^{8} + 6 \, a b^{2} c d e^{3} x^{8} + 3 \, a^{2} c^{2} d e^{3} x^{8} + \frac {1}{2} \, a b^{3} e^{4} x^{8} + \frac {3}{2} \, a^{2} b c e^{4} x^{8} + \frac {6}{7} \, b^{2} c^{2} d^{4} x^{7} + \frac {4}{7} \, a c^{3} d^{4} x^{7} + \frac {16}{7} \, b^{3} c d^{3} e x^{7} + \frac {48}{7} \, a b c^{2} d^{3} e x^{7} + \frac {6}{7} \, b^{4} d^{2} e^{2} x^{7} + \frac {72}{7} \, a b^{2} c d^{2} e^{2} x^{7} + \frac {36}{7} \, a^{2} c^{2} d^{2} e^{2} x^{7} + \frac {16}{7} \, a b^{3} d e^{3} x^{7} + \frac {48}{7} \, a^{2} b c d e^{3} x^{7} + \frac {6}{7} \, a^{2} b^{2} e^{4} x^{7} + \frac {4}{7} \, a^{3} c e^{4} x^{7} + \frac {2}{3} \, b^{3} c d^{4} x^{6} + 2 \, a b c^{2} d^{4} x^{6} + \frac {2}{3} \, b^{4} d^{3} e x^{6} + 8 \, a b^{2} c d^{3} e x^{6} + 4 \, a^{2} c^{2} d^{3} e x^{6} + 4 \, a b^{3} d^{2} e^{2} x^{6} + 12 \, a^{2} b c d^{2} e^{2} x^{6} + 4 \, a^{2} b^{2} d e^{3} x^{6} + \frac {8}{3} \, a^{3} c d e^{3} x^{6} + \frac {2}{3} \, a^{3} b e^{4} x^{6} + \frac {1}{5} \, b^{4} d^{4} x^{5} + \frac {12}{5} \, a b^{2} c d^{4} x^{5} + \frac {6}{5} \, a^{2} c^{2} d^{4} x^{5} + \frac {16}{5} \, a b^{3} d^{3} e x^{5} + \frac {48}{5} \, a^{2} b c d^{3} e x^{5} + \frac {36}{5} \, a^{2} b^{2} d^{2} e^{2} x^{5} + \frac {24}{5} \, a^{3} c d^{2} e^{2} x^{5} + \frac {16}{5} \, a^{3} b d e^{3} x^{5} + \frac {1}{5} \, a^{4} e^{4} x^{5} + a b^{3} d^{4} x^{4} + 3 \, a^{2} b c d^{4} x^{4} + 6 \, a^{2} b^{2} d^{3} e x^{4} + 4 \, a^{3} c d^{3} e x^{4} + 6 \, a^{3} b d^{2} e^{2} x^{4} + a^{4} d e^{3} x^{4} + 2 \, a^{2} b^{2} d^{4} x^{3} + \frac {4}{3} \, a^{3} c d^{4} x^{3} + \frac {16}{3} \, a^{3} b d^{3} e x^{3} + 2 \, a^{4} d^{2} e^{2} x^{3} + 2 \, a^{3} b d^{4} x^{2} + 2 \, a^{4} d^{3} e x^{2} + a^{4} d^{4} x \]
1/13*c^4*e^4*x^13 + 1/3*c^4*d*e^3*x^12 + 1/3*b*c^3*e^4*x^12 + 6/11*c^4*d^2 *e^2*x^11 + 16/11*b*c^3*d*e^3*x^11 + 6/11*b^2*c^2*e^4*x^11 + 4/11*a*c^3*e^ 4*x^11 + 2/5*c^4*d^3*e*x^10 + 12/5*b*c^3*d^2*e^2*x^10 + 12/5*b^2*c^2*d*e^3 *x^10 + 8/5*a*c^3*d*e^3*x^10 + 2/5*b^3*c*e^4*x^10 + 6/5*a*b*c^2*e^4*x^10 + 1/9*c^4*d^4*x^9 + 16/9*b*c^3*d^3*e*x^9 + 4*b^2*c^2*d^2*e^2*x^9 + 8/3*a*c^ 3*d^2*e^2*x^9 + 16/9*b^3*c*d*e^3*x^9 + 16/3*a*b*c^2*d*e^3*x^9 + 1/9*b^4*e^ 4*x^9 + 4/3*a*b^2*c*e^4*x^9 + 2/3*a^2*c^2*e^4*x^9 + 1/2*b*c^3*d^4*x^8 + 3* b^2*c^2*d^3*e*x^8 + 2*a*c^3*d^3*e*x^8 + 3*b^3*c*d^2*e^2*x^8 + 9*a*b*c^2*d^ 2*e^2*x^8 + 1/2*b^4*d*e^3*x^8 + 6*a*b^2*c*d*e^3*x^8 + 3*a^2*c^2*d*e^3*x^8 + 1/2*a*b^3*e^4*x^8 + 3/2*a^2*b*c*e^4*x^8 + 6/7*b^2*c^2*d^4*x^7 + 4/7*a*c^ 3*d^4*x^7 + 16/7*b^3*c*d^3*e*x^7 + 48/7*a*b*c^2*d^3*e*x^7 + 6/7*b^4*d^2*e^ 2*x^7 + 72/7*a*b^2*c*d^2*e^2*x^7 + 36/7*a^2*c^2*d^2*e^2*x^7 + 16/7*a*b^3*d *e^3*x^7 + 48/7*a^2*b*c*d*e^3*x^7 + 6/7*a^2*b^2*e^4*x^7 + 4/7*a^3*c*e^4*x^ 7 + 2/3*b^3*c*d^4*x^6 + 2*a*b*c^2*d^4*x^6 + 2/3*b^4*d^3*e*x^6 + 8*a*b^2*c* d^3*e*x^6 + 4*a^2*c^2*d^3*e*x^6 + 4*a*b^3*d^2*e^2*x^6 + 12*a^2*b*c*d^2*e^2 *x^6 + 4*a^2*b^2*d*e^3*x^6 + 8/3*a^3*c*d*e^3*x^6 + 2/3*a^3*b*e^4*x^6 + 1/5 *b^4*d^4*x^5 + 12/5*a*b^2*c*d^4*x^5 + 6/5*a^2*c^2*d^4*x^5 + 16/5*a*b^3*d^3 *e*x^5 + 48/5*a^2*b*c*d^3*e*x^5 + 36/5*a^2*b^2*d^2*e^2*x^5 + 24/5*a^3*c*d^ 2*e^2*x^5 + 16/5*a^3*b*d*e^3*x^5 + 1/5*a^4*e^4*x^5 + a*b^3*d^4*x^4 + 3*a^2 *b*c*d^4*x^4 + 6*a^2*b^2*d^3*e*x^4 + 4*a^3*c*d^3*e*x^4 + 6*a^3*b*d^2*e^...
Time = 10.00 (sec) , antiderivative size = 817, normalized size of antiderivative = 1.84 \[ \int (d+e x)^4 \left (a+b x+c x^2\right )^4 \, dx=x^6\,\left (\frac {2\,a^3\,b\,e^4}{3}+\frac {8\,a^3\,c\,d\,e^3}{3}+4\,a^2\,b^2\,d\,e^3+12\,a^2\,b\,c\,d^2\,e^2+4\,a^2\,c^2\,d^3\,e+4\,a\,b^3\,d^2\,e^2+8\,a\,b^2\,c\,d^3\,e+2\,a\,b\,c^2\,d^4+\frac {2\,b^4\,d^3\,e}{3}+\frac {2\,b^3\,c\,d^4}{3}\right )+x^8\,\left (\frac {3\,a^2\,b\,c\,e^4}{2}+3\,a^2\,c^2\,d\,e^3+\frac {a\,b^3\,e^4}{2}+6\,a\,b^2\,c\,d\,e^3+9\,a\,b\,c^2\,d^2\,e^2+2\,a\,c^3\,d^3\,e+\frac {b^4\,d\,e^3}{2}+3\,b^3\,c\,d^2\,e^2+3\,b^2\,c^2\,d^3\,e+\frac {b\,c^3\,d^4}{2}\right )+x^7\,\left (\frac {4\,a^3\,c\,e^4}{7}+\frac {6\,a^2\,b^2\,e^4}{7}+\frac {48\,a^2\,b\,c\,d\,e^3}{7}+\frac {36\,a^2\,c^2\,d^2\,e^2}{7}+\frac {16\,a\,b^3\,d\,e^3}{7}+\frac {72\,a\,b^2\,c\,d^2\,e^2}{7}+\frac {48\,a\,b\,c^2\,d^3\,e}{7}+\frac {4\,a\,c^3\,d^4}{7}+\frac {6\,b^4\,d^2\,e^2}{7}+\frac {16\,b^3\,c\,d^3\,e}{7}+\frac {6\,b^2\,c^2\,d^4}{7}\right )+x^5\,\left (\frac {a^4\,e^4}{5}+\frac {16\,a^3\,b\,d\,e^3}{5}+\frac {24\,a^3\,c\,d^2\,e^2}{5}+\frac {36\,a^2\,b^2\,d^2\,e^2}{5}+\frac {48\,a^2\,b\,c\,d^3\,e}{5}+\frac {6\,a^2\,c^2\,d^4}{5}+\frac {16\,a\,b^3\,d^3\,e}{5}+\frac {12\,a\,b^2\,c\,d^4}{5}+\frac {b^4\,d^4}{5}\right )+x^4\,\left (a^4\,d\,e^3+6\,a^3\,b\,d^2\,e^2+4\,c\,a^3\,d^3\,e+6\,a^2\,b^2\,d^3\,e+3\,c\,a^2\,b\,d^4+a\,b^3\,d^4\right )+x^9\,\left (\frac {2\,a^2\,c^2\,e^4}{3}+\frac {4\,a\,b^2\,c\,e^4}{3}+\frac {16\,a\,b\,c^2\,d\,e^3}{3}+\frac {8\,a\,c^3\,d^2\,e^2}{3}+\frac {b^4\,e^4}{9}+\frac {16\,b^3\,c\,d\,e^3}{9}+4\,b^2\,c^2\,d^2\,e^2+\frac {16\,b\,c^3\,d^3\,e}{9}+\frac {c^4\,d^4}{9}\right )+x^{10}\,\left (\frac {2\,b^3\,c\,e^4}{5}+\frac {12\,b^2\,c^2\,d\,e^3}{5}+\frac {12\,b\,c^3\,d^2\,e^2}{5}+\frac {6\,a\,b\,c^2\,e^4}{5}+\frac {2\,c^4\,d^3\,e}{5}+\frac {8\,a\,c^3\,d\,e^3}{5}\right )+a^4\,d^4\,x+\frac {c^4\,e^4\,x^{13}}{13}+2\,a^3\,d^3\,x^2\,\left (a\,e+b\,d\right )+\frac {c^3\,e^3\,x^{12}\,\left (b\,e+c\,d\right )}{3}+\frac {2\,a^2\,d^2\,x^3\,\left (3\,a^2\,e^2+8\,a\,b\,d\,e+2\,c\,a\,d^2+3\,b^2\,d^2\right )}{3}+\frac {2\,c^2\,e^2\,x^{11}\,\left (3\,b^2\,e^2+8\,b\,c\,d\,e+3\,c^2\,d^2+2\,a\,c\,e^2\right )}{11} \]
x^6*((2*a^3*b*e^4)/3 + (2*b^3*c*d^4)/3 + (2*b^4*d^3*e)/3 + 4*a*b^3*d^2*e^2 + 4*a^2*b^2*d*e^3 + 4*a^2*c^2*d^3*e + 2*a*b*c^2*d^4 + (8*a^3*c*d*e^3)/3 + 8*a*b^2*c*d^3*e + 12*a^2*b*c*d^2*e^2) + x^8*((a*b^3*e^4)/2 + (b*c^3*d^4)/ 2 + (b^4*d*e^3)/2 + 3*a^2*c^2*d*e^3 + 3*b^2*c^2*d^3*e + 3*b^3*c*d^2*e^2 + (3*a^2*b*c*e^4)/2 + 2*a*c^3*d^3*e + 6*a*b^2*c*d*e^3 + 9*a*b*c^2*d^2*e^2) + x^7*((4*a*c^3*d^4)/7 + (4*a^3*c*e^4)/7 + (6*a^2*b^2*e^4)/7 + (6*b^2*c^2*d ^4)/7 + (6*b^4*d^2*e^2)/7 + (36*a^2*c^2*d^2*e^2)/7 + (16*a*b^3*d*e^3)/7 + (16*b^3*c*d^3*e)/7 + (48*a*b*c^2*d^3*e)/7 + (48*a^2*b*c*d*e^3)/7 + (72*a*b ^2*c*d^2*e^2)/7) + x^5*((a^4*e^4)/5 + (b^4*d^4)/5 + (6*a^2*c^2*d^4)/5 + (2 4*a^3*c*d^2*e^2)/5 + (36*a^2*b^2*d^2*e^2)/5 + (12*a*b^2*c*d^4)/5 + (16*a*b ^3*d^3*e)/5 + (16*a^3*b*d*e^3)/5 + (48*a^2*b*c*d^3*e)/5) + x^4*(a*b^3*d^4 + a^4*d*e^3 + 6*a^2*b^2*d^3*e + 6*a^3*b*d^2*e^2 + 3*a^2*b*c*d^4 + 4*a^3*c* d^3*e) + x^9*((b^4*e^4)/9 + (c^4*d^4)/9 + (2*a^2*c^2*e^4)/3 + (8*a*c^3*d^2 *e^2)/3 + 4*b^2*c^2*d^2*e^2 + (4*a*b^2*c*e^4)/3 + (16*b*c^3*d^3*e)/9 + (16 *b^3*c*d*e^3)/9 + (16*a*b*c^2*d*e^3)/3) + x^10*((2*b^3*c*e^4)/5 + (2*c^4*d ^3*e)/5 + (12*b*c^3*d^2*e^2)/5 + (12*b^2*c^2*d*e^3)/5 + (6*a*b*c^2*e^4)/5 + (8*a*c^3*d*e^3)/5) + a^4*d^4*x + (c^4*e^4*x^13)/13 + 2*a^3*d^3*x^2*(a*e + b*d) + (c^3*e^3*x^12*(b*e + c*d))/3 + (2*a^2*d^2*x^3*(3*a^2*e^2 + 3*b^2* d^2 + 2*a*c*d^2 + 8*a*b*d*e))/3 + (2*c^2*e^2*x^11*(3*b^2*e^2 + 3*c^2*d^2 + 2*a*c*e^2 + 8*b*c*d*e))/11