Integrand size = 20, antiderivative size = 443 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^4 \, dx=\frac {\left (c d^2-b d e+a e^2\right )^4 (d+e x)^4}{4 e^9}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}{5 e^9}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^6}{3 e^9}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^7}{7 e^9}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^8}{8 e^9}-\frac {4 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^9}{9 e^9}+\frac {c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{10}}{5 e^9}-\frac {4 c^3 (2 c d-b e) (d+e x)^{11}}{11 e^9}+\frac {c^4 (d+e x)^{12}}{12 e^9} \]
1/4*(a*e^2-b*d*e+c*d^2)^4*(e*x+d)^4/e^9-4/5*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^ 2)^3*(e*x+d)^5/e^9+1/3*(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)^6/e^9-4/7*(-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)^7/e^9+1/8*(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)^8/e^9-4/9*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)^9/e^9+1/5*c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b *d))*(e*x+d)^10/e^9-4/11*c^3*(-b*e+2*c*d)*(e*x+d)^11/e^9+1/12*c^4*(e*x+d)^ 12/e^9
Time = 0.12 (sec) , antiderivative size = 611, normalized size of antiderivative = 1.38 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^4 \, dx=a^4 d^3 x+\frac {1}{2} a^3 d^2 (4 b d+3 a e) x^2+\frac {1}{3} a^2 d \left (6 b^2 d^2+12 a b d e+a \left (4 c d^2+3 a e^2\right )\right ) x^3+\frac {1}{4} a \left (4 b^3 d^3+18 a b^2 d^2 e+12 a b d \left (c d^2+a e^2\right )+a^2 e \left (12 c d^2+a e^2\right )\right ) x^4+\frac {1}{5} \left (b^4 d^3+12 a b^3 d^2 e+4 a^2 b e \left (9 c d^2+a e^2\right )+6 a^2 c d \left (c d^2+2 a e^2\right )+6 a b^2 d \left (2 c d^2+3 a e^2\right )\right ) x^5+\frac {1}{6} \left (3 b^4 d^2 e+6 a b^2 e \left (6 c d^2+a e^2\right )+2 a^2 c e \left (9 c d^2+2 a e^2\right )+12 a b c d \left (c d^2+3 a e^2\right )+4 b^3 \left (c d^3+3 a d e^2\right )\right ) x^6+\frac {1}{7} \left (3 b^4 d e^2+12 a b c e \left (3 c d^2+a e^2\right )+6 b^2 c d \left (c d^2+6 a e^2\right )+2 a c^2 d \left (2 c d^2+9 a e^2\right )+4 b^3 \left (3 c d^2 e+a e^3\right )\right ) x^7+\frac {1}{8} \left (12 b^3 c d e^2+b^4 e^3+6 a c^2 e \left (2 c d^2+a e^2\right )+6 b^2 c e \left (3 c d^2+2 a e^2\right )+4 b c^2 d \left (c d^2+9 a e^2\right )\right ) x^8+\frac {1}{9} c \left (c^3 d^3+4 b^3 e^3+12 c^2 d e (b d+a e)+6 b c e^2 (3 b d+2 a e)\right ) x^9+\frac {1}{10} c^2 e \left (3 c^2 d^2+6 b^2 e^2+4 c e (3 b d+a e)\right ) x^{10}+\frac {1}{11} c^3 e^2 (3 c d+4 b e) x^{11}+\frac {1}{12} c^4 e^3 x^{12} \]
a^4*d^3*x + (a^3*d^2*(4*b*d + 3*a*e)*x^2)/2 + (a^2*d*(6*b^2*d^2 + 12*a*b*d *e + a*(4*c*d^2 + 3*a*e^2))*x^3)/3 + (a*(4*b^3*d^3 + 18*a*b^2*d^2*e + 12*a *b*d*(c*d^2 + a*e^2) + a^2*e*(12*c*d^2 + a*e^2))*x^4)/4 + ((b^4*d^3 + 12*a *b^3*d^2*e + 4*a^2*b*e*(9*c*d^2 + a*e^2) + 6*a^2*c*d*(c*d^2 + 2*a*e^2) + 6 *a*b^2*d*(2*c*d^2 + 3*a*e^2))*x^5)/5 + ((3*b^4*d^2*e + 6*a*b^2*e*(6*c*d^2 + a*e^2) + 2*a^2*c*e*(9*c*d^2 + 2*a*e^2) + 12*a*b*c*d*(c*d^2 + 3*a*e^2) + 4*b^3*(c*d^3 + 3*a*d*e^2))*x^6)/6 + ((3*b^4*d*e^2 + 12*a*b*c*e*(3*c*d^2 + a*e^2) + 6*b^2*c*d*(c*d^2 + 6*a*e^2) + 2*a*c^2*d*(2*c*d^2 + 9*a*e^2) + 4*b ^3*(3*c*d^2*e + a*e^3))*x^7)/7 + ((12*b^3*c*d*e^2 + b^4*e^3 + 6*a*c^2*e*(2 *c*d^2 + a*e^2) + 6*b^2*c*e*(3*c*d^2 + 2*a*e^2) + 4*b*c^2*d*(c*d^2 + 9*a*e ^2))*x^8)/8 + (c*(c^3*d^3 + 4*b^3*e^3 + 12*c^2*d*e*(b*d + a*e) + 6*b*c*e^2 *(3*b*d + 2*a*e))*x^9)/9 + (c^2*e*(3*c^2*d^2 + 6*b^2*e^2 + 4*c*e*(3*b*d + a*e))*x^10)/10 + (c^3*e^2*(3*c*d + 4*b*e)*x^11)/11 + (c^4*e^3*x^12)/12
Time = 1.00 (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.
| \(\displaystyle \int (d+e x)^3 \left (a+b x+c x^2\right )^4 \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {(d+e x)^7 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^8}+\frac {2 c^2 (d+e x)^9 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8}+\frac {4 c (d+e x)^8 (2 c d-b e) \left (c e (7 b d-3 a e)-b^2 e^2-7 c^2 d^2\right )}{e^8}+\frac {4 (d+e x)^6 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-3 a c e^2-b^2 e^2+7 b c d e-7 c^2 d^2\right )}{e^8}+\frac {2 (d+e x)^5 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8}+\frac {4 (d+e x)^4 (b e-2 c d) \left (a e^2-b d e+c d^2\right )^3}{e^8}+\frac {(d+e x)^3 \left (a e^2-b d e+c d^2\right )^4}{e^8}-\frac {4 c^3 (d+e x)^{10} (2 c d-b e)}{e^8}+\frac {c^4 (d+e x)^{11}}{e^8}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^8 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{8 e^9}+\frac {c^2 (d+e x)^{10} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9}-\frac {4 c (d+e x)^9 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{9 e^9}-\frac {4 (d+e x)^7 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^9}+\frac {(d+e x)^6 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^9}-\frac {4 (d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{5 e^9}+\frac {(d+e x)^4 \left (a e^2-b d e+c d^2\right )^4}{4 e^9}-\frac {4 c^3 (d+e x)^{11} (2 c d-b e)}{11 e^9}+\frac {c^4 (d+e x)^{12}}{12 e^9}\) |
((c*d^2 - b*d*e + a*e^2)^4*(d + e*x)^4)/(4*e^9) - (4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^5)/(5*e^9) + ((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)^6)/(3*e^9) - (4*(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)^7)/(7*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)^8)/(8*e^9) - (4*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)^9)/(9*e^9) + (c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^10)/(5*e^9) - (4*c^3*(2*c*d - b*e)*(d + e *x)^11)/(11*e^9) + (c^4*(d + e*x)^12)/(12*e^9)
3.22.47.3.1 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 = 3.02 (sec) , antiderivative size = 646, normalized size of antiderivative = 1.46
| method | result | size |
| norman | \(\frac {c^{4} e^{3} x^{12}}{12}+\left (\frac {4}{11} e^{3} c^{3} b +\frac {3}{11} d \,e^{2} c^{4}\right ) x^{11}+\left (\frac {2}{5} a \,c^{3} e^{3}+\frac {3}{5} b^{2} c^{2} e^{3}+\frac {6}{5} d \,e^{2} c^{3} b +\frac {3}{10} d^{2} e \,c^{4}\right ) x^{10}+\left (\frac {4}{3} a b \,c^{2} e^{3}+\frac {4}{3} a \,c^{3} d \,e^{2}+\frac {4}{9} b^{3} c \,e^{3}+2 b^{2} c^{2} d \,e^{2}+\frac {4}{3} b \,c^{3} d^{2} e +\frac {1}{9} d^{3} c^{4}\right ) x^{9}+\left (\frac {3}{4} a^{2} c^{2} e^{3}+\frac {3}{2} a \,b^{2} c \,e^{3}+\frac {9}{2} a b \,c^{2} d \,e^{2}+\frac {3}{2} a \,c^{3} d^{2} e +\frac {1}{8} b^{4} e^{3}+\frac {3}{2} b^{3} c d \,e^{2}+\frac {9}{4} b^{2} c^{2} d^{2} e +\frac {1}{2} c^{3} b \,d^{3}\right ) x^{8}+\left (\frac {12}{7} a^{2} b c \,e^{3}+\frac {18}{7} a^{2} c^{2} d \,e^{2}+\frac {4}{7} a \,b^{3} e^{3}+\frac {36}{7} a \,b^{2} c d \,e^{2}+\frac {36}{7} a b \,c^{2} d^{2} e +\frac {4}{7} a \,c^{3} d^{3}+\frac {3}{7} b^{4} d \,e^{2}+\frac {12}{7} b^{3} c \,d^{2} e +\frac {6}{7} b^{2} c^{2} d^{3}\right ) x^{7}+\left (\frac {2}{3} a^{3} c \,e^{3}+a^{2} b^{2} e^{3}+6 a^{2} b c d \,e^{2}+3 a^{2} c^{2} d^{2} e +2 a \,b^{3} d \,e^{2}+6 a \,b^{2} c \,d^{2} e +2 a b \,c^{2} d^{3}+\frac {1}{2} b^{4} d^{2} e +\frac {2}{3} b^{3} c \,d^{3}\right ) x^{6}+\left (\frac {4}{5} a^{3} b \,e^{3}+\frac {12}{5} a^{3} c d \,e^{2}+\frac {18}{5} a^{2} b^{2} d \,e^{2}+\frac {36}{5} a^{2} b c \,d^{2} e +\frac {6}{5} a^{2} c^{2} d^{3}+\frac {12}{5} a \,b^{3} d^{2} e +\frac {12}{5} b^{2} c \,d^{3} a +\frac {1}{5} b^{4} d^{3}\right ) x^{5}+\left (\frac {1}{4} e^{3} a^{4}+3 d \,e^{2} a^{3} b +3 a^{3} c \,d^{2} e +\frac {9}{2} a^{2} b^{2} d^{2} e +3 a^{2} c \,d^{3} b +a \,b^{3} d^{3}\right ) x^{4}+\left (d \,e^{2} a^{4}+4 d^{2} e \,a^{3} b +\frac {4}{3} a^{3} c \,d^{3}+2 a^{2} b^{2} d^{3}\right ) x^{3}+\left (\frac {3}{2} d^{2} e \,a^{4}+2 a^{3} b \,d^{3}\right ) x^{2}+a^{4} d^{3} x\) | \(646\) |
| default | \(\frac {c^{4} e^{3} x^{12}}{12}+\frac {\left (4 e^{3} c^{3} b +3 d \,e^{2} c^{4}\right ) x^{11}}{11}+\frac {\left (3 d^{2} e \,c^{4}+12 d \,e^{2} c^{3} b +e^{3} \left (2 \left (2 a c +b^{2}\right ) c^{2}+4 b^{2} c^{2}\right )\right ) x^{10}}{10}+\frac {\left (d^{3} c^{4}+12 b \,c^{3} d^{2} e +3 d \,e^{2} \left (2 \left (2 a c +b^{2}\right ) c^{2}+4 b^{2} c^{2}\right )+e^{3} \left (4 b \,c^{2} a +4 \left (2 a c +b^{2}\right ) b c \right )\right ) x^{9}}{9}+\frac {\left (4 c^{3} b \,d^{3}+3 d^{2} e \left (2 \left (2 a c +b^{2}\right ) c^{2}+4 b^{2} c^{2}\right )+3 d \,e^{2} \left (4 b \,c^{2} a +4 \left (2 a c +b^{2}\right ) b c \right )+e^{3} \left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right )\right ) x^{8}}{8}+\frac {\left (d^{3} \left (2 \left (2 a c +b^{2}\right ) c^{2}+4 b^{2} c^{2}\right )+3 d^{2} e \left (4 b \,c^{2} a +4 \left (2 a c +b^{2}\right ) b c \right )+3 d \,e^{2} \left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right )+e^{3} \left (4 a^{2} b c +4 a b \left (2 a c +b^{2}\right )\right )\right ) x^{7}}{7}+\frac {\left (d^{3} \left (4 b \,c^{2} a +4 \left (2 a c +b^{2}\right ) b c \right )+3 d^{2} e \left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right )+3 d \,e^{2} \left (4 a^{2} b c +4 a b \left (2 a c +b^{2}\right )\right )+e^{3} \left (2 a^{2} \left (2 a c +b^{2}\right )+4 a^{2} b^{2}\right )\right ) x^{6}}{6}+\frac {\left (d^{3} \left (2 a^{2} c^{2}+8 a \,b^{2} c +\left (2 a c +b^{2}\right )^{2}\right )+3 d^{2} e \left (4 a^{2} b c +4 a b \left (2 a c +b^{2}\right )\right )+3 d \,e^{2} \left (2 a^{2} \left (2 a c +b^{2}\right )+4 a^{2} b^{2}\right )+4 a^{3} b \,e^{3}\right ) x^{5}}{5}+\frac {\left (d^{3} \left (4 a^{2} b c +4 a b \left (2 a c +b^{2}\right )\right )+3 d^{2} e \left (2 a^{2} \left (2 a c +b^{2}\right )+4 a^{2} b^{2}\right )+12 d \,e^{2} a^{3} b +e^{3} a^{4}\right ) x^{4}}{4}+\frac {\left (d^{3} \left (2 a^{2} \left (2 a c +b^{2}\right )+4 a^{2} b^{2}\right )+12 d^{2} e \,a^{3} b +3 d \,e^{2} a^{4}\right ) x^{3}}{3}+\frac {\left (3 d^{2} e \,a^{4}+4 a^{3} b \,d^{3}\right ) x^{2}}{2}+a^{4} d^{3} x\) | \(747\) |
| gosper | \(\frac {36}{5} x^{5} a^{2} b c \,d^{2} e +6 x^{6} a^{2} b c d \,e^{2}+3 x^{6} a^{2} c^{2} d^{2} e +2 x^{6} a \,b^{3} d \,e^{2}+2 x^{6} a b \,c^{2} d^{3}+\frac {12}{5} x^{5} a^{3} c d \,e^{2}+\frac {18}{5} x^{5} a^{2} b^{2} d \,e^{2}+\frac {12}{5} x^{5} a \,b^{3} d^{2} e +3 x^{4} d \,e^{2} a^{3} b +3 x^{4} a^{3} c \,d^{2} e +\frac {9}{2} x^{4} a^{2} b^{2} d^{2} e +\frac {9}{4} x^{8} b^{2} c^{2} d^{2} e +\frac {12}{7} x^{7} b^{3} c \,d^{2} e +a^{4} d^{3} x +\frac {1}{5} d^{3} x^{5} b^{4}+2 a^{2} b^{2} d^{3} x^{3}+2 a^{3} b \,d^{3} x^{2}+a \,b^{3} d^{3} x^{4}+\frac {36}{7} x^{7} a b \,c^{2} d^{2} e +\frac {4}{3} x^{9} a b \,c^{2} e^{3}+\frac {4}{3} x^{9} a \,c^{3} d \,e^{2}+2 x^{9} b^{2} c^{2} d \,e^{2}+\frac {4}{3} x^{9} b \,c^{3} d^{2} e +\frac {3}{2} x^{8} a \,b^{2} c \,e^{3}+\frac {3}{2} x^{8} a \,c^{3} d^{2} e +\frac {3}{2} x^{8} b^{3} c d \,e^{2}+3 x^{4} a^{2} c \,d^{3} b +4 x^{3} d^{2} e \,a^{3} b +\frac {18}{7} x^{7} a^{2} c^{2} d \,e^{2}+\frac {12}{7} x^{7} a^{2} b c \,e^{3}+\frac {6}{5} x^{10} d \,e^{2} c^{3} b +\frac {3}{5} x^{10} b^{2} c^{2} e^{3}+\frac {3}{10} x^{10} d^{2} e \,c^{4}+\frac {4}{9} x^{9} b^{3} c \,e^{3}+\frac {1}{2} x^{8} c^{3} b \,d^{3}+6 x^{6} a \,b^{2} c \,d^{2} e +\frac {1}{9} x^{9} d^{3} c^{4}+\frac {36}{7} x^{7} a \,b^{2} c d \,e^{2}+\frac {1}{8} e^{3} b^{4} x^{8}+\frac {9}{2} x^{8} a b \,c^{2} d \,e^{2}+\frac {3}{2} d^{2} e \,a^{4} x^{2}+\frac {6}{5} x^{5} a^{2} c^{2} d^{3}+x^{3} d \,e^{2} a^{4}+x^{6} a^{2} b^{2} e^{3}+\frac {1}{2} x^{6} b^{4} d^{2} e +\frac {2}{3} x^{6} b^{3} c \,d^{3}+\frac {1}{4} a^{4} e^{3} x^{4}+\frac {1}{12} c^{4} e^{3} x^{12}+\frac {4}{5} x^{5} a^{3} b \,e^{3}+\frac {4}{11} x^{11} e^{3} c^{3} b +\frac {4}{7} x^{7} a \,b^{3} e^{3}+\frac {4}{7} x^{7} a \,c^{3} d^{3}+\frac {3}{7} x^{7} b^{4} d \,e^{2}+\frac {6}{7} x^{7} b^{2} c^{2} d^{3}+\frac {4}{3} a^{3} c \,d^{3} x^{3}+\frac {2}{5} a \,c^{3} e^{3} x^{10}+\frac {3}{11} c^{4} d \,e^{2} x^{11}+\frac {12}{5} a \,b^{2} c \,d^{3} x^{5}+\frac {2}{3} a^{3} c \,e^{3} x^{6}+\frac {3}{4} a^{2} c^{2} e^{3} x^{8}\) | \(770\) |
| risch | \(\frac {36}{5} x^{5} a^{2} b c \,d^{2} e +6 x^{6} a^{2} b c d \,e^{2}+3 x^{6} a^{2} c^{2} d^{2} e +2 x^{6} a \,b^{3} d \,e^{2}+2 x^{6} a b \,c^{2} d^{3}+\frac {12}{5} x^{5} a^{3} c d \,e^{2}+\frac {18}{5} x^{5} a^{2} b^{2} d \,e^{2}+\frac {12}{5} x^{5} a \,b^{3} d^{2} e +3 x^{4} d \,e^{2} a^{3} b +3 x^{4} a^{3} c \,d^{2} e +\frac {9}{2} x^{4} a^{2} b^{2} d^{2} e +\frac {9}{4} x^{8} b^{2} c^{2} d^{2} e +\frac {12}{7} x^{7} b^{3} c \,d^{2} e +a^{4} d^{3} x +\frac {1}{5} d^{3} x^{5} b^{4}+2 a^{2} b^{2} d^{3} x^{3}+2 a^{3} b \,d^{3} x^{2}+a \,b^{3} d^{3} x^{4}+\frac {36}{7} x^{7} a b \,c^{2} d^{2} e +\frac {4}{3} x^{9} a b \,c^{2} e^{3}+\frac {4}{3} x^{9} a \,c^{3} d \,e^{2}+2 x^{9} b^{2} c^{2} d \,e^{2}+\frac {4}{3} x^{9} b \,c^{3} d^{2} e +\frac {3}{2} x^{8} a \,b^{2} c \,e^{3}+\frac {3}{2} x^{8} a \,c^{3} d^{2} e +\frac {3}{2} x^{8} b^{3} c d \,e^{2}+3 x^{4} a^{2} c \,d^{3} b +4 x^{3} d^{2} e \,a^{3} b +\frac {18}{7} x^{7} a^{2} c^{2} d \,e^{2}+\frac {12}{7} x^{7} a^{2} b c \,e^{3}+\frac {6}{5} x^{10} d \,e^{2} c^{3} b +\frac {3}{5} x^{10} b^{2} c^{2} e^{3}+\frac {3}{10} x^{10} d^{2} e \,c^{4}+\frac {4}{9} x^{9} b^{3} c \,e^{3}+\frac {1}{2} x^{8} c^{3} b \,d^{3}+6 x^{6} a \,b^{2} c \,d^{2} e +\frac {1}{9} x^{9} d^{3} c^{4}+\frac {36}{7} x^{7} a \,b^{2} c d \,e^{2}+\frac {1}{8} e^{3} b^{4} x^{8}+\frac {9}{2} x^{8} a b \,c^{2} d \,e^{2}+\frac {3}{2} d^{2} e \,a^{4} x^{2}+\frac {6}{5} x^{5} a^{2} c^{2} d^{3}+x^{3} d \,e^{2} a^{4}+x^{6} a^{2} b^{2} e^{3}+\frac {1}{2} x^{6} b^{4} d^{2} e +\frac {2}{3} x^{6} b^{3} c \,d^{3}+\frac {1}{4} a^{4} e^{3} x^{4}+\frac {1}{12} c^{4} e^{3} x^{12}+\frac {4}{5} x^{5} a^{3} b \,e^{3}+\frac {4}{11} x^{11} e^{3} c^{3} b +\frac {4}{7} x^{7} a \,b^{3} e^{3}+\frac {4}{7} x^{7} a \,c^{3} d^{3}+\frac {3}{7} x^{7} b^{4} d \,e^{2}+\frac {6}{7} x^{7} b^{2} c^{2} d^{3}+\frac {4}{3} a^{3} c \,d^{3} x^{3}+\frac {2}{5} a \,c^{3} e^{3} x^{10}+\frac {3}{11} c^{4} d \,e^{2} x^{11}+\frac {12}{5} a \,b^{2} c \,d^{3} x^{5}+\frac {2}{3} a^{3} c \,e^{3} x^{6}+\frac {3}{4} a^{2} c^{2} e^{3} x^{8}\) | \(770\) |
| parallelrisch | \(\frac {36}{5} x^{5} a^{2} b c \,d^{2} e +6 x^{6} a^{2} b c d \,e^{2}+3 x^{6} a^{2} c^{2} d^{2} e +2 x^{6} a \,b^{3} d \,e^{2}+2 x^{6} a b \,c^{2} d^{3}+\frac {12}{5} x^{5} a^{3} c d \,e^{2}+\frac {18}{5} x^{5} a^{2} b^{2} d \,e^{2}+\frac {12}{5} x^{5} a \,b^{3} d^{2} e +3 x^{4} d \,e^{2} a^{3} b +3 x^{4} a^{3} c \,d^{2} e +\frac {9}{2} x^{4} a^{2} b^{2} d^{2} e +\frac {9}{4} x^{8} b^{2} c^{2} d^{2} e +\frac {12}{7} x^{7} b^{3} c \,d^{2} e +a^{4} d^{3} x +\frac {1}{5} d^{3} x^{5} b^{4}+2 a^{2} b^{2} d^{3} x^{3}+2 a^{3} b \,d^{3} x^{2}+a \,b^{3} d^{3} x^{4}+\frac {36}{7} x^{7} a b \,c^{2} d^{2} e +\frac {4}{3} x^{9} a b \,c^{2} e^{3}+\frac {4}{3} x^{9} a \,c^{3} d \,e^{2}+2 x^{9} b^{2} c^{2} d \,e^{2}+\frac {4}{3} x^{9} b \,c^{3} d^{2} e +\frac {3}{2} x^{8} a \,b^{2} c \,e^{3}+\frac {3}{2} x^{8} a \,c^{3} d^{2} e +\frac {3}{2} x^{8} b^{3} c d \,e^{2}+3 x^{4} a^{2} c \,d^{3} b +4 x^{3} d^{2} e \,a^{3} b +\frac {18}{7} x^{7} a^{2} c^{2} d \,e^{2}+\frac {12}{7} x^{7} a^{2} b c \,e^{3}+\frac {6}{5} x^{10} d \,e^{2} c^{3} b +\frac {3}{5} x^{10} b^{2} c^{2} e^{3}+\frac {3}{10} x^{10} d^{2} e \,c^{4}+\frac {4}{9} x^{9} b^{3} c \,e^{3}+\frac {1}{2} x^{8} c^{3} b \,d^{3}+6 x^{6} a \,b^{2} c \,d^{2} e +\frac {1}{9} x^{9} d^{3} c^{4}+\frac {36}{7} x^{7} a \,b^{2} c d \,e^{2}+\frac {1}{8} e^{3} b^{4} x^{8}+\frac {9}{2} x^{8} a b \,c^{2} d \,e^{2}+\frac {3}{2} d^{2} e \,a^{4} x^{2}+\frac {6}{5} x^{5} a^{2} c^{2} d^{3}+x^{3} d \,e^{2} a^{4}+x^{6} a^{2} b^{2} e^{3}+\frac {1}{2} x^{6} b^{4} d^{2} e +\frac {2}{3} x^{6} b^{3} c \,d^{3}+\frac {1}{4} a^{4} e^{3} x^{4}+\frac {1}{12} c^{4} e^{3} x^{12}+\frac {4}{5} x^{5} a^{3} b \,e^{3}+\frac {4}{11} x^{11} e^{3} c^{3} b +\frac {4}{7} x^{7} a \,b^{3} e^{3}+\frac {4}{7} x^{7} a \,c^{3} d^{3}+\frac {3}{7} x^{7} b^{4} d \,e^{2}+\frac {6}{7} x^{7} b^{2} c^{2} d^{3}+\frac {4}{3} a^{3} c \,d^{3} x^{3}+\frac {2}{5} a \,c^{3} e^{3} x^{10}+\frac {3}{11} c^{4} d \,e^{2} x^{11}+\frac {12}{5} a \,b^{2} c \,d^{3} x^{5}+\frac {2}{3} a^{3} c \,e^{3} x^{6}+\frac {3}{4} a^{2} c^{2} e^{3} x^{8}\) | \(770\) |
1/12*c^4*e^3*x^12+(4/11*e^3*c^3*b+3/11*d*e^2*c^4)*x^11+(2/5*a*c^3*e^3+3/5* b^2*c^2*e^3+6/5*d*e^2*c^3*b+3/10*d^2*e*c^4)*x^10+(4/3*a*b*c^2*e^3+4/3*a*c^ 3*d*e^2+4/9*b^3*c*e^3+2*b^2*c^2*d*e^2+4/3*b*c^3*d^2*e+1/9*d^3*c^4)*x^9+(3/ 4*a^2*c^2*e^3+3/2*a*b^2*c*e^3+9/2*a*b*c^2*d*e^2+3/2*a*c^3*d^2*e+1/8*b^4*e^ 3+3/2*b^3*c*d*e^2+9/4*b^2*c^2*d^2*e+1/2*c^3*b*d^3)*x^8+(12/7*a^2*b*c*e^3+1 8/7*a^2*c^2*d*e^2+4/7*a*b^3*e^3+36/7*a*b^2*c*d*e^2+36/7*a*b*c^2*d^2*e+4/7* a*c^3*d^3+3/7*b^4*d*e^2+12/7*b^3*c*d^2*e+6/7*b^2*c^2*d^3)*x^7+(2/3*a^3*c*e ^3+a^2*b^2*e^3+6*a^2*b*c*d*e^2+3*a^2*c^2*d^2*e+2*a*b^3*d*e^2+6*a*b^2*c*d^2 *e+2*a*b*c^2*d^3+1/2*b^4*d^2*e+2/3*b^3*c*d^3)*x^6+(4/5*a^3*b*e^3+12/5*a^3* c*d*e^2+18/5*a^2*b^2*d*e^2+36/5*a^2*b*c*d^2*e+6/5*a^2*c^2*d^3+12/5*a*b^3*d ^2*e+12/5*b^2*c*d^3*a+1/5*b^4*d^3)*x^5+(1/4*e^3*a^4+3*d*e^2*a^3*b+3*a^3*c* d^2*e+9/2*a^2*b^2*d^2*e+3*a^2*c*d^3*b+a*b^3*d^3)*x^4+(d*e^2*a^4+4*d^2*e*a^ 3*b+4/3*a^3*c*d^3+2*a^2*b^2*d^3)*x^3+(3/2*d^2*e*a^4+2*a^3*b*d^3)*x^2+a^4*d ^3*x
Time = 0.27 (sec) , antiderivative size = 614, normalized size of antiderivative = 1.39 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^4 \, dx=\frac {1}{12} \, c^{4} e^{3} x^{12} + \frac {1}{11} \, {\left (3 \, c^{4} d e^{2} + 4 \, b c^{3} e^{3}\right )} x^{11} + \frac {1}{10} \, {\left (3 \, c^{4} d^{2} e + 12 \, b c^{3} d e^{2} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{3}\right )} x^{10} + \frac {1}{9} \, {\left (c^{4} d^{3} + 12 \, b c^{3} d^{2} e + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{2} + 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{3}\right )} x^{9} + \frac {1}{8} \, {\left (4 \, b c^{3} d^{3} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e + 12 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{2} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{3}\right )} x^{8} + a^{4} d^{3} x + \frac {1}{7} \, {\left (2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} + 12 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{2} + 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{3}\right )} x^{7} + \frac {1}{6} \, {\left (4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e + 12 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{2} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (4 \, a^{3} b e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} + 12 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e + 6 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (12 \, a^{3} b d e^{2} + a^{4} e^{3} + 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} + 6 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e\right )} x^{4} + \frac {1}{3} \, {\left (12 \, a^{3} b d^{2} e + 3 \, a^{4} d e^{2} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{3}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d^{3} + 3 \, a^{4} d^{2} e\right )} x^{2} \]
1/12*c^4*e^3*x^12 + 1/11*(3*c^4*d*e^2 + 4*b*c^3*e^3)*x^11 + 1/10*(3*c^4*d^ 2*e + 12*b*c^3*d*e^2 + 2*(3*b^2*c^2 + 2*a*c^3)*e^3)*x^10 + 1/9*(c^4*d^3 + 12*b*c^3*d^2*e + 6*(3*b^2*c^2 + 2*a*c^3)*d*e^2 + 4*(b^3*c + 3*a*b*c^2)*e^3 )*x^9 + 1/8*(4*b*c^3*d^3 + 6*(3*b^2*c^2 + 2*a*c^3)*d^2*e + 12*(b^3*c + 3*a *b*c^2)*d*e^2 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^3)*x^8 + a^4*d^3*x + 1/7* (2*(3*b^2*c^2 + 2*a*c^3)*d^3 + 12*(b^3*c + 3*a*b*c^2)*d^2*e + 3*(b^4 + 12* a*b^2*c + 6*a^2*c^2)*d*e^2 + 4*(a*b^3 + 3*a^2*b*c)*e^3)*x^7 + 1/6*(4*(b^3* c + 3*a*b*c^2)*d^3 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e + 12*(a*b^3 + 3*a^2*b*c)*d*e^2 + 2*(3*a^2*b^2 + 2*a^3*c)*e^3)*x^6 + 1/5*(4*a^3*b*e^3 + ( b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3 + 12*(a*b^3 + 3*a^2*b*c)*d^2*e + 6*(3*a^ 2*b^2 + 2*a^3*c)*d*e^2)*x^5 + 1/4*(12*a^3*b*d*e^2 + a^4*e^3 + 4*(a*b^3 + 3 *a^2*b*c)*d^3 + 6*(3*a^2*b^2 + 2*a^3*c)*d^2*e)*x^4 + 1/3*(12*a^3*b*d^2*e + 3*a^4*d*e^2 + 2*(3*a^2*b^2 + 2*a^3*c)*d^3)*x^3 + 1/2*(4*a^3*b*d^3 + 3*a^4 *d^2*e)*x^2
Time = 0.07 (sec) , antiderivative size = 777, normalized size of antiderivative = 1.75 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^4 \, dx=a^{4} d^{3} x + \frac {c^{4} e^{3} x^{12}}{12} + x^{11} \cdot \left (\frac {4 b c^{3} e^{3}}{11} + \frac {3 c^{4} d e^{2}}{11}\right ) + x^{10} \cdot \left (\frac {2 a c^{3} e^{3}}{5} + \frac {3 b^{2} c^{2} e^{3}}{5} + \frac {6 b c^{3} d e^{2}}{5} + \frac {3 c^{4} d^{2} e}{10}\right ) + x^{9} \cdot \left (\frac {4 a b c^{2} e^{3}}{3} + \frac {4 a c^{3} d e^{2}}{3} + \frac {4 b^{3} c e^{3}}{9} + 2 b^{2} c^{2} d e^{2} + \frac {4 b c^{3} d^{2} e}{3} + \frac {c^{4} d^{3}}{9}\right ) + x^{8} \cdot \left (\frac {3 a^{2} c^{2} e^{3}}{4} + \frac {3 a b^{2} c e^{3}}{2} + \frac {9 a b c^{2} d e^{2}}{2} + \frac {3 a c^{3} d^{2} e}{2} + \frac {b^{4} e^{3}}{8} + \frac {3 b^{3} c d e^{2}}{2} + \frac {9 b^{2} c^{2} d^{2} e}{4} + \frac {b c^{3} d^{3}}{2}\right ) + x^{7} \cdot \left (\frac {12 a^{2} b c e^{3}}{7} + \frac {18 a^{2} c^{2} d e^{2}}{7} + \frac {4 a b^{3} e^{3}}{7} + \frac {36 a b^{2} c d e^{2}}{7} + \frac {36 a b c^{2} d^{2} e}{7} + \frac {4 a c^{3} d^{3}}{7} + \frac {3 b^{4} d e^{2}}{7} + \frac {12 b^{3} c d^{2} e}{7} + \frac {6 b^{2} c^{2} d^{3}}{7}\right ) + x^{6} \cdot \left (\frac {2 a^{3} c e^{3}}{3} + a^{2} b^{2} e^{3} + 6 a^{2} b c d e^{2} + 3 a^{2} c^{2} d^{2} e + 2 a b^{3} d e^{2} + 6 a b^{2} c d^{2} e + 2 a b c^{2} d^{3} + \frac {b^{4} d^{2} e}{2} + \frac {2 b^{3} c d^{3}}{3}\right ) + x^{5} \cdot \left (\frac {4 a^{3} b e^{3}}{5} + \frac {12 a^{3} c d e^{2}}{5} + \frac {18 a^{2} b^{2} d e^{2}}{5} + \frac {36 a^{2} b c d^{2} e}{5} + \frac {6 a^{2} c^{2} d^{3}}{5} + \frac {12 a b^{3} d^{2} e}{5} + \frac {12 a b^{2} c d^{3}}{5} + \frac {b^{4} d^{3}}{5}\right ) + x^{4} \left (\frac {a^{4} e^{3}}{4} + 3 a^{3} b d e^{2} + 3 a^{3} c d^{2} e + \frac {9 a^{2} b^{2} d^{2} e}{2} + 3 a^{2} b c d^{3} + a b^{3} d^{3}\right ) + x^{3} \left (a^{4} d e^{2} + 4 a^{3} b d^{2} e + \frac {4 a^{3} c d^{3}}{3} + 2 a^{2} b^{2} d^{3}\right ) + x^{2} \cdot \left (\frac {3 a^{4} d^{2} e}{2} + 2 a^{3} b d^{3}\right ) \]
a**4*d**3*x + c**4*e**3*x**12/12 + x**11*(4*b*c**3*e**3/11 + 3*c**4*d*e**2 /11) + x**10*(2*a*c**3*e**3/5 + 3*b**2*c**2*e**3/5 + 6*b*c**3*d*e**2/5 + 3 *c**4*d**2*e/10) + x**9*(4*a*b*c**2*e**3/3 + 4*a*c**3*d*e**2/3 + 4*b**3*c* e**3/9 + 2*b**2*c**2*d*e**2 + 4*b*c**3*d**2*e/3 + c**4*d**3/9) + x**8*(3*a **2*c**2*e**3/4 + 3*a*b**2*c*e**3/2 + 9*a*b*c**2*d*e**2/2 + 3*a*c**3*d**2* e/2 + b**4*e**3/8 + 3*b**3*c*d*e**2/2 + 9*b**2*c**2*d**2*e/4 + b*c**3*d**3 /2) + x**7*(12*a**2*b*c*e**3/7 + 18*a**2*c**2*d*e**2/7 + 4*a*b**3*e**3/7 + 36*a*b**2*c*d*e**2/7 + 36*a*b*c**2*d**2*e/7 + 4*a*c**3*d**3/7 + 3*b**4*d* e**2/7 + 12*b**3*c*d**2*e/7 + 6*b**2*c**2*d**3/7) + x**6*(2*a**3*c*e**3/3 + a**2*b**2*e**3 + 6*a**2*b*c*d*e**2 + 3*a**2*c**2*d**2*e + 2*a*b**3*d*e** 2 + 6*a*b**2*c*d**2*e + 2*a*b*c**2*d**3 + b**4*d**2*e/2 + 2*b**3*c*d**3/3) + x**5*(4*a**3*b*e**3/5 + 12*a**3*c*d*e**2/5 + 18*a**2*b**2*d*e**2/5 + 36 *a**2*b*c*d**2*e/5 + 6*a**2*c**2*d**3/5 + 12*a*b**3*d**2*e/5 + 12*a*b**2*c *d**3/5 + b**4*d**3/5) + x**4*(a**4*e**3/4 + 3*a**3*b*d*e**2 + 3*a**3*c*d* *2*e + 9*a**2*b**2*d**2*e/2 + 3*a**2*b*c*d**3 + a*b**3*d**3) + x**3*(a**4* d*e**2 + 4*a**3*b*d**2*e + 4*a**3*c*d**3/3 + 2*a**2*b**2*d**3) + x**2*(3*a **4*d**2*e/2 + 2*a**3*b*d**3)
Time = 0.19 (sec) , antiderivative size = 614, normalized size of antiderivative = 1.39 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^4 \, dx=\frac {1}{12} \, c^{4} e^{3} x^{12} + \frac {1}{11} \, {\left (3 \, c^{4} d e^{2} + 4 \, b c^{3} e^{3}\right )} x^{11} + \frac {1}{10} \, {\left (3 \, c^{4} d^{2} e + 12 \, b c^{3} d e^{2} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{3}\right )} x^{10} + \frac {1}{9} \, {\left (c^{4} d^{3} + 12 \, b c^{3} d^{2} e + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{2} + 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{3}\right )} x^{9} + \frac {1}{8} \, {\left (4 \, b c^{3} d^{3} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e + 12 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{2} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{3}\right )} x^{8} + a^{4} d^{3} x + \frac {1}{7} \, {\left (2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} + 12 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{2} + 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{3}\right )} x^{7} + \frac {1}{6} \, {\left (4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e + 12 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{2} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (4 \, a^{3} b e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} + 12 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e + 6 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (12 \, a^{3} b d e^{2} + a^{4} e^{3} + 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} + 6 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e\right )} x^{4} + \frac {1}{3} \, {\left (12 \, a^{3} b d^{2} e + 3 \, a^{4} d e^{2} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{3}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d^{3} + 3 \, a^{4} d^{2} e\right )} x^{2} \]
1/12*c^4*e^3*x^12 + 1/11*(3*c^4*d*e^2 + 4*b*c^3*e^3)*x^11 + 1/10*(3*c^4*d^ 2*e + 12*b*c^3*d*e^2 + 2*(3*b^2*c^2 + 2*a*c^3)*e^3)*x^10 + 1/9*(c^4*d^3 + 12*b*c^3*d^2*e + 6*(3*b^2*c^2 + 2*a*c^3)*d*e^2 + 4*(b^3*c + 3*a*b*c^2)*e^3 )*x^9 + 1/8*(4*b*c^3*d^3 + 6*(3*b^2*c^2 + 2*a*c^3)*d^2*e + 12*(b^3*c + 3*a *b*c^2)*d*e^2 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^3)*x^8 + a^4*d^3*x + 1/7* (2*(3*b^2*c^2 + 2*a*c^3)*d^3 + 12*(b^3*c + 3*a*b*c^2)*d^2*e + 3*(b^4 + 12* a*b^2*c + 6*a^2*c^2)*d*e^2 + 4*(a*b^3 + 3*a^2*b*c)*e^3)*x^7 + 1/6*(4*(b^3* c + 3*a*b*c^2)*d^3 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e + 12*(a*b^3 + 3*a^2*b*c)*d*e^2 + 2*(3*a^2*b^2 + 2*a^3*c)*e^3)*x^6 + 1/5*(4*a^3*b*e^3 + ( b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3 + 12*(a*b^3 + 3*a^2*b*c)*d^2*e + 6*(3*a^ 2*b^2 + 2*a^3*c)*d*e^2)*x^5 + 1/4*(12*a^3*b*d*e^2 + a^4*e^3 + 4*(a*b^3 + 3 *a^2*b*c)*d^3 + 6*(3*a^2*b^2 + 2*a^3*c)*d^2*e)*x^4 + 1/3*(12*a^3*b*d^2*e + 3*a^4*d*e^2 + 2*(3*a^2*b^2 + 2*a^3*c)*d^3)*x^3 + 1/2*(4*a^3*b*d^3 + 3*a^4 *d^2*e)*x^2
Time = 0.27 (sec) , antiderivative size = 769, normalized size of antiderivative = 1.74 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^4 \, dx=\frac {1}{12} \, c^{4} e^{3} x^{12} + \frac {3}{11} \, c^{4} d e^{2} x^{11} + \frac {4}{11} \, b c^{3} e^{3} x^{11} + \frac {3}{10} \, c^{4} d^{2} e x^{10} + \frac {6}{5} \, b c^{3} d e^{2} x^{10} + \frac {3}{5} \, b^{2} c^{2} e^{3} x^{10} + \frac {2}{5} \, a c^{3} e^{3} x^{10} + \frac {1}{9} \, c^{4} d^{3} x^{9} + \frac {4}{3} \, b c^{3} d^{2} e x^{9} + 2 \, b^{2} c^{2} d e^{2} x^{9} + \frac {4}{3} \, a c^{3} d e^{2} x^{9} + \frac {4}{9} \, b^{3} c e^{3} x^{9} + \frac {4}{3} \, a b c^{2} e^{3} x^{9} + \frac {1}{2} \, b c^{3} d^{3} x^{8} + \frac {9}{4} \, b^{2} c^{2} d^{2} e x^{8} + \frac {3}{2} \, a c^{3} d^{2} e x^{8} + \frac {3}{2} \, b^{3} c d e^{2} x^{8} + \frac {9}{2} \, a b c^{2} d e^{2} x^{8} + \frac {1}{8} \, b^{4} e^{3} x^{8} + \frac {3}{2} \, a b^{2} c e^{3} x^{8} + \frac {3}{4} \, a^{2} c^{2} e^{3} x^{8} + \frac {6}{7} \, b^{2} c^{2} d^{3} x^{7} + \frac {4}{7} \, a c^{3} d^{3} x^{7} + \frac {12}{7} \, b^{3} c d^{2} e x^{7} + \frac {36}{7} \, a b c^{2} d^{2} e x^{7} + \frac {3}{7} \, b^{4} d e^{2} x^{7} + \frac {36}{7} \, a b^{2} c d e^{2} x^{7} + \frac {18}{7} \, a^{2} c^{2} d e^{2} x^{7} + \frac {4}{7} \, a b^{3} e^{3} x^{7} + \frac {12}{7} \, a^{2} b c e^{3} x^{7} + \frac {2}{3} \, b^{3} c d^{3} x^{6} + 2 \, a b c^{2} d^{3} x^{6} + \frac {1}{2} \, b^{4} d^{2} e x^{6} + 6 \, a b^{2} c d^{2} e x^{6} + 3 \, a^{2} c^{2} d^{2} e x^{6} + 2 \, a b^{3} d e^{2} x^{6} + 6 \, a^{2} b c d e^{2} x^{6} + a^{2} b^{2} e^{3} x^{6} + \frac {2}{3} \, a^{3} c e^{3} x^{6} + \frac {1}{5} \, b^{4} d^{3} x^{5} + \frac {12}{5} \, a b^{2} c d^{3} x^{5} + \frac {6}{5} \, a^{2} c^{2} d^{3} x^{5} + \frac {12}{5} \, a b^{3} d^{2} e x^{5} + \frac {36}{5} \, a^{2} b c d^{2} e x^{5} + \frac {18}{5} \, a^{2} b^{2} d e^{2} x^{5} + \frac {12}{5} \, a^{3} c d e^{2} x^{5} + \frac {4}{5} \, a^{3} b e^{3} x^{5} + a b^{3} d^{3} x^{4} + 3 \, a^{2} b c d^{3} x^{4} + \frac {9}{2} \, a^{2} b^{2} d^{2} e x^{4} + 3 \, a^{3} c d^{2} e x^{4} + 3 \, a^{3} b d e^{2} x^{4} + \frac {1}{4} \, a^{4} e^{3} x^{4} + 2 \, a^{2} b^{2} d^{3} x^{3} + \frac {4}{3} \, a^{3} c d^{3} x^{3} + 4 \, a^{3} b d^{2} e x^{3} + a^{4} d e^{2} x^{3} + 2 \, a^{3} b d^{3} x^{2} + \frac {3}{2} \, a^{4} d^{2} e x^{2} + a^{4} d^{3} x \]
1/12*c^4*e^3*x^12 + 3/11*c^4*d*e^2*x^11 + 4/11*b*c^3*e^3*x^11 + 3/10*c^4*d ^2*e*x^10 + 6/5*b*c^3*d*e^2*x^10 + 3/5*b^2*c^2*e^3*x^10 + 2/5*a*c^3*e^3*x^ 10 + 1/9*c^4*d^3*x^9 + 4/3*b*c^3*d^2*e*x^9 + 2*b^2*c^2*d*e^2*x^9 + 4/3*a*c ^3*d*e^2*x^9 + 4/9*b^3*c*e^3*x^9 + 4/3*a*b*c^2*e^3*x^9 + 1/2*b*c^3*d^3*x^8 + 9/4*b^2*c^2*d^2*e*x^8 + 3/2*a*c^3*d^2*e*x^8 + 3/2*b^3*c*d*e^2*x^8 + 9/2 *a*b*c^2*d*e^2*x^8 + 1/8*b^4*e^3*x^8 + 3/2*a*b^2*c*e^3*x^8 + 3/4*a^2*c^2*e ^3*x^8 + 6/7*b^2*c^2*d^3*x^7 + 4/7*a*c^3*d^3*x^7 + 12/7*b^3*c*d^2*e*x^7 + 36/7*a*b*c^2*d^2*e*x^7 + 3/7*b^4*d*e^2*x^7 + 36/7*a*b^2*c*d*e^2*x^7 + 18/7 *a^2*c^2*d*e^2*x^7 + 4/7*a*b^3*e^3*x^7 + 12/7*a^2*b*c*e^3*x^7 + 2/3*b^3*c* d^3*x^6 + 2*a*b*c^2*d^3*x^6 + 1/2*b^4*d^2*e*x^6 + 6*a*b^2*c*d^2*e*x^6 + 3* a^2*c^2*d^2*e*x^6 + 2*a*b^3*d*e^2*x^6 + 6*a^2*b*c*d*e^2*x^6 + a^2*b^2*e^3* x^6 + 2/3*a^3*c*e^3*x^6 + 1/5*b^4*d^3*x^5 + 12/5*a*b^2*c*d^3*x^5 + 6/5*a^2 *c^2*d^3*x^5 + 12/5*a*b^3*d^2*e*x^5 + 36/5*a^2*b*c*d^2*e*x^5 + 18/5*a^2*b^ 2*d*e^2*x^5 + 12/5*a^3*c*d*e^2*x^5 + 4/5*a^3*b*e^3*x^5 + a*b^3*d^3*x^4 + 3 *a^2*b*c*d^3*x^4 + 9/2*a^2*b^2*d^2*e*x^4 + 3*a^3*c*d^2*e*x^4 + 3*a^3*b*d*e ^2*x^4 + 1/4*a^4*e^3*x^4 + 2*a^2*b^2*d^3*x^3 + 4/3*a^3*c*d^3*x^3 + 4*a^3*b *d^2*e*x^3 + a^4*d*e^2*x^3 + 2*a^3*b*d^3*x^2 + 3/2*a^4*d^2*e*x^2 + a^4*d^3 *x
Time = 0.22 (sec) , antiderivative size = 630, normalized size of antiderivative = 1.42 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^4 \, dx=x^5\,\left (\frac {4\,a^3\,b\,e^3}{5}+\frac {12\,a^3\,c\,d\,e^2}{5}+\frac {18\,a^2\,b^2\,d\,e^2}{5}+\frac {36\,a^2\,b\,c\,d^2\,e}{5}+\frac {6\,a^2\,c^2\,d^3}{5}+\frac {12\,a\,b^3\,d^2\,e}{5}+\frac {12\,a\,b^2\,c\,d^3}{5}+\frac {b^4\,d^3}{5}\right )+x^8\,\left (\frac {3\,a^2\,c^2\,e^3}{4}+\frac {3\,a\,b^2\,c\,e^3}{2}+\frac {9\,a\,b\,c^2\,d\,e^2}{2}+\frac {3\,a\,c^3\,d^2\,e}{2}+\frac {b^4\,e^3}{8}+\frac {3\,b^3\,c\,d\,e^2}{2}+\frac {9\,b^2\,c^2\,d^2\,e}{4}+\frac {b\,c^3\,d^3}{2}\right )+x^6\,\left (\frac {2\,a^3\,c\,e^3}{3}+a^2\,b^2\,e^3+6\,a^2\,b\,c\,d\,e^2+3\,a^2\,c^2\,d^2\,e+2\,a\,b^3\,d\,e^2+6\,a\,b^2\,c\,d^2\,e+2\,a\,b\,c^2\,d^3+\frac {b^4\,d^2\,e}{2}+\frac {2\,b^3\,c\,d^3}{3}\right )+x^7\,\left (\frac {12\,a^2\,b\,c\,e^3}{7}+\frac {18\,a^2\,c^2\,d\,e^2}{7}+\frac {4\,a\,b^3\,e^3}{7}+\frac {36\,a\,b^2\,c\,d\,e^2}{7}+\frac {36\,a\,b\,c^2\,d^2\,e}{7}+\frac {4\,a\,c^3\,d^3}{7}+\frac {3\,b^4\,d\,e^2}{7}+\frac {12\,b^3\,c\,d^2\,e}{7}+\frac {6\,b^2\,c^2\,d^3}{7}\right )+x^4\,\left (\frac {a^4\,e^3}{4}+3\,a^3\,b\,d\,e^2+3\,c\,a^3\,d^2\,e+\frac {9\,a^2\,b^2\,d^2\,e}{2}+3\,c\,a^2\,b\,d^3+a\,b^3\,d^3\right )+x^9\,\left (\frac {4\,b^3\,c\,e^3}{9}+2\,b^2\,c^2\,d\,e^2+\frac {4\,b\,c^3\,d^2\,e}{3}+\frac {4\,a\,b\,c^2\,e^3}{3}+\frac {c^4\,d^3}{9}+\frac {4\,a\,c^3\,d\,e^2}{3}\right )+a^4\,d^3\,x+\frac {c^4\,e^3\,x^{12}}{12}+\frac {a^2\,d\,x^3\,\left (3\,a^2\,e^2+12\,a\,b\,d\,e+4\,c\,a\,d^2+6\,b^2\,d^2\right )}{3}+\frac {c^2\,e\,x^{10}\,\left (6\,b^2\,e^2+12\,b\,c\,d\,e+3\,c^2\,d^2+4\,a\,c\,e^2\right )}{10}+\frac {a^3\,d^2\,x^2\,\left (3\,a\,e+4\,b\,d\right )}{2}+\frac {c^3\,e^2\,x^{11}\,\left (4\,b\,e+3\,c\,d\right )}{11} \]
x^5*((b^4*d^3)/5 + (4*a^3*b*e^3)/5 + (6*a^2*c^2*d^3)/5 + (18*a^2*b^2*d*e^2 )/5 + (12*a*b^2*c*d^3)/5 + (12*a*b^3*d^2*e)/5 + (12*a^3*c*d*e^2)/5 + (36*a ^2*b*c*d^2*e)/5) + x^8*((b^4*e^3)/8 + (b*c^3*d^3)/2 + (3*a^2*c^2*e^3)/4 + (9*b^2*c^2*d^2*e)/4 + (3*a*b^2*c*e^3)/2 + (3*a*c^3*d^2*e)/2 + (3*b^3*c*d*e ^2)/2 + (9*a*b*c^2*d*e^2)/2) + x^6*((2*a^3*c*e^3)/3 + (2*b^3*c*d^3)/3 + (b ^4*d^2*e)/2 + a^2*b^2*e^3 + 3*a^2*c^2*d^2*e + 2*a*b*c^2*d^3 + 2*a*b^3*d*e^ 2 + 6*a*b^2*c*d^2*e + 6*a^2*b*c*d*e^2) + x^7*((4*a*b^3*e^3)/7 + (4*a*c^3*d ^3)/7 + (3*b^4*d*e^2)/7 + (6*b^2*c^2*d^3)/7 + (18*a^2*c^2*d*e^2)/7 + (12*a ^2*b*c*e^3)/7 + (12*b^3*c*d^2*e)/7 + (36*a*b*c^2*d^2*e)/7 + (36*a*b^2*c*d* e^2)/7) + x^4*((a^4*e^3)/4 + a*b^3*d^3 + (9*a^2*b^2*d^2*e)/2 + 3*a^2*b*c*d ^3 + 3*a^3*b*d*e^2 + 3*a^3*c*d^2*e) + x^9*((c^4*d^3)/9 + (4*b^3*c*e^3)/9 + 2*b^2*c^2*d*e^2 + (4*a*b*c^2*e^3)/3 + (4*a*c^3*d*e^2)/3 + (4*b*c^3*d^2*e) /3) + a^4*d^3*x + (c^4*e^3*x^12)/12 + (a^2*d*x^3*(3*a^2*e^2 + 6*b^2*d^2 + 4*a*c*d^2 + 12*a*b*d*e))/3 + (c^2*e*x^10*(6*b^2*e^2 + 3*c^2*d^2 + 4*a*c*e^ 2 + 12*b*c*d*e))/10 + (a^3*d^2*x^2*(3*a*e + 4*b*d))/2 + (c^3*e^2*x^11*(4*b *e + 3*c*d))/11