Integrand size = 24, antiderivative size = 412 \[ \int (A+B x) (d+e x)^4 \left (b x+c x^2\right )^3 \, dx=\frac {1}{4} A b^3 d^4 x^4+\frac {1}{5} b^2 d^3 (b B d+3 A c d+4 A b e) x^5+\frac {1}{6} b d^2 \left (3 A c^2 d^2+2 b^2 e (2 B d+3 A e)+3 b c d (B d+4 A e)\right ) x^6+\frac {1}{7} d \left (A c^3 d^3+2 b^3 e^2 (3 B d+2 A e)+6 b^2 c d e (2 B d+3 A e)+3 b c^2 d^2 (B d+4 A e)\right ) x^7+\frac {1}{8} \left (A e \left (4 c^3 d^3+18 b c^2 d^2 e+12 b^2 c d e^2+b^3 e^3\right )+B d \left (c^3 d^3+12 b c^2 d^2 e+18 b^2 c d e^2+4 b^3 e^3\right )\right ) x^8+\frac {1}{9} e \left (3 A c e \left (2 c^2 d^2+4 b c d e+b^2 e^2\right )+B \left (4 c^3 d^3+18 b c^2 d^2 e+12 b^2 c d e^2+b^3 e^3\right )\right ) x^9+\frac {1}{10} c e^2 \left (A c e (4 c d+3 b e)+3 B \left (2 c^2 d^2+4 b c d e+b^2 e^2\right )\right ) x^{10}+\frac {1}{11} c^2 e^3 (4 B c d+3 b B e+A c e) x^{11}+\frac {1}{12} B c^3 e^4 x^{12} \]
1/4*A*b^3*d^4*x^4+1/5*b^2*d^3*(4*A*b*e+3*A*c*d+B*b*d)*x^5+1/6*b*d^2*(3*A*c ^2*d^2+2*b^2*e*(3*A*e+2*B*d)+3*b*c*d*(4*A*e+B*d))*x^6+1/7*d*(A*c^3*d^3+2*b ^3*e^2*(2*A*e+3*B*d)+6*b^2*c*d*e*(3*A*e+2*B*d)+3*b*c^2*d^2*(4*A*e+B*d))*x^ 7+1/8*(A*e*(b^3*e^3+12*b^2*c*d*e^2+18*b*c^2*d^2*e+4*c^3*d^3)+B*d*(4*b^3*e^ 3+18*b^2*c*d*e^2+12*b*c^2*d^2*e+c^3*d^3))*x^8+1/9*e*(3*A*c*e*(b^2*e^2+4*b* c*d*e+2*c^2*d^2)+B*(b^3*e^3+12*b^2*c*d*e^2+18*b*c^2*d^2*e+4*c^3*d^3))*x^9+ 1/10*c*e^2*(A*c*e*(3*b*e+4*c*d)+3*B*(b^2*e^2+4*b*c*d*e+2*c^2*d^2))*x^10+1/ 11*c^2*e^3*(A*c*e+3*B*b*e+4*B*c*d)*x^11+1/12*B*c^3*e^4*x^12
Time = 0.11 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.00 \[ \int (A+B x) (d+e x)^4 \left (b x+c x^2\right )^3 \, dx=\frac {1}{4} A b^3 d^4 x^4+\frac {1}{5} b^2 d^3 (b B d+3 A c d+4 A b e) x^5+\frac {1}{6} b d^2 \left (3 A c^2 d^2+2 b^2 e (2 B d+3 A e)+3 b c d (B d+4 A e)\right ) x^6+\frac {1}{7} d \left (A c^3 d^3+2 b^3 e^2 (3 B d+2 A e)+6 b^2 c d e (2 B d+3 A e)+3 b c^2 d^2 (B d+4 A e)\right ) x^7+\frac {1}{8} \left (A e \left (4 c^3 d^3+18 b c^2 d^2 e+12 b^2 c d e^2+b^3 e^3\right )+B d \left (c^3 d^3+12 b c^2 d^2 e+18 b^2 c d e^2+4 b^3 e^3\right )\right ) x^8+\frac {1}{9} e \left (3 A c e \left (2 c^2 d^2+4 b c d e+b^2 e^2\right )+B \left (4 c^3 d^3+18 b c^2 d^2 e+12 b^2 c d e^2+b^3 e^3\right )\right ) x^9+\frac {1}{10} c e^2 \left (A c e (4 c d+3 b e)+3 B \left (2 c^2 d^2+4 b c d e+b^2 e^2\right )\right ) x^{10}+\frac {1}{11} c^2 e^3 (4 B c d+3 b B e+A c e) x^{11}+\frac {1}{12} B c^3 e^4 x^{12} \]
(A*b^3*d^4*x^4)/4 + (b^2*d^3*(b*B*d + 3*A*c*d + 4*A*b*e)*x^5)/5 + (b*d^2*( 3*A*c^2*d^2 + 2*b^2*e*(2*B*d + 3*A*e) + 3*b*c*d*(B*d + 4*A*e))*x^6)/6 + (d *(A*c^3*d^3 + 2*b^3*e^2*(3*B*d + 2*A*e) + 6*b^2*c*d*e*(2*B*d + 3*A*e) + 3* b*c^2*d^2*(B*d + 4*A*e))*x^7)/7 + ((A*e*(4*c^3*d^3 + 18*b*c^2*d^2*e + 12*b ^2*c*d*e^2 + b^3*e^3) + B*d*(c^3*d^3 + 12*b*c^2*d^2*e + 18*b^2*c*d*e^2 + 4 *b^3*e^3))*x^8)/8 + (e*(3*A*c*e*(2*c^2*d^2 + 4*b*c*d*e + b^2*e^2) + B*(4*c ^3*d^3 + 18*b*c^2*d^2*e + 12*b^2*c*d*e^2 + b^3*e^3))*x^9)/9 + (c*e^2*(A*c* e*(4*c*d + 3*b*e) + 3*B*(2*c^2*d^2 + 4*b*c*d*e + b^2*e^2))*x^10)/10 + (c^2 *e^3*(4*B*c*d + 3*b*B*e + A*c*e)*x^11)/11 + (B*c^3*e^4*x^12)/12
Time = 0.76 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, 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.
| \(\displaystyle \int (A+B x) \left (b x+c x^2\right )^3 (d+e x)^4 \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (A b^3 d^4 x^3+c e^2 x^9 \left (A c e (3 b e+4 c d)+3 B \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )\right )+b d^2 x^5 \left (2 b^2 e (3 A e+2 B d)+3 b c d (4 A e+B d)+3 A c^2 d^2\right )+b^2 d^3 x^4 (4 A b e+3 A c d+b B d)+d x^6 \left (2 b^3 e^2 (2 A e+3 B d)+6 b^2 c d e (3 A e+2 B d)+3 b c^2 d^2 (4 A e+B d)+A c^3 d^3\right )+e x^8 \left (3 A c e \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )+B \left (b^3 e^3+12 b^2 c d e^2+18 b c^2 d^2 e+4 c^3 d^3\right )\right )+x^7 \left (A e \left (b^3 e^3+12 b^2 c d e^2+18 b c^2 d^2 e+4 c^3 d^3\right )+B d \left (4 b^3 e^3+18 b^2 c d e^2+12 b c^2 d^2 e+c^3 d^3\right )\right )+c^2 e^3 x^{10} (A c e+3 b B e+4 B c d)+B c^3 e^4 x^{11}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} A b^3 d^4 x^4+\frac {1}{10} c e^2 x^{10} \left (A c e (3 b e+4 c d)+3 B \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )\right )+\frac {1}{6} b d^2 x^6 \left (2 b^2 e (3 A e+2 B d)+3 b c d (4 A e+B d)+3 A c^2 d^2\right )+\frac {1}{5} b^2 d^3 x^5 (4 A b e+3 A c d+b B d)+\frac {1}{7} d x^7 \left (2 b^3 e^2 (2 A e+3 B d)+6 b^2 c d e (3 A e+2 B d)+3 b c^2 d^2 (4 A e+B d)+A c^3 d^3\right )+\frac {1}{9} e x^9 \left (3 A c e \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )+B \left (b^3 e^3+12 b^2 c d e^2+18 b c^2 d^2 e+4 c^3 d^3\right )\right )+\frac {1}{8} x^8 \left (A e \left (b^3 e^3+12 b^2 c d e^2+18 b c^2 d^2 e+4 c^3 d^3\right )+B d \left (4 b^3 e^3+18 b^2 c d e^2+12 b c^2 d^2 e+c^3 d^3\right )\right )+\frac {1}{11} c^2 e^3 x^{11} (A c e+3 b B e+4 B c d)+\frac {1}{12} B c^3 e^4 x^{12}\) |
(A*b^3*d^4*x^4)/4 + (b^2*d^3*(b*B*d + 3*A*c*d + 4*A*b*e)*x^5)/5 + (b*d^2*( 3*A*c^2*d^2 + 2*b^2*e*(2*B*d + 3*A*e) + 3*b*c*d*(B*d + 4*A*e))*x^6)/6 + (d *(A*c^3*d^3 + 2*b^3*e^2*(3*B*d + 2*A*e) + 6*b^2*c*d*e*(2*B*d + 3*A*e) + 3* b*c^2*d^2*(B*d + 4*A*e))*x^7)/7 + ((A*e*(4*c^3*d^3 + 18*b*c^2*d^2*e + 12*b ^2*c*d*e^2 + b^3*e^3) + B*d*(c^3*d^3 + 12*b*c^2*d^2*e + 18*b^2*c*d*e^2 + 4 *b^3*e^3))*x^8)/8 + (e*(3*A*c*e*(2*c^2*d^2 + 4*b*c*d*e + b^2*e^2) + B*(4*c ^3*d^3 + 18*b*c^2*d^2*e + 12*b^2*c*d*e^2 + b^3*e^3))*x^9)/9 + (c*e^2*(A*c* e*(4*c*d + 3*b*e) + 3*B*(2*c^2*d^2 + 4*b*c*d*e + b^2*e^2))*x^10)/10 + (c^2 *e^3*(4*B*c*d + 3*b*B*e + A*c*e)*x^11)/11 + (B*c^3*e^4*x^12)/12
3.12.28.3.1 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 = 0.19 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.08
| method | result | size |
| default | \(\frac {B \,c^{3} e^{4} x^{12}}{12}+\frac {\left (\left (A \,e^{4}+4 B d \,e^{3}\right ) c^{3}+3 B \,e^{4} b \,c^{2}\right ) x^{11}}{11}+\frac {\left (\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) c^{3}+3 \left (A \,e^{4}+4 B d \,e^{3}\right ) b \,c^{2}+3 B \,e^{4} b^{2} c \right ) x^{10}}{10}+\frac {\left (\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) c^{3}+3 \left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) b \,c^{2}+3 \left (A \,e^{4}+4 B d \,e^{3}\right ) b^{2} c +B \,e^{4} b^{3}\right ) x^{9}}{9}+\frac {\left (\left (4 A \,d^{3} e +B \,d^{4}\right ) c^{3}+3 \left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) b \,c^{2}+3 \left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) b^{2} c +\left (A \,e^{4}+4 B d \,e^{3}\right ) b^{3}\right ) x^{8}}{8}+\frac {\left (A \,d^{4} c^{3}+3 \left (4 A \,d^{3} e +B \,d^{4}\right ) b \,c^{2}+3 \left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) b^{2} c +\left (4 A d \,e^{3}+6 B \,d^{2} e^{2}\right ) b^{3}\right ) x^{7}}{7}+\frac {\left (3 A \,d^{4} b \,c^{2}+3 \left (4 A \,d^{3} e +B \,d^{4}\right ) b^{2} c +\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) b^{3}\right ) x^{6}}{6}+\frac {\left (3 A \,d^{4} b^{2} c +\left (4 A \,d^{3} e +B \,d^{4}\right ) b^{3}\right ) x^{5}}{5}+\frac {A \,b^{3} d^{4} x^{4}}{4}\) | \(444\) |
| norman | \(\frac {B \,c^{3} e^{4} x^{12}}{12}+\left (\frac {1}{11} A \,c^{3} e^{4}+\frac {3}{11} B \,e^{4} b \,c^{2}+\frac {4}{11} B \,c^{3} d \,e^{3}\right ) x^{11}+\left (\frac {3}{10} A b \,c^{2} e^{4}+\frac {2}{5} A \,c^{3} d \,e^{3}+\frac {3}{10} B \,e^{4} b^{2} c +\frac {6}{5} B b \,c^{2} d \,e^{3}+\frac {3}{5} B \,c^{3} d^{2} e^{2}\right ) x^{10}+\left (\frac {1}{3} A \,b^{2} c \,e^{4}+\frac {4}{3} A b \,c^{2} d \,e^{3}+\frac {2}{3} A \,c^{3} d^{2} e^{2}+\frac {1}{9} B \,e^{4} b^{3}+\frac {4}{3} B \,b^{2} c d \,e^{3}+2 B b \,c^{2} d^{2} e^{2}+\frac {4}{9} B \,c^{3} d^{3} e \right ) x^{9}+\left (\frac {1}{8} A \,b^{3} e^{4}+\frac {3}{2} A \,b^{2} c d \,e^{3}+\frac {9}{4} A b \,c^{2} d^{2} e^{2}+\frac {1}{2} A \,c^{3} d^{3} e +\frac {1}{2} B \,b^{3} d \,e^{3}+\frac {9}{4} B \,b^{2} c \,d^{2} e^{2}+\frac {3}{2} B b \,c^{2} d^{3} e +\frac {1}{8} B \,c^{3} d^{4}\right ) x^{8}+\left (\frac {4}{7} A \,b^{3} d \,e^{3}+\frac {18}{7} A \,b^{2} c \,d^{2} e^{2}+\frac {12}{7} A b \,c^{2} d^{3} e +\frac {1}{7} A \,d^{4} c^{3}+\frac {6}{7} B \,b^{3} d^{2} e^{2}+\frac {12}{7} B \,b^{2} c \,d^{3} e +\frac {3}{7} B b \,c^{2} d^{4}\right ) x^{7}+\left (A \,b^{3} d^{2} e^{2}+2 A \,b^{2} c \,d^{3} e +\frac {1}{2} A \,d^{4} b \,c^{2}+\frac {2}{3} B \,b^{3} d^{3} e +\frac {1}{2} B \,b^{2} c \,d^{4}\right ) x^{6}+\left (\frac {4}{5} A \,b^{3} d^{3} e +\frac {3}{5} A \,d^{4} b^{2} c +\frac {1}{5} B \,b^{3} d^{4}\right ) x^{5}+\frac {A \,b^{3} d^{4} x^{4}}{4}\) | \(462\) |
| gosper | \(\frac {x^{4} \left (2310 B \,c^{3} e^{4} x^{8}+2520 x^{7} A \,c^{3} e^{4}+7560 x^{7} B \,e^{4} b \,c^{2}+10080 x^{7} B \,c^{3} d \,e^{3}+8316 x^{6} A b \,c^{2} e^{4}+11088 x^{6} A \,c^{3} d \,e^{3}+8316 x^{6} B \,e^{4} b^{2} c +33264 x^{6} B b \,c^{2} d \,e^{3}+16632 x^{6} B \,c^{3} d^{2} e^{2}+9240 x^{5} A \,b^{2} c \,e^{4}+36960 x^{5} A b \,c^{2} d \,e^{3}+18480 x^{5} A \,c^{3} d^{2} e^{2}+3080 x^{5} B \,e^{4} b^{3}+36960 x^{5} B \,b^{2} c d \,e^{3}+55440 x^{5} B b \,c^{2} d^{2} e^{2}+12320 x^{5} B \,c^{3} d^{3} e +3465 x^{4} A \,b^{3} e^{4}+41580 x^{4} A \,b^{2} c d \,e^{3}+62370 x^{4} A b \,c^{2} d^{2} e^{2}+13860 x^{4} A \,c^{3} d^{3} e +13860 x^{4} B \,b^{3} d \,e^{3}+62370 x^{4} B \,b^{2} c \,d^{2} e^{2}+41580 x^{4} B b \,c^{2} d^{3} e +3465 x^{4} B \,c^{3} d^{4}+15840 x^{3} A \,b^{3} d \,e^{3}+71280 x^{3} A \,b^{2} c \,d^{2} e^{2}+47520 x^{3} A b \,c^{2} d^{3} e +3960 x^{3} A \,d^{4} c^{3}+23760 x^{3} B \,b^{3} d^{2} e^{2}+47520 x^{3} B \,b^{2} c \,d^{3} e +11880 x^{3} B b \,c^{2} d^{4}+27720 x^{2} A \,b^{3} d^{2} e^{2}+55440 x^{2} A \,b^{2} c \,d^{3} e +13860 x^{2} A \,d^{4} b \,c^{2}+18480 x^{2} B \,b^{3} d^{3} e +13860 x^{2} B \,b^{2} c \,d^{4}+22176 x A \,b^{3} d^{3} e +16632 x A \,d^{4} b^{2} c +5544 x B \,b^{3} d^{4}+6930 A \,d^{4} b^{3}\right )}{27720}\) | \(538\) |
| risch | \(\frac {1}{12} B \,c^{3} e^{4} x^{12}+\frac {1}{4} A \,b^{3} d^{4} x^{4}+\frac {3}{2} x^{8} A \,b^{2} c d \,e^{3}+\frac {9}{4} x^{8} A b \,c^{2} d^{2} e^{2}+\frac {9}{4} x^{8} B \,b^{2} c \,d^{2} e^{2}+\frac {3}{2} x^{8} B b \,c^{2} d^{3} e +\frac {18}{7} x^{7} A \,b^{2} c \,d^{2} e^{2}+\frac {12}{7} x^{7} A b \,c^{2} d^{3} e +\frac {12}{7} x^{7} B \,b^{2} c \,d^{3} e +2 x^{6} A \,b^{2} c \,d^{3} e +\frac {4}{3} x^{9} B \,b^{2} c d \,e^{3}+2 x^{9} B b \,c^{2} d^{2} e^{2}+\frac {4}{3} x^{9} A b \,c^{2} d \,e^{3}+\frac {6}{5} x^{10} B b \,c^{2} d \,e^{3}+\frac {1}{11} x^{11} A \,c^{3} e^{4}+\frac {1}{9} x^{9} B \,e^{4} b^{3}+\frac {1}{8} x^{8} A \,b^{3} e^{4}+\frac {1}{8} x^{8} B \,c^{3} d^{4}+\frac {1}{7} x^{7} A \,d^{4} c^{3}+\frac {1}{5} x^{5} B \,b^{3} d^{4}+\frac {2}{3} x^{9} A \,c^{3} d^{2} e^{2}+\frac {4}{9} x^{9} B \,c^{3} d^{3} e +\frac {1}{2} x^{8} A \,c^{3} d^{3} e +\frac {1}{2} x^{8} B \,b^{3} d \,e^{3}+\frac {4}{7} x^{7} A \,b^{3} d \,e^{3}+\frac {6}{7} x^{7} B \,b^{3} d^{2} e^{2}+\frac {3}{7} x^{7} B b \,c^{2} d^{4}+x^{6} A \,b^{3} d^{2} e^{2}+\frac {1}{2} x^{6} A \,d^{4} b \,c^{2}+\frac {2}{3} x^{6} B \,b^{3} d^{3} e +\frac {1}{2} x^{6} B \,b^{2} c \,d^{4}+\frac {4}{5} x^{5} A \,b^{3} d^{3} e +\frac {3}{5} x^{5} A \,d^{4} b^{2} c +\frac {4}{11} x^{11} B \,c^{3} d \,e^{3}+\frac {3}{10} x^{10} A b \,c^{2} e^{4}+\frac {2}{5} x^{10} A \,c^{3} d \,e^{3}+\frac {3}{10} x^{10} B \,e^{4} b^{2} c +\frac {3}{5} x^{10} B \,c^{3} d^{2} e^{2}+\frac {1}{3} x^{9} A \,b^{2} c \,e^{4}+\frac {3}{11} x^{11} B \,e^{4} b \,c^{2}\) | \(541\) |
| parallelrisch | \(\frac {1}{12} B \,c^{3} e^{4} x^{12}+\frac {1}{4} A \,b^{3} d^{4} x^{4}+\frac {3}{2} x^{8} A \,b^{2} c d \,e^{3}+\frac {9}{4} x^{8} A b \,c^{2} d^{2} e^{2}+\frac {9}{4} x^{8} B \,b^{2} c \,d^{2} e^{2}+\frac {3}{2} x^{8} B b \,c^{2} d^{3} e +\frac {18}{7} x^{7} A \,b^{2} c \,d^{2} e^{2}+\frac {12}{7} x^{7} A b \,c^{2} d^{3} e +\frac {12}{7} x^{7} B \,b^{2} c \,d^{3} e +2 x^{6} A \,b^{2} c \,d^{3} e +\frac {4}{3} x^{9} B \,b^{2} c d \,e^{3}+2 x^{9} B b \,c^{2} d^{2} e^{2}+\frac {4}{3} x^{9} A b \,c^{2} d \,e^{3}+\frac {6}{5} x^{10} B b \,c^{2} d \,e^{3}+\frac {1}{11} x^{11} A \,c^{3} e^{4}+\frac {1}{9} x^{9} B \,e^{4} b^{3}+\frac {1}{8} x^{8} A \,b^{3} e^{4}+\frac {1}{8} x^{8} B \,c^{3} d^{4}+\frac {1}{7} x^{7} A \,d^{4} c^{3}+\frac {1}{5} x^{5} B \,b^{3} d^{4}+\frac {2}{3} x^{9} A \,c^{3} d^{2} e^{2}+\frac {4}{9} x^{9} B \,c^{3} d^{3} e +\frac {1}{2} x^{8} A \,c^{3} d^{3} e +\frac {1}{2} x^{8} B \,b^{3} d \,e^{3}+\frac {4}{7} x^{7} A \,b^{3} d \,e^{3}+\frac {6}{7} x^{7} B \,b^{3} d^{2} e^{2}+\frac {3}{7} x^{7} B b \,c^{2} d^{4}+x^{6} A \,b^{3} d^{2} e^{2}+\frac {1}{2} x^{6} A \,d^{4} b \,c^{2}+\frac {2}{3} x^{6} B \,b^{3} d^{3} e +\frac {1}{2} x^{6} B \,b^{2} c \,d^{4}+\frac {4}{5} x^{5} A \,b^{3} d^{3} e +\frac {3}{5} x^{5} A \,d^{4} b^{2} c +\frac {4}{11} x^{11} B \,c^{3} d \,e^{3}+\frac {3}{10} x^{10} A b \,c^{2} e^{4}+\frac {2}{5} x^{10} A \,c^{3} d \,e^{3}+\frac {3}{10} x^{10} B \,e^{4} b^{2} c +\frac {3}{5} x^{10} B \,c^{3} d^{2} e^{2}+\frac {1}{3} x^{9} A \,b^{2} c \,e^{4}+\frac {3}{11} x^{11} B \,e^{4} b \,c^{2}\) | \(541\) |
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+3*B*e^4*b^2*c)*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+3*(A*e^4+ 4*B*d*e^3)*b^2*c+B*e^4*b^3)*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+3*(4*A*d*e^3+6*B*d^2*e^2)*b^2*c+(A*e^4+4*B*d*e^3)*b^3)*x^ 8+1/7*(A*d^4*c^3+3*(4*A*d^3*e+B*d^4)*b*c^2+3*(6*A*d^2*e^2+4*B*d^3*e)*b^2*c +(4*A*d*e^3+6*B*d^2*e^2)*b^3)*x^7+1/6*(3*A*d^4*b*c^2+3*(4*A*d^3*e+B*d^4)*b ^2*c+(6*A*d^2*e^2+4*B*d^3*e)*b^3)*x^6+1/5*(3*A*d^4*b^2*c+(4*A*d^3*e+B*d^4) *b^3)*x^5+1/4*A*b^3*d^4*x^4
Time = 0.26 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.04 \[ \int (A+B x) (d+e x)^4 \left (b x+c x^2\right )^3 \, dx=\frac {1}{12} \, B c^{3} e^{4} x^{12} + \frac {1}{4} \, A b^{3} d^{4} x^{4} + \frac {1}{11} \, {\left (4 \, B c^{3} d e^{3} + {\left (3 \, B b c^{2} + A c^{3}\right )} e^{4}\right )} x^{11} + \frac {1}{10} \, {\left (6 \, B c^{3} d^{2} e^{2} + 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{3} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} e^{4}\right )} x^{10} + \frac {1}{9} \, {\left (4 \, B c^{3} d^{3} e + 6 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{2} + 12 \, {\left (B b^{2} c + A b c^{2}\right )} d e^{3} + {\left (B b^{3} + 3 \, A b^{2} c\right )} e^{4}\right )} x^{9} + \frac {1}{8} \, {\left (B c^{3} d^{4} + A b^{3} e^{4} + 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e + 18 \, {\left (B b^{2} c + A b c^{2}\right )} d^{2} e^{2} + 4 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (4 \, A b^{3} d e^{3} + {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} + 12 \, {\left (B b^{2} c + A b c^{2}\right )} d^{3} e + 6 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (6 \, A b^{3} d^{2} e^{2} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{4} + 4 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e\right )} x^{6} + \frac {1}{5} \, {\left (4 \, A b^{3} d^{3} e + {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4}\right )} x^{5} \]
1/12*B*c^3*e^4*x^12 + 1/4*A*b^3*d^4*x^4 + 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 + 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 + A*b*c^2)*d*e^3 + (B*b^3 + 3*A*b^2*c)*e^4)*x^9 + 1/8*(B*c^3*d^4 + A*b^3*e^4 + 4*(3*B*b*c^2 + A*c^3)*d^3*e + 18*(B*b^2*c + A*b*c^2)*d^2*e^2 + 4*(B*b^3 + 3*A*b^2*c)*d*e^3)*x^8 + 1/7*(4*A*b^3*d*e^3 + (3*B*b*c^2 + A*c^3)*d^4 + 12*(B*b^2*c + A*b*c^2)*d^3*e + 6*(B*b^3 + 3*A *b^2*c)*d^2*e^2)*x^7 + 1/6*(6*A*b^3*d^2*e^2 + 3*(B*b^2*c + A*b*c^2)*d^4 + 4*(B*b^3 + 3*A*b^2*c)*d^3*e)*x^6 + 1/5*(4*A*b^3*d^3*e + (B*b^3 + 3*A*b^2*c )*d^4)*x^5
Time = 0.05 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.37 \[ \int (A+B x) (d+e x)^4 \left (b x+c x^2\right )^3 \, dx=\frac {A b^{3} d^{4} x^{4}}{4} + \frac {B c^{3} e^{4} x^{12}}{12} + x^{11} \left (\frac {A c^{3} e^{4}}{11} + \frac {3 B b c^{2} e^{4}}{11} + \frac {4 B c^{3} d e^{3}}{11}\right ) + x^{10} \cdot \left (\frac {3 A b c^{2} e^{4}}{10} + \frac {2 A c^{3} d e^{3}}{5} + \frac {3 B b^{2} c e^{4}}{10} + \frac {6 B b c^{2} d e^{3}}{5} + \frac {3 B c^{3} d^{2} e^{2}}{5}\right ) + x^{9} \left (\frac {A b^{2} c e^{4}}{3} + \frac {4 A b c^{2} d e^{3}}{3} + \frac {2 A c^{3} d^{2} e^{2}}{3} + \frac {B b^{3} e^{4}}{9} + \frac {4 B b^{2} c d e^{3}}{3} + 2 B b c^{2} d^{2} e^{2} + \frac {4 B c^{3} d^{3} e}{9}\right ) + x^{8} \left (\frac {A b^{3} e^{4}}{8} + \frac {3 A b^{2} c d e^{3}}{2} + \frac {9 A b c^{2} d^{2} e^{2}}{4} + \frac {A c^{3} d^{3} e}{2} + \frac {B b^{3} d e^{3}}{2} + \frac {9 B b^{2} c d^{2} e^{2}}{4} + \frac {3 B b c^{2} d^{3} e}{2} + \frac {B c^{3} d^{4}}{8}\right ) + x^{7} \cdot \left (\frac {4 A b^{3} d e^{3}}{7} + \frac {18 A b^{2} c d^{2} e^{2}}{7} + \frac {12 A b c^{2} d^{3} e}{7} + \frac {A c^{3} d^{4}}{7} + \frac {6 B b^{3} d^{2} e^{2}}{7} + \frac {12 B b^{2} c d^{3} e}{7} + \frac {3 B b c^{2} d^{4}}{7}\right ) + x^{6} \left (A b^{3} d^{2} e^{2} + 2 A b^{2} c d^{3} e + \frac {A b c^{2} d^{4}}{2} + \frac {2 B b^{3} d^{3} e}{3} + \frac {B b^{2} c d^{4}}{2}\right ) + x^{5} \cdot \left (\frac {4 A b^{3} d^{3} e}{5} + \frac {3 A b^{2} c d^{4}}{5} + \frac {B b^{3} d^{4}}{5}\right ) \]
A*b**3*d**4*x**4/4 + 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*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*b**2*c*e**4/3 + 4*A*b*c**2*d*e**3/3 + 2*A*c**3*d**2*e**2/3 + 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*(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 + 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*(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 + 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*b**3*d**2*e**2 + 2*A*b**2*c* d**3*e + A*b*c**2*d**4/2 + 2*B*b**3*d**3*e/3 + B*b**2*c*d**4/2) + x**5*(4* A*b**3*d**3*e/5 + 3*A*b**2*c*d**4/5 + B*b**3*d**4/5)
Time = 0.21 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.04 \[ \int (A+B x) (d+e x)^4 \left (b x+c x^2\right )^3 \, dx=\frac {1}{12} \, B c^{3} e^{4} x^{12} + \frac {1}{4} \, A b^{3} d^{4} x^{4} + \frac {1}{11} \, {\left (4 \, B c^{3} d e^{3} + {\left (3 \, B b c^{2} + A c^{3}\right )} e^{4}\right )} x^{11} + \frac {1}{10} \, {\left (6 \, B c^{3} d^{2} e^{2} + 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{3} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} e^{4}\right )} x^{10} + \frac {1}{9} \, {\left (4 \, B c^{3} d^{3} e + 6 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{2} + 12 \, {\left (B b^{2} c + A b c^{2}\right )} d e^{3} + {\left (B b^{3} + 3 \, A b^{2} c\right )} e^{4}\right )} x^{9} + \frac {1}{8} \, {\left (B c^{3} d^{4} + A b^{3} e^{4} + 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e + 18 \, {\left (B b^{2} c + A b c^{2}\right )} d^{2} e^{2} + 4 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (4 \, A b^{3} d e^{3} + {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} + 12 \, {\left (B b^{2} c + A b c^{2}\right )} d^{3} e + 6 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (6 \, A b^{3} d^{2} e^{2} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{4} + 4 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e\right )} x^{6} + \frac {1}{5} \, {\left (4 \, A b^{3} d^{3} e + {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4}\right )} x^{5} \]
1/12*B*c^3*e^4*x^12 + 1/4*A*b^3*d^4*x^4 + 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 + 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 + A*b*c^2)*d*e^3 + (B*b^3 + 3*A*b^2*c)*e^4)*x^9 + 1/8*(B*c^3*d^4 + A*b^3*e^4 + 4*(3*B*b*c^2 + A*c^3)*d^3*e + 18*(B*b^2*c + A*b*c^2)*d^2*e^2 + 4*(B*b^3 + 3*A*b^2*c)*d*e^3)*x^8 + 1/7*(4*A*b^3*d*e^3 + (3*B*b*c^2 + A*c^3)*d^4 + 12*(B*b^2*c + A*b*c^2)*d^3*e + 6*(B*b^3 + 3*A *b^2*c)*d^2*e^2)*x^7 + 1/6*(6*A*b^3*d^2*e^2 + 3*(B*b^2*c + A*b*c^2)*d^4 + 4*(B*b^3 + 3*A*b^2*c)*d^3*e)*x^6 + 1/5*(4*A*b^3*d^3*e + (B*b^3 + 3*A*b^2*c )*d^4)*x^5
Time = 0.27 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.31 \[ \int (A+B x) (d+e x)^4 \left (b x+c x^2\right )^3 \, dx=\frac {1}{12} \, B c^{3} e^{4} x^{12} + \frac {4}{11} \, B c^{3} d e^{3} x^{11} + \frac {3}{11} \, B b c^{2} e^{4} x^{11} + \frac {1}{11} \, A c^{3} e^{4} x^{11} + \frac {3}{5} \, B c^{3} d^{2} e^{2} x^{10} + \frac {6}{5} \, B b c^{2} d e^{3} x^{10} + \frac {2}{5} \, A c^{3} d e^{3} x^{10} + \frac {3}{10} \, B b^{2} c e^{4} x^{10} + \frac {3}{10} \, A b c^{2} e^{4} x^{10} + \frac {4}{9} \, B c^{3} d^{3} e x^{9} + 2 \, B b c^{2} d^{2} e^{2} x^{9} + \frac {2}{3} \, A c^{3} d^{2} e^{2} x^{9} + \frac {4}{3} \, B b^{2} c d e^{3} x^{9} + \frac {4}{3} \, A b c^{2} d e^{3} x^{9} + \frac {1}{9} \, B b^{3} e^{4} x^{9} + \frac {1}{3} \, A b^{2} c e^{4} x^{9} + \frac {1}{8} \, B c^{3} d^{4} x^{8} + \frac {3}{2} \, B b c^{2} d^{3} e x^{8} + \frac {1}{2} \, A c^{3} d^{3} e x^{8} + \frac {9}{4} \, B b^{2} c d^{2} e^{2} x^{8} + \frac {9}{4} \, A b c^{2} d^{2} e^{2} x^{8} + \frac {1}{2} \, B b^{3} d e^{3} x^{8} + \frac {3}{2} \, A b^{2} c d e^{3} x^{8} + \frac {1}{8} \, A b^{3} e^{4} x^{8} + \frac {3}{7} \, B b c^{2} d^{4} x^{7} + \frac {1}{7} \, A c^{3} d^{4} x^{7} + \frac {12}{7} \, B b^{2} c d^{3} e x^{7} + \frac {12}{7} \, A b c^{2} d^{3} e x^{7} + \frac {6}{7} \, B b^{3} d^{2} e^{2} x^{7} + \frac {18}{7} \, A b^{2} c d^{2} e^{2} x^{7} + \frac {4}{7} \, A b^{3} d e^{3} x^{7} + \frac {1}{2} \, B b^{2} c d^{4} x^{6} + \frac {1}{2} \, A b c^{2} d^{4} x^{6} + \frac {2}{3} \, B b^{3} d^{3} e x^{6} + 2 \, A b^{2} c d^{3} e x^{6} + A b^{3} d^{2} e^{2} x^{6} + \frac {1}{5} \, B b^{3} d^{4} x^{5} + \frac {3}{5} \, A b^{2} c d^{4} x^{5} + \frac {4}{5} \, A b^{3} d^{3} e x^{5} + \frac {1}{4} \, A b^{3} d^{4} x^{4} \]
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*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*A*b*c^2*d*e^3*x^9 + 1/9*B*b^3*e^4*x^9 + 1/3*A*b^2*c*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*A*b*c^2*d^2*e^2*x^8 + 1/2*B*b^3*d*e^3*x^8 + 3/2*A *b^2*c*d*e^3*x^8 + 1/8*A*b^3*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*A*b*c^2*d^3*e*x^7 + 6/7*B*b^3*d^2*e^2 *x^7 + 18/7*A*b^2*c*d^2*e^2*x^7 + 4/7*A*b^3*d*e^3*x^7 + 1/2*B*b^2*c*d^4*x^ 6 + 1/2*A*b*c^2*d^4*x^6 + 2/3*B*b^3*d^3*e*x^6 + 2*A*b^2*c*d^3*e*x^6 + A*b^ 3*d^2*e^2*x^6 + 1/5*B*b^3*d^4*x^5 + 3/5*A*b^2*c*d^4*x^5 + 4/5*A*b^3*d^3*e* x^5 + 1/4*A*b^3*d^4*x^4
Time = 0.17 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.08 \[ \int (A+B x) (d+e x)^4 \left (b x+c x^2\right )^3 \, dx=x^6\,\left (\frac {2\,B\,b^3\,d^3\,e}{3}+A\,b^3\,d^2\,e^2+\frac {B\,b^2\,c\,d^4}{2}+2\,A\,b^2\,c\,d^3\,e+\frac {A\,b\,c^2\,d^4}{2}\right )+x^{10}\,\left (\frac {3\,B\,b^2\,c\,e^4}{10}+\frac {6\,B\,b\,c^2\,d\,e^3}{5}+\frac {3\,A\,b\,c^2\,e^4}{10}+\frac {3\,B\,c^3\,d^2\,e^2}{5}+\frac {2\,A\,c^3\,d\,e^3}{5}\right )+x^8\,\left (\frac {B\,b^3\,d\,e^3}{2}+\frac {A\,b^3\,e^4}{8}+\frac {9\,B\,b^2\,c\,d^2\,e^2}{4}+\frac {3\,A\,b^2\,c\,d\,e^3}{2}+\frac {3\,B\,b\,c^2\,d^3\,e}{2}+\frac {9\,A\,b\,c^2\,d^2\,e^2}{4}+\frac {B\,c^3\,d^4}{8}+\frac {A\,c^3\,d^3\,e}{2}\right )+x^7\,\left (\frac {6\,B\,b^3\,d^2\,e^2}{7}+\frac {4\,A\,b^3\,d\,e^3}{7}+\frac {12\,B\,b^2\,c\,d^3\,e}{7}+\frac {18\,A\,b^2\,c\,d^2\,e^2}{7}+\frac {3\,B\,b\,c^2\,d^4}{7}+\frac {12\,A\,b\,c^2\,d^3\,e}{7}+\frac {A\,c^3\,d^4}{7}\right )+x^9\,\left (\frac {B\,b^3\,e^4}{9}+\frac {4\,B\,b^2\,c\,d\,e^3}{3}+\frac {A\,b^2\,c\,e^4}{3}+2\,B\,b\,c^2\,d^2\,e^2+\frac {4\,A\,b\,c^2\,d\,e^3}{3}+\frac {4\,B\,c^3\,d^3\,e}{9}+\frac {2\,A\,c^3\,d^2\,e^2}{3}\right )+\frac {b^2\,d^3\,x^5\,\left (4\,A\,b\,e+3\,A\,c\,d+B\,b\,d\right )}{5}+\frac {c^2\,e^3\,x^{11}\,\left (A\,c\,e+3\,B\,b\,e+4\,B\,c\,d\right )}{11}+\frac {A\,b^3\,d^4\,x^4}{4}+\frac {B\,c^3\,e^4\,x^{12}}{12} \]
x^6*((A*b*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 + 2*A*b^2*c*d^3*e) + x^10*((3*A*b*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^8*((A*b^3*e ^4)/8 + (B*c^3*d^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*b^2*c*d^2*e^2)/4 + (3*A*b^2*c*d*e^3)/2 + (3*B*b*c^2*d^3*e)/ 2) + x^7*((A*c^3*d^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*b^2*c*d^2*e^2)/7 + (12*A*b*c^2*d^3*e)/7 + (12*B*b^2*c* d^3*e)/7) + x^9*((B*b^3*e^4)/9 + (A*b^2*c*e^4)/3 + (4*B*c^3*d^3*e)/9 + (2* A*c^3*d^2*e^2)/3 + 2*B*b*c^2*d^2*e^2 + (4*A*b*c^2*d*e^3)/3 + (4*B*b^2*c*d* e^3)/3) + (b^2*d^3*x^5*(4*A*b*e + 3*A*c*d + B*b*d))/5 + (c^2*e^3*x^11*(A*c *e + 3*B*b*e + 4*B*c*d))/11 + (A*b^3*d^4*x^4)/4 + (B*c^3*e^4*x^12)/12