Integrand size = 26, antiderivative size = 411 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}{5 e^8}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^6}{6 e^8}-\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^7}{7 e^8}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^8}{8 e^8}-\frac {5 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^9}{9 e^8}+\frac {3 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{10}}{10 e^8}-\frac {7 c^3 (2 c d-b e) (d+e x)^{11}}{11 e^8}+\frac {c^4 (d+e x)^{12}}{6 e^8} \]
-1/5*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^5/e^8+1/6*(a*e^2-b*d*e+c*d ^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))*(e*x+d)^6/e^8-3/7*(-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^8+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^8-5/9*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^8+3/10*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^8-7/11*c^3*(-b*e+2*c *d)*(e*x+d)^11/e^8+1/6*c^4*(e*x+d)^12/e^8
Time = 0.25 (sec) , antiderivative size = 735, normalized size of antiderivative = 1.79 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=a^3 b d^4 x+\frac {1}{2} a^2 d^3 \left (3 b^2 d+2 a c d+4 a b e\right ) x^2+\frac {1}{3} a d^2 \left (3 b^3 d^2+12 a b^2 d e+8 a^2 c d e+3 a b \left (3 c d^2+2 a e^2\right )\right ) x^3+\frac {1}{4} d \left (b^4 d^3+12 a b^3 d^2 e+4 a^2 b e \left (9 c d^2+a e^2\right )+6 a^2 c d \left (c d^2+2 a e^2\right )+6 a b^2 d \left (2 c d^2+3 a e^2\right )\right ) x^4+\frac {1}{5} \left (4 b^4 d^3 e+8 a^2 c d e \left (3 c d^2+a e^2\right )+12 a b^2 d e \left (4 c d^2+a e^2\right )+b^3 \left (5 c d^4+18 a d^2 e^2\right )+a b \left (15 c^2 d^4+54 a c d^2 e^2+a^2 e^4\right )\right ) x^5+\frac {1}{6} \left (6 b^4 d^2 e^2+12 a b c d e \left (5 c d^2+3 a e^2\right )+4 b^3 \left (5 c d^3 e+3 a d e^3\right )+2 a c \left (3 c^2 d^4+18 a c d^2 e^2+a^2 e^4\right )+3 b^2 \left (3 c^2 d^4+24 a c d^2 e^2+a^2 e^4\right )\right ) x^6+\frac {1}{7} \left (4 b^4 d e^3+24 a c^2 d e \left (c d^2+a e^2\right )+12 b^2 c d e \left (3 c d^2+4 a e^2\right )+3 b^3 \left (10 c d^2 e^2+a e^4\right )+b c \left (7 c^2 d^4+90 a c d^2 e^2+9 a^2 e^4\right )\right ) x^7+\frac {1}{8} \left (2 c^4 d^4+b^4 e^4+4 b^2 c e^3 (5 b d+3 a e)+4 c^3 d^2 e (7 b d+9 a e)+6 c^2 e^2 \left (9 b^2 d^2+10 a b d e+a^2 e^2\right )\right ) x^8+\frac {1}{9} c e \left (8 c^3 d^3+5 b^3 e^3+6 c^2 d e (7 b d+4 a e)+3 b c e^2 (12 b d+5 a e)\right ) x^9+\frac {1}{10} c^2 e^2 \left (12 c^2 d^2+9 b^2 e^2+2 c e (14 b d+3 a e)\right ) x^{10}+\frac {1}{11} c^3 e^3 (8 c d+7 b e) x^{11}+\frac {1}{6} c^4 e^4 x^{12} \]
a^3*b*d^4*x + (a^2*d^3*(3*b^2*d + 2*a*c*d + 4*a*b*e)*x^2)/2 + (a*d^2*(3*b^ 3*d^2 + 12*a*b^2*d*e + 8*a^2*c*d*e + 3*a*b*(3*c*d^2 + 2*a*e^2))*x^3)/3 + ( d*(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^4)/4 + ((4*b^4*d^3*e + 8* a^2*c*d*e*(3*c*d^2 + a*e^2) + 12*a*b^2*d*e*(4*c*d^2 + a*e^2) + b^3*(5*c*d^ 4 + 18*a*d^2*e^2) + a*b*(15*c^2*d^4 + 54*a*c*d^2*e^2 + a^2*e^4))*x^5)/5 + ((6*b^4*d^2*e^2 + 12*a*b*c*d*e*(5*c*d^2 + 3*a*e^2) + 4*b^3*(5*c*d^3*e + 3* a*d*e^3) + 2*a*c*(3*c^2*d^4 + 18*a*c*d^2*e^2 + a^2*e^4) + 3*b^2*(3*c^2*d^4 + 24*a*c*d^2*e^2 + a^2*e^4))*x^6)/6 + ((4*b^4*d*e^3 + 24*a*c^2*d*e*(c*d^2 + a*e^2) + 12*b^2*c*d*e*(3*c*d^2 + 4*a*e^2) + 3*b^3*(10*c*d^2*e^2 + a*e^4 ) + b*c*(7*c^2*d^4 + 90*a*c*d^2*e^2 + 9*a^2*e^4))*x^7)/7 + ((2*c^4*d^4 + b ^4*e^4 + 4*b^2*c*e^3*(5*b*d + 3*a*e) + 4*c^3*d^2*e*(7*b*d + 9*a*e) + 6*c^2 *e^2*(9*b^2*d^2 + 10*a*b*d*e + a^2*e^2))*x^8)/8 + (c*e*(8*c^3*d^3 + 5*b^3* e^3 + 6*c^2*d*e*(7*b*d + 4*a*e) + 3*b*c*e^2*(12*b*d + 5*a*e))*x^9)/9 + (c^ 2*e^2*(12*c^2*d^2 + 9*b^2*e^2 + 2*c*e*(14*b*d + 3*a*e))*x^10)/10 + (c^3*e^ 3*(8*c*d + 7*b*e)*x^11)/11 + (c^4*e^4*x^12)/6
Time = 1.03 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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.
| \(\displaystyle \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {(d+e x)^7 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^7}+\frac {3 c^2 (d+e x)^9 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^7}+\frac {5 c (d+e x)^8 (2 c d-b e) \left (c e (7 b d-3 a e)-b^2 e^2-7 c^2 d^2\right )}{e^7}+\frac {3 (d+e x)^6 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-3 a c e^2-b^2 e^2+7 b c d e-7 c^2 d^2\right )}{e^7}+\frac {(d+e x)^5 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^7}+\frac {(d+e x)^4 (b e-2 c d) \left (a e^2-b d e+c d^2\right )^3}{e^7}-\frac {7 c^3 (d+e x)^{10} (2 c d-b e)}{e^7}+\frac {2 c^4 (d+e x)^{11}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^8 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{8 e^8}+\frac {3 c^2 (d+e x)^{10} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{10 e^8}-\frac {5 c (d+e x)^9 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{9 e^8}-\frac {3 (d+e x)^7 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^8}+\frac {(d+e x)^6 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{6 e^8}-\frac {(d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{5 e^8}-\frac {7 c^3 (d+e x)^{11} (2 c d-b e)}{11 e^8}+\frac {c^4 (d+e x)^{12}}{6 e^8}\) |
-1/5*((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^5)/e^8 + ((c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x) ^6)/(6*e^8) - (3*(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^8) + ((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^8) - (5*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^8) + (3*c^2 *(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^10)/(10*e^8) - ( 7*c^3*(2*c*d - b*e)*(d + e*x)^11)/(11*e^8) + (c^4*(d + e*x)^12)/(6*e^8)
3.16.14.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 0.38 (sec) , antiderivative size = 783, normalized size of antiderivative = 1.91
| method | result | size |
| norman | \(\frac {c^{4} e^{4} x^{12}}{6}+\left (\frac {7}{11} c^{3} e^{4} b +\frac {8}{11} c^{4} d \,e^{3}\right ) x^{11}+\left (\frac {3}{5} a \,c^{3} e^{4}+\frac {9}{10} b^{2} c^{2} e^{4}+\frac {14}{5} b \,c^{3} d \,e^{3}+\frac {6}{5} c^{4} d^{2} e^{2}\right ) x^{10}+\left (\frac {5}{3} a b \,c^{2} e^{4}+\frac {8}{3} a \,c^{3} d \,e^{3}+\frac {5}{9} b^{3} c \,e^{4}+4 b^{2} c^{2} d \,e^{3}+\frac {14}{3} b \,c^{3} d^{2} e^{2}+\frac {8}{9} c^{4} d^{3} e \right ) x^{9}+\left (\frac {3}{4} a^{2} c^{2} e^{4}+\frac {3}{2} a \,b^{2} c \,e^{4}+\frac {15}{2} a b \,c^{2} d \,e^{3}+\frac {9}{2} a \,c^{3} d^{2} e^{2}+\frac {1}{8} b^{4} e^{4}+\frac {5}{2} b^{3} c d \,e^{3}+\frac {27}{4} b^{2} c^{2} d^{2} e^{2}+\frac {7}{2} b \,c^{3} d^{3} e +\frac {1}{4} c^{4} d^{4}\right ) x^{8}+\left (\frac {9}{7} c \,e^{4} b \,a^{2}+\frac {24}{7} a^{2} c^{2} d \,e^{3}+\frac {3}{7} a \,b^{3} e^{4}+\frac {48}{7} a \,b^{2} c d \,e^{3}+\frac {90}{7} a b \,c^{2} d^{2} e^{2}+\frac {24}{7} a \,c^{3} d^{3} e +\frac {4}{7} b^{4} d \,e^{3}+\frac {30}{7} b^{3} c \,d^{2} e^{2}+\frac {36}{7} b^{2} c^{2} d^{3} e +b \,d^{4} c^{3}\right ) x^{7}+\left (\frac {1}{3} c \,e^{4} a^{3}+\frac {1}{2} a^{2} b^{2} e^{4}+6 a^{2} b c d \,e^{3}+6 a^{2} c^{2} d^{2} e^{2}+2 a \,b^{3} d \,e^{3}+12 a \,b^{2} c \,d^{2} e^{2}+10 a b \,c^{2} d^{3} e +a \,c^{3} d^{4}+b^{4} d^{2} e^{2}+\frac {10}{3} b^{3} c \,d^{3} e +\frac {3}{2} b^{2} d^{4} c^{2}\right ) x^{6}+\left (\frac {1}{5} a^{3} b \,e^{4}+\frac {8}{5} a^{3} c d \,e^{3}+\frac {12}{5} a^{2} b^{2} d \,e^{3}+\frac {54}{5} a^{2} b c \,d^{2} e^{2}+\frac {24}{5} a^{2} c^{2} d^{3} e +\frac {18}{5} a \,b^{3} d^{2} e^{2}+\frac {48}{5} a \,b^{2} c \,d^{3} e +3 a b \,c^{2} d^{4}+\frac {4}{5} b^{4} d^{3} e +b^{3} c \,d^{4}\right ) x^{5}+\left (a^{3} b d \,e^{3}+3 a^{3} c \,d^{2} e^{2}+\frac {9}{2} a^{2} b^{2} d^{2} e^{2}+9 a^{2} b c \,d^{3} e +\frac {3}{2} a^{2} c^{2} d^{4}+3 a \,b^{3} d^{3} e +3 a \,b^{2} c \,d^{4}+\frac {1}{4} b^{4} d^{4}\right ) x^{4}+\left (2 a^{3} b \,d^{2} e^{2}+\frac {8}{3} a^{3} c \,d^{3} e +4 a^{2} b^{2} d^{3} e +3 a^{2} b c \,d^{4}+a \,b^{3} d^{4}\right ) x^{3}+\left (2 a^{3} b \,d^{3} e +a^{3} c \,d^{4}+\frac {3}{2} b^{2} d^{4} a^{2}\right ) x^{2}+b \,d^{4} a^{3} x\) | \(783\) |
| gosper | \(x^{3} a \,b^{3} d^{4}+x^{2} a^{3} c \,d^{4}+\frac {3}{2} x^{2} b^{2} d^{4} a^{2}+x^{5} b^{3} c \,d^{4}+\frac {3}{2} x^{4} a^{2} c^{2} d^{4}+\frac {4}{5} x^{5} b^{4} d^{3} e +x^{6} a \,c^{3} d^{4}+x^{6} b^{4} d^{2} e^{2}+\frac {3}{2} x^{6} b^{2} d^{4} c^{2}+\frac {1}{5} x^{5} a^{3} b \,e^{4}+\frac {3}{7} x^{7} a \,b^{3} e^{4}+\frac {4}{7} x^{7} b^{4} d \,e^{3}+x^{7} b \,d^{4} c^{3}+\frac {1}{3} x^{6} c \,e^{4} a^{3}+\frac {1}{2} x^{6} a^{2} b^{2} e^{4}+\frac {3}{5} x^{10} a \,c^{3} e^{4}+\frac {9}{10} x^{10} b^{2} c^{2} e^{4}+\frac {6}{5} x^{10} c^{4} d^{2} e^{2}+\frac {5}{9} x^{9} b^{3} c \,e^{4}+\frac {8}{9} x^{9} c^{4} d^{3} e +\frac {3}{4} x^{8} a^{2} c^{2} e^{4}+\frac {7}{11} x^{11} c^{3} e^{4} b +\frac {8}{11} x^{11} c^{4} d \,e^{3}+b \,d^{4} a^{3} x +\frac {48}{5} x^{5} a \,b^{2} c \,d^{3} e +9 x^{4} a^{2} b c \,d^{3} e +12 x^{6} a \,b^{2} c \,d^{2} e^{2}+10 x^{6} a b \,c^{2} d^{3} e +\frac {54}{5} x^{5} a^{2} b c \,d^{2} e^{2}+6 x^{6} a^{2} b c d \,e^{3}+\frac {15}{2} x^{8} a b \,c^{2} d \,e^{3}+\frac {48}{7} x^{7} a \,b^{2} c d \,e^{3}+\frac {90}{7} x^{7} a b \,c^{2} d^{2} e^{2}+\frac {1}{6} c^{4} e^{4} x^{12}+\frac {1}{8} x^{8} b^{4} e^{4}+\frac {1}{4} x^{8} c^{4} d^{4}+\frac {1}{4} x^{4} b^{4} d^{4}+\frac {5}{2} x^{8} b^{3} c d \,e^{3}+\frac {27}{4} x^{8} b^{2} c^{2} d^{2} e^{2}+\frac {7}{2} x^{8} b \,c^{3} d^{3} e +\frac {9}{7} x^{7} c \,e^{4} b \,a^{2}+\frac {24}{7} x^{7} a^{2} c^{2} d \,e^{3}+\frac {24}{7} x^{7} a \,c^{3} d^{3} e +3 x^{4} a \,b^{3} d^{3} e +3 x^{4} a \,b^{2} c \,d^{4}+\frac {18}{5} x^{5} a \,b^{3} d^{2} e^{2}+3 x^{5} a b \,c^{2} d^{4}+x^{4} a^{3} b d \,e^{3}+3 x^{4} a^{3} c \,d^{2} e^{2}+\frac {9}{2} x^{4} a^{2} b^{2} d^{2} e^{2}+\frac {14}{5} x^{10} b \,c^{3} d \,e^{3}+\frac {5}{3} x^{9} a b \,c^{2} e^{4}+\frac {8}{3} x^{9} a \,c^{3} d \,e^{3}+4 x^{9} b^{2} c^{2} d \,e^{3}+\frac {14}{3} x^{9} b \,c^{3} d^{2} e^{2}+\frac {3}{2} x^{8} a \,b^{2} c \,e^{4}+\frac {9}{2} x^{8} a \,c^{3} d^{2} e^{2}+2 x^{2} a^{3} b \,d^{3} e +2 x^{3} a^{3} b \,d^{2} e^{2}+\frac {8}{3} x^{3} a^{3} c \,d^{3} e +4 x^{3} a^{2} b^{2} d^{3} e +3 x^{3} a^{2} b c \,d^{4}+6 x^{6} a^{2} c^{2} d^{2} e^{2}+2 x^{6} a \,b^{3} d \,e^{3}+\frac {10}{3} x^{6} b^{3} c \,d^{3} e +\frac {8}{5} x^{5} a^{3} c d \,e^{3}+\frac {12}{5} x^{5} a^{2} b^{2} d \,e^{3}+\frac {24}{5} x^{5} a^{2} c^{2} d^{3} e +\frac {30}{7} x^{7} b^{3} c \,d^{2} e^{2}+\frac {36}{7} x^{7} b^{2} c^{2} d^{3} e\) | \(937\) |
| risch | \(x^{3} a \,b^{3} d^{4}+x^{2} a^{3} c \,d^{4}+\frac {3}{2} x^{2} b^{2} d^{4} a^{2}+x^{5} b^{3} c \,d^{4}+\frac {3}{2} x^{4} a^{2} c^{2} d^{4}+\frac {4}{5} x^{5} b^{4} d^{3} e +x^{6} a \,c^{3} d^{4}+x^{6} b^{4} d^{2} e^{2}+\frac {3}{2} x^{6} b^{2} d^{4} c^{2}+\frac {1}{5} x^{5} a^{3} b \,e^{4}+\frac {3}{7} x^{7} a \,b^{3} e^{4}+\frac {4}{7} x^{7} b^{4} d \,e^{3}+x^{7} b \,d^{4} c^{3}+\frac {1}{3} x^{6} c \,e^{4} a^{3}+\frac {1}{2} x^{6} a^{2} b^{2} e^{4}+\frac {3}{5} x^{10} a \,c^{3} e^{4}+\frac {9}{10} x^{10} b^{2} c^{2} e^{4}+\frac {6}{5} x^{10} c^{4} d^{2} e^{2}+\frac {5}{9} x^{9} b^{3} c \,e^{4}+\frac {8}{9} x^{9} c^{4} d^{3} e +\frac {3}{4} x^{8} a^{2} c^{2} e^{4}+\frac {7}{11} x^{11} c^{3} e^{4} b +\frac {8}{11} x^{11} c^{4} d \,e^{3}+b \,d^{4} a^{3} x +\frac {48}{5} x^{5} a \,b^{2} c \,d^{3} e +9 x^{4} a^{2} b c \,d^{3} e +12 x^{6} a \,b^{2} c \,d^{2} e^{2}+10 x^{6} a b \,c^{2} d^{3} e +\frac {54}{5} x^{5} a^{2} b c \,d^{2} e^{2}+6 x^{6} a^{2} b c d \,e^{3}+\frac {15}{2} x^{8} a b \,c^{2} d \,e^{3}+\frac {48}{7} x^{7} a \,b^{2} c d \,e^{3}+\frac {90}{7} x^{7} a b \,c^{2} d^{2} e^{2}+\frac {1}{6} c^{4} e^{4} x^{12}+\frac {1}{8} x^{8} b^{4} e^{4}+\frac {1}{4} x^{8} c^{4} d^{4}+\frac {1}{4} x^{4} b^{4} d^{4}+\frac {5}{2} x^{8} b^{3} c d \,e^{3}+\frac {27}{4} x^{8} b^{2} c^{2} d^{2} e^{2}+\frac {7}{2} x^{8} b \,c^{3} d^{3} e +\frac {9}{7} x^{7} c \,e^{4} b \,a^{2}+\frac {24}{7} x^{7} a^{2} c^{2} d \,e^{3}+\frac {24}{7} x^{7} a \,c^{3} d^{3} e +3 x^{4} a \,b^{3} d^{3} e +3 x^{4} a \,b^{2} c \,d^{4}+\frac {18}{5} x^{5} a \,b^{3} d^{2} e^{2}+3 x^{5} a b \,c^{2} d^{4}+x^{4} a^{3} b d \,e^{3}+3 x^{4} a^{3} c \,d^{2} e^{2}+\frac {9}{2} x^{4} a^{2} b^{2} d^{2} e^{2}+\frac {14}{5} x^{10} b \,c^{3} d \,e^{3}+\frac {5}{3} x^{9} a b \,c^{2} e^{4}+\frac {8}{3} x^{9} a \,c^{3} d \,e^{3}+4 x^{9} b^{2} c^{2} d \,e^{3}+\frac {14}{3} x^{9} b \,c^{3} d^{2} e^{2}+\frac {3}{2} x^{8} a \,b^{2} c \,e^{4}+\frac {9}{2} x^{8} a \,c^{3} d^{2} e^{2}+2 x^{2} a^{3} b \,d^{3} e +2 x^{3} a^{3} b \,d^{2} e^{2}+\frac {8}{3} x^{3} a^{3} c \,d^{3} e +4 x^{3} a^{2} b^{2} d^{3} e +3 x^{3} a^{2} b c \,d^{4}+6 x^{6} a^{2} c^{2} d^{2} e^{2}+2 x^{6} a \,b^{3} d \,e^{3}+\frac {10}{3} x^{6} b^{3} c \,d^{3} e +\frac {8}{5} x^{5} a^{3} c d \,e^{3}+\frac {12}{5} x^{5} a^{2} b^{2} d \,e^{3}+\frac {24}{5} x^{5} a^{2} c^{2} d^{3} e +\frac {30}{7} x^{7} b^{3} c \,d^{2} e^{2}+\frac {36}{7} x^{7} b^{2} c^{2} d^{3} e\) | \(937\) |
| parallelrisch | \(x^{3} a \,b^{3} d^{4}+x^{2} a^{3} c \,d^{4}+\frac {3}{2} x^{2} b^{2} d^{4} a^{2}+x^{5} b^{3} c \,d^{4}+\frac {3}{2} x^{4} a^{2} c^{2} d^{4}+\frac {4}{5} x^{5} b^{4} d^{3} e +x^{6} a \,c^{3} d^{4}+x^{6} b^{4} d^{2} e^{2}+\frac {3}{2} x^{6} b^{2} d^{4} c^{2}+\frac {1}{5} x^{5} a^{3} b \,e^{4}+\frac {3}{7} x^{7} a \,b^{3} e^{4}+\frac {4}{7} x^{7} b^{4} d \,e^{3}+x^{7} b \,d^{4} c^{3}+\frac {1}{3} x^{6} c \,e^{4} a^{3}+\frac {1}{2} x^{6} a^{2} b^{2} e^{4}+\frac {3}{5} x^{10} a \,c^{3} e^{4}+\frac {9}{10} x^{10} b^{2} c^{2} e^{4}+\frac {6}{5} x^{10} c^{4} d^{2} e^{2}+\frac {5}{9} x^{9} b^{3} c \,e^{4}+\frac {8}{9} x^{9} c^{4} d^{3} e +\frac {3}{4} x^{8} a^{2} c^{2} e^{4}+\frac {7}{11} x^{11} c^{3} e^{4} b +\frac {8}{11} x^{11} c^{4} d \,e^{3}+b \,d^{4} a^{3} x +\frac {48}{5} x^{5} a \,b^{2} c \,d^{3} e +9 x^{4} a^{2} b c \,d^{3} e +12 x^{6} a \,b^{2} c \,d^{2} e^{2}+10 x^{6} a b \,c^{2} d^{3} e +\frac {54}{5} x^{5} a^{2} b c \,d^{2} e^{2}+6 x^{6} a^{2} b c d \,e^{3}+\frac {15}{2} x^{8} a b \,c^{2} d \,e^{3}+\frac {48}{7} x^{7} a \,b^{2} c d \,e^{3}+\frac {90}{7} x^{7} a b \,c^{2} d^{2} e^{2}+\frac {1}{6} c^{4} e^{4} x^{12}+\frac {1}{8} x^{8} b^{4} e^{4}+\frac {1}{4} x^{8} c^{4} d^{4}+\frac {1}{4} x^{4} b^{4} d^{4}+\frac {5}{2} x^{8} b^{3} c d \,e^{3}+\frac {27}{4} x^{8} b^{2} c^{2} d^{2} e^{2}+\frac {7}{2} x^{8} b \,c^{3} d^{3} e +\frac {9}{7} x^{7} c \,e^{4} b \,a^{2}+\frac {24}{7} x^{7} a^{2} c^{2} d \,e^{3}+\frac {24}{7} x^{7} a \,c^{3} d^{3} e +3 x^{4} a \,b^{3} d^{3} e +3 x^{4} a \,b^{2} c \,d^{4}+\frac {18}{5} x^{5} a \,b^{3} d^{2} e^{2}+3 x^{5} a b \,c^{2} d^{4}+x^{4} a^{3} b d \,e^{3}+3 x^{4} a^{3} c \,d^{2} e^{2}+\frac {9}{2} x^{4} a^{2} b^{2} d^{2} e^{2}+\frac {14}{5} x^{10} b \,c^{3} d \,e^{3}+\frac {5}{3} x^{9} a b \,c^{2} e^{4}+\frac {8}{3} x^{9} a \,c^{3} d \,e^{3}+4 x^{9} b^{2} c^{2} d \,e^{3}+\frac {14}{3} x^{9} b \,c^{3} d^{2} e^{2}+\frac {3}{2} x^{8} a \,b^{2} c \,e^{4}+\frac {9}{2} x^{8} a \,c^{3} d^{2} e^{2}+2 x^{2} a^{3} b \,d^{3} e +2 x^{3} a^{3} b \,d^{2} e^{2}+\frac {8}{3} x^{3} a^{3} c \,d^{3} e +4 x^{3} a^{2} b^{2} d^{3} e +3 x^{3} a^{2} b c \,d^{4}+6 x^{6} a^{2} c^{2} d^{2} e^{2}+2 x^{6} a \,b^{3} d \,e^{3}+\frac {10}{3} x^{6} b^{3} c \,d^{3} e +\frac {8}{5} x^{5} a^{3} c d \,e^{3}+\frac {12}{5} x^{5} a^{2} b^{2} d \,e^{3}+\frac {24}{5} x^{5} a^{2} c^{2} d^{3} e +\frac {30}{7} x^{7} b^{3} c \,d^{2} e^{2}+\frac {36}{7} x^{7} b^{2} c^{2} d^{3} e\) | \(937\) |
| default | \(\text {Expression too large to display}\) | \(1052\) |
1/6*c^4*e^4*x^12+(7/11*c^3*e^4*b+8/11*c^4*d*e^3)*x^11+(3/5*a*c^3*e^4+9/10* b^2*c^2*e^4+14/5*b*c^3*d*e^3+6/5*c^4*d^2*e^2)*x^10+(5/3*a*b*c^2*e^4+8/3*a* c^3*d*e^3+5/9*b^3*c*e^4+4*b^2*c^2*d*e^3+14/3*b*c^3*d^2*e^2+8/9*c^4*d^3*e)* x^9+(3/4*a^2*c^2*e^4+3/2*a*b^2*c*e^4+15/2*a*b*c^2*d*e^3+9/2*a*c^3*d^2*e^2+ 1/8*b^4*e^4+5/2*b^3*c*d*e^3+27/4*b^2*c^2*d^2*e^2+7/2*b*c^3*d^3*e+1/4*c^4*d ^4)*x^8+(9/7*c*e^4*b*a^2+24/7*a^2*c^2*d*e^3+3/7*a*b^3*e^4+48/7*a*b^2*c*d*e ^3+90/7*a*b*c^2*d^2*e^2+24/7*a*c^3*d^3*e+4/7*b^4*d*e^3+30/7*b^3*c*d^2*e^2+ 36/7*b^2*c^2*d^3*e+b*d^4*c^3)*x^7+(1/3*c*e^4*a^3+1/2*a^2*b^2*e^4+6*a^2*b*c *d*e^3+6*a^2*c^2*d^2*e^2+2*a*b^3*d*e^3+12*a*b^2*c*d^2*e^2+10*a*b*c^2*d^3*e +a*c^3*d^4+b^4*d^2*e^2+10/3*b^3*c*d^3*e+3/2*b^2*d^4*c^2)*x^6+(1/5*a^3*b*e^ 4+8/5*a^3*c*d*e^3+12/5*a^2*b^2*d*e^3+54/5*a^2*b*c*d^2*e^2+24/5*a^2*c^2*d^3 *e+18/5*a*b^3*d^2*e^2+48/5*a*b^2*c*d^3*e+3*a*b*c^2*d^4+4/5*b^4*d^3*e+b^3*c *d^4)*x^5+(a^3*b*d*e^3+3*a^3*c*d^2*e^2+9/2*a^2*b^2*d^2*e^2+9*a^2*b*c*d^3*e +3/2*a^2*c^2*d^4+3*a*b^3*d^3*e+3*a*b^2*c*d^4+1/4*b^4*d^4)*x^4+(2*a^3*b*d^2 *e^2+8/3*a^3*c*d^3*e+4*a^2*b^2*d^3*e+3*a^2*b*c*d^4+a*b^3*d^4)*x^3+(2*a^3*b *d^3*e+a^3*c*d^4+3/2*b^2*d^4*a^2)*x^2+b*d^4*a^3*x
Time = 0.29 (sec) , antiderivative size = 729, normalized size of antiderivative = 1.77 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{6} \, c^{4} e^{4} x^{12} + \frac {1}{11} \, {\left (8 \, c^{4} d e^{3} + 7 \, b c^{3} e^{4}\right )} x^{11} + \frac {1}{10} \, {\left (12 \, c^{4} d^{2} e^{2} + 28 \, b c^{3} d e^{3} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{4}\right )} x^{10} + \frac {1}{9} \, {\left (8 \, c^{4} d^{3} e + 42 \, b c^{3} d^{2} e^{2} + 12 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{3} + 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{4}\right )} x^{9} + a^{3} b d^{4} x + \frac {1}{8} \, {\left (2 \, c^{4} d^{4} + 28 \, b c^{3} d^{3} e + 18 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{2} + 20 \, {\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^{8} + \frac {1}{7} \, {\left (7 \, b c^{3} d^{4} + 12 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e + 30 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{2} + 4 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{3} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} + 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e + 6 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{2} + 12 \, {\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^{6} + \frac {1}{5} \, {\left (a^{3} b e^{4} + 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} + 4 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e + 18 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{2} + 4 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, a^{3} b d e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} + 12 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e + 6 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{3} b d^{2} e^{2} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{4} + 4 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d^{3} e + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{4}\right )} x^{2} \]
1/6*c^4*e^4*x^12 + 1/11*(8*c^4*d*e^3 + 7*b*c^3*e^4)*x^11 + 1/10*(12*c^4*d^ 2*e^2 + 28*b*c^3*d*e^3 + 3*(3*b^2*c^2 + 2*a*c^3)*e^4)*x^10 + 1/9*(8*c^4*d^ 3*e + 42*b*c^3*d^2*e^2 + 12*(3*b^2*c^2 + 2*a*c^3)*d*e^3 + 5*(b^3*c + 3*a*b *c^2)*e^4)*x^9 + a^3*b*d^4*x + 1/8*(2*c^4*d^4 + 28*b*c^3*d^3*e + 18*(3*b^2 *c^2 + 2*a*c^3)*d^2*e^2 + 20*(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^8 + 1/7*(7*b*c^3*d^4 + 12*(3*b^2*c^2 + 2*a*c^3)*d^3*e + 30*(b^3*c + 3*a*b*c^2)*d^2*e^2 + 4*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^3 + 3*(a*b^3 + 3*a^2*b*c)*e^4)*x^7 + 1/6*(3*(3*b^2*c^2 + 2*a*c^3)*d^4 + 20* (b^3*c + 3*a*b*c^2)*d^3*e + 6*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^2 + 12* (a*b^3 + 3*a^2*b*c)*d*e^3 + (3*a^2*b^2 + 2*a^3*c)*e^4)*x^6 + 1/5*(a^3*b*e^ 4 + 5*(b^3*c + 3*a*b*c^2)*d^4 + 4*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e + 1 8*(a*b^3 + 3*a^2*b*c)*d^2*e^2 + 4*(3*a^2*b^2 + 2*a^3*c)*d*e^3)*x^5 + 1/4*( 4*a^3*b*d*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4 + 12*(a*b^3 + 3*a^2*b*c )*d^3*e + 6*(3*a^2*b^2 + 2*a^3*c)*d^2*e^2)*x^4 + 1/3*(6*a^3*b*d^2*e^2 + 3* (a*b^3 + 3*a^2*b*c)*d^4 + 4*(3*a^2*b^2 + 2*a^3*c)*d^3*e)*x^3 + 1/2*(4*a^3* b*d^3*e + (3*a^2*b^2 + 2*a^3*c)*d^4)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 935 vs. \(2 (411) = 822\).
Time = 0.07 (sec) , antiderivative size = 935, normalized size of antiderivative = 2.27 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=a^{3} b d^{4} x + \frac {c^{4} e^{4} x^{12}}{6} + x^{11} \cdot \left (\frac {7 b c^{3} e^{4}}{11} + \frac {8 c^{4} d e^{3}}{11}\right ) + x^{10} \cdot \left (\frac {3 a c^{3} e^{4}}{5} + \frac {9 b^{2} c^{2} e^{4}}{10} + \frac {14 b c^{3} d e^{3}}{5} + \frac {6 c^{4} d^{2} e^{2}}{5}\right ) + x^{9} \cdot \left (\frac {5 a b c^{2} e^{4}}{3} + \frac {8 a c^{3} d e^{3}}{3} + \frac {5 b^{3} c e^{4}}{9} + 4 b^{2} c^{2} d e^{3} + \frac {14 b c^{3} d^{2} e^{2}}{3} + \frac {8 c^{4} d^{3} e}{9}\right ) + x^{8} \cdot \left (\frac {3 a^{2} c^{2} e^{4}}{4} + \frac {3 a b^{2} c e^{4}}{2} + \frac {15 a b c^{2} d e^{3}}{2} + \frac {9 a c^{3} d^{2} e^{2}}{2} + \frac {b^{4} e^{4}}{8} + \frac {5 b^{3} c d e^{3}}{2} + \frac {27 b^{2} c^{2} d^{2} e^{2}}{4} + \frac {7 b c^{3} d^{3} e}{2} + \frac {c^{4} d^{4}}{4}\right ) + x^{7} \cdot \left (\frac {9 a^{2} b c e^{4}}{7} + \frac {24 a^{2} c^{2} d e^{3}}{7} + \frac {3 a b^{3} e^{4}}{7} + \frac {48 a b^{2} c d e^{3}}{7} + \frac {90 a b c^{2} d^{2} e^{2}}{7} + \frac {24 a c^{3} d^{3} e}{7} + \frac {4 b^{4} d e^{3}}{7} + \frac {30 b^{3} c d^{2} e^{2}}{7} + \frac {36 b^{2} c^{2} d^{3} e}{7} + b c^{3} d^{4}\right ) + x^{6} \left (\frac {a^{3} c e^{4}}{3} + \frac {a^{2} b^{2} e^{4}}{2} + 6 a^{2} b c d e^{3} + 6 a^{2} c^{2} d^{2} e^{2} + 2 a b^{3} d e^{3} + 12 a b^{2} c d^{2} e^{2} + 10 a b c^{2} d^{3} e + a c^{3} d^{4} + b^{4} d^{2} e^{2} + \frac {10 b^{3} c d^{3} e}{3} + \frac {3 b^{2} c^{2} d^{4}}{2}\right ) + x^{5} \left (\frac {a^{3} b e^{4}}{5} + \frac {8 a^{3} c d e^{3}}{5} + \frac {12 a^{2} b^{2} d e^{3}}{5} + \frac {54 a^{2} b c d^{2} e^{2}}{5} + \frac {24 a^{2} c^{2} d^{3} e}{5} + \frac {18 a b^{3} d^{2} e^{2}}{5} + \frac {48 a b^{2} c d^{3} e}{5} + 3 a b c^{2} d^{4} + \frac {4 b^{4} d^{3} e}{5} + b^{3} c d^{4}\right ) + x^{4} \left (a^{3} b d e^{3} + 3 a^{3} c d^{2} e^{2} + \frac {9 a^{2} b^{2} d^{2} e^{2}}{2} + 9 a^{2} b c d^{3} e + \frac {3 a^{2} c^{2} d^{4}}{2} + 3 a b^{3} d^{3} e + 3 a b^{2} c d^{4} + \frac {b^{4} d^{4}}{4}\right ) + x^{3} \cdot \left (2 a^{3} b d^{2} e^{2} + \frac {8 a^{3} c d^{3} e}{3} + 4 a^{2} b^{2} d^{3} e + 3 a^{2} b c d^{4} + a b^{3} d^{4}\right ) + x^{2} \cdot \left (2 a^{3} b d^{3} e + a^{3} c d^{4} + \frac {3 a^{2} b^{2} d^{4}}{2}\right ) \]
a**3*b*d**4*x + c**4*e**4*x**12/6 + x**11*(7*b*c**3*e**4/11 + 8*c**4*d*e** 3/11) + x**10*(3*a*c**3*e**4/5 + 9*b**2*c**2*e**4/10 + 14*b*c**3*d*e**3/5 + 6*c**4*d**2*e**2/5) + x**9*(5*a*b*c**2*e**4/3 + 8*a*c**3*d*e**3/3 + 5*b* *3*c*e**4/9 + 4*b**2*c**2*d*e**3 + 14*b*c**3*d**2*e**2/3 + 8*c**4*d**3*e/9 ) + x**8*(3*a**2*c**2*e**4/4 + 3*a*b**2*c*e**4/2 + 15*a*b*c**2*d*e**3/2 + 9*a*c**3*d**2*e**2/2 + b**4*e**4/8 + 5*b**3*c*d*e**3/2 + 27*b**2*c**2*d**2 *e**2/4 + 7*b*c**3*d**3*e/2 + c**4*d**4/4) + x**7*(9*a**2*b*c*e**4/7 + 24* a**2*c**2*d*e**3/7 + 3*a*b**3*e**4/7 + 48*a*b**2*c*d*e**3/7 + 90*a*b*c**2* d**2*e**2/7 + 24*a*c**3*d**3*e/7 + 4*b**4*d*e**3/7 + 30*b**3*c*d**2*e**2/7 + 36*b**2*c**2*d**3*e/7 + b*c**3*d**4) + x**6*(a**3*c*e**4/3 + a**2*b**2* e**4/2 + 6*a**2*b*c*d*e**3 + 6*a**2*c**2*d**2*e**2 + 2*a*b**3*d*e**3 + 12* a*b**2*c*d**2*e**2 + 10*a*b*c**2*d**3*e + a*c**3*d**4 + b**4*d**2*e**2 + 1 0*b**3*c*d**3*e/3 + 3*b**2*c**2*d**4/2) + x**5*(a**3*b*e**4/5 + 8*a**3*c*d *e**3/5 + 12*a**2*b**2*d*e**3/5 + 54*a**2*b*c*d**2*e**2/5 + 24*a**2*c**2*d **3*e/5 + 18*a*b**3*d**2*e**2/5 + 48*a*b**2*c*d**3*e/5 + 3*a*b*c**2*d**4 + 4*b**4*d**3*e/5 + b**3*c*d**4) + x**4*(a**3*b*d*e**3 + 3*a**3*c*d**2*e**2 + 9*a**2*b**2*d**2*e**2/2 + 9*a**2*b*c*d**3*e + 3*a**2*c**2*d**4/2 + 3*a* b**3*d**3*e + 3*a*b**2*c*d**4 + b**4*d**4/4) + x**3*(2*a**3*b*d**2*e**2 + 8*a**3*c*d**3*e/3 + 4*a**2*b**2*d**3*e + 3*a**2*b*c*d**4 + a*b**3*d**4) + x**2*(2*a**3*b*d**3*e + a**3*c*d**4 + 3*a**2*b**2*d**4/2)
Time = 0.19 (sec) , antiderivative size = 729, normalized size of antiderivative = 1.77 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{6} \, c^{4} e^{4} x^{12} + \frac {1}{11} \, {\left (8 \, c^{4} d e^{3} + 7 \, b c^{3} e^{4}\right )} x^{11} + \frac {1}{10} \, {\left (12 \, c^{4} d^{2} e^{2} + 28 \, b c^{3} d e^{3} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{4}\right )} x^{10} + \frac {1}{9} \, {\left (8 \, c^{4} d^{3} e + 42 \, b c^{3} d^{2} e^{2} + 12 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{3} + 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{4}\right )} x^{9} + a^{3} b d^{4} x + \frac {1}{8} \, {\left (2 \, c^{4} d^{4} + 28 \, b c^{3} d^{3} e + 18 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{2} + 20 \, {\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^{8} + \frac {1}{7} \, {\left (7 \, b c^{3} d^{4} + 12 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e + 30 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{2} + 4 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{3} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} + 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e + 6 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{2} + 12 \, {\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^{6} + \frac {1}{5} \, {\left (a^{3} b e^{4} + 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} + 4 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e + 18 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{2} + 4 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, a^{3} b d e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} + 12 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e + 6 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{3} b d^{2} e^{2} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{4} + 4 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b d^{3} e + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{4}\right )} x^{2} \]
1/6*c^4*e^4*x^12 + 1/11*(8*c^4*d*e^3 + 7*b*c^3*e^4)*x^11 + 1/10*(12*c^4*d^ 2*e^2 + 28*b*c^3*d*e^3 + 3*(3*b^2*c^2 + 2*a*c^3)*e^4)*x^10 + 1/9*(8*c^4*d^ 3*e + 42*b*c^3*d^2*e^2 + 12*(3*b^2*c^2 + 2*a*c^3)*d*e^3 + 5*(b^3*c + 3*a*b *c^2)*e^4)*x^9 + a^3*b*d^4*x + 1/8*(2*c^4*d^4 + 28*b*c^3*d^3*e + 18*(3*b^2 *c^2 + 2*a*c^3)*d^2*e^2 + 20*(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^8 + 1/7*(7*b*c^3*d^4 + 12*(3*b^2*c^2 + 2*a*c^3)*d^3*e + 30*(b^3*c + 3*a*b*c^2)*d^2*e^2 + 4*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^3 + 3*(a*b^3 + 3*a^2*b*c)*e^4)*x^7 + 1/6*(3*(3*b^2*c^2 + 2*a*c^3)*d^4 + 20* (b^3*c + 3*a*b*c^2)*d^3*e + 6*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^2 + 12* (a*b^3 + 3*a^2*b*c)*d*e^3 + (3*a^2*b^2 + 2*a^3*c)*e^4)*x^6 + 1/5*(a^3*b*e^ 4 + 5*(b^3*c + 3*a*b*c^2)*d^4 + 4*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e + 1 8*(a*b^3 + 3*a^2*b*c)*d^2*e^2 + 4*(3*a^2*b^2 + 2*a^3*c)*d*e^3)*x^5 + 1/4*( 4*a^3*b*d*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4 + 12*(a*b^3 + 3*a^2*b*c )*d^3*e + 6*(3*a^2*b^2 + 2*a^3*c)*d^2*e^2)*x^4 + 1/3*(6*a^3*b*d^2*e^2 + 3* (a*b^3 + 3*a^2*b*c)*d^4 + 4*(3*a^2*b^2 + 2*a^3*c)*d^3*e)*x^3 + 1/2*(4*a^3* b*d^3*e + (3*a^2*b^2 + 2*a^3*c)*d^4)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 936 vs. \(2 (395) = 790\).
Time = 0.28 (sec) , antiderivative size = 936, normalized size of antiderivative = 2.28 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{6} \, c^{4} e^{4} x^{12} + \frac {8}{11} \, c^{4} d e^{3} x^{11} + \frac {7}{11} \, b c^{3} e^{4} x^{11} + \frac {6}{5} \, c^{4} d^{2} e^{2} x^{10} + \frac {14}{5} \, b c^{3} d e^{3} x^{10} + \frac {9}{10} \, b^{2} c^{2} e^{4} x^{10} + \frac {3}{5} \, a c^{3} e^{4} x^{10} + \frac {8}{9} \, c^{4} d^{3} e x^{9} + \frac {14}{3} \, b c^{3} d^{2} e^{2} x^{9} + 4 \, b^{2} c^{2} d e^{3} x^{9} + \frac {8}{3} \, a c^{3} d e^{3} x^{9} + \frac {5}{9} \, b^{3} c e^{4} x^{9} + \frac {5}{3} \, a b c^{2} e^{4} x^{9} + \frac {1}{4} \, c^{4} d^{4} x^{8} + \frac {7}{2} \, b c^{3} d^{3} e x^{8} + \frac {27}{4} \, b^{2} c^{2} d^{2} e^{2} x^{8} + \frac {9}{2} \, a c^{3} d^{2} e^{2} x^{8} + \frac {5}{2} \, b^{3} c d e^{3} x^{8} + \frac {15}{2} \, a b c^{2} d e^{3} x^{8} + \frac {1}{8} \, b^{4} e^{4} x^{8} + \frac {3}{2} \, a b^{2} c e^{4} x^{8} + \frac {3}{4} \, a^{2} c^{2} e^{4} x^{8} + b c^{3} d^{4} x^{7} + \frac {36}{7} \, b^{2} c^{2} d^{3} e x^{7} + \frac {24}{7} \, a c^{3} d^{3} e x^{7} + \frac {30}{7} \, b^{3} c d^{2} e^{2} x^{7} + \frac {90}{7} \, a b c^{2} d^{2} e^{2} x^{7} + \frac {4}{7} \, b^{4} d e^{3} x^{7} + \frac {48}{7} \, a b^{2} c d e^{3} x^{7} + \frac {24}{7} \, a^{2} c^{2} d e^{3} x^{7} + \frac {3}{7} \, a b^{3} e^{4} x^{7} + \frac {9}{7} \, a^{2} b c e^{4} x^{7} + \frac {3}{2} \, b^{2} c^{2} d^{4} x^{6} + a c^{3} d^{4} x^{6} + \frac {10}{3} \, b^{3} c d^{3} e x^{6} + 10 \, a b c^{2} d^{3} e x^{6} + b^{4} d^{2} e^{2} x^{6} + 12 \, a b^{2} c d^{2} e^{2} x^{6} + 6 \, a^{2} c^{2} d^{2} e^{2} x^{6} + 2 \, a b^{3} d e^{3} x^{6} + 6 \, a^{2} b c d e^{3} x^{6} + \frac {1}{2} \, a^{2} b^{2} e^{4} x^{6} + \frac {1}{3} \, a^{3} c e^{4} x^{6} + b^{3} c d^{4} x^{5} + 3 \, a b c^{2} d^{4} x^{5} + \frac {4}{5} \, b^{4} d^{3} e x^{5} + \frac {48}{5} \, a b^{2} c d^{3} e x^{5} + \frac {24}{5} \, a^{2} c^{2} d^{3} e x^{5} + \frac {18}{5} \, a b^{3} d^{2} e^{2} x^{5} + \frac {54}{5} \, a^{2} b c d^{2} e^{2} x^{5} + \frac {12}{5} \, a^{2} b^{2} d e^{3} x^{5} + \frac {8}{5} \, a^{3} c d e^{3} x^{5} + \frac {1}{5} \, a^{3} b e^{4} x^{5} + \frac {1}{4} \, b^{4} d^{4} x^{4} + 3 \, a b^{2} c d^{4} x^{4} + \frac {3}{2} \, a^{2} c^{2} d^{4} x^{4} + 3 \, a b^{3} d^{3} e x^{4} + 9 \, a^{2} b c d^{3} e x^{4} + \frac {9}{2} \, a^{2} b^{2} d^{2} e^{2} x^{4} + 3 \, a^{3} c d^{2} e^{2} x^{4} + a^{3} b d e^{3} x^{4} + a b^{3} d^{4} x^{3} + 3 \, a^{2} b c d^{4} x^{3} + 4 \, a^{2} b^{2} d^{3} e x^{3} + \frac {8}{3} \, a^{3} c d^{3} e x^{3} + 2 \, a^{3} b d^{2} e^{2} x^{3} + \frac {3}{2} \, a^{2} b^{2} d^{4} x^{2} + a^{3} c d^{4} x^{2} + 2 \, a^{3} b d^{3} e x^{2} + a^{3} b d^{4} x \]
1/6*c^4*e^4*x^12 + 8/11*c^4*d*e^3*x^11 + 7/11*b*c^3*e^4*x^11 + 6/5*c^4*d^2 *e^2*x^10 + 14/5*b*c^3*d*e^3*x^10 + 9/10*b^2*c^2*e^4*x^10 + 3/5*a*c^3*e^4* x^10 + 8/9*c^4*d^3*e*x^9 + 14/3*b*c^3*d^2*e^2*x^9 + 4*b^2*c^2*d*e^3*x^9 + 8/3*a*c^3*d*e^3*x^9 + 5/9*b^3*c*e^4*x^9 + 5/3*a*b*c^2*e^4*x^9 + 1/4*c^4*d^ 4*x^8 + 7/2*b*c^3*d^3*e*x^8 + 27/4*b^2*c^2*d^2*e^2*x^8 + 9/2*a*c^3*d^2*e^2 *x^8 + 5/2*b^3*c*d*e^3*x^8 + 15/2*a*b*c^2*d*e^3*x^8 + 1/8*b^4*e^4*x^8 + 3/ 2*a*b^2*c*e^4*x^8 + 3/4*a^2*c^2*e^4*x^8 + b*c^3*d^4*x^7 + 36/7*b^2*c^2*d^3 *e*x^7 + 24/7*a*c^3*d^3*e*x^7 + 30/7*b^3*c*d^2*e^2*x^7 + 90/7*a*b*c^2*d^2* e^2*x^7 + 4/7*b^4*d*e^3*x^7 + 48/7*a*b^2*c*d*e^3*x^7 + 24/7*a^2*c^2*d*e^3* x^7 + 3/7*a*b^3*e^4*x^7 + 9/7*a^2*b*c*e^4*x^7 + 3/2*b^2*c^2*d^4*x^6 + a*c^ 3*d^4*x^6 + 10/3*b^3*c*d^3*e*x^6 + 10*a*b*c^2*d^3*e*x^6 + b^4*d^2*e^2*x^6 + 12*a*b^2*c*d^2*e^2*x^6 + 6*a^2*c^2*d^2*e^2*x^6 + 2*a*b^3*d*e^3*x^6 + 6*a ^2*b*c*d*e^3*x^6 + 1/2*a^2*b^2*e^4*x^6 + 1/3*a^3*c*e^4*x^6 + b^3*c*d^4*x^5 + 3*a*b*c^2*d^4*x^5 + 4/5*b^4*d^3*e*x^5 + 48/5*a*b^2*c*d^3*e*x^5 + 24/5*a ^2*c^2*d^3*e*x^5 + 18/5*a*b^3*d^2*e^2*x^5 + 54/5*a^2*b*c*d^2*e^2*x^5 + 12/ 5*a^2*b^2*d*e^3*x^5 + 8/5*a^3*c*d*e^3*x^5 + 1/5*a^3*b*e^4*x^5 + 1/4*b^4*d^ 4*x^4 + 3*a*b^2*c*d^4*x^4 + 3/2*a^2*c^2*d^4*x^4 + 3*a*b^3*d^3*e*x^4 + 9*a^ 2*b*c*d^3*e*x^4 + 9/2*a^2*b^2*d^2*e^2*x^4 + 3*a^3*c*d^2*e^2*x^4 + a^3*b*d* e^3*x^4 + a*b^3*d^4*x^3 + 3*a^2*b*c*d^4*x^3 + 4*a^2*b^2*d^3*e*x^3 + 8/3*a^ 3*c*d^3*e*x^3 + 2*a^3*b*d^2*e^2*x^3 + 3/2*a^2*b^2*d^4*x^2 + a^3*c*d^4*x...
Time = 0.26 (sec) , antiderivative size = 768, normalized size of antiderivative = 1.87 \[ \int (b+2 c x) (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx=x^3\,\left (2\,a^3\,b\,d^2\,e^2+\frac {8\,c\,a^3\,d^3\,e}{3}+4\,a^2\,b^2\,d^3\,e+3\,c\,a^2\,b\,d^4+a\,b^3\,d^4\right )+x^5\,\left (\frac {a^3\,b\,e^4}{5}+\frac {8\,a^3\,c\,d\,e^3}{5}+\frac {12\,a^2\,b^2\,d\,e^3}{5}+\frac {54\,a^2\,b\,c\,d^2\,e^2}{5}+\frac {24\,a^2\,c^2\,d^3\,e}{5}+\frac {18\,a\,b^3\,d^2\,e^2}{5}+\frac {48\,a\,b^2\,c\,d^3\,e}{5}+3\,a\,b\,c^2\,d^4+\frac {4\,b^4\,d^3\,e}{5}+b^3\,c\,d^4\right )+x^7\,\left (\frac {9\,a^2\,b\,c\,e^4}{7}+\frac {24\,a^2\,c^2\,d\,e^3}{7}+\frac {3\,a\,b^3\,e^4}{7}+\frac {48\,a\,b^2\,c\,d\,e^3}{7}+\frac {90\,a\,b\,c^2\,d^2\,e^2}{7}+\frac {24\,a\,c^3\,d^3\,e}{7}+\frac {4\,b^4\,d\,e^3}{7}+\frac {30\,b^3\,c\,d^2\,e^2}{7}+\frac {36\,b^2\,c^2\,d^3\,e}{7}+b\,c^3\,d^4\right )+x^6\,\left (\frac {a^3\,c\,e^4}{3}+\frac {a^2\,b^2\,e^4}{2}+6\,a^2\,b\,c\,d\,e^3+6\,a^2\,c^2\,d^2\,e^2+2\,a\,b^3\,d\,e^3+12\,a\,b^2\,c\,d^2\,e^2+10\,a\,b\,c^2\,d^3\,e+a\,c^3\,d^4+b^4\,d^2\,e^2+\frac {10\,b^3\,c\,d^3\,e}{3}+\frac {3\,b^2\,c^2\,d^4}{2}\right )+x^8\,\left (\frac {3\,a^2\,c^2\,e^4}{4}+\frac {3\,a\,b^2\,c\,e^4}{2}+\frac {15\,a\,b\,c^2\,d\,e^3}{2}+\frac {9\,a\,c^3\,d^2\,e^2}{2}+\frac {b^4\,e^4}{8}+\frac {5\,b^3\,c\,d\,e^3}{2}+\frac {27\,b^2\,c^2\,d^2\,e^2}{4}+\frac {7\,b\,c^3\,d^3\,e}{2}+\frac {c^4\,d^4}{4}\right )+x^9\,\left (\frac {5\,b^3\,c\,e^4}{9}+4\,b^2\,c^2\,d\,e^3+\frac {14\,b\,c^3\,d^2\,e^2}{3}+\frac {5\,a\,b\,c^2\,e^4}{3}+\frac {8\,c^4\,d^3\,e}{9}+\frac {8\,a\,c^3\,d\,e^3}{3}\right )+x^4\,\left (a^3\,b\,d\,e^3+3\,a^3\,c\,d^2\,e^2+\frac {9\,a^2\,b^2\,d^2\,e^2}{2}+9\,a^2\,b\,c\,d^3\,e+\frac {3\,a^2\,c^2\,d^4}{2}+3\,a\,b^3\,d^3\,e+3\,a\,b^2\,c\,d^4+\frac {b^4\,d^4}{4}\right )+\frac {c^4\,e^4\,x^{12}}{6}+\frac {c^3\,e^3\,x^{11}\,\left (7\,b\,e+8\,c\,d\right )}{11}+\frac {a^2\,d^3\,x^2\,\left (3\,d\,b^2+4\,a\,e\,b+2\,a\,c\,d\right )}{2}+\frac {c^2\,e^2\,x^{10}\,\left (9\,b^2\,e^2+28\,b\,c\,d\,e+12\,c^2\,d^2+6\,a\,c\,e^2\right )}{10}+a^3\,b\,d^4\,x \]
x^3*(a*b^3*d^4 + 4*a^2*b^2*d^3*e + 2*a^3*b*d^2*e^2 + 3*a^2*b*c*d^4 + (8*a^ 3*c*d^3*e)/3) + x^5*((a^3*b*e^4)/5 + b^3*c*d^4 + (4*b^4*d^3*e)/5 + (18*a*b ^3*d^2*e^2)/5 + (12*a^2*b^2*d*e^3)/5 + (24*a^2*c^2*d^3*e)/5 + 3*a*b*c^2*d^ 4 + (8*a^3*c*d*e^3)/5 + (48*a*b^2*c*d^3*e)/5 + (54*a^2*b*c*d^2*e^2)/5) + x ^7*((3*a*b^3*e^4)/7 + b*c^3*d^4 + (4*b^4*d*e^3)/7 + (24*a^2*c^2*d*e^3)/7 + (36*b^2*c^2*d^3*e)/7 + (30*b^3*c*d^2*e^2)/7 + (9*a^2*b*c*e^4)/7 + (24*a*c ^3*d^3*e)/7 + (48*a*b^2*c*d*e^3)/7 + (90*a*b*c^2*d^2*e^2)/7) + x^6*(a*c^3* d^4 + (a^3*c*e^4)/3 + (a^2*b^2*e^4)/2 + (3*b^2*c^2*d^4)/2 + b^4*d^2*e^2 + 6*a^2*c^2*d^2*e^2 + 2*a*b^3*d*e^3 + (10*b^3*c*d^3*e)/3 + 10*a*b*c^2*d^3*e + 6*a^2*b*c*d*e^3 + 12*a*b^2*c*d^2*e^2) + x^8*((b^4*e^4)/8 + (c^4*d^4)/4 + (3*a^2*c^2*e^4)/4 + (9*a*c^3*d^2*e^2)/2 + (27*b^2*c^2*d^2*e^2)/4 + (3*a*b ^2*c*e^4)/2 + (7*b*c^3*d^3*e)/2 + (5*b^3*c*d*e^3)/2 + (15*a*b*c^2*d*e^3)/2 ) + x^9*((5*b^3*c*e^4)/9 + (8*c^4*d^3*e)/9 + (14*b*c^3*d^2*e^2)/3 + 4*b^2* c^2*d*e^3 + (5*a*b*c^2*e^4)/3 + (8*a*c^3*d*e^3)/3) + x^4*((b^4*d^4)/4 + (3 *a^2*c^2*d^4)/2 + 3*a^3*c*d^2*e^2 + (9*a^2*b^2*d^2*e^2)/2 + 3*a*b^2*c*d^4 + 3*a*b^3*d^3*e + a^3*b*d*e^3 + 9*a^2*b*c*d^3*e) + (c^4*e^4*x^12)/6 + (c^3 *e^3*x^11*(7*b*e + 8*c*d))/11 + (a^2*d^3*x^2*(3*b^2*d + 4*a*b*e + 2*a*c*d) )/2 + (c^2*e^2*x^10*(9*b^2*e^2 + 12*c^2*d^2 + 6*a*c*e^2 + 28*b*c*d*e))/10 + a^3*b*d^4*x