Integrand size = 26, antiderivative size = 173 \[ \int (d+e x)^8 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {(b d-a e)^6 (d+e x)^9}{9 e^7}-\frac {3 b (b d-a e)^5 (d+e x)^{10}}{5 e^7}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{11}}{11 e^7}-\frac {5 b^3 (b d-a e)^3 (d+e x)^{12}}{3 e^7}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{13}}{13 e^7}-\frac {3 b^5 (b d-a e) (d+e x)^{14}}{7 e^7}+\frac {b^6 (d+e x)^{15}}{15 e^7} \]
1/9*(-a*e+b*d)^6*(e*x+d)^9/e^7-3/5*b*(-a*e+b*d)^5*(e*x+d)^10/e^7+15/11*b^2 *(-a*e+b*d)^4*(e*x+d)^11/e^7-5/3*b^3*(-a*e+b*d)^3*(e*x+d)^12/e^7+15/13*b^4 *(-a*e+b*d)^2*(e*x+d)^13/e^7-3/7*b^5*(-a*e+b*d)*(e*x+d)^14/e^7+1/15*b^6*(e *x+d)^15/e^7
Leaf count is larger than twice the leaf count of optimal. \(771\) vs. \(2(173)=346\).
Time = 0.07 (sec) , antiderivative size = 771, normalized size of antiderivative = 4.46 \[ \int (d+e x)^8 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=a^6 d^8 x+a^5 d^7 (3 b d+4 a e) x^2+\frac {1}{3} a^4 d^6 \left (15 b^2 d^2+48 a b d e+28 a^2 e^2\right ) x^3+a^3 d^5 \left (5 b^3 d^3+30 a b^2 d^2 e+42 a^2 b d e^2+14 a^3 e^3\right ) x^4+\frac {1}{5} a^2 d^4 \left (15 b^4 d^4+160 a b^3 d^3 e+420 a^2 b^2 d^2 e^2+336 a^3 b d e^3+70 a^4 e^4\right ) x^5+\frac {1}{3} a d^3 \left (3 b^5 d^5+60 a b^4 d^4 e+280 a^2 b^3 d^3 e^2+420 a^3 b^2 d^2 e^3+210 a^4 b d e^4+28 a^5 e^5\right ) x^6+\frac {1}{7} d^2 \left (b^6 d^6+48 a b^5 d^5 e+420 a^2 b^4 d^4 e^2+1120 a^3 b^3 d^3 e^3+1050 a^4 b^2 d^2 e^4+336 a^5 b d e^5+28 a^6 e^6\right ) x^7+d e \left (b^6 d^6+21 a b^5 d^5 e+105 a^2 b^4 d^4 e^2+175 a^3 b^3 d^3 e^3+105 a^4 b^2 d^2 e^4+21 a^5 b d e^5+a^6 e^6\right ) x^8+\frac {1}{9} e^2 \left (28 b^6 d^6+336 a b^5 d^5 e+1050 a^2 b^4 d^4 e^2+1120 a^3 b^3 d^3 e^3+420 a^4 b^2 d^2 e^4+48 a^5 b d e^5+a^6 e^6\right ) x^9+\frac {1}{5} b e^3 \left (28 b^5 d^5+210 a b^4 d^4 e+420 a^2 b^3 d^3 e^2+280 a^3 b^2 d^2 e^3+60 a^4 b d e^4+3 a^5 e^5\right ) x^{10}+\frac {1}{11} b^2 e^4 \left (70 b^4 d^4+336 a b^3 d^3 e+420 a^2 b^2 d^2 e^2+160 a^3 b d e^3+15 a^4 e^4\right ) x^{11}+\frac {1}{3} b^3 e^5 \left (14 b^3 d^3+42 a b^2 d^2 e+30 a^2 b d e^2+5 a^3 e^3\right ) x^{12}+\frac {1}{13} b^4 e^6 \left (28 b^2 d^2+48 a b d e+15 a^2 e^2\right ) x^{13}+\frac {1}{7} b^5 e^7 (4 b d+3 a e) x^{14}+\frac {1}{15} b^6 e^8 x^{15} \]
a^6*d^8*x + a^5*d^7*(3*b*d + 4*a*e)*x^2 + (a^4*d^6*(15*b^2*d^2 + 48*a*b*d* e + 28*a^2*e^2)*x^3)/3 + a^3*d^5*(5*b^3*d^3 + 30*a*b^2*d^2*e + 42*a^2*b*d* e^2 + 14*a^3*e^3)*x^4 + (a^2*d^4*(15*b^4*d^4 + 160*a*b^3*d^3*e + 420*a^2*b ^2*d^2*e^2 + 336*a^3*b*d*e^3 + 70*a^4*e^4)*x^5)/5 + (a*d^3*(3*b^5*d^5 + 60 *a*b^4*d^4*e + 280*a^2*b^3*d^3*e^2 + 420*a^3*b^2*d^2*e^3 + 210*a^4*b*d*e^4 + 28*a^5*e^5)*x^6)/3 + (d^2*(b^6*d^6 + 48*a*b^5*d^5*e + 420*a^2*b^4*d^4*e ^2 + 1120*a^3*b^3*d^3*e^3 + 1050*a^4*b^2*d^2*e^4 + 336*a^5*b*d*e^5 + 28*a^ 6*e^6)*x^7)/7 + d*e*(b^6*d^6 + 21*a*b^5*d^5*e + 105*a^2*b^4*d^4*e^2 + 175* a^3*b^3*d^3*e^3 + 105*a^4*b^2*d^2*e^4 + 21*a^5*b*d*e^5 + a^6*e^6)*x^8 + (e ^2*(28*b^6*d^6 + 336*a*b^5*d^5*e + 1050*a^2*b^4*d^4*e^2 + 1120*a^3*b^3*d^3 *e^3 + 420*a^4*b^2*d^2*e^4 + 48*a^5*b*d*e^5 + a^6*e^6)*x^9)/9 + (b*e^3*(28 *b^5*d^5 + 210*a*b^4*d^4*e + 420*a^2*b^3*d^3*e^2 + 280*a^3*b^2*d^2*e^3 + 6 0*a^4*b*d*e^4 + 3*a^5*e^5)*x^10)/5 + (b^2*e^4*(70*b^4*d^4 + 336*a*b^3*d^3* e + 420*a^2*b^2*d^2*e^2 + 160*a^3*b*d*e^3 + 15*a^4*e^4)*x^11)/11 + (b^3*e^ 5*(14*b^3*d^3 + 42*a*b^2*d^2*e + 30*a^2*b*d*e^2 + 5*a^3*e^3)*x^12)/3 + (b^ 4*e^6*(28*b^2*d^2 + 48*a*b*d*e + 15*a^2*e^2)*x^13)/13 + (b^5*e^7*(4*b*d + 3*a*e)*x^14)/7 + (b^6*e^8*x^15)/15
Time = 0.67 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1098, 27, 49, 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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^3 (d+e x)^8 \, dx\) |
| \(\Big \downarrow \) 1098 |
| \(\displaystyle \frac {\int b^6 (a+b x)^6 (d+e x)^8dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^6 (d+e x)^8dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (-\frac {6 b^5 (d+e x)^{13} (b d-a e)}{e^6}+\frac {15 b^4 (d+e x)^{12} (b d-a e)^2}{e^6}-\frac {20 b^3 (d+e x)^{11} (b d-a e)^3}{e^6}+\frac {15 b^2 (d+e x)^{10} (b d-a e)^4}{e^6}-\frac {6 b (d+e x)^9 (b d-a e)^5}{e^6}+\frac {(d+e x)^8 (a e-b d)^6}{e^6}+\frac {b^6 (d+e x)^{14}}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 b^5 (d+e x)^{14} (b d-a e)}{7 e^7}+\frac {15 b^4 (d+e x)^{13} (b d-a e)^2}{13 e^7}-\frac {5 b^3 (d+e x)^{12} (b d-a e)^3}{3 e^7}+\frac {15 b^2 (d+e x)^{11} (b d-a e)^4}{11 e^7}-\frac {3 b (d+e x)^{10} (b d-a e)^5}{5 e^7}+\frac {(d+e x)^9 (b d-a e)^6}{9 e^7}+\frac {b^6 (d+e x)^{15}}{15 e^7}\) |
((b*d - a*e)^6*(d + e*x)^9)/(9*e^7) - (3*b*(b*d - a*e)^5*(d + e*x)^10)/(5* e^7) + (15*b^2*(b*d - a*e)^4*(d + e*x)^11)/(11*e^7) - (5*b^3*(b*d - a*e)^3 *(d + e*x)^12)/(3*e^7) + (15*b^4*(b*d - a*e)^2*(d + e*x)^13)/(13*e^7) - (3 *b^5*(b*d - a*e)*(d + e*x)^14)/(7*e^7) + (b^6*(d + e*x)^15)/(15*e^7)
3.15.81.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[ {a, b, c, d, e, m}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(788\) vs. \(2(159)=318\).
Time = 2.28 (sec) , antiderivative size = 789, normalized size of antiderivative = 4.56
| method | result | size |
| norman | \(a^{6} d^{8} x +\left (4 a^{6} d^{7} e +3 a^{5} b \,d^{8}\right ) x^{2}+\left (\frac {28}{3} a^{6} d^{6} e^{2}+16 a^{5} b \,d^{7} e +5 a^{4} b^{2} d^{8}\right ) x^{3}+\left (14 a^{6} d^{5} e^{3}+42 a^{5} b \,d^{6} e^{2}+30 a^{4} b^{2} d^{7} e +5 a^{3} b^{3} d^{8}\right ) x^{4}+\left (14 a^{6} d^{4} e^{4}+\frac {336}{5} a^{5} b \,d^{5} e^{3}+84 a^{4} b^{2} d^{6} e^{2}+32 a^{3} b^{3} d^{7} e +3 a^{2} b^{4} d^{8}\right ) x^{5}+\left (\frac {28}{3} a^{6} d^{3} e^{5}+70 a^{5} b \,d^{4} e^{4}+140 a^{4} b^{2} d^{5} e^{3}+\frac {280}{3} a^{3} b^{3} d^{6} e^{2}+20 a^{2} b^{4} d^{7} e +a \,b^{5} d^{8}\right ) x^{6}+\left (4 a^{6} d^{2} e^{6}+48 a^{5} b \,d^{3} e^{5}+150 a^{4} b^{2} d^{4} e^{4}+160 a^{3} b^{3} d^{5} e^{3}+60 a^{2} b^{4} d^{6} e^{2}+\frac {48}{7} a \,b^{5} d^{7} e +\frac {1}{7} b^{6} d^{8}\right ) x^{7}+\left (a^{6} d \,e^{7}+21 a^{5} b \,d^{2} e^{6}+105 a^{4} b^{2} d^{3} e^{5}+175 a^{3} b^{3} d^{4} e^{4}+105 a^{2} b^{4} d^{5} e^{3}+21 a \,b^{5} d^{6} e^{2}+b^{6} d^{7} e \right ) x^{8}+\left (\frac {1}{9} a^{6} e^{8}+\frac {16}{3} a^{5} b d \,e^{7}+\frac {140}{3} a^{4} b^{2} d^{2} e^{6}+\frac {1120}{9} a^{3} b^{3} d^{3} e^{5}+\frac {350}{3} a^{2} b^{4} d^{4} e^{4}+\frac {112}{3} a \,b^{5} d^{5} e^{3}+\frac {28}{9} b^{6} d^{6} e^{2}\right ) x^{9}+\left (\frac {3}{5} a^{5} b \,e^{8}+12 a^{4} b^{2} d \,e^{7}+56 a^{3} b^{3} d^{2} e^{6}+84 a^{2} b^{4} d^{3} e^{5}+42 a \,b^{5} d^{4} e^{4}+\frac {28}{5} b^{6} d^{5} e^{3}\right ) x^{10}+\left (\frac {15}{11} a^{4} b^{2} e^{8}+\frac {160}{11} a^{3} b^{3} d \,e^{7}+\frac {420}{11} a^{2} b^{4} d^{2} e^{6}+\frac {336}{11} a \,b^{5} d^{3} e^{5}+\frac {70}{11} b^{6} d^{4} e^{4}\right ) x^{11}+\left (\frac {5}{3} a^{3} b^{3} e^{8}+10 a^{2} b^{4} d \,e^{7}+14 a \,b^{5} d^{2} e^{6}+\frac {14}{3} b^{6} d^{3} e^{5}\right ) x^{12}+\left (\frac {15}{13} a^{2} b^{4} e^{8}+\frac {48}{13} a \,b^{5} d \,e^{7}+\frac {28}{13} b^{6} d^{2} e^{6}\right ) x^{13}+\left (\frac {3}{7} a \,b^{5} e^{8}+\frac {4}{7} b^{6} d \,e^{7}\right ) x^{14}+\frac {b^{6} e^{8} x^{15}}{15}\) | \(789\) |
| default | \(\frac {b^{6} e^{8} x^{15}}{15}+\frac {\left (6 a \,b^{5} e^{8}+8 b^{6} d \,e^{7}\right ) x^{14}}{14}+\frac {\left (15 a^{2} b^{4} e^{8}+48 a \,b^{5} d \,e^{7}+28 b^{6} d^{2} e^{6}\right ) x^{13}}{13}+\frac {\left (20 a^{3} b^{3} e^{8}+120 a^{2} b^{4} d \,e^{7}+168 a \,b^{5} d^{2} e^{6}+56 b^{6} d^{3} e^{5}\right ) x^{12}}{12}+\frac {\left (15 a^{4} b^{2} e^{8}+160 a^{3} b^{3} d \,e^{7}+420 a^{2} b^{4} d^{2} e^{6}+336 a \,b^{5} d^{3} e^{5}+70 b^{6} d^{4} e^{4}\right ) x^{11}}{11}+\frac {\left (6 a^{5} b \,e^{8}+120 a^{4} b^{2} d \,e^{7}+560 a^{3} b^{3} d^{2} e^{6}+840 a^{2} b^{4} d^{3} e^{5}+420 a \,b^{5} d^{4} e^{4}+56 b^{6} d^{5} e^{3}\right ) x^{10}}{10}+\frac {\left (a^{6} e^{8}+48 a^{5} b d \,e^{7}+420 a^{4} b^{2} d^{2} e^{6}+1120 a^{3} b^{3} d^{3} e^{5}+1050 a^{2} b^{4} d^{4} e^{4}+336 a \,b^{5} d^{5} e^{3}+28 b^{6} d^{6} e^{2}\right ) x^{9}}{9}+\frac {\left (8 a^{6} d \,e^{7}+168 a^{5} b \,d^{2} e^{6}+840 a^{4} b^{2} d^{3} e^{5}+1400 a^{3} b^{3} d^{4} e^{4}+840 a^{2} b^{4} d^{5} e^{3}+168 a \,b^{5} d^{6} e^{2}+8 b^{6} d^{7} e \right ) x^{8}}{8}+\frac {\left (28 a^{6} d^{2} e^{6}+336 a^{5} b \,d^{3} e^{5}+1050 a^{4} b^{2} d^{4} e^{4}+1120 a^{3} b^{3} d^{5} e^{3}+420 a^{2} b^{4} d^{6} e^{2}+48 a \,b^{5} d^{7} e +b^{6} d^{8}\right ) x^{7}}{7}+\frac {\left (56 a^{6} d^{3} e^{5}+420 a^{5} b \,d^{4} e^{4}+840 a^{4} b^{2} d^{5} e^{3}+560 a^{3} b^{3} d^{6} e^{2}+120 a^{2} b^{4} d^{7} e +6 a \,b^{5} d^{8}\right ) x^{6}}{6}+\frac {\left (70 a^{6} d^{4} e^{4}+336 a^{5} b \,d^{5} e^{3}+420 a^{4} b^{2} d^{6} e^{2}+160 a^{3} b^{3} d^{7} e +15 a^{2} b^{4} d^{8}\right ) x^{5}}{5}+\frac {\left (56 a^{6} d^{5} e^{3}+168 a^{5} b \,d^{6} e^{2}+120 a^{4} b^{2} d^{7} e +20 a^{3} b^{3} d^{8}\right ) x^{4}}{4}+\frac {\left (28 a^{6} d^{6} e^{2}+48 a^{5} b \,d^{7} e +15 a^{4} b^{2} d^{8}\right ) x^{3}}{3}+\frac {\left (8 a^{6} d^{7} e +6 a^{5} b \,d^{8}\right ) x^{2}}{2}+a^{6} d^{8} x\) | \(803\) |
| risch | \(\frac {1}{7} x^{7} b^{6} d^{8}+16 x^{3} a^{5} b \,d^{7} e +105 a^{4} b^{2} d^{3} e^{5} x^{8}+\frac {15}{13} x^{13} a^{2} b^{4} e^{8}+\frac {28}{13} x^{13} b^{6} d^{2} e^{6}+\frac {5}{3} x^{12} a^{3} b^{3} e^{8}+\frac {3}{7} x^{14} a \,b^{5} e^{8}+\frac {4}{7} x^{14} b^{6} d \,e^{7}+5 a^{3} b^{3} d^{8} x^{4}+4 a^{6} d^{7} e \,x^{2}+3 a^{5} b \,d^{8} x^{2}+\frac {1}{9} x^{9} a^{6} e^{8}+\frac {112}{3} x^{9} a \,b^{5} d^{5} e^{3}+48 x^{7} a^{5} b \,d^{3} e^{5}+150 x^{7} a^{4} b^{2} d^{4} e^{4}+160 x^{7} a^{3} b^{3} d^{5} e^{3}+60 x^{7} a^{2} b^{4} d^{6} e^{2}+\frac {48}{7} x^{7} a \,b^{5} d^{7} e +70 x^{6} a^{5} b \,d^{4} e^{4}+140 x^{6} a^{4} b^{2} d^{5} e^{3}+\frac {280}{3} x^{6} a^{3} b^{3} d^{6} e^{2}+20 x^{6} a^{2} b^{4} d^{7} e +5 x^{3} a^{4} b^{2} d^{8}+a^{6} d \,e^{7} x^{8}+b^{6} d^{7} e \,x^{8}+\frac {336}{5} x^{5} a^{5} b \,d^{5} e^{3}+84 x^{5} a^{4} b^{2} d^{6} e^{2}+32 x^{5} a^{3} b^{3} d^{7} e +a^{6} d^{8} x +\frac {1}{15} b^{6} e^{8} x^{15}+\frac {160}{11} x^{11} a^{3} b^{3} d \,e^{7}+\frac {420}{11} x^{11} a^{2} b^{4} d^{2} e^{6}+\frac {336}{11} x^{11} a \,b^{5} d^{3} e^{5}+12 x^{10} a^{4} b^{2} d \,e^{7}+56 x^{10} a^{3} b^{3} d^{2} e^{6}+84 x^{10} a^{2} b^{4} d^{3} e^{5}+42 x^{10} a \,b^{5} d^{4} e^{4}+\frac {16}{3} x^{9} a^{5} b d \,e^{7}+\frac {140}{3} x^{9} a^{4} b^{2} d^{2} e^{6}+\frac {1120}{9} x^{9} a^{3} b^{3} d^{3} e^{5}+\frac {350}{3} x^{9} a^{2} b^{4} d^{4} e^{4}+4 x^{7} a^{6} d^{2} e^{6}+\frac {28}{3} x^{6} a^{6} d^{3} e^{5}+x^{6} a \,b^{5} d^{8}+14 x^{5} a^{6} d^{4} e^{4}+3 x^{5} a^{2} b^{4} d^{8}+\frac {28}{3} x^{3} a^{6} d^{6} e^{2}+21 a^{5} b \,d^{2} e^{6} x^{8}+\frac {48}{13} x^{13} a \,b^{5} d \,e^{7}+10 x^{12} a^{2} b^{4} d \,e^{7}+14 x^{12} a \,b^{5} d^{2} e^{6}+\frac {14}{3} x^{12} b^{6} d^{3} e^{5}+\frac {15}{11} x^{11} a^{4} b^{2} e^{8}+\frac {70}{11} x^{11} b^{6} d^{4} e^{4}+\frac {3}{5} x^{10} a^{5} b \,e^{8}+\frac {28}{5} x^{10} b^{6} d^{5} e^{3}+\frac {28}{9} x^{9} b^{6} d^{6} e^{2}+175 a^{3} b^{3} d^{4} e^{4} x^{8}+105 a^{2} b^{4} d^{5} e^{3} x^{8}+21 a \,b^{5} d^{6} e^{2} x^{8}+42 a^{5} b \,d^{6} e^{2} x^{4}+30 a^{4} b^{2} d^{7} e \,x^{4}+14 a^{6} d^{5} e^{3} x^{4}\) | \(907\) |
| parallelrisch | \(\frac {1}{7} x^{7} b^{6} d^{8}+16 x^{3} a^{5} b \,d^{7} e +105 a^{4} b^{2} d^{3} e^{5} x^{8}+\frac {15}{13} x^{13} a^{2} b^{4} e^{8}+\frac {28}{13} x^{13} b^{6} d^{2} e^{6}+\frac {5}{3} x^{12} a^{3} b^{3} e^{8}+\frac {3}{7} x^{14} a \,b^{5} e^{8}+\frac {4}{7} x^{14} b^{6} d \,e^{7}+5 a^{3} b^{3} d^{8} x^{4}+4 a^{6} d^{7} e \,x^{2}+3 a^{5} b \,d^{8} x^{2}+\frac {1}{9} x^{9} a^{6} e^{8}+\frac {112}{3} x^{9} a \,b^{5} d^{5} e^{3}+48 x^{7} a^{5} b \,d^{3} e^{5}+150 x^{7} a^{4} b^{2} d^{4} e^{4}+160 x^{7} a^{3} b^{3} d^{5} e^{3}+60 x^{7} a^{2} b^{4} d^{6} e^{2}+\frac {48}{7} x^{7} a \,b^{5} d^{7} e +70 x^{6} a^{5} b \,d^{4} e^{4}+140 x^{6} a^{4} b^{2} d^{5} e^{3}+\frac {280}{3} x^{6} a^{3} b^{3} d^{6} e^{2}+20 x^{6} a^{2} b^{4} d^{7} e +5 x^{3} a^{4} b^{2} d^{8}+a^{6} d \,e^{7} x^{8}+b^{6} d^{7} e \,x^{8}+\frac {336}{5} x^{5} a^{5} b \,d^{5} e^{3}+84 x^{5} a^{4} b^{2} d^{6} e^{2}+32 x^{5} a^{3} b^{3} d^{7} e +a^{6} d^{8} x +\frac {1}{15} b^{6} e^{8} x^{15}+\frac {160}{11} x^{11} a^{3} b^{3} d \,e^{7}+\frac {420}{11} x^{11} a^{2} b^{4} d^{2} e^{6}+\frac {336}{11} x^{11} a \,b^{5} d^{3} e^{5}+12 x^{10} a^{4} b^{2} d \,e^{7}+56 x^{10} a^{3} b^{3} d^{2} e^{6}+84 x^{10} a^{2} b^{4} d^{3} e^{5}+42 x^{10} a \,b^{5} d^{4} e^{4}+\frac {16}{3} x^{9} a^{5} b d \,e^{7}+\frac {140}{3} x^{9} a^{4} b^{2} d^{2} e^{6}+\frac {1120}{9} x^{9} a^{3} b^{3} d^{3} e^{5}+\frac {350}{3} x^{9} a^{2} b^{4} d^{4} e^{4}+4 x^{7} a^{6} d^{2} e^{6}+\frac {28}{3} x^{6} a^{6} d^{3} e^{5}+x^{6} a \,b^{5} d^{8}+14 x^{5} a^{6} d^{4} e^{4}+3 x^{5} a^{2} b^{4} d^{8}+\frac {28}{3} x^{3} a^{6} d^{6} e^{2}+21 a^{5} b \,d^{2} e^{6} x^{8}+\frac {48}{13} x^{13} a \,b^{5} d \,e^{7}+10 x^{12} a^{2} b^{4} d \,e^{7}+14 x^{12} a \,b^{5} d^{2} e^{6}+\frac {14}{3} x^{12} b^{6} d^{3} e^{5}+\frac {15}{11} x^{11} a^{4} b^{2} e^{8}+\frac {70}{11} x^{11} b^{6} d^{4} e^{4}+\frac {3}{5} x^{10} a^{5} b \,e^{8}+\frac {28}{5} x^{10} b^{6} d^{5} e^{3}+\frac {28}{9} x^{9} b^{6} d^{6} e^{2}+175 a^{3} b^{3} d^{4} e^{4} x^{8}+105 a^{2} b^{4} d^{5} e^{3} x^{8}+21 a \,b^{5} d^{6} e^{2} x^{8}+42 a^{5} b \,d^{6} e^{2} x^{4}+30 a^{4} b^{2} d^{7} e \,x^{4}+14 a^{6} d^{5} e^{3} x^{4}\) | \(907\) |
| gosper | \(\frac {x \left (3003 b^{6} e^{8} x^{14}+19305 x^{13} a \,b^{5} e^{8}+25740 x^{13} b^{6} d \,e^{7}+51975 x^{12} a^{2} b^{4} e^{8}+166320 x^{12} a \,b^{5} d \,e^{7}+97020 x^{12} b^{6} d^{2} e^{6}+75075 x^{11} a^{3} b^{3} e^{8}+450450 x^{11} a^{2} b^{4} d \,e^{7}+630630 x^{11} a \,b^{5} d^{2} e^{6}+210210 x^{11} b^{6} d^{3} e^{5}+61425 x^{10} a^{4} b^{2} e^{8}+655200 x^{10} a^{3} b^{3} d \,e^{7}+1719900 x^{10} a^{2} b^{4} d^{2} e^{6}+1375920 x^{10} a \,b^{5} d^{3} e^{5}+286650 x^{10} b^{6} d^{4} e^{4}+27027 x^{9} a^{5} b \,e^{8}+540540 x^{9} a^{4} b^{2} d \,e^{7}+2522520 x^{9} a^{3} b^{3} d^{2} e^{6}+3783780 x^{9} a^{2} b^{4} d^{3} e^{5}+1891890 x^{9} a \,b^{5} d^{4} e^{4}+252252 x^{9} b^{6} d^{5} e^{3}+5005 x^{8} a^{6} e^{8}+240240 x^{8} a^{5} b d \,e^{7}+2102100 x^{8} a^{4} b^{2} d^{2} e^{6}+5605600 x^{8} a^{3} b^{3} d^{3} e^{5}+5255250 x^{8} a^{2} b^{4} d^{4} e^{4}+1681680 x^{8} a \,b^{5} d^{5} e^{3}+140140 x^{8} b^{6} d^{6} e^{2}+45045 a^{6} d \,e^{7} x^{7}+945945 a^{5} b \,d^{2} e^{6} x^{7}+4729725 a^{4} b^{2} d^{3} e^{5} x^{7}+7882875 a^{3} b^{3} d^{4} e^{4} x^{7}+4729725 a^{2} b^{4} d^{5} e^{3} x^{7}+945945 a \,b^{5} d^{6} e^{2} x^{7}+45045 b^{6} d^{7} e \,x^{7}+180180 x^{6} a^{6} d^{2} e^{6}+2162160 x^{6} a^{5} b \,d^{3} e^{5}+6756750 x^{6} a^{4} b^{2} d^{4} e^{4}+7207200 x^{6} a^{3} b^{3} d^{5} e^{3}+2702700 x^{6} a^{2} b^{4} d^{6} e^{2}+308880 x^{6} a \,b^{5} d^{7} e +6435 x^{6} b^{6} d^{8}+420420 x^{5} a^{6} d^{3} e^{5}+3153150 x^{5} a^{5} b \,d^{4} e^{4}+6306300 x^{5} a^{4} b^{2} d^{5} e^{3}+4204200 x^{5} a^{3} b^{3} d^{6} e^{2}+900900 x^{5} a^{2} b^{4} d^{7} e +45045 x^{5} a \,b^{5} d^{8}+630630 x^{4} a^{6} d^{4} e^{4}+3027024 x^{4} a^{5} b \,d^{5} e^{3}+3783780 x^{4} a^{4} b^{2} d^{6} e^{2}+1441440 x^{4} a^{3} b^{3} d^{7} e +135135 x^{4} a^{2} b^{4} d^{8}+630630 a^{6} d^{5} e^{3} x^{3}+1891890 a^{5} b \,d^{6} e^{2} x^{3}+1351350 a^{4} b^{2} d^{7} e \,x^{3}+225225 a^{3} b^{3} d^{8} x^{3}+420420 x^{2} a^{6} d^{6} e^{2}+720720 x^{2} a^{5} b \,d^{7} e +225225 x^{2} a^{4} b^{2} d^{8}+180180 a^{6} d^{7} e x +135135 a^{5} b \,d^{8} x +45045 a^{6} d^{8}\right )}{45045}\) | \(909\) |
a^6*d^8*x+(4*a^6*d^7*e+3*a^5*b*d^8)*x^2+(28/3*a^6*d^6*e^2+16*a^5*b*d^7*e+5 *a^4*b^2*d^8)*x^3+(14*a^6*d^5*e^3+42*a^5*b*d^6*e^2+30*a^4*b^2*d^7*e+5*a^3* b^3*d^8)*x^4+(14*a^6*d^4*e^4+336/5*a^5*b*d^5*e^3+84*a^4*b^2*d^6*e^2+32*a^3 *b^3*d^7*e+3*a^2*b^4*d^8)*x^5+(28/3*a^6*d^3*e^5+70*a^5*b*d^4*e^4+140*a^4*b ^2*d^5*e^3+280/3*a^3*b^3*d^6*e^2+20*a^2*b^4*d^7*e+a*b^5*d^8)*x^6+(4*a^6*d^ 2*e^6+48*a^5*b*d^3*e^5+150*a^4*b^2*d^4*e^4+160*a^3*b^3*d^5*e^3+60*a^2*b^4* d^6*e^2+48/7*a*b^5*d^7*e+1/7*b^6*d^8)*x^7+(a^6*d*e^7+21*a^5*b*d^2*e^6+105* a^4*b^2*d^3*e^5+175*a^3*b^3*d^4*e^4+105*a^2*b^4*d^5*e^3+21*a*b^5*d^6*e^2+b ^6*d^7*e)*x^8+(1/9*a^6*e^8+16/3*a^5*b*d*e^7+140/3*a^4*b^2*d^2*e^6+1120/9*a ^3*b^3*d^3*e^5+350/3*a^2*b^4*d^4*e^4+112/3*a*b^5*d^5*e^3+28/9*b^6*d^6*e^2) *x^9+(3/5*a^5*b*e^8+12*a^4*b^2*d*e^7+56*a^3*b^3*d^2*e^6+84*a^2*b^4*d^3*e^5 +42*a*b^5*d^4*e^4+28/5*b^6*d^5*e^3)*x^10+(15/11*a^4*b^2*e^8+160/11*a^3*b^3 *d*e^7+420/11*a^2*b^4*d^2*e^6+336/11*a*b^5*d^3*e^5+70/11*b^6*d^4*e^4)*x^11 +(5/3*a^3*b^3*e^8+10*a^2*b^4*d*e^7+14*a*b^5*d^2*e^6+14/3*b^6*d^3*e^5)*x^12 +(15/13*a^2*b^4*e^8+48/13*a*b^5*d*e^7+28/13*b^6*d^2*e^6)*x^13+(3/7*a*b^5*e ^8+4/7*b^6*d*e^7)*x^14+1/15*b^6*e^8*x^15
Leaf count of result is larger than twice the leaf count of optimal. 797 vs. \(2 (159) = 318\).
Time = 0.25 (sec) , antiderivative size = 797, normalized size of antiderivative = 4.61 \[ \int (d+e x)^8 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{15} \, b^{6} e^{8} x^{15} + a^{6} d^{8} x + \frac {1}{7} \, {\left (4 \, b^{6} d e^{7} + 3 \, a b^{5} e^{8}\right )} x^{14} + \frac {1}{13} \, {\left (28 \, b^{6} d^{2} e^{6} + 48 \, a b^{5} d e^{7} + 15 \, a^{2} b^{4} e^{8}\right )} x^{13} + \frac {1}{3} \, {\left (14 \, b^{6} d^{3} e^{5} + 42 \, a b^{5} d^{2} e^{6} + 30 \, a^{2} b^{4} d e^{7} + 5 \, a^{3} b^{3} e^{8}\right )} x^{12} + \frac {1}{11} \, {\left (70 \, b^{6} d^{4} e^{4} + 336 \, a b^{5} d^{3} e^{5} + 420 \, a^{2} b^{4} d^{2} e^{6} + 160 \, a^{3} b^{3} d e^{7} + 15 \, a^{4} b^{2} e^{8}\right )} x^{11} + \frac {1}{5} \, {\left (28 \, b^{6} d^{5} e^{3} + 210 \, a b^{5} d^{4} e^{4} + 420 \, a^{2} b^{4} d^{3} e^{5} + 280 \, a^{3} b^{3} d^{2} e^{6} + 60 \, a^{4} b^{2} d e^{7} + 3 \, a^{5} b e^{8}\right )} x^{10} + \frac {1}{9} \, {\left (28 \, b^{6} d^{6} e^{2} + 336 \, a b^{5} d^{5} e^{3} + 1050 \, a^{2} b^{4} d^{4} e^{4} + 1120 \, a^{3} b^{3} d^{3} e^{5} + 420 \, a^{4} b^{2} d^{2} e^{6} + 48 \, a^{5} b d e^{7} + a^{6} e^{8}\right )} x^{9} + {\left (b^{6} d^{7} e + 21 \, a b^{5} d^{6} e^{2} + 105 \, a^{2} b^{4} d^{5} e^{3} + 175 \, a^{3} b^{3} d^{4} e^{4} + 105 \, a^{4} b^{2} d^{3} e^{5} + 21 \, a^{5} b d^{2} e^{6} + a^{6} d e^{7}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{8} + 48 \, a b^{5} d^{7} e + 420 \, a^{2} b^{4} d^{6} e^{2} + 1120 \, a^{3} b^{3} d^{5} e^{3} + 1050 \, a^{4} b^{2} d^{4} e^{4} + 336 \, a^{5} b d^{3} e^{5} + 28 \, a^{6} d^{2} e^{6}\right )} x^{7} + \frac {1}{3} \, {\left (3 \, a b^{5} d^{8} + 60 \, a^{2} b^{4} d^{7} e + 280 \, a^{3} b^{3} d^{6} e^{2} + 420 \, a^{4} b^{2} d^{5} e^{3} + 210 \, a^{5} b d^{4} e^{4} + 28 \, a^{6} d^{3} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (15 \, a^{2} b^{4} d^{8} + 160 \, a^{3} b^{3} d^{7} e + 420 \, a^{4} b^{2} d^{6} e^{2} + 336 \, a^{5} b d^{5} e^{3} + 70 \, a^{6} d^{4} e^{4}\right )} x^{5} + {\left (5 \, a^{3} b^{3} d^{8} + 30 \, a^{4} b^{2} d^{7} e + 42 \, a^{5} b d^{6} e^{2} + 14 \, a^{6} d^{5} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (15 \, a^{4} b^{2} d^{8} + 48 \, a^{5} b d^{7} e + 28 \, a^{6} d^{6} e^{2}\right )} x^{3} + {\left (3 \, a^{5} b d^{8} + 4 \, a^{6} d^{7} e\right )} x^{2} \]
1/15*b^6*e^8*x^15 + a^6*d^8*x + 1/7*(4*b^6*d*e^7 + 3*a*b^5*e^8)*x^14 + 1/1 3*(28*b^6*d^2*e^6 + 48*a*b^5*d*e^7 + 15*a^2*b^4*e^8)*x^13 + 1/3*(14*b^6*d^ 3*e^5 + 42*a*b^5*d^2*e^6 + 30*a^2*b^4*d*e^7 + 5*a^3*b^3*e^8)*x^12 + 1/11*( 70*b^6*d^4*e^4 + 336*a*b^5*d^3*e^5 + 420*a^2*b^4*d^2*e^6 + 160*a^3*b^3*d*e ^7 + 15*a^4*b^2*e^8)*x^11 + 1/5*(28*b^6*d^5*e^3 + 210*a*b^5*d^4*e^4 + 420* a^2*b^4*d^3*e^5 + 280*a^3*b^3*d^2*e^6 + 60*a^4*b^2*d*e^7 + 3*a^5*b*e^8)*x^ 10 + 1/9*(28*b^6*d^6*e^2 + 336*a*b^5*d^5*e^3 + 1050*a^2*b^4*d^4*e^4 + 1120 *a^3*b^3*d^3*e^5 + 420*a^4*b^2*d^2*e^6 + 48*a^5*b*d*e^7 + a^6*e^8)*x^9 + ( b^6*d^7*e + 21*a*b^5*d^6*e^2 + 105*a^2*b^4*d^5*e^3 + 175*a^3*b^3*d^4*e^4 + 105*a^4*b^2*d^3*e^5 + 21*a^5*b*d^2*e^6 + a^6*d*e^7)*x^8 + 1/7*(b^6*d^8 + 48*a*b^5*d^7*e + 420*a^2*b^4*d^6*e^2 + 1120*a^3*b^3*d^5*e^3 + 1050*a^4*b^2 *d^4*e^4 + 336*a^5*b*d^3*e^5 + 28*a^6*d^2*e^6)*x^7 + 1/3*(3*a*b^5*d^8 + 60 *a^2*b^4*d^7*e + 280*a^3*b^3*d^6*e^2 + 420*a^4*b^2*d^5*e^3 + 210*a^5*b*d^4 *e^4 + 28*a^6*d^3*e^5)*x^6 + 1/5*(15*a^2*b^4*d^8 + 160*a^3*b^3*d^7*e + 420 *a^4*b^2*d^6*e^2 + 336*a^5*b*d^5*e^3 + 70*a^6*d^4*e^4)*x^5 + (5*a^3*b^3*d^ 8 + 30*a^4*b^2*d^7*e + 42*a^5*b*d^6*e^2 + 14*a^6*d^5*e^3)*x^4 + 1/3*(15*a^ 4*b^2*d^8 + 48*a^5*b*d^7*e + 28*a^6*d^6*e^2)*x^3 + (3*a^5*b*d^8 + 4*a^6*d^ 7*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 884 vs. \(2 (158) = 316\).
Time = 0.08 (sec) , antiderivative size = 884, normalized size of antiderivative = 5.11 \[ \int (d+e x)^8 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=a^{6} d^{8} x + \frac {b^{6} e^{8} x^{15}}{15} + x^{14} \cdot \left (\frac {3 a b^{5} e^{8}}{7} + \frac {4 b^{6} d e^{7}}{7}\right ) + x^{13} \cdot \left (\frac {15 a^{2} b^{4} e^{8}}{13} + \frac {48 a b^{5} d e^{7}}{13} + \frac {28 b^{6} d^{2} e^{6}}{13}\right ) + x^{12} \cdot \left (\frac {5 a^{3} b^{3} e^{8}}{3} + 10 a^{2} b^{4} d e^{7} + 14 a b^{5} d^{2} e^{6} + \frac {14 b^{6} d^{3} e^{5}}{3}\right ) + x^{11} \cdot \left (\frac {15 a^{4} b^{2} e^{8}}{11} + \frac {160 a^{3} b^{3} d e^{7}}{11} + \frac {420 a^{2} b^{4} d^{2} e^{6}}{11} + \frac {336 a b^{5} d^{3} e^{5}}{11} + \frac {70 b^{6} d^{4} e^{4}}{11}\right ) + x^{10} \cdot \left (\frac {3 a^{5} b e^{8}}{5} + 12 a^{4} b^{2} d e^{7} + 56 a^{3} b^{3} d^{2} e^{6} + 84 a^{2} b^{4} d^{3} e^{5} + 42 a b^{5} d^{4} e^{4} + \frac {28 b^{6} d^{5} e^{3}}{5}\right ) + x^{9} \left (\frac {a^{6} e^{8}}{9} + \frac {16 a^{5} b d e^{7}}{3} + \frac {140 a^{4} b^{2} d^{2} e^{6}}{3} + \frac {1120 a^{3} b^{3} d^{3} e^{5}}{9} + \frac {350 a^{2} b^{4} d^{4} e^{4}}{3} + \frac {112 a b^{5} d^{5} e^{3}}{3} + \frac {28 b^{6} d^{6} e^{2}}{9}\right ) + x^{8} \left (a^{6} d e^{7} + 21 a^{5} b d^{2} e^{6} + 105 a^{4} b^{2} d^{3} e^{5} + 175 a^{3} b^{3} d^{4} e^{4} + 105 a^{2} b^{4} d^{5} e^{3} + 21 a b^{5} d^{6} e^{2} + b^{6} d^{7} e\right ) + x^{7} \cdot \left (4 a^{6} d^{2} e^{6} + 48 a^{5} b d^{3} e^{5} + 150 a^{4} b^{2} d^{4} e^{4} + 160 a^{3} b^{3} d^{5} e^{3} + 60 a^{2} b^{4} d^{6} e^{2} + \frac {48 a b^{5} d^{7} e}{7} + \frac {b^{6} d^{8}}{7}\right ) + x^{6} \cdot \left (\frac {28 a^{6} d^{3} e^{5}}{3} + 70 a^{5} b d^{4} e^{4} + 140 a^{4} b^{2} d^{5} e^{3} + \frac {280 a^{3} b^{3} d^{6} e^{2}}{3} + 20 a^{2} b^{4} d^{7} e + a b^{5} d^{8}\right ) + x^{5} \cdot \left (14 a^{6} d^{4} e^{4} + \frac {336 a^{5} b d^{5} e^{3}}{5} + 84 a^{4} b^{2} d^{6} e^{2} + 32 a^{3} b^{3} d^{7} e + 3 a^{2} b^{4} d^{8}\right ) + x^{4} \cdot \left (14 a^{6} d^{5} e^{3} + 42 a^{5} b d^{6} e^{2} + 30 a^{4} b^{2} d^{7} e + 5 a^{3} b^{3} d^{8}\right ) + x^{3} \cdot \left (\frac {28 a^{6} d^{6} e^{2}}{3} + 16 a^{5} b d^{7} e + 5 a^{4} b^{2} d^{8}\right ) + x^{2} \cdot \left (4 a^{6} d^{7} e + 3 a^{5} b d^{8}\right ) \]
a**6*d**8*x + b**6*e**8*x**15/15 + x**14*(3*a*b**5*e**8/7 + 4*b**6*d*e**7/ 7) + x**13*(15*a**2*b**4*e**8/13 + 48*a*b**5*d*e**7/13 + 28*b**6*d**2*e**6 /13) + x**12*(5*a**3*b**3*e**8/3 + 10*a**2*b**4*d*e**7 + 14*a*b**5*d**2*e* *6 + 14*b**6*d**3*e**5/3) + x**11*(15*a**4*b**2*e**8/11 + 160*a**3*b**3*d* e**7/11 + 420*a**2*b**4*d**2*e**6/11 + 336*a*b**5*d**3*e**5/11 + 70*b**6*d **4*e**4/11) + x**10*(3*a**5*b*e**8/5 + 12*a**4*b**2*d*e**7 + 56*a**3*b**3 *d**2*e**6 + 84*a**2*b**4*d**3*e**5 + 42*a*b**5*d**4*e**4 + 28*b**6*d**5*e **3/5) + x**9*(a**6*e**8/9 + 16*a**5*b*d*e**7/3 + 140*a**4*b**2*d**2*e**6/ 3 + 1120*a**3*b**3*d**3*e**5/9 + 350*a**2*b**4*d**4*e**4/3 + 112*a*b**5*d* *5*e**3/3 + 28*b**6*d**6*e**2/9) + x**8*(a**6*d*e**7 + 21*a**5*b*d**2*e**6 + 105*a**4*b**2*d**3*e**5 + 175*a**3*b**3*d**4*e**4 + 105*a**2*b**4*d**5* e**3 + 21*a*b**5*d**6*e**2 + b**6*d**7*e) + x**7*(4*a**6*d**2*e**6 + 48*a* *5*b*d**3*e**5 + 150*a**4*b**2*d**4*e**4 + 160*a**3*b**3*d**5*e**3 + 60*a* *2*b**4*d**6*e**2 + 48*a*b**5*d**7*e/7 + b**6*d**8/7) + x**6*(28*a**6*d**3 *e**5/3 + 70*a**5*b*d**4*e**4 + 140*a**4*b**2*d**5*e**3 + 280*a**3*b**3*d* *6*e**2/3 + 20*a**2*b**4*d**7*e + a*b**5*d**8) + x**5*(14*a**6*d**4*e**4 + 336*a**5*b*d**5*e**3/5 + 84*a**4*b**2*d**6*e**2 + 32*a**3*b**3*d**7*e + 3 *a**2*b**4*d**8) + x**4*(14*a**6*d**5*e**3 + 42*a**5*b*d**6*e**2 + 30*a**4 *b**2*d**7*e + 5*a**3*b**3*d**8) + x**3*(28*a**6*d**6*e**2/3 + 16*a**5*b*d **7*e + 5*a**4*b**2*d**8) + x**2*(4*a**6*d**7*e + 3*a**5*b*d**8)
Leaf count of result is larger than twice the leaf count of optimal. 797 vs. \(2 (159) = 318\).
Time = 0.20 (sec) , antiderivative size = 797, normalized size of antiderivative = 4.61 \[ \int (d+e x)^8 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{15} \, b^{6} e^{8} x^{15} + a^{6} d^{8} x + \frac {1}{7} \, {\left (4 \, b^{6} d e^{7} + 3 \, a b^{5} e^{8}\right )} x^{14} + \frac {1}{13} \, {\left (28 \, b^{6} d^{2} e^{6} + 48 \, a b^{5} d e^{7} + 15 \, a^{2} b^{4} e^{8}\right )} x^{13} + \frac {1}{3} \, {\left (14 \, b^{6} d^{3} e^{5} + 42 \, a b^{5} d^{2} e^{6} + 30 \, a^{2} b^{4} d e^{7} + 5 \, a^{3} b^{3} e^{8}\right )} x^{12} + \frac {1}{11} \, {\left (70 \, b^{6} d^{4} e^{4} + 336 \, a b^{5} d^{3} e^{5} + 420 \, a^{2} b^{4} d^{2} e^{6} + 160 \, a^{3} b^{3} d e^{7} + 15 \, a^{4} b^{2} e^{8}\right )} x^{11} + \frac {1}{5} \, {\left (28 \, b^{6} d^{5} e^{3} + 210 \, a b^{5} d^{4} e^{4} + 420 \, a^{2} b^{4} d^{3} e^{5} + 280 \, a^{3} b^{3} d^{2} e^{6} + 60 \, a^{4} b^{2} d e^{7} + 3 \, a^{5} b e^{8}\right )} x^{10} + \frac {1}{9} \, {\left (28 \, b^{6} d^{6} e^{2} + 336 \, a b^{5} d^{5} e^{3} + 1050 \, a^{2} b^{4} d^{4} e^{4} + 1120 \, a^{3} b^{3} d^{3} e^{5} + 420 \, a^{4} b^{2} d^{2} e^{6} + 48 \, a^{5} b d e^{7} + a^{6} e^{8}\right )} x^{9} + {\left (b^{6} d^{7} e + 21 \, a b^{5} d^{6} e^{2} + 105 \, a^{2} b^{4} d^{5} e^{3} + 175 \, a^{3} b^{3} d^{4} e^{4} + 105 \, a^{4} b^{2} d^{3} e^{5} + 21 \, a^{5} b d^{2} e^{6} + a^{6} d e^{7}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{8} + 48 \, a b^{5} d^{7} e + 420 \, a^{2} b^{4} d^{6} e^{2} + 1120 \, a^{3} b^{3} d^{5} e^{3} + 1050 \, a^{4} b^{2} d^{4} e^{4} + 336 \, a^{5} b d^{3} e^{5} + 28 \, a^{6} d^{2} e^{6}\right )} x^{7} + \frac {1}{3} \, {\left (3 \, a b^{5} d^{8} + 60 \, a^{2} b^{4} d^{7} e + 280 \, a^{3} b^{3} d^{6} e^{2} + 420 \, a^{4} b^{2} d^{5} e^{3} + 210 \, a^{5} b d^{4} e^{4} + 28 \, a^{6} d^{3} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (15 \, a^{2} b^{4} d^{8} + 160 \, a^{3} b^{3} d^{7} e + 420 \, a^{4} b^{2} d^{6} e^{2} + 336 \, a^{5} b d^{5} e^{3} + 70 \, a^{6} d^{4} e^{4}\right )} x^{5} + {\left (5 \, a^{3} b^{3} d^{8} + 30 \, a^{4} b^{2} d^{7} e + 42 \, a^{5} b d^{6} e^{2} + 14 \, a^{6} d^{5} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (15 \, a^{4} b^{2} d^{8} + 48 \, a^{5} b d^{7} e + 28 \, a^{6} d^{6} e^{2}\right )} x^{3} + {\left (3 \, a^{5} b d^{8} + 4 \, a^{6} d^{7} e\right )} x^{2} \]
1/15*b^6*e^8*x^15 + a^6*d^8*x + 1/7*(4*b^6*d*e^7 + 3*a*b^5*e^8)*x^14 + 1/1 3*(28*b^6*d^2*e^6 + 48*a*b^5*d*e^7 + 15*a^2*b^4*e^8)*x^13 + 1/3*(14*b^6*d^ 3*e^5 + 42*a*b^5*d^2*e^6 + 30*a^2*b^4*d*e^7 + 5*a^3*b^3*e^8)*x^12 + 1/11*( 70*b^6*d^4*e^4 + 336*a*b^5*d^3*e^5 + 420*a^2*b^4*d^2*e^6 + 160*a^3*b^3*d*e ^7 + 15*a^4*b^2*e^8)*x^11 + 1/5*(28*b^6*d^5*e^3 + 210*a*b^5*d^4*e^4 + 420* a^2*b^4*d^3*e^5 + 280*a^3*b^3*d^2*e^6 + 60*a^4*b^2*d*e^7 + 3*a^5*b*e^8)*x^ 10 + 1/9*(28*b^6*d^6*e^2 + 336*a*b^5*d^5*e^3 + 1050*a^2*b^4*d^4*e^4 + 1120 *a^3*b^3*d^3*e^5 + 420*a^4*b^2*d^2*e^6 + 48*a^5*b*d*e^7 + a^6*e^8)*x^9 + ( b^6*d^7*e + 21*a*b^5*d^6*e^2 + 105*a^2*b^4*d^5*e^3 + 175*a^3*b^3*d^4*e^4 + 105*a^4*b^2*d^3*e^5 + 21*a^5*b*d^2*e^6 + a^6*d*e^7)*x^8 + 1/7*(b^6*d^8 + 48*a*b^5*d^7*e + 420*a^2*b^4*d^6*e^2 + 1120*a^3*b^3*d^5*e^3 + 1050*a^4*b^2 *d^4*e^4 + 336*a^5*b*d^3*e^5 + 28*a^6*d^2*e^6)*x^7 + 1/3*(3*a*b^5*d^8 + 60 *a^2*b^4*d^7*e + 280*a^3*b^3*d^6*e^2 + 420*a^4*b^2*d^5*e^3 + 210*a^5*b*d^4 *e^4 + 28*a^6*d^3*e^5)*x^6 + 1/5*(15*a^2*b^4*d^8 + 160*a^3*b^3*d^7*e + 420 *a^4*b^2*d^6*e^2 + 336*a^5*b*d^5*e^3 + 70*a^6*d^4*e^4)*x^5 + (5*a^3*b^3*d^ 8 + 30*a^4*b^2*d^7*e + 42*a^5*b*d^6*e^2 + 14*a^6*d^5*e^3)*x^4 + 1/3*(15*a^ 4*b^2*d^8 + 48*a^5*b*d^7*e + 28*a^6*d^6*e^2)*x^3 + (3*a^5*b*d^8 + 4*a^6*d^ 7*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 906 vs. \(2 (159) = 318\).
Time = 0.27 (sec) , antiderivative size = 906, normalized size of antiderivative = 5.24 \[ \int (d+e x)^8 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{15} \, b^{6} e^{8} x^{15} + \frac {4}{7} \, b^{6} d e^{7} x^{14} + \frac {3}{7} \, a b^{5} e^{8} x^{14} + \frac {28}{13} \, b^{6} d^{2} e^{6} x^{13} + \frac {48}{13} \, a b^{5} d e^{7} x^{13} + \frac {15}{13} \, a^{2} b^{4} e^{8} x^{13} + \frac {14}{3} \, b^{6} d^{3} e^{5} x^{12} + 14 \, a b^{5} d^{2} e^{6} x^{12} + 10 \, a^{2} b^{4} d e^{7} x^{12} + \frac {5}{3} \, a^{3} b^{3} e^{8} x^{12} + \frac {70}{11} \, b^{6} d^{4} e^{4} x^{11} + \frac {336}{11} \, a b^{5} d^{3} e^{5} x^{11} + \frac {420}{11} \, a^{2} b^{4} d^{2} e^{6} x^{11} + \frac {160}{11} \, a^{3} b^{3} d e^{7} x^{11} + \frac {15}{11} \, a^{4} b^{2} e^{8} x^{11} + \frac {28}{5} \, b^{6} d^{5} e^{3} x^{10} + 42 \, a b^{5} d^{4} e^{4} x^{10} + 84 \, a^{2} b^{4} d^{3} e^{5} x^{10} + 56 \, a^{3} b^{3} d^{2} e^{6} x^{10} + 12 \, a^{4} b^{2} d e^{7} x^{10} + \frac {3}{5} \, a^{5} b e^{8} x^{10} + \frac {28}{9} \, b^{6} d^{6} e^{2} x^{9} + \frac {112}{3} \, a b^{5} d^{5} e^{3} x^{9} + \frac {350}{3} \, a^{2} b^{4} d^{4} e^{4} x^{9} + \frac {1120}{9} \, a^{3} b^{3} d^{3} e^{5} x^{9} + \frac {140}{3} \, a^{4} b^{2} d^{2} e^{6} x^{9} + \frac {16}{3} \, a^{5} b d e^{7} x^{9} + \frac {1}{9} \, a^{6} e^{8} x^{9} + b^{6} d^{7} e x^{8} + 21 \, a b^{5} d^{6} e^{2} x^{8} + 105 \, a^{2} b^{4} d^{5} e^{3} x^{8} + 175 \, a^{3} b^{3} d^{4} e^{4} x^{8} + 105 \, a^{4} b^{2} d^{3} e^{5} x^{8} + 21 \, a^{5} b d^{2} e^{6} x^{8} + a^{6} d e^{7} x^{8} + \frac {1}{7} \, b^{6} d^{8} x^{7} + \frac {48}{7} \, a b^{5} d^{7} e x^{7} + 60 \, a^{2} b^{4} d^{6} e^{2} x^{7} + 160 \, a^{3} b^{3} d^{5} e^{3} x^{7} + 150 \, a^{4} b^{2} d^{4} e^{4} x^{7} + 48 \, a^{5} b d^{3} e^{5} x^{7} + 4 \, a^{6} d^{2} e^{6} x^{7} + a b^{5} d^{8} x^{6} + 20 \, a^{2} b^{4} d^{7} e x^{6} + \frac {280}{3} \, a^{3} b^{3} d^{6} e^{2} x^{6} + 140 \, a^{4} b^{2} d^{5} e^{3} x^{6} + 70 \, a^{5} b d^{4} e^{4} x^{6} + \frac {28}{3} \, a^{6} d^{3} e^{5} x^{6} + 3 \, a^{2} b^{4} d^{8} x^{5} + 32 \, a^{3} b^{3} d^{7} e x^{5} + 84 \, a^{4} b^{2} d^{6} e^{2} x^{5} + \frac {336}{5} \, a^{5} b d^{5} e^{3} x^{5} + 14 \, a^{6} d^{4} e^{4} x^{5} + 5 \, a^{3} b^{3} d^{8} x^{4} + 30 \, a^{4} b^{2} d^{7} e x^{4} + 42 \, a^{5} b d^{6} e^{2} x^{4} + 14 \, a^{6} d^{5} e^{3} x^{4} + 5 \, a^{4} b^{2} d^{8} x^{3} + 16 \, a^{5} b d^{7} e x^{3} + \frac {28}{3} \, a^{6} d^{6} e^{2} x^{3} + 3 \, a^{5} b d^{8} x^{2} + 4 \, a^{6} d^{7} e x^{2} + a^{6} d^{8} x \]
1/15*b^6*e^8*x^15 + 4/7*b^6*d*e^7*x^14 + 3/7*a*b^5*e^8*x^14 + 28/13*b^6*d^ 2*e^6*x^13 + 48/13*a*b^5*d*e^7*x^13 + 15/13*a^2*b^4*e^8*x^13 + 14/3*b^6*d^ 3*e^5*x^12 + 14*a*b^5*d^2*e^6*x^12 + 10*a^2*b^4*d*e^7*x^12 + 5/3*a^3*b^3*e ^8*x^12 + 70/11*b^6*d^4*e^4*x^11 + 336/11*a*b^5*d^3*e^5*x^11 + 420/11*a^2* b^4*d^2*e^6*x^11 + 160/11*a^3*b^3*d*e^7*x^11 + 15/11*a^4*b^2*e^8*x^11 + 28 /5*b^6*d^5*e^3*x^10 + 42*a*b^5*d^4*e^4*x^10 + 84*a^2*b^4*d^3*e^5*x^10 + 56 *a^3*b^3*d^2*e^6*x^10 + 12*a^4*b^2*d*e^7*x^10 + 3/5*a^5*b*e^8*x^10 + 28/9* b^6*d^6*e^2*x^9 + 112/3*a*b^5*d^5*e^3*x^9 + 350/3*a^2*b^4*d^4*e^4*x^9 + 11 20/9*a^3*b^3*d^3*e^5*x^9 + 140/3*a^4*b^2*d^2*e^6*x^9 + 16/3*a^5*b*d*e^7*x^ 9 + 1/9*a^6*e^8*x^9 + b^6*d^7*e*x^8 + 21*a*b^5*d^6*e^2*x^8 + 105*a^2*b^4*d ^5*e^3*x^8 + 175*a^3*b^3*d^4*e^4*x^8 + 105*a^4*b^2*d^3*e^5*x^8 + 21*a^5*b* d^2*e^6*x^8 + a^6*d*e^7*x^8 + 1/7*b^6*d^8*x^7 + 48/7*a*b^5*d^7*e*x^7 + 60* a^2*b^4*d^6*e^2*x^7 + 160*a^3*b^3*d^5*e^3*x^7 + 150*a^4*b^2*d^4*e^4*x^7 + 48*a^5*b*d^3*e^5*x^7 + 4*a^6*d^2*e^6*x^7 + a*b^5*d^8*x^6 + 20*a^2*b^4*d^7* e*x^6 + 280/3*a^3*b^3*d^6*e^2*x^6 + 140*a^4*b^2*d^5*e^3*x^6 + 70*a^5*b*d^4 *e^4*x^6 + 28/3*a^6*d^3*e^5*x^6 + 3*a^2*b^4*d^8*x^5 + 32*a^3*b^3*d^7*e*x^5 + 84*a^4*b^2*d^6*e^2*x^5 + 336/5*a^5*b*d^5*e^3*x^5 + 14*a^6*d^4*e^4*x^5 + 5*a^3*b^3*d^8*x^4 + 30*a^4*b^2*d^7*e*x^4 + 42*a^5*b*d^6*e^2*x^4 + 14*a^6* d^5*e^3*x^4 + 5*a^4*b^2*d^8*x^3 + 16*a^5*b*d^7*e*x^3 + 28/3*a^6*d^6*e^2*x^ 3 + 3*a^5*b*d^8*x^2 + 4*a^6*d^7*e*x^2 + a^6*d^8*x
Time = 10.25 (sec) , antiderivative size = 768, normalized size of antiderivative = 4.44 \[ \int (d+e x)^8 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=x^7\,\left (4\,a^6\,d^2\,e^6+48\,a^5\,b\,d^3\,e^5+150\,a^4\,b^2\,d^4\,e^4+160\,a^3\,b^3\,d^5\,e^3+60\,a^2\,b^4\,d^6\,e^2+\frac {48\,a\,b^5\,d^7\,e}{7}+\frac {b^6\,d^8}{7}\right )+x^9\,\left (\frac {a^6\,e^8}{9}+\frac {16\,a^5\,b\,d\,e^7}{3}+\frac {140\,a^4\,b^2\,d^2\,e^6}{3}+\frac {1120\,a^3\,b^3\,d^3\,e^5}{9}+\frac {350\,a^2\,b^4\,d^4\,e^4}{3}+\frac {112\,a\,b^5\,d^5\,e^3}{3}+\frac {28\,b^6\,d^6\,e^2}{9}\right )+x^5\,\left (14\,a^6\,d^4\,e^4+\frac {336\,a^5\,b\,d^5\,e^3}{5}+84\,a^4\,b^2\,d^6\,e^2+32\,a^3\,b^3\,d^7\,e+3\,a^2\,b^4\,d^8\right )+x^{11}\,\left (\frac {15\,a^4\,b^2\,e^8}{11}+\frac {160\,a^3\,b^3\,d\,e^7}{11}+\frac {420\,a^2\,b^4\,d^2\,e^6}{11}+\frac {336\,a\,b^5\,d^3\,e^5}{11}+\frac {70\,b^6\,d^4\,e^4}{11}\right )+x^6\,\left (\frac {28\,a^6\,d^3\,e^5}{3}+70\,a^5\,b\,d^4\,e^4+140\,a^4\,b^2\,d^5\,e^3+\frac {280\,a^3\,b^3\,d^6\,e^2}{3}+20\,a^2\,b^4\,d^7\,e+a\,b^5\,d^8\right )+x^{10}\,\left (\frac {3\,a^5\,b\,e^8}{5}+12\,a^4\,b^2\,d\,e^7+56\,a^3\,b^3\,d^2\,e^6+84\,a^2\,b^4\,d^3\,e^5+42\,a\,b^5\,d^4\,e^4+\frac {28\,b^6\,d^5\,e^3}{5}\right )+x^8\,\left (a^6\,d\,e^7+21\,a^5\,b\,d^2\,e^6+105\,a^4\,b^2\,d^3\,e^5+175\,a^3\,b^3\,d^4\,e^4+105\,a^2\,b^4\,d^5\,e^3+21\,a\,b^5\,d^6\,e^2+b^6\,d^7\,e\right )+a^6\,d^8\,x+\frac {b^6\,e^8\,x^{15}}{15}+a^3\,d^5\,x^4\,\left (14\,a^3\,e^3+42\,a^2\,b\,d\,e^2+30\,a\,b^2\,d^2\,e+5\,b^3\,d^3\right )+\frac {b^3\,e^5\,x^{12}\,\left (5\,a^3\,e^3+30\,a^2\,b\,d\,e^2+42\,a\,b^2\,d^2\,e+14\,b^3\,d^3\right )}{3}+a^5\,d^7\,x^2\,\left (4\,a\,e+3\,b\,d\right )+\frac {b^5\,e^7\,x^{14}\,\left (3\,a\,e+4\,b\,d\right )}{7}+\frac {a^4\,d^6\,x^3\,\left (28\,a^2\,e^2+48\,a\,b\,d\,e+15\,b^2\,d^2\right )}{3}+\frac {b^4\,e^6\,x^{13}\,\left (15\,a^2\,e^2+48\,a\,b\,d\,e+28\,b^2\,d^2\right )}{13} \]
x^7*((b^6*d^8)/7 + 4*a^6*d^2*e^6 + 48*a^5*b*d^3*e^5 + 60*a^2*b^4*d^6*e^2 + 160*a^3*b^3*d^5*e^3 + 150*a^4*b^2*d^4*e^4 + (48*a*b^5*d^7*e)/7) + x^9*((a ^6*e^8)/9 + (28*b^6*d^6*e^2)/9 + (112*a*b^5*d^5*e^3)/3 + (350*a^2*b^4*d^4* e^4)/3 + (1120*a^3*b^3*d^3*e^5)/9 + (140*a^4*b^2*d^2*e^6)/3 + (16*a^5*b*d* e^7)/3) + x^5*(3*a^2*b^4*d^8 + 14*a^6*d^4*e^4 + 32*a^3*b^3*d^7*e + (336*a^ 5*b*d^5*e^3)/5 + 84*a^4*b^2*d^6*e^2) + x^11*((15*a^4*b^2*e^8)/11 + (70*b^6 *d^4*e^4)/11 + (336*a*b^5*d^3*e^5)/11 + (160*a^3*b^3*d*e^7)/11 + (420*a^2* b^4*d^2*e^6)/11) + x^6*(a*b^5*d^8 + (28*a^6*d^3*e^5)/3 + 20*a^2*b^4*d^7*e + 70*a^5*b*d^4*e^4 + (280*a^3*b^3*d^6*e^2)/3 + 140*a^4*b^2*d^5*e^3) + x^10 *((3*a^5*b*e^8)/5 + (28*b^6*d^5*e^3)/5 + 42*a*b^5*d^4*e^4 + 12*a^4*b^2*d*e ^7 + 84*a^2*b^4*d^3*e^5 + 56*a^3*b^3*d^2*e^6) + x^8*(a^6*d*e^7 + b^6*d^7*e + 21*a*b^5*d^6*e^2 + 21*a^5*b*d^2*e^6 + 105*a^2*b^4*d^5*e^3 + 175*a^3*b^3 *d^4*e^4 + 105*a^4*b^2*d^3*e^5) + a^6*d^8*x + (b^6*e^8*x^15)/15 + a^3*d^5* x^4*(14*a^3*e^3 + 5*b^3*d^3 + 30*a*b^2*d^2*e + 42*a^2*b*d*e^2) + (b^3*e^5* x^12*(5*a^3*e^3 + 14*b^3*d^3 + 42*a*b^2*d^2*e + 30*a^2*b*d*e^2))/3 + a^5*d ^7*x^2*(4*a*e + 3*b*d) + (b^5*e^7*x^14*(3*a*e + 4*b*d))/7 + (a^4*d^6*x^3*( 28*a^2*e^2 + 15*b^2*d^2 + 48*a*b*d*e))/3 + (b^4*e^6*x^13*(15*a^2*e^2 + 28* b^2*d^2 + 48*a*b*d*e))/13