Integrand size = 26, antiderivative size = 119 \[ \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {(b d-a e)^4 (d+e x)^7}{7 e^5}-\frac {b (b d-a e)^3 (d+e x)^8}{2 e^5}+\frac {2 b^2 (b d-a e)^2 (d+e x)^9}{3 e^5}-\frac {2 b^3 (b d-a e) (d+e x)^{10}}{5 e^5}+\frac {b^4 (d+e x)^{11}}{11 e^5} \]
1/7*(-a*e+b*d)^4*(e*x+d)^7/e^5-1/2*b*(-a*e+b*d)^3*(e*x+d)^8/e^5+2/3*b^2*(- a*e+b*d)^2*(e*x+d)^9/e^5-2/5*b^3*(-a*e+b*d)*(e*x+d)^10/e^5+1/11*b^4*(e*x+d )^11/e^5
Leaf count is larger than twice the leaf count of optimal. \(398\) vs. \(2(119)=238\).
Time = 0.04 (sec) , antiderivative size = 398, normalized size of antiderivative = 3.34 \[ \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=a^4 d^6 x+a^3 d^5 (2 b d+3 a e) x^2+a^2 d^4 \left (2 b^2 d^2+8 a b d e+5 a^2 e^2\right ) x^3+a d^3 \left (b^3 d^3+9 a b^2 d^2 e+15 a^2 b d e^2+5 a^3 e^3\right ) x^4+\frac {1}{5} d^2 \left (b^4 d^4+24 a b^3 d^3 e+90 a^2 b^2 d^2 e^2+80 a^3 b d e^3+15 a^4 e^4\right ) x^5+d e \left (b^4 d^4+10 a b^3 d^3 e+20 a^2 b^2 d^2 e^2+10 a^3 b d e^3+a^4 e^4\right ) x^6+\frac {1}{7} e^2 \left (15 b^4 d^4+80 a b^3 d^3 e+90 a^2 b^2 d^2 e^2+24 a^3 b d e^3+a^4 e^4\right ) x^7+\frac {1}{2} b e^3 \left (5 b^3 d^3+15 a b^2 d^2 e+9 a^2 b d e^2+a^3 e^3\right ) x^8+\frac {1}{3} b^2 e^4 \left (5 b^2 d^2+8 a b d e+2 a^2 e^2\right ) x^9+\frac {1}{5} b^3 e^5 (3 b d+2 a e) x^{10}+\frac {1}{11} b^4 e^6 x^{11} \]
a^4*d^6*x + a^3*d^5*(2*b*d + 3*a*e)*x^2 + a^2*d^4*(2*b^2*d^2 + 8*a*b*d*e + 5*a^2*e^2)*x^3 + a*d^3*(b^3*d^3 + 9*a*b^2*d^2*e + 15*a^2*b*d*e^2 + 5*a^3* e^3)*x^4 + (d^2*(b^4*d^4 + 24*a*b^3*d^3*e + 90*a^2*b^2*d^2*e^2 + 80*a^3*b* d*e^3 + 15*a^4*e^4)*x^5)/5 + d*e*(b^4*d^4 + 10*a*b^3*d^3*e + 20*a^2*b^2*d^ 2*e^2 + 10*a^3*b*d*e^3 + a^4*e^4)*x^6 + (e^2*(15*b^4*d^4 + 80*a*b^3*d^3*e + 90*a^2*b^2*d^2*e^2 + 24*a^3*b*d*e^3 + a^4*e^4)*x^7)/7 + (b*e^3*(5*b^3*d^ 3 + 15*a*b^2*d^2*e + 9*a^2*b*d*e^2 + a^3*e^3)*x^8)/2 + (b^2*e^4*(5*b^2*d^2 + 8*a*b*d*e + 2*a^2*e^2)*x^9)/3 + (b^3*e^5*(3*b*d + 2*a*e)*x^10)/5 + (b^4 *e^6*x^11)/11
Time = 0.44 (sec) , antiderivative size = 119, 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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^2 (d+e x)^6 \, dx\) |
| \(\Big \downarrow \) 1098 |
| \(\displaystyle \frac {\int b^4 (a+b x)^4 (d+e x)^6dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^4 (d+e x)^6dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (-\frac {4 b^3 (d+e x)^9 (b d-a e)}{e^4}+\frac {6 b^2 (d+e x)^8 (b d-a e)^2}{e^4}-\frac {4 b (d+e x)^7 (b d-a e)^3}{e^4}+\frac {(d+e x)^6 (a e-b d)^4}{e^4}+\frac {b^4 (d+e x)^{10}}{e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b^3 (d+e x)^{10} (b d-a e)}{5 e^5}+\frac {2 b^2 (d+e x)^9 (b d-a e)^2}{3 e^5}-\frac {b (d+e x)^8 (b d-a e)^3}{2 e^5}+\frac {(d+e x)^7 (b d-a e)^4}{7 e^5}+\frac {b^4 (d+e x)^{11}}{11 e^5}\) |
((b*d - a*e)^4*(d + e*x)^7)/(7*e^5) - (b*(b*d - a*e)^3*(d + e*x)^8)/(2*e^5 ) + (2*b^2*(b*d - a*e)^2*(d + e*x)^9)/(3*e^5) - (2*b^3*(b*d - a*e)*(d + e* x)^10)/(5*e^5) + (b^4*(d + e*x)^11)/(11*e^5)
3.15.63.3.1 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. \(416\) vs. \(2(109)=218\).
Time = 5.69 (sec) , antiderivative size = 417, normalized size of antiderivative = 3.50
| method | result | size |
| norman | \(\frac {b^{4} e^{6} x^{11}}{11}+\left (\frac {2}{5} a \,b^{3} e^{6}+\frac {3}{5} b^{4} d \,e^{5}\right ) x^{10}+\left (\frac {2}{3} a^{2} b^{2} e^{6}+\frac {8}{3} a \,b^{3} d \,e^{5}+\frac {5}{3} b^{4} d^{2} e^{4}\right ) x^{9}+\left (\frac {1}{2} a^{3} b \,e^{6}+\frac {9}{2} a^{2} b^{2} d \,e^{5}+\frac {15}{2} a \,b^{3} d^{2} e^{4}+\frac {5}{2} b^{4} d^{3} e^{3}\right ) x^{8}+\left (\frac {1}{7} a^{4} e^{6}+\frac {24}{7} a^{3} b d \,e^{5}+\frac {90}{7} a^{2} b^{2} d^{2} e^{4}+\frac {80}{7} a \,b^{3} d^{3} e^{3}+\frac {15}{7} b^{4} d^{4} e^{2}\right ) x^{7}+\left (a^{4} d \,e^{5}+10 a^{3} b \,d^{2} e^{4}+20 a^{2} b^{2} d^{3} e^{3}+10 a \,b^{3} d^{4} e^{2}+b^{4} d^{5} e \right ) x^{6}+\left (3 a^{4} d^{2} e^{4}+16 a^{3} b \,d^{3} e^{3}+18 a^{2} b^{2} d^{4} e^{2}+\frac {24}{5} a \,b^{3} d^{5} e +\frac {1}{5} b^{4} d^{6}\right ) x^{5}+\left (5 a^{4} d^{3} e^{3}+15 a^{3} b \,d^{4} e^{2}+9 a^{2} b^{2} d^{5} e +a \,b^{3} d^{6}\right ) x^{4}+\left (5 a^{4} d^{4} e^{2}+8 a^{3} b \,d^{5} e +2 a^{2} b^{2} d^{6}\right ) x^{3}+\left (3 a^{4} d^{5} e +2 a^{3} b \,d^{6}\right ) x^{2}+d^{6} a^{4} x\) | \(417\) |
| default | \(\frac {b^{4} e^{6} x^{11}}{11}+\frac {\left (4 a \,b^{3} e^{6}+6 b^{4} d \,e^{5}\right ) x^{10}}{10}+\frac {\left (6 a^{2} b^{2} e^{6}+24 a \,b^{3} d \,e^{5}+15 b^{4} d^{2} e^{4}\right ) x^{9}}{9}+\frac {\left (4 a^{3} b \,e^{6}+36 a^{2} b^{2} d \,e^{5}+60 a \,b^{3} d^{2} e^{4}+20 b^{4} d^{3} e^{3}\right ) x^{8}}{8}+\frac {\left (a^{4} e^{6}+24 a^{3} b d \,e^{5}+90 a^{2} b^{2} d^{2} e^{4}+80 a \,b^{3} d^{3} e^{3}+15 b^{4} d^{4} e^{2}\right ) x^{7}}{7}+\frac {\left (6 a^{4} d \,e^{5}+60 a^{3} b \,d^{2} e^{4}+120 a^{2} b^{2} d^{3} e^{3}+60 a \,b^{3} d^{4} e^{2}+6 b^{4} d^{5} e \right ) x^{6}}{6}+\frac {\left (15 a^{4} d^{2} e^{4}+80 a^{3} b \,d^{3} e^{3}+90 a^{2} b^{2} d^{4} e^{2}+24 a \,b^{3} d^{5} e +b^{4} d^{6}\right ) x^{5}}{5}+\frac {\left (20 a^{4} d^{3} e^{3}+60 a^{3} b \,d^{4} e^{2}+36 a^{2} b^{2} d^{5} e +4 a \,b^{3} d^{6}\right ) x^{4}}{4}+\frac {\left (15 a^{4} d^{4} e^{2}+24 a^{3} b \,d^{5} e +6 a^{2} b^{2} d^{6}\right ) x^{3}}{3}+\frac {\left (6 a^{4} d^{5} e +4 a^{3} b \,d^{6}\right ) x^{2}}{2}+d^{6} a^{4} x\) | \(427\) |
| risch | \(\frac {1}{2} x^{8} a^{3} b \,e^{6}+\frac {5}{2} x^{8} b^{4} d^{3} e^{3}+\frac {15}{7} x^{7} b^{4} d^{4} e^{2}+3 x^{5} a^{4} d^{2} e^{4}+a^{4} d \,e^{5} x^{6}+b^{4} d^{5} e \,x^{6}+5 a^{4} d^{3} e^{3} x^{4}+a \,b^{3} d^{6} x^{4}+5 a^{4} d^{4} e^{2} x^{3}+2 a^{2} b^{2} d^{6} x^{3}+2 a^{3} b \,d^{6} x^{2}+3 d^{5} e \,a^{4} x^{2}+\frac {1}{11} b^{4} e^{6} x^{11}+\frac {8}{3} x^{9} a \,b^{3} d \,e^{5}+\frac {9}{2} x^{8} a^{2} b^{2} d \,e^{5}+\frac {1}{7} x^{7} a^{4} e^{6}+\frac {1}{5} x^{5} b^{4} d^{6}+\frac {5}{3} x^{9} b^{4} d^{2} e^{4}+\frac {24}{7} x^{7} a^{3} b d \,e^{5}+\frac {90}{7} x^{7} a^{2} b^{2} d^{2} e^{4}+\frac {80}{7} x^{7} a \,b^{3} d^{3} e^{3}+d^{6} a^{4} x +\frac {2}{3} x^{9} a^{2} b^{2} e^{6}+\frac {2}{5} x^{10} a \,b^{3} e^{6}+\frac {3}{5} x^{10} b^{4} d \,e^{5}+\frac {15}{2} x^{8} a \,b^{3} d^{2} e^{4}+16 x^{5} a^{3} b \,d^{3} e^{3}+18 x^{5} a^{2} b^{2} d^{4} e^{2}+\frac {24}{5} x^{5} a \,b^{3} d^{5} e +10 a^{3} b \,d^{2} e^{4} x^{6}+20 a^{2} b^{2} d^{3} e^{3} x^{6}+10 a \,b^{3} d^{4} e^{2} x^{6}+15 a^{3} b \,d^{4} e^{2} x^{4}+9 a^{2} b^{2} d^{5} e \,x^{4}+8 a^{3} b \,d^{5} e \,x^{3}\) | \(471\) |
| parallelrisch | \(\frac {1}{2} x^{8} a^{3} b \,e^{6}+\frac {5}{2} x^{8} b^{4} d^{3} e^{3}+\frac {15}{7} x^{7} b^{4} d^{4} e^{2}+3 x^{5} a^{4} d^{2} e^{4}+a^{4} d \,e^{5} x^{6}+b^{4} d^{5} e \,x^{6}+5 a^{4} d^{3} e^{3} x^{4}+a \,b^{3} d^{6} x^{4}+5 a^{4} d^{4} e^{2} x^{3}+2 a^{2} b^{2} d^{6} x^{3}+2 a^{3} b \,d^{6} x^{2}+3 d^{5} e \,a^{4} x^{2}+\frac {1}{11} b^{4} e^{6} x^{11}+\frac {8}{3} x^{9} a \,b^{3} d \,e^{5}+\frac {9}{2} x^{8} a^{2} b^{2} d \,e^{5}+\frac {1}{7} x^{7} a^{4} e^{6}+\frac {1}{5} x^{5} b^{4} d^{6}+\frac {5}{3} x^{9} b^{4} d^{2} e^{4}+\frac {24}{7} x^{7} a^{3} b d \,e^{5}+\frac {90}{7} x^{7} a^{2} b^{2} d^{2} e^{4}+\frac {80}{7} x^{7} a \,b^{3} d^{3} e^{3}+d^{6} a^{4} x +\frac {2}{3} x^{9} a^{2} b^{2} e^{6}+\frac {2}{5} x^{10} a \,b^{3} e^{6}+\frac {3}{5} x^{10} b^{4} d \,e^{5}+\frac {15}{2} x^{8} a \,b^{3} d^{2} e^{4}+16 x^{5} a^{3} b \,d^{3} e^{3}+18 x^{5} a^{2} b^{2} d^{4} e^{2}+\frac {24}{5} x^{5} a \,b^{3} d^{5} e +10 a^{3} b \,d^{2} e^{4} x^{6}+20 a^{2} b^{2} d^{3} e^{3} x^{6}+10 a \,b^{3} d^{4} e^{2} x^{6}+15 a^{3} b \,d^{4} e^{2} x^{4}+9 a^{2} b^{2} d^{5} e \,x^{4}+8 a^{3} b \,d^{5} e \,x^{3}\) | \(471\) |
| gosper | \(\frac {x \left (210 b^{4} e^{6} x^{10}+924 x^{9} a \,b^{3} e^{6}+1386 x^{9} b^{4} d \,e^{5}+1540 x^{8} a^{2} b^{2} e^{6}+6160 x^{8} a \,b^{3} d \,e^{5}+3850 x^{8} b^{4} d^{2} e^{4}+1155 x^{7} a^{3} b \,e^{6}+10395 x^{7} a^{2} b^{2} d \,e^{5}+17325 x^{7} a \,b^{3} d^{2} e^{4}+5775 x^{7} b^{4} d^{3} e^{3}+330 x^{6} a^{4} e^{6}+7920 x^{6} a^{3} b d \,e^{5}+29700 x^{6} a^{2} b^{2} d^{2} e^{4}+26400 x^{6} a \,b^{3} d^{3} e^{3}+4950 x^{6} b^{4} d^{4} e^{2}+2310 a^{4} d \,e^{5} x^{5}+23100 a^{3} b \,d^{2} e^{4} x^{5}+46200 a^{2} b^{2} d^{3} e^{3} x^{5}+23100 a \,b^{3} d^{4} e^{2} x^{5}+2310 b^{4} d^{5} e \,x^{5}+6930 x^{4} a^{4} d^{2} e^{4}+36960 x^{4} a^{3} b \,d^{3} e^{3}+41580 x^{4} a^{2} b^{2} d^{4} e^{2}+11088 x^{4} a \,b^{3} d^{5} e +462 x^{4} b^{4} d^{6}+11550 a^{4} d^{3} e^{3} x^{3}+34650 a^{3} b \,d^{4} e^{2} x^{3}+20790 a^{2} b^{2} d^{5} e \,x^{3}+2310 a \,b^{3} d^{6} x^{3}+11550 a^{4} d^{4} e^{2} x^{2}+18480 a^{3} b \,d^{5} e \,x^{2}+4620 a^{2} b^{2} d^{6} x^{2}+6930 a^{4} d^{5} e x +4620 a^{3} b \,d^{6} x +2310 a^{4} d^{6}\right )}{2310}\) | \(473\) |
1/11*b^4*e^6*x^11+(2/5*a*b^3*e^6+3/5*b^4*d*e^5)*x^10+(2/3*a^2*b^2*e^6+8/3* a*b^3*d*e^5+5/3*b^4*d^2*e^4)*x^9+(1/2*a^3*b*e^6+9/2*a^2*b^2*d*e^5+15/2*a*b ^3*d^2*e^4+5/2*b^4*d^3*e^3)*x^8+(1/7*a^4*e^6+24/7*a^3*b*d*e^5+90/7*a^2*b^2 *d^2*e^4+80/7*a*b^3*d^3*e^3+15/7*b^4*d^4*e^2)*x^7+(a^4*d*e^5+10*a^3*b*d^2* e^4+20*a^2*b^2*d^3*e^3+10*a*b^3*d^4*e^2+b^4*d^5*e)*x^6+(3*a^4*d^2*e^4+16*a ^3*b*d^3*e^3+18*a^2*b^2*d^4*e^2+24/5*a*b^3*d^5*e+1/5*b^4*d^6)*x^5+(5*a^4*d ^3*e^3+15*a^3*b*d^4*e^2+9*a^2*b^2*d^5*e+a*b^3*d^6)*x^4+(5*a^4*d^4*e^2+8*a^ 3*b*d^5*e+2*a^2*b^2*d^6)*x^3+(3*a^4*d^5*e+2*a^3*b*d^6)*x^2+d^6*a^4*x
Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (109) = 218\).
Time = 0.28 (sec) , antiderivative size = 418, normalized size of antiderivative = 3.51 \[ \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{11} \, b^{4} e^{6} x^{11} + a^{4} d^{6} x + \frac {1}{5} \, {\left (3 \, b^{4} d e^{5} + 2 \, a b^{3} e^{6}\right )} x^{10} + \frac {1}{3} \, {\left (5 \, b^{4} d^{2} e^{4} + 8 \, a b^{3} d e^{5} + 2 \, a^{2} b^{2} e^{6}\right )} x^{9} + \frac {1}{2} \, {\left (5 \, b^{4} d^{3} e^{3} + 15 \, a b^{3} d^{2} e^{4} + 9 \, a^{2} b^{2} d e^{5} + a^{3} b e^{6}\right )} x^{8} + \frac {1}{7} \, {\left (15 \, b^{4} d^{4} e^{2} + 80 \, a b^{3} d^{3} e^{3} + 90 \, a^{2} b^{2} d^{2} e^{4} + 24 \, a^{3} b d e^{5} + a^{4} e^{6}\right )} x^{7} + {\left (b^{4} d^{5} e + 10 \, a b^{3} d^{4} e^{2} + 20 \, a^{2} b^{2} d^{3} e^{3} + 10 \, a^{3} b d^{2} e^{4} + a^{4} d e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{6} + 24 \, a b^{3} d^{5} e + 90 \, a^{2} b^{2} d^{4} e^{2} + 80 \, a^{3} b d^{3} e^{3} + 15 \, a^{4} d^{2} e^{4}\right )} x^{5} + {\left (a b^{3} d^{6} + 9 \, a^{2} b^{2} d^{5} e + 15 \, a^{3} b d^{4} e^{2} + 5 \, a^{4} d^{3} e^{3}\right )} x^{4} + {\left (2 \, a^{2} b^{2} d^{6} + 8 \, a^{3} b d^{5} e + 5 \, a^{4} d^{4} e^{2}\right )} x^{3} + {\left (2 \, a^{3} b d^{6} + 3 \, a^{4} d^{5} e\right )} x^{2} \]
1/11*b^4*e^6*x^11 + a^4*d^6*x + 1/5*(3*b^4*d*e^5 + 2*a*b^3*e^6)*x^10 + 1/3 *(5*b^4*d^2*e^4 + 8*a*b^3*d*e^5 + 2*a^2*b^2*e^6)*x^9 + 1/2*(5*b^4*d^3*e^3 + 15*a*b^3*d^2*e^4 + 9*a^2*b^2*d*e^5 + a^3*b*e^6)*x^8 + 1/7*(15*b^4*d^4*e^ 2 + 80*a*b^3*d^3*e^3 + 90*a^2*b^2*d^2*e^4 + 24*a^3*b*d*e^5 + a^4*e^6)*x^7 + (b^4*d^5*e + 10*a*b^3*d^4*e^2 + 20*a^2*b^2*d^3*e^3 + 10*a^3*b*d^2*e^4 + a^4*d*e^5)*x^6 + 1/5*(b^4*d^6 + 24*a*b^3*d^5*e + 90*a^2*b^2*d^4*e^2 + 80*a ^3*b*d^3*e^3 + 15*a^4*d^2*e^4)*x^5 + (a*b^3*d^6 + 9*a^2*b^2*d^5*e + 15*a^3 *b*d^4*e^2 + 5*a^4*d^3*e^3)*x^4 + (2*a^2*b^2*d^6 + 8*a^3*b*d^5*e + 5*a^4*d ^4*e^2)*x^3 + (2*a^3*b*d^6 + 3*a^4*d^5*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 462 vs. \(2 (105) = 210\).
Time = 0.05 (sec) , antiderivative size = 462, normalized size of antiderivative = 3.88 \[ \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=a^{4} d^{6} x + \frac {b^{4} e^{6} x^{11}}{11} + x^{10} \cdot \left (\frac {2 a b^{3} e^{6}}{5} + \frac {3 b^{4} d e^{5}}{5}\right ) + x^{9} \cdot \left (\frac {2 a^{2} b^{2} e^{6}}{3} + \frac {8 a b^{3} d e^{5}}{3} + \frac {5 b^{4} d^{2} e^{4}}{3}\right ) + x^{8} \left (\frac {a^{3} b e^{6}}{2} + \frac {9 a^{2} b^{2} d e^{5}}{2} + \frac {15 a b^{3} d^{2} e^{4}}{2} + \frac {5 b^{4} d^{3} e^{3}}{2}\right ) + x^{7} \left (\frac {a^{4} e^{6}}{7} + \frac {24 a^{3} b d e^{5}}{7} + \frac {90 a^{2} b^{2} d^{2} e^{4}}{7} + \frac {80 a b^{3} d^{3} e^{3}}{7} + \frac {15 b^{4} d^{4} e^{2}}{7}\right ) + x^{6} \left (a^{4} d e^{5} + 10 a^{3} b d^{2} e^{4} + 20 a^{2} b^{2} d^{3} e^{3} + 10 a b^{3} d^{4} e^{2} + b^{4} d^{5} e\right ) + x^{5} \cdot \left (3 a^{4} d^{2} e^{4} + 16 a^{3} b d^{3} e^{3} + 18 a^{2} b^{2} d^{4} e^{2} + \frac {24 a b^{3} d^{5} e}{5} + \frac {b^{4} d^{6}}{5}\right ) + x^{4} \cdot \left (5 a^{4} d^{3} e^{3} + 15 a^{3} b d^{4} e^{2} + 9 a^{2} b^{2} d^{5} e + a b^{3} d^{6}\right ) + x^{3} \cdot \left (5 a^{4} d^{4} e^{2} + 8 a^{3} b d^{5} e + 2 a^{2} b^{2} d^{6}\right ) + x^{2} \cdot \left (3 a^{4} d^{5} e + 2 a^{3} b d^{6}\right ) \]
a**4*d**6*x + b**4*e**6*x**11/11 + x**10*(2*a*b**3*e**6/5 + 3*b**4*d*e**5/ 5) + x**9*(2*a**2*b**2*e**6/3 + 8*a*b**3*d*e**5/3 + 5*b**4*d**2*e**4/3) + x**8*(a**3*b*e**6/2 + 9*a**2*b**2*d*e**5/2 + 15*a*b**3*d**2*e**4/2 + 5*b** 4*d**3*e**3/2) + x**7*(a**4*e**6/7 + 24*a**3*b*d*e**5/7 + 90*a**2*b**2*d** 2*e**4/7 + 80*a*b**3*d**3*e**3/7 + 15*b**4*d**4*e**2/7) + x**6*(a**4*d*e** 5 + 10*a**3*b*d**2*e**4 + 20*a**2*b**2*d**3*e**3 + 10*a*b**3*d**4*e**2 + b **4*d**5*e) + x**5*(3*a**4*d**2*e**4 + 16*a**3*b*d**3*e**3 + 18*a**2*b**2* d**4*e**2 + 24*a*b**3*d**5*e/5 + b**4*d**6/5) + x**4*(5*a**4*d**3*e**3 + 1 5*a**3*b*d**4*e**2 + 9*a**2*b**2*d**5*e + a*b**3*d**6) + x**3*(5*a**4*d**4 *e**2 + 8*a**3*b*d**5*e + 2*a**2*b**2*d**6) + x**2*(3*a**4*d**5*e + 2*a**3 *b*d**6)
Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (109) = 218\).
Time = 0.19 (sec) , antiderivative size = 418, normalized size of antiderivative = 3.51 \[ \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{11} \, b^{4} e^{6} x^{11} + a^{4} d^{6} x + \frac {1}{5} \, {\left (3 \, b^{4} d e^{5} + 2 \, a b^{3} e^{6}\right )} x^{10} + \frac {1}{3} \, {\left (5 \, b^{4} d^{2} e^{4} + 8 \, a b^{3} d e^{5} + 2 \, a^{2} b^{2} e^{6}\right )} x^{9} + \frac {1}{2} \, {\left (5 \, b^{4} d^{3} e^{3} + 15 \, a b^{3} d^{2} e^{4} + 9 \, a^{2} b^{2} d e^{5} + a^{3} b e^{6}\right )} x^{8} + \frac {1}{7} \, {\left (15 \, b^{4} d^{4} e^{2} + 80 \, a b^{3} d^{3} e^{3} + 90 \, a^{2} b^{2} d^{2} e^{4} + 24 \, a^{3} b d e^{5} + a^{4} e^{6}\right )} x^{7} + {\left (b^{4} d^{5} e + 10 \, a b^{3} d^{4} e^{2} + 20 \, a^{2} b^{2} d^{3} e^{3} + 10 \, a^{3} b d^{2} e^{4} + a^{4} d e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{6} + 24 \, a b^{3} d^{5} e + 90 \, a^{2} b^{2} d^{4} e^{2} + 80 \, a^{3} b d^{3} e^{3} + 15 \, a^{4} d^{2} e^{4}\right )} x^{5} + {\left (a b^{3} d^{6} + 9 \, a^{2} b^{2} d^{5} e + 15 \, a^{3} b d^{4} e^{2} + 5 \, a^{4} d^{3} e^{3}\right )} x^{4} + {\left (2 \, a^{2} b^{2} d^{6} + 8 \, a^{3} b d^{5} e + 5 \, a^{4} d^{4} e^{2}\right )} x^{3} + {\left (2 \, a^{3} b d^{6} + 3 \, a^{4} d^{5} e\right )} x^{2} \]
1/11*b^4*e^6*x^11 + a^4*d^6*x + 1/5*(3*b^4*d*e^5 + 2*a*b^3*e^6)*x^10 + 1/3 *(5*b^4*d^2*e^4 + 8*a*b^3*d*e^5 + 2*a^2*b^2*e^6)*x^9 + 1/2*(5*b^4*d^3*e^3 + 15*a*b^3*d^2*e^4 + 9*a^2*b^2*d*e^5 + a^3*b*e^6)*x^8 + 1/7*(15*b^4*d^4*e^ 2 + 80*a*b^3*d^3*e^3 + 90*a^2*b^2*d^2*e^4 + 24*a^3*b*d*e^5 + a^4*e^6)*x^7 + (b^4*d^5*e + 10*a*b^3*d^4*e^2 + 20*a^2*b^2*d^3*e^3 + 10*a^3*b*d^2*e^4 + a^4*d*e^5)*x^6 + 1/5*(b^4*d^6 + 24*a*b^3*d^5*e + 90*a^2*b^2*d^4*e^2 + 80*a ^3*b*d^3*e^3 + 15*a^4*d^2*e^4)*x^5 + (a*b^3*d^6 + 9*a^2*b^2*d^5*e + 15*a^3 *b*d^4*e^2 + 5*a^4*d^3*e^3)*x^4 + (2*a^2*b^2*d^6 + 8*a^3*b*d^5*e + 5*a^4*d ^4*e^2)*x^3 + (2*a^3*b*d^6 + 3*a^4*d^5*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 470 vs. \(2 (109) = 218\).
Time = 0.26 (sec) , antiderivative size = 470, normalized size of antiderivative = 3.95 \[ \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{11} \, b^{4} e^{6} x^{11} + \frac {3}{5} \, b^{4} d e^{5} x^{10} + \frac {2}{5} \, a b^{3} e^{6} x^{10} + \frac {5}{3} \, b^{4} d^{2} e^{4} x^{9} + \frac {8}{3} \, a b^{3} d e^{5} x^{9} + \frac {2}{3} \, a^{2} b^{2} e^{6} x^{9} + \frac {5}{2} \, b^{4} d^{3} e^{3} x^{8} + \frac {15}{2} \, a b^{3} d^{2} e^{4} x^{8} + \frac {9}{2} \, a^{2} b^{2} d e^{5} x^{8} + \frac {1}{2} \, a^{3} b e^{6} x^{8} + \frac {15}{7} \, b^{4} d^{4} e^{2} x^{7} + \frac {80}{7} \, a b^{3} d^{3} e^{3} x^{7} + \frac {90}{7} \, a^{2} b^{2} d^{2} e^{4} x^{7} + \frac {24}{7} \, a^{3} b d e^{5} x^{7} + \frac {1}{7} \, a^{4} e^{6} x^{7} + b^{4} d^{5} e x^{6} + 10 \, a b^{3} d^{4} e^{2} x^{6} + 20 \, a^{2} b^{2} d^{3} e^{3} x^{6} + 10 \, a^{3} b d^{2} e^{4} x^{6} + a^{4} d e^{5} x^{6} + \frac {1}{5} \, b^{4} d^{6} x^{5} + \frac {24}{5} \, a b^{3} d^{5} e x^{5} + 18 \, a^{2} b^{2} d^{4} e^{2} x^{5} + 16 \, a^{3} b d^{3} e^{3} x^{5} + 3 \, a^{4} d^{2} e^{4} x^{5} + a b^{3} d^{6} x^{4} + 9 \, a^{2} b^{2} d^{5} e x^{4} + 15 \, a^{3} b d^{4} e^{2} x^{4} + 5 \, a^{4} d^{3} e^{3} x^{4} + 2 \, a^{2} b^{2} d^{6} x^{3} + 8 \, a^{3} b d^{5} e x^{3} + 5 \, a^{4} d^{4} e^{2} x^{3} + 2 \, a^{3} b d^{6} x^{2} + 3 \, a^{4} d^{5} e x^{2} + a^{4} d^{6} x \]
1/11*b^4*e^6*x^11 + 3/5*b^4*d*e^5*x^10 + 2/5*a*b^3*e^6*x^10 + 5/3*b^4*d^2* e^4*x^9 + 8/3*a*b^3*d*e^5*x^9 + 2/3*a^2*b^2*e^6*x^9 + 5/2*b^4*d^3*e^3*x^8 + 15/2*a*b^3*d^2*e^4*x^8 + 9/2*a^2*b^2*d*e^5*x^8 + 1/2*a^3*b*e^6*x^8 + 15/ 7*b^4*d^4*e^2*x^7 + 80/7*a*b^3*d^3*e^3*x^7 + 90/7*a^2*b^2*d^2*e^4*x^7 + 24 /7*a^3*b*d*e^5*x^7 + 1/7*a^4*e^6*x^7 + b^4*d^5*e*x^6 + 10*a*b^3*d^4*e^2*x^ 6 + 20*a^2*b^2*d^3*e^3*x^6 + 10*a^3*b*d^2*e^4*x^6 + a^4*d*e^5*x^6 + 1/5*b^ 4*d^6*x^5 + 24/5*a*b^3*d^5*e*x^5 + 18*a^2*b^2*d^4*e^2*x^5 + 16*a^3*b*d^3*e ^3*x^5 + 3*a^4*d^2*e^4*x^5 + a*b^3*d^6*x^4 + 9*a^2*b^2*d^5*e*x^4 + 15*a^3* b*d^4*e^2*x^4 + 5*a^4*d^3*e^3*x^4 + 2*a^2*b^2*d^6*x^3 + 8*a^3*b*d^5*e*x^3 + 5*a^4*d^4*e^2*x^3 + 2*a^3*b*d^6*x^2 + 3*a^4*d^5*e*x^2 + a^4*d^6*x
Time = 9.94 (sec) , antiderivative size = 402, normalized size of antiderivative = 3.38 \[ \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=x^5\,\left (3\,a^4\,d^2\,e^4+16\,a^3\,b\,d^3\,e^3+18\,a^2\,b^2\,d^4\,e^2+\frac {24\,a\,b^3\,d^5\,e}{5}+\frac {b^4\,d^6}{5}\right )+x^7\,\left (\frac {a^4\,e^6}{7}+\frac {24\,a^3\,b\,d\,e^5}{7}+\frac {90\,a^2\,b^2\,d^2\,e^4}{7}+\frac {80\,a\,b^3\,d^3\,e^3}{7}+\frac {15\,b^4\,d^4\,e^2}{7}\right )+x^4\,\left (5\,a^4\,d^3\,e^3+15\,a^3\,b\,d^4\,e^2+9\,a^2\,b^2\,d^5\,e+a\,b^3\,d^6\right )+x^8\,\left (\frac {a^3\,b\,e^6}{2}+\frac {9\,a^2\,b^2\,d\,e^5}{2}+\frac {15\,a\,b^3\,d^2\,e^4}{2}+\frac {5\,b^4\,d^3\,e^3}{2}\right )+x^6\,\left (a^4\,d\,e^5+10\,a^3\,b\,d^2\,e^4+20\,a^2\,b^2\,d^3\,e^3+10\,a\,b^3\,d^4\,e^2+b^4\,d^5\,e\right )+a^4\,d^6\,x+\frac {b^4\,e^6\,x^{11}}{11}+a^3\,d^5\,x^2\,\left (3\,a\,e+2\,b\,d\right )+\frac {b^3\,e^5\,x^{10}\,\left (2\,a\,e+3\,b\,d\right )}{5}+a^2\,d^4\,x^3\,\left (5\,a^2\,e^2+8\,a\,b\,d\,e+2\,b^2\,d^2\right )+\frac {b^2\,e^4\,x^9\,\left (2\,a^2\,e^2+8\,a\,b\,d\,e+5\,b^2\,d^2\right )}{3} \]
x^5*((b^4*d^6)/5 + 3*a^4*d^2*e^4 + 16*a^3*b*d^3*e^3 + 18*a^2*b^2*d^4*e^2 + (24*a*b^3*d^5*e)/5) + x^7*((a^4*e^6)/7 + (15*b^4*d^4*e^2)/7 + (80*a*b^3*d ^3*e^3)/7 + (90*a^2*b^2*d^2*e^4)/7 + (24*a^3*b*d*e^5)/7) + x^4*(a*b^3*d^6 + 5*a^4*d^3*e^3 + 9*a^2*b^2*d^5*e + 15*a^3*b*d^4*e^2) + x^8*((a^3*b*e^6)/2 + (5*b^4*d^3*e^3)/2 + (15*a*b^3*d^2*e^4)/2 + (9*a^2*b^2*d*e^5)/2) + x^6*( a^4*d*e^5 + b^4*d^5*e + 10*a*b^3*d^4*e^2 + 10*a^3*b*d^2*e^4 + 20*a^2*b^2*d ^3*e^3) + a^4*d^6*x + (b^4*e^6*x^11)/11 + a^3*d^5*x^2*(3*a*e + 2*b*d) + (b ^3*e^5*x^10*(2*a*e + 3*b*d))/5 + a^2*d^4*x^3*(5*a^2*e^2 + 2*b^2*d^2 + 8*a* b*d*e) + (b^2*e^4*x^9*(2*a^2*e^2 + 5*b^2*d^2 + 8*a*b*d*e))/3