Integrand size = 31, antiderivative size = 146 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {(b d-a e)^5 (a+b x)^6}{6 b^6}+\frac {5 e (b d-a e)^4 (a+b x)^7}{7 b^6}+\frac {5 e^2 (b d-a e)^3 (a+b x)^8}{4 b^6}+\frac {10 e^3 (b d-a e)^2 (a+b x)^9}{9 b^6}+\frac {e^4 (b d-a e) (a+b x)^{10}}{2 b^6}+\frac {e^5 (a+b x)^{11}}{11 b^6} \] Output:
1/6*(-a*e+b*d)^5*(b*x+a)^6/b^6+5/7*e*(-a*e+b*d)^4*(b*x+a)^7/b^6+5/4*e^2*(- a*e+b*d)^3*(b*x+a)^8/b^6+10/9*e^3*(-a*e+b*d)^2*(b*x+a)^9/b^6+1/2*e^4*(-a*e +b*d)*(b*x+a)^10/b^6+1/11*e^5*(b*x+a)^11/b^6
Leaf count is larger than twice the leaf count of optimal. \(413\) vs. \(2(146)=292\).
Time = 0.06 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.83 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=a^5 d^5 x+\frac {5}{2} a^4 d^4 (b d+a e) x^2+\frac {5}{3} a^3 d^3 \left (2 b^2 d^2+5 a b d e+2 a^2 e^2\right ) x^3+\frac {5}{2} a^2 d^2 \left (b^3 d^3+5 a b^2 d^2 e+5 a^2 b d e^2+a^3 e^3\right ) x^4+a d \left (b^4 d^4+10 a b^3 d^3 e+20 a^2 b^2 d^2 e^2+10 a^3 b d e^3+a^4 e^4\right ) x^5+\frac {1}{6} \left (b^5 d^5+25 a b^4 d^4 e+100 a^2 b^3 d^3 e^2+100 a^3 b^2 d^2 e^3+25 a^4 b d e^4+a^5 e^5\right ) x^6+\frac {5}{7} b e \left (b^4 d^4+10 a b^3 d^3 e+20 a^2 b^2 d^2 e^2+10 a^3 b d e^3+a^4 e^4\right ) x^7+\frac {5}{4} b^2 e^2 \left (b^3 d^3+5 a b^2 d^2 e+5 a^2 b d e^2+a^3 e^3\right ) x^8+\frac {5}{9} b^3 e^3 \left (2 b^2 d^2+5 a b d e+2 a^2 e^2\right ) x^9+\frac {1}{2} b^4 e^4 (b d+a e) x^{10}+\frac {1}{11} b^5 e^5 x^{11} \] Input:
Integrate[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
Output:
a^5*d^5*x + (5*a^4*d^4*(b*d + a*e)*x^2)/2 + (5*a^3*d^3*(2*b^2*d^2 + 5*a*b* d*e + 2*a^2*e^2)*x^3)/3 + (5*a^2*d^2*(b^3*d^3 + 5*a*b^2*d^2*e + 5*a^2*b*d* e^2 + a^3*e^3)*x^4)/2 + a*d*(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^5 + ((b^5*d^5 + 25*a*b^4*d^4*e + 100*a^2*b^ 3*d^3*e^2 + 100*a^3*b^2*d^2*e^3 + 25*a^4*b*d*e^4 + a^5*e^5)*x^6)/6 + (5*b* 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^7)/7 + (5*b^2*e^2*(b^3*d^3 + 5*a*b^2*d^2*e + 5*a^2*b*d*e^2 + a^3*e^3) *x^8)/4 + (5*b^3*e^3*(2*b^2*d^2 + 5*a*b*d*e + 2*a^2*e^2)*x^9)/9 + (b^4*e^4 *(b*d + a*e)*x^10)/2 + (b^5*e^5*x^11)/11
Time = 0.76 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1184, 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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2 (d+e x)^5 \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int b^4 (a+b x)^5 (d+e x)^5dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^5 (d+e x)^5dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (\frac {5 e^4 (a+b x)^9 (b d-a e)}{b^5}+\frac {10 e^3 (a+b x)^8 (b d-a e)^2}{b^5}+\frac {10 e^2 (a+b x)^7 (b d-a e)^3}{b^5}+\frac {5 e (a+b x)^6 (b d-a e)^4}{b^5}+\frac {(a+b x)^5 (b d-a e)^5}{b^5}+\frac {e^5 (a+b x)^{10}}{b^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^4 (a+b x)^{10} (b d-a e)}{2 b^6}+\frac {10 e^3 (a+b x)^9 (b d-a e)^2}{9 b^6}+\frac {5 e^2 (a+b x)^8 (b d-a e)^3}{4 b^6}+\frac {5 e (a+b x)^7 (b d-a e)^4}{7 b^6}+\frac {(a+b x)^6 (b d-a e)^5}{6 b^6}+\frac {e^5 (a+b x)^{11}}{11 b^6}\) |
Input:
Int[(a + b*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
Output:
((b*d - a*e)^5*(a + b*x)^6)/(6*b^6) + (5*e*(b*d - a*e)^4*(a + b*x)^7)/(7*b ^6) + (5*e^2*(b*d - a*e)^3*(a + b*x)^8)/(4*b^6) + (10*e^3*(b*d - a*e)^2*(a + b*x)^9)/(9*b^6) + (e^4*(b*d - a*e)*(a + b*x)^10)/(2*b^6) + (e^5*(a + b* x)^11)/(11*b^6)
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_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(431\) vs. \(2(134)=268\).
Time = 1.29 (sec) , antiderivative size = 432, normalized size of antiderivative = 2.96
| method | result | size |
| norman | \(\frac {b^{5} e^{5} x^{11}}{11}+\left (\frac {1}{2} e^{5} a \,b^{4}+\frac {1}{2} d \,e^{4} b^{5}\right ) x^{10}+\left (\frac {10}{9} e^{5} a^{2} b^{3}+\frac {25}{9} d \,e^{4} a \,b^{4}+\frac {10}{9} d^{2} e^{3} b^{5}\right ) x^{9}+\left (\frac {5}{4} e^{5} a^{3} b^{2}+\frac {25}{4} d \,e^{4} a^{2} b^{3}+\frac {25}{4} d^{2} e^{3} a \,b^{4}+\frac {5}{4} d^{3} e^{2} b^{5}\right ) x^{8}+\left (\frac {5}{7} a^{4} b \,e^{5}+\frac {50}{7} d \,e^{4} a^{3} b^{2}+\frac {100}{7} a^{2} b^{3} d^{2} e^{3}+\frac {50}{7} d^{3} e^{2} a \,b^{4}+\frac {5}{7} d^{4} e \,b^{5}\right ) x^{7}+\left (\frac {1}{6} e^{5} a^{5}+\frac {25}{6} a^{4} b d \,e^{4}+\frac {50}{3} a^{3} b^{2} d^{2} e^{3}+\frac {50}{3} a^{2} b^{3} d^{3} e^{2}+\frac {25}{6} a \,b^{4} d^{4} e +\frac {1}{6} b^{5} d^{5}\right ) x^{6}+\left (d \,e^{4} a^{5}+10 a^{4} b \,d^{2} e^{3}+20 a^{3} b^{2} d^{3} e^{2}+10 a^{2} b^{3} d^{4} e +d^{5} a \,b^{4}\right ) x^{5}+\left (\frac {5}{2} d^{2} e^{3} a^{5}+\frac {25}{2} d^{3} e^{2} a^{4} b +\frac {25}{2} d^{4} e \,a^{3} b^{2}+\frac {5}{2} d^{5} a^{2} b^{3}\right ) x^{4}+\left (\frac {10}{3} d^{3} e^{2} a^{5}+\frac {25}{3} d^{4} e \,a^{4} b +\frac {10}{3} d^{5} a^{3} b^{2}\right ) x^{3}+\left (\frac {5}{2} d^{4} e \,a^{5}+\frac {5}{2} a^{4} b \,d^{5}\right ) x^{2}+a^{5} d^{5} x\) | \(432\) |
| risch | \(\frac {5}{2} x^{4} d^{2} e^{3} a^{5}+\frac {5}{2} x^{4} d^{5} a^{2} b^{3}+\frac {10}{3} x^{3} d^{3} e^{2} a^{5}+\frac {10}{3} x^{3} d^{5} a^{3} b^{2}+\frac {5}{2} x^{2} d^{4} e \,a^{5}+\frac {5}{2} x^{2} a^{4} b \,d^{5}+a^{5} d \,e^{4} x^{5}+a \,b^{4} d^{5} x^{5}+\frac {1}{2} x^{10} e^{5} a \,b^{4}+\frac {1}{2} x^{10} d \,e^{4} b^{5}+\frac {10}{9} x^{9} e^{5} a^{2} b^{3}+\frac {10}{9} x^{9} d^{2} e^{3} b^{5}+\frac {5}{4} x^{8} e^{5} a^{3} b^{2}+\frac {5}{4} x^{8} d^{3} e^{2} b^{5}+\frac {5}{7} x^{7} a^{4} b \,e^{5}+\frac {5}{7} x^{7} d^{4} e \,b^{5}+\frac {25}{9} x^{9} d \,e^{4} a \,b^{4}+\frac {25}{4} x^{8} d \,e^{4} a^{2} b^{3}+\frac {25}{4} x^{8} d^{2} e^{3} a \,b^{4}+\frac {50}{7} x^{7} d \,e^{4} a^{3} b^{2}+\frac {100}{7} x^{7} a^{2} b^{3} d^{2} e^{3}+\frac {50}{7} x^{7} d^{3} e^{2} a \,b^{4}+\frac {25}{6} x^{6} a^{4} b d \,e^{4}+\frac {50}{3} x^{6} a^{3} b^{2} d^{2} e^{3}+\frac {50}{3} x^{6} a^{2} b^{3} d^{3} e^{2}+\frac {25}{6} x^{6} a \,b^{4} d^{4} e +\frac {25}{2} x^{4} d^{3} e^{2} a^{4} b +\frac {25}{2} x^{4} d^{4} e \,a^{3} b^{2}+\frac {25}{3} x^{3} d^{4} e \,a^{4} b +10 a^{4} b \,d^{2} e^{3} x^{5}+20 a^{3} b^{2} d^{3} e^{2} x^{5}+10 a^{2} b^{3} d^{4} e \,x^{5}+\frac {1}{6} x^{6} e^{5} a^{5}+\frac {1}{6} x^{6} b^{5} d^{5}+\frac {1}{11} b^{5} e^{5} x^{11}+a^{5} d^{5} x\) | \(489\) |
| parallelrisch | \(\frac {5}{2} x^{4} d^{2} e^{3} a^{5}+\frac {5}{2} x^{4} d^{5} a^{2} b^{3}+\frac {10}{3} x^{3} d^{3} e^{2} a^{5}+\frac {10}{3} x^{3} d^{5} a^{3} b^{2}+\frac {5}{2} x^{2} d^{4} e \,a^{5}+\frac {5}{2} x^{2} a^{4} b \,d^{5}+a^{5} d \,e^{4} x^{5}+a \,b^{4} d^{5} x^{5}+\frac {1}{2} x^{10} e^{5} a \,b^{4}+\frac {1}{2} x^{10} d \,e^{4} b^{5}+\frac {10}{9} x^{9} e^{5} a^{2} b^{3}+\frac {10}{9} x^{9} d^{2} e^{3} b^{5}+\frac {5}{4} x^{8} e^{5} a^{3} b^{2}+\frac {5}{4} x^{8} d^{3} e^{2} b^{5}+\frac {5}{7} x^{7} a^{4} b \,e^{5}+\frac {5}{7} x^{7} d^{4} e \,b^{5}+\frac {25}{9} x^{9} d \,e^{4} a \,b^{4}+\frac {25}{4} x^{8} d \,e^{4} a^{2} b^{3}+\frac {25}{4} x^{8} d^{2} e^{3} a \,b^{4}+\frac {50}{7} x^{7} d \,e^{4} a^{3} b^{2}+\frac {100}{7} x^{7} a^{2} b^{3} d^{2} e^{3}+\frac {50}{7} x^{7} d^{3} e^{2} a \,b^{4}+\frac {25}{6} x^{6} a^{4} b d \,e^{4}+\frac {50}{3} x^{6} a^{3} b^{2} d^{2} e^{3}+\frac {50}{3} x^{6} a^{2} b^{3} d^{3} e^{2}+\frac {25}{6} x^{6} a \,b^{4} d^{4} e +\frac {25}{2} x^{4} d^{3} e^{2} a^{4} b +\frac {25}{2} x^{4} d^{4} e \,a^{3} b^{2}+\frac {25}{3} x^{3} d^{4} e \,a^{4} b +10 a^{4} b \,d^{2} e^{3} x^{5}+20 a^{3} b^{2} d^{3} e^{2} x^{5}+10 a^{2} b^{3} d^{4} e \,x^{5}+\frac {1}{6} x^{6} e^{5} a^{5}+\frac {1}{6} x^{6} b^{5} d^{5}+\frac {1}{11} b^{5} e^{5} x^{11}+a^{5} d^{5} x\) | \(489\) |
| gosper | \(\frac {x \left (252 b^{5} e^{5} x^{10}+1386 x^{9} e^{5} a \,b^{4}+1386 x^{9} d \,e^{4} b^{5}+3080 x^{8} e^{5} a^{2} b^{3}+7700 x^{8} d \,e^{4} a \,b^{4}+3080 x^{8} d^{2} e^{3} b^{5}+3465 x^{7} e^{5} a^{3} b^{2}+17325 x^{7} d \,e^{4} a^{2} b^{3}+17325 x^{7} d^{2} e^{3} a \,b^{4}+3465 x^{7} d^{3} e^{2} b^{5}+1980 x^{6} e^{5} a^{4} b +19800 x^{6} d \,e^{4} a^{3} b^{2}+39600 x^{6} d^{2} e^{3} a^{2} b^{3}+19800 x^{6} d^{3} e^{2} a \,b^{4}+1980 x^{6} d^{4} e \,b^{5}+462 x^{5} e^{5} a^{5}+11550 x^{5} a^{4} b d \,e^{4}+46200 x^{5} a^{3} b^{2} d^{2} e^{3}+46200 x^{5} a^{2} b^{3} d^{3} e^{2}+11550 x^{5} a \,b^{4} d^{4} e +462 b^{5} d^{5} x^{5}+2772 a^{5} d \,e^{4} x^{4}+27720 a^{4} b \,d^{2} e^{3} x^{4}+55440 a^{3} b^{2} d^{3} e^{2} x^{4}+27720 a^{2} b^{3} d^{4} e \,x^{4}+2772 a \,b^{4} d^{5} x^{4}+6930 x^{3} d^{2} e^{3} a^{5}+34650 x^{3} d^{3} e^{2} a^{4} b +34650 x^{3} d^{4} e \,a^{3} b^{2}+6930 a^{2} b^{3} d^{5} x^{3}+9240 x^{2} d^{3} e^{2} a^{5}+23100 x^{2} d^{4} e \,a^{4} b +9240 a^{3} b^{2} d^{5} x^{2}+6930 x \,d^{4} e \,a^{5}+6930 a^{4} b \,d^{5} x +2772 a^{5} d^{5}\right )}{2772}\) | \(490\) |
| orering | \(\frac {x \left (252 b^{5} e^{5} x^{10}+1386 x^{9} e^{5} a \,b^{4}+1386 x^{9} d \,e^{4} b^{5}+3080 x^{8} e^{5} a^{2} b^{3}+7700 x^{8} d \,e^{4} a \,b^{4}+3080 x^{8} d^{2} e^{3} b^{5}+3465 x^{7} e^{5} a^{3} b^{2}+17325 x^{7} d \,e^{4} a^{2} b^{3}+17325 x^{7} d^{2} e^{3} a \,b^{4}+3465 x^{7} d^{3} e^{2} b^{5}+1980 x^{6} e^{5} a^{4} b +19800 x^{6} d \,e^{4} a^{3} b^{2}+39600 x^{6} d^{2} e^{3} a^{2} b^{3}+19800 x^{6} d^{3} e^{2} a \,b^{4}+1980 x^{6} d^{4} e \,b^{5}+462 x^{5} e^{5} a^{5}+11550 x^{5} a^{4} b d \,e^{4}+46200 x^{5} a^{3} b^{2} d^{2} e^{3}+46200 x^{5} a^{2} b^{3} d^{3} e^{2}+11550 x^{5} a \,b^{4} d^{4} e +462 b^{5} d^{5} x^{5}+2772 a^{5} d \,e^{4} x^{4}+27720 a^{4} b \,d^{2} e^{3} x^{4}+55440 a^{3} b^{2} d^{3} e^{2} x^{4}+27720 a^{2} b^{3} d^{4} e \,x^{4}+2772 a \,b^{4} d^{5} x^{4}+6930 x^{3} d^{2} e^{3} a^{5}+34650 x^{3} d^{3} e^{2} a^{4} b +34650 x^{3} d^{4} e \,a^{3} b^{2}+6930 a^{2} b^{3} d^{5} x^{3}+9240 x^{2} d^{3} e^{2} a^{5}+23100 x^{2} d^{4} e \,a^{4} b +9240 a^{3} b^{2} d^{5} x^{2}+6930 x \,d^{4} e \,a^{5}+6930 a^{4} b \,d^{5} x +2772 a^{5} d^{5}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}{2772 \left (b x +a \right )^{4}}\) | \(515\) |
| default | \(\frac {b^{5} e^{5} x^{11}}{11}+\frac {\left (\left (a \,e^{5}+5 b d \,e^{4}\right ) b^{4}+4 e^{5} a \,b^{4}\right ) x^{10}}{10}+\frac {\left (\left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) b^{4}+4 \left (a \,e^{5}+5 b d \,e^{4}\right ) a \,b^{3}+6 e^{5} a^{2} b^{3}\right ) x^{9}}{9}+\frac {\left (\left (10 a \,e^{3} d^{2}+10 b \,d^{3} e^{2}\right ) b^{4}+4 \left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a \,b^{3}+6 \left (a \,e^{5}+5 b d \,e^{4}\right ) a^{2} b^{2}+4 e^{5} a^{3} b^{2}\right ) x^{8}}{8}+\frac {\left (\left (10 a \,e^{2} d^{3}+5 b \,d^{4} e \right ) b^{4}+4 \left (10 a \,e^{3} d^{2}+10 b \,d^{3} e^{2}\right ) a \,b^{3}+6 \left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a^{2} b^{2}+4 \left (a \,e^{5}+5 b d \,e^{4}\right ) a^{3} b +a^{4} b \,e^{5}\right ) x^{7}}{7}+\frac {\left (\left (5 a \,d^{4} e +b \,d^{5}\right ) b^{4}+4 \left (10 a \,e^{2} d^{3}+5 b \,d^{4} e \right ) a \,b^{3}+6 \left (10 a \,e^{3} d^{2}+10 b \,d^{3} e^{2}\right ) a^{2} b^{2}+4 \left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a^{3} b +\left (a \,e^{5}+5 b d \,e^{4}\right ) a^{4}\right ) x^{6}}{6}+\frac {\left (d^{5} a \,b^{4}+4 \left (5 a \,d^{4} e +b \,d^{5}\right ) a \,b^{3}+6 \left (10 a \,e^{2} d^{3}+5 b \,d^{4} e \right ) a^{2} b^{2}+4 \left (10 a \,e^{3} d^{2}+10 b \,d^{3} e^{2}\right ) a^{3} b +\left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a^{4}\right ) x^{5}}{5}+\frac {\left (4 d^{5} a^{2} b^{3}+6 \left (5 a \,d^{4} e +b \,d^{5}\right ) a^{2} b^{2}+4 \left (10 a \,e^{2} d^{3}+5 b \,d^{4} e \right ) a^{3} b +\left (10 a \,e^{3} d^{2}+10 b \,d^{3} e^{2}\right ) a^{4}\right ) x^{4}}{4}+\frac {\left (6 d^{5} a^{3} b^{2}+4 \left (5 a \,d^{4} e +b \,d^{5}\right ) a^{3} b +\left (10 a \,e^{2} d^{3}+5 b \,d^{4} e \right ) a^{4}\right ) x^{3}}{3}+\frac {\left (4 a^{4} b \,d^{5}+\left (5 a \,d^{4} e +b \,d^{5}\right ) a^{4}\right ) x^{2}}{2}+a^{5} d^{5} x\) | \(688\) |
Input:
int((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x,method=_RETURNVERBOSE)
Output:
1/11*b^5*e^5*x^11+(1/2*e^5*a*b^4+1/2*d*e^4*b^5)*x^10+(10/9*e^5*a^2*b^3+25/ 9*d*e^4*a*b^4+10/9*d^2*e^3*b^5)*x^9+(5/4*e^5*a^3*b^2+25/4*d*e^4*a^2*b^3+25 /4*d^2*e^3*a*b^4+5/4*d^3*e^2*b^5)*x^8+(5/7*a^4*b*e^5+50/7*d*e^4*a^3*b^2+10 0/7*a^2*b^3*d^2*e^3+50/7*d^3*e^2*a*b^4+5/7*d^4*e*b^5)*x^7+(1/6*e^5*a^5+25/ 6*a^4*b*d*e^4+50/3*a^3*b^2*d^2*e^3+50/3*a^2*b^3*d^3*e^2+25/6*a*b^4*d^4*e+1 /6*b^5*d^5)*x^6+(a^5*d*e^4+10*a^4*b*d^2*e^3+20*a^3*b^2*d^3*e^2+10*a^2*b^3* d^4*e+a*b^4*d^5)*x^5+(5/2*d^2*e^3*a^5+25/2*d^3*e^2*a^4*b+25/2*d^4*e*a^3*b^ 2+5/2*d^5*a^2*b^3)*x^4+(10/3*d^3*e^2*a^5+25/3*d^4*e*a^4*b+10/3*d^5*a^3*b^2 )*x^3+(5/2*d^4*e*a^5+5/2*a^4*b*d^5)*x^2+a^5*d^5*x
Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (134) = 268\).
Time = 0.07 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.92 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{11} \, b^{5} e^{5} x^{11} + a^{5} d^{5} x + \frac {1}{2} \, {\left (b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{10} + \frac {5}{9} \, {\left (2 \, b^{5} d^{2} e^{3} + 5 \, a b^{4} d e^{4} + 2 \, a^{2} b^{3} e^{5}\right )} x^{9} + \frac {5}{4} \, {\left (b^{5} d^{3} e^{2} + 5 \, a b^{4} d^{2} e^{3} + 5 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{8} + \frac {5}{7} \, {\left (b^{5} d^{4} e + 10 \, a b^{4} d^{3} e^{2} + 20 \, a^{2} b^{3} d^{2} e^{3} + 10 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} d^{5} + 25 \, a b^{4} d^{4} e + 100 \, a^{2} b^{3} d^{3} e^{2} + 100 \, a^{3} b^{2} d^{2} e^{3} + 25 \, a^{4} b d e^{4} + a^{5} e^{5}\right )} x^{6} + {\left (a b^{4} d^{5} + 10 \, a^{2} b^{3} d^{4} e + 20 \, a^{3} b^{2} d^{3} e^{2} + 10 \, a^{4} b d^{2} e^{3} + a^{5} d e^{4}\right )} x^{5} + \frac {5}{2} \, {\left (a^{2} b^{3} d^{5} + 5 \, a^{3} b^{2} d^{4} e + 5 \, a^{4} b d^{3} e^{2} + a^{5} d^{2} e^{3}\right )} x^{4} + \frac {5}{3} \, {\left (2 \, a^{3} b^{2} d^{5} + 5 \, a^{4} b d^{4} e + 2 \, a^{5} d^{3} e^{2}\right )} x^{3} + \frac {5}{2} \, {\left (a^{4} b d^{5} + a^{5} d^{4} e\right )} x^{2} \] Input:
integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")
Output:
1/11*b^5*e^5*x^11 + a^5*d^5*x + 1/2*(b^5*d*e^4 + a*b^4*e^5)*x^10 + 5/9*(2* b^5*d^2*e^3 + 5*a*b^4*d*e^4 + 2*a^2*b^3*e^5)*x^9 + 5/4*(b^5*d^3*e^2 + 5*a* b^4*d^2*e^3 + 5*a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^8 + 5/7*(b^5*d^4*e + 10*a*b ^4*d^3*e^2 + 20*a^2*b^3*d^2*e^3 + 10*a^3*b^2*d*e^4 + a^4*b*e^5)*x^7 + 1/6* (b^5*d^5 + 25*a*b^4*d^4*e + 100*a^2*b^3*d^3*e^2 + 100*a^3*b^2*d^2*e^3 + 25 *a^4*b*d*e^4 + a^5*e^5)*x^6 + (a*b^4*d^5 + 10*a^2*b^3*d^4*e + 20*a^3*b^2*d ^3*e^2 + 10*a^4*b*d^2*e^3 + a^5*d*e^4)*x^5 + 5/2*(a^2*b^3*d^5 + 5*a^3*b^2* d^4*e + 5*a^4*b*d^3*e^2 + a^5*d^2*e^3)*x^4 + 5/3*(2*a^3*b^2*d^5 + 5*a^4*b* d^4*e + 2*a^5*d^3*e^2)*x^3 + 5/2*(a^4*b*d^5 + a^5*d^4*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (131) = 262\).
Time = 0.06 (sec) , antiderivative size = 500, normalized size of antiderivative = 3.42 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=a^{5} d^{5} x + \frac {b^{5} e^{5} x^{11}}{11} + x^{10} \left (\frac {a b^{4} e^{5}}{2} + \frac {b^{5} d e^{4}}{2}\right ) + x^{9} \cdot \left (\frac {10 a^{2} b^{3} e^{5}}{9} + \frac {25 a b^{4} d e^{4}}{9} + \frac {10 b^{5} d^{2} e^{3}}{9}\right ) + x^{8} \cdot \left (\frac {5 a^{3} b^{2} e^{5}}{4} + \frac {25 a^{2} b^{3} d e^{4}}{4} + \frac {25 a b^{4} d^{2} e^{3}}{4} + \frac {5 b^{5} d^{3} e^{2}}{4}\right ) + x^{7} \cdot \left (\frac {5 a^{4} b e^{5}}{7} + \frac {50 a^{3} b^{2} d e^{4}}{7} + \frac {100 a^{2} b^{3} d^{2} e^{3}}{7} + \frac {50 a b^{4} d^{3} e^{2}}{7} + \frac {5 b^{5} d^{4} e}{7}\right ) + x^{6} \left (\frac {a^{5} e^{5}}{6} + \frac {25 a^{4} b d e^{4}}{6} + \frac {50 a^{3} b^{2} d^{2} e^{3}}{3} + \frac {50 a^{2} b^{3} d^{3} e^{2}}{3} + \frac {25 a b^{4} d^{4} e}{6} + \frac {b^{5} d^{5}}{6}\right ) + x^{5} \left (a^{5} d e^{4} + 10 a^{4} b d^{2} e^{3} + 20 a^{3} b^{2} d^{3} e^{2} + 10 a^{2} b^{3} d^{4} e + a b^{4} d^{5}\right ) + x^{4} \cdot \left (\frac {5 a^{5} d^{2} e^{3}}{2} + \frac {25 a^{4} b d^{3} e^{2}}{2} + \frac {25 a^{3} b^{2} d^{4} e}{2} + \frac {5 a^{2} b^{3} d^{5}}{2}\right ) + x^{3} \cdot \left (\frac {10 a^{5} d^{3} e^{2}}{3} + \frac {25 a^{4} b d^{4} e}{3} + \frac {10 a^{3} b^{2} d^{5}}{3}\right ) + x^{2} \cdot \left (\frac {5 a^{5} d^{4} e}{2} + \frac {5 a^{4} b d^{5}}{2}\right ) \] Input:
integrate((b*x+a)*(e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**2,x)
Output:
a**5*d**5*x + b**5*e**5*x**11/11 + x**10*(a*b**4*e**5/2 + b**5*d*e**4/2) + x**9*(10*a**2*b**3*e**5/9 + 25*a*b**4*d*e**4/9 + 10*b**5*d**2*e**3/9) + x **8*(5*a**3*b**2*e**5/4 + 25*a**2*b**3*d*e**4/4 + 25*a*b**4*d**2*e**3/4 + 5*b**5*d**3*e**2/4) + x**7*(5*a**4*b*e**5/7 + 50*a**3*b**2*d*e**4/7 + 100* a**2*b**3*d**2*e**3/7 + 50*a*b**4*d**3*e**2/7 + 5*b**5*d**4*e/7) + x**6*(a **5*e**5/6 + 25*a**4*b*d*e**4/6 + 50*a**3*b**2*d**2*e**3/3 + 50*a**2*b**3* d**3*e**2/3 + 25*a*b**4*d**4*e/6 + b**5*d**5/6) + x**5*(a**5*d*e**4 + 10*a **4*b*d**2*e**3 + 20*a**3*b**2*d**3*e**2 + 10*a**2*b**3*d**4*e + a*b**4*d* *5) + x**4*(5*a**5*d**2*e**3/2 + 25*a**4*b*d**3*e**2/2 + 25*a**3*b**2*d**4 *e/2 + 5*a**2*b**3*d**5/2) + x**3*(10*a**5*d**3*e**2/3 + 25*a**4*b*d**4*e/ 3 + 10*a**3*b**2*d**5/3) + x**2*(5*a**5*d**4*e/2 + 5*a**4*b*d**5/2)
Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (134) = 268\).
Time = 0.03 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.92 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{11} \, b^{5} e^{5} x^{11} + a^{5} d^{5} x + \frac {1}{2} \, {\left (b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{10} + \frac {5}{9} \, {\left (2 \, b^{5} d^{2} e^{3} + 5 \, a b^{4} d e^{4} + 2 \, a^{2} b^{3} e^{5}\right )} x^{9} + \frac {5}{4} \, {\left (b^{5} d^{3} e^{2} + 5 \, a b^{4} d^{2} e^{3} + 5 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{8} + \frac {5}{7} \, {\left (b^{5} d^{4} e + 10 \, a b^{4} d^{3} e^{2} + 20 \, a^{2} b^{3} d^{2} e^{3} + 10 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} d^{5} + 25 \, a b^{4} d^{4} e + 100 \, a^{2} b^{3} d^{3} e^{2} + 100 \, a^{3} b^{2} d^{2} e^{3} + 25 \, a^{4} b d e^{4} + a^{5} e^{5}\right )} x^{6} + {\left (a b^{4} d^{5} + 10 \, a^{2} b^{3} d^{4} e + 20 \, a^{3} b^{2} d^{3} e^{2} + 10 \, a^{4} b d^{2} e^{3} + a^{5} d e^{4}\right )} x^{5} + \frac {5}{2} \, {\left (a^{2} b^{3} d^{5} + 5 \, a^{3} b^{2} d^{4} e + 5 \, a^{4} b d^{3} e^{2} + a^{5} d^{2} e^{3}\right )} x^{4} + \frac {5}{3} \, {\left (2 \, a^{3} b^{2} d^{5} + 5 \, a^{4} b d^{4} e + 2 \, a^{5} d^{3} e^{2}\right )} x^{3} + \frac {5}{2} \, {\left (a^{4} b d^{5} + a^{5} d^{4} e\right )} x^{2} \] Input:
integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")
Output:
1/11*b^5*e^5*x^11 + a^5*d^5*x + 1/2*(b^5*d*e^4 + a*b^4*e^5)*x^10 + 5/9*(2* b^5*d^2*e^3 + 5*a*b^4*d*e^4 + 2*a^2*b^3*e^5)*x^9 + 5/4*(b^5*d^3*e^2 + 5*a* b^4*d^2*e^3 + 5*a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^8 + 5/7*(b^5*d^4*e + 10*a*b ^4*d^3*e^2 + 20*a^2*b^3*d^2*e^3 + 10*a^3*b^2*d*e^4 + a^4*b*e^5)*x^7 + 1/6* (b^5*d^5 + 25*a*b^4*d^4*e + 100*a^2*b^3*d^3*e^2 + 100*a^3*b^2*d^2*e^3 + 25 *a^4*b*d*e^4 + a^5*e^5)*x^6 + (a*b^4*d^5 + 10*a^2*b^3*d^4*e + 20*a^3*b^2*d ^3*e^2 + 10*a^4*b*d^2*e^3 + a^5*d*e^4)*x^5 + 5/2*(a^2*b^3*d^5 + 5*a^3*b^2* d^4*e + 5*a^4*b*d^3*e^2 + a^5*d^2*e^3)*x^4 + 5/3*(2*a^3*b^2*d^5 + 5*a^4*b* d^4*e + 2*a^5*d^3*e^2)*x^3 + 5/2*(a^4*b*d^5 + a^5*d^4*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 488 vs. \(2 (134) = 268\).
Time = 0.14 (sec) , antiderivative size = 488, normalized size of antiderivative = 3.34 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{11} \, b^{5} e^{5} x^{11} + \frac {1}{2} \, b^{5} d e^{4} x^{10} + \frac {1}{2} \, a b^{4} e^{5} x^{10} + \frac {10}{9} \, b^{5} d^{2} e^{3} x^{9} + \frac {25}{9} \, a b^{4} d e^{4} x^{9} + \frac {10}{9} \, a^{2} b^{3} e^{5} x^{9} + \frac {5}{4} \, b^{5} d^{3} e^{2} x^{8} + \frac {25}{4} \, a b^{4} d^{2} e^{3} x^{8} + \frac {25}{4} \, a^{2} b^{3} d e^{4} x^{8} + \frac {5}{4} \, a^{3} b^{2} e^{5} x^{8} + \frac {5}{7} \, b^{5} d^{4} e x^{7} + \frac {50}{7} \, a b^{4} d^{3} e^{2} x^{7} + \frac {100}{7} \, a^{2} b^{3} d^{2} e^{3} x^{7} + \frac {50}{7} \, a^{3} b^{2} d e^{4} x^{7} + \frac {5}{7} \, a^{4} b e^{5} x^{7} + \frac {1}{6} \, b^{5} d^{5} x^{6} + \frac {25}{6} \, a b^{4} d^{4} e x^{6} + \frac {50}{3} \, a^{2} b^{3} d^{3} e^{2} x^{6} + \frac {50}{3} \, a^{3} b^{2} d^{2} e^{3} x^{6} + \frac {25}{6} \, a^{4} b d e^{4} x^{6} + \frac {1}{6} \, a^{5} e^{5} x^{6} + a b^{4} d^{5} x^{5} + 10 \, a^{2} b^{3} d^{4} e x^{5} + 20 \, a^{3} b^{2} d^{3} e^{2} x^{5} + 10 \, a^{4} b d^{2} e^{3} x^{5} + a^{5} d e^{4} x^{5} + \frac {5}{2} \, a^{2} b^{3} d^{5} x^{4} + \frac {25}{2} \, a^{3} b^{2} d^{4} e x^{4} + \frac {25}{2} \, a^{4} b d^{3} e^{2} x^{4} + \frac {5}{2} \, a^{5} d^{2} e^{3} x^{4} + \frac {10}{3} \, a^{3} b^{2} d^{5} x^{3} + \frac {25}{3} \, a^{4} b d^{4} e x^{3} + \frac {10}{3} \, a^{5} d^{3} e^{2} x^{3} + \frac {5}{2} \, a^{4} b d^{5} x^{2} + \frac {5}{2} \, a^{5} d^{4} e x^{2} + a^{5} d^{5} x \] Input:
integrate((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")
Output:
1/11*b^5*e^5*x^11 + 1/2*b^5*d*e^4*x^10 + 1/2*a*b^4*e^5*x^10 + 10/9*b^5*d^2 *e^3*x^9 + 25/9*a*b^4*d*e^4*x^9 + 10/9*a^2*b^3*e^5*x^9 + 5/4*b^5*d^3*e^2*x ^8 + 25/4*a*b^4*d^2*e^3*x^8 + 25/4*a^2*b^3*d*e^4*x^8 + 5/4*a^3*b^2*e^5*x^8 + 5/7*b^5*d^4*e*x^7 + 50/7*a*b^4*d^3*e^2*x^7 + 100/7*a^2*b^3*d^2*e^3*x^7 + 50/7*a^3*b^2*d*e^4*x^7 + 5/7*a^4*b*e^5*x^7 + 1/6*b^5*d^5*x^6 + 25/6*a*b^ 4*d^4*e*x^6 + 50/3*a^2*b^3*d^3*e^2*x^6 + 50/3*a^3*b^2*d^2*e^3*x^6 + 25/6*a ^4*b*d*e^4*x^6 + 1/6*a^5*e^5*x^6 + a*b^4*d^5*x^5 + 10*a^2*b^3*d^4*e*x^5 + 20*a^3*b^2*d^3*e^2*x^5 + 10*a^4*b*d^2*e^3*x^5 + a^5*d*e^4*x^5 + 5/2*a^2*b^ 3*d^5*x^4 + 25/2*a^3*b^2*d^4*e*x^4 + 25/2*a^4*b*d^3*e^2*x^4 + 5/2*a^5*d^2* e^3*x^4 + 10/3*a^3*b^2*d^5*x^3 + 25/3*a^4*b*d^4*e*x^3 + 10/3*a^5*d^3*e^2*x ^3 + 5/2*a^4*b*d^5*x^2 + 5/2*a^5*d^4*e*x^2 + a^5*d^5*x
Time = 11.13 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.77 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=x^6\,\left (\frac {a^5\,e^5}{6}+\frac {25\,a^4\,b\,d\,e^4}{6}+\frac {50\,a^3\,b^2\,d^2\,e^3}{3}+\frac {50\,a^2\,b^3\,d^3\,e^2}{3}+\frac {25\,a\,b^4\,d^4\,e}{6}+\frac {b^5\,d^5}{6}\right )+x^5\,\left (a^5\,d\,e^4+10\,a^4\,b\,d^2\,e^3+20\,a^3\,b^2\,d^3\,e^2+10\,a^2\,b^3\,d^4\,e+a\,b^4\,d^5\right )+x^7\,\left (\frac {5\,a^4\,b\,e^5}{7}+\frac {50\,a^3\,b^2\,d\,e^4}{7}+\frac {100\,a^2\,b^3\,d^2\,e^3}{7}+\frac {50\,a\,b^4\,d^3\,e^2}{7}+\frac {5\,b^5\,d^4\,e}{7}\right )+a^5\,d^5\,x+\frac {b^5\,e^5\,x^{11}}{11}+\frac {5\,a^2\,d^2\,x^4\,\left (a^3\,e^3+5\,a^2\,b\,d\,e^2+5\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{2}+\frac {5\,b^2\,e^2\,x^8\,\left (a^3\,e^3+5\,a^2\,b\,d\,e^2+5\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{4}+\frac {5\,a^4\,d^4\,x^2\,\left (a\,e+b\,d\right )}{2}+\frac {b^4\,e^4\,x^{10}\,\left (a\,e+b\,d\right )}{2}+\frac {5\,a^3\,d^3\,x^3\,\left (2\,a^2\,e^2+5\,a\,b\,d\,e+2\,b^2\,d^2\right )}{3}+\frac {5\,b^3\,e^3\,x^9\,\left (2\,a^2\,e^2+5\,a\,b\,d\,e+2\,b^2\,d^2\right )}{9} \] Input:
int((a + b*x)*(d + e*x)^5*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)
Output:
x^6*((a^5*e^5)/6 + (b^5*d^5)/6 + (50*a^2*b^3*d^3*e^2)/3 + (50*a^3*b^2*d^2* e^3)/3 + (25*a*b^4*d^4*e)/6 + (25*a^4*b*d*e^4)/6) + x^5*(a*b^4*d^5 + a^5*d *e^4 + 10*a^2*b^3*d^4*e + 10*a^4*b*d^2*e^3 + 20*a^3*b^2*d^3*e^2) + x^7*((5 *a^4*b*e^5)/7 + (5*b^5*d^4*e)/7 + (50*a*b^4*d^3*e^2)/7 + (50*a^3*b^2*d*e^4 )/7 + (100*a^2*b^3*d^2*e^3)/7) + a^5*d^5*x + (b^5*e^5*x^11)/11 + (5*a^2*d^ 2*x^4*(a^3*e^3 + b^3*d^3 + 5*a*b^2*d^2*e + 5*a^2*b*d*e^2))/2 + (5*b^2*e^2* x^8*(a^3*e^3 + b^3*d^3 + 5*a*b^2*d^2*e + 5*a^2*b*d*e^2))/4 + (5*a^4*d^4*x^ 2*(a*e + b*d))/2 + (b^4*e^4*x^10*(a*e + b*d))/2 + (5*a^3*d^3*x^3*(2*a^2*e^ 2 + 2*b^2*d^2 + 5*a*b*d*e))/3 + (5*b^3*e^3*x^9*(2*a^2*e^2 + 2*b^2*d^2 + 5* a*b*d*e))/9
Time = 0.22 (sec) , antiderivative size = 489, normalized size of antiderivative = 3.35 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {x \left (252 b^{5} e^{5} x^{10}+1386 a \,b^{4} e^{5} x^{9}+1386 b^{5} d \,e^{4} x^{9}+3080 a^{2} b^{3} e^{5} x^{8}+7700 a \,b^{4} d \,e^{4} x^{8}+3080 b^{5} d^{2} e^{3} x^{8}+3465 a^{3} b^{2} e^{5} x^{7}+17325 a^{2} b^{3} d \,e^{4} x^{7}+17325 a \,b^{4} d^{2} e^{3} x^{7}+3465 b^{5} d^{3} e^{2} x^{7}+1980 a^{4} b \,e^{5} x^{6}+19800 a^{3} b^{2} d \,e^{4} x^{6}+39600 a^{2} b^{3} d^{2} e^{3} x^{6}+19800 a \,b^{4} d^{3} e^{2} x^{6}+1980 b^{5} d^{4} e \,x^{6}+462 a^{5} e^{5} x^{5}+11550 a^{4} b d \,e^{4} x^{5}+46200 a^{3} b^{2} d^{2} e^{3} x^{5}+46200 a^{2} b^{3} d^{3} e^{2} x^{5}+11550 a \,b^{4} d^{4} e \,x^{5}+462 b^{5} d^{5} x^{5}+2772 a^{5} d \,e^{4} x^{4}+27720 a^{4} b \,d^{2} e^{3} x^{4}+55440 a^{3} b^{2} d^{3} e^{2} x^{4}+27720 a^{2} b^{3} d^{4} e \,x^{4}+2772 a \,b^{4} d^{5} x^{4}+6930 a^{5} d^{2} e^{3} x^{3}+34650 a^{4} b \,d^{3} e^{2} x^{3}+34650 a^{3} b^{2} d^{4} e \,x^{3}+6930 a^{2} b^{3} d^{5} x^{3}+9240 a^{5} d^{3} e^{2} x^{2}+23100 a^{4} b \,d^{4} e \,x^{2}+9240 a^{3} b^{2} d^{5} x^{2}+6930 a^{5} d^{4} e x +6930 a^{4} b \,d^{5} x +2772 a^{5} d^{5}\right )}{2772} \] Input:
int((b*x+a)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x)
Output:
(x*(2772*a**5*d**5 + 6930*a**5*d**4*e*x + 9240*a**5*d**3*e**2*x**2 + 6930* a**5*d**2*e**3*x**3 + 2772*a**5*d*e**4*x**4 + 462*a**5*e**5*x**5 + 6930*a* *4*b*d**5*x + 23100*a**4*b*d**4*e*x**2 + 34650*a**4*b*d**3*e**2*x**3 + 277 20*a**4*b*d**2*e**3*x**4 + 11550*a**4*b*d*e**4*x**5 + 1980*a**4*b*e**5*x** 6 + 9240*a**3*b**2*d**5*x**2 + 34650*a**3*b**2*d**4*e*x**3 + 55440*a**3*b* *2*d**3*e**2*x**4 + 46200*a**3*b**2*d**2*e**3*x**5 + 19800*a**3*b**2*d*e** 4*x**6 + 3465*a**3*b**2*e**5*x**7 + 6930*a**2*b**3*d**5*x**3 + 27720*a**2* b**3*d**4*e*x**4 + 46200*a**2*b**3*d**3*e**2*x**5 + 39600*a**2*b**3*d**2*e **3*x**6 + 17325*a**2*b**3*d*e**4*x**7 + 3080*a**2*b**3*e**5*x**8 + 2772*a *b**4*d**5*x**4 + 11550*a*b**4*d**4*e*x**5 + 19800*a*b**4*d**3*e**2*x**6 + 17325*a*b**4*d**2*e**3*x**7 + 7700*a*b**4*d*e**4*x**8 + 1386*a*b**4*e**5* x**9 + 462*b**5*d**5*x**5 + 1980*b**5*d**4*e*x**6 + 3465*b**5*d**3*e**2*x* *7 + 3080*b**5*d**2*e**3*x**8 + 1386*b**5*d*e**4*x**9 + 252*b**5*e**5*x**1 0))/2772