Integrand size = 26, antiderivative size = 143 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {(b d-a e)^5 (a+b x)^7}{7 b^6}+\frac {5 e (b d-a e)^4 (a+b x)^8}{8 b^6}+\frac {10 e^2 (b d-a e)^3 (a+b x)^9}{9 b^6}+\frac {e^3 (b d-a e)^2 (a+b x)^{10}}{b^6}+\frac {5 e^4 (b d-a e) (a+b x)^{11}}{11 b^6}+\frac {e^5 (a+b x)^{12}}{12 b^6} \] Output:
1/7*(-a*e+b*d)^5*(b*x+a)^7/b^6+5/8*e*(-a*e+b*d)^4*(b*x+a)^8/b^6+10/9*e^2*( -a*e+b*d)^3*(b*x+a)^9/b^6+e^3*(-a*e+b*d)^2*(b*x+a)^10/b^6+5/11*e^4*(-a*e+b *d)*(b*x+a)^11/b^6+1/12*e^5*(b*x+a)^12/b^6
Leaf count is larger than twice the leaf count of optimal. \(501\) vs. \(2(143)=286\).
Time = 0.05 (sec) , antiderivative size = 501, normalized size of antiderivative = 3.50 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=a^6 d^5 x+\frac {1}{2} a^5 d^4 (6 b d+5 a e) x^2+\frac {5}{3} a^4 d^3 \left (3 b^2 d^2+6 a b d e+2 a^2 e^2\right ) x^3+\frac {5}{4} a^3 d^2 \left (4 b^3 d^3+15 a b^2 d^2 e+12 a^2 b d e^2+2 a^3 e^3\right ) x^4+a^2 d \left (3 b^4 d^4+20 a b^3 d^3 e+30 a^2 b^2 d^2 e^2+12 a^3 b d e^3+a^4 e^4\right ) x^5+\frac {1}{6} a \left (6 b^5 d^5+75 a b^4 d^4 e+200 a^2 b^3 d^3 e^2+150 a^3 b^2 d^2 e^3+30 a^4 b d e^4+a^5 e^5\right ) x^6+\frac {1}{7} b \left (b^5 d^5+30 a b^4 d^4 e+150 a^2 b^3 d^3 e^2+200 a^3 b^2 d^2 e^3+75 a^4 b d e^4+6 a^5 e^5\right ) x^7+\frac {5}{8} b^2 e \left (b^4 d^4+12 a b^3 d^3 e+30 a^2 b^2 d^2 e^2+20 a^3 b d e^3+3 a^4 e^4\right ) x^8+\frac {5}{9} b^3 e^2 \left (2 b^3 d^3+12 a b^2 d^2 e+15 a^2 b d e^2+4 a^3 e^3\right ) x^9+\frac {1}{2} b^4 e^3 \left (2 b^2 d^2+6 a b d e+3 a^2 e^2\right ) x^{10}+\frac {1}{11} b^5 e^4 (5 b d+6 a e) x^{11}+\frac {1}{12} b^6 e^5 x^{12} \] Input:
Integrate[(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
Output:
a^6*d^5*x + (a^5*d^4*(6*b*d + 5*a*e)*x^2)/2 + (5*a^4*d^3*(3*b^2*d^2 + 6*a* b*d*e + 2*a^2*e^2)*x^3)/3 + (5*a^3*d^2*(4*b^3*d^3 + 15*a*b^2*d^2*e + 12*a^ 2*b*d*e^2 + 2*a^3*e^3)*x^4)/4 + a^2*d*(3*b^4*d^4 + 20*a*b^3*d^3*e + 30*a^2 *b^2*d^2*e^2 + 12*a^3*b*d*e^3 + a^4*e^4)*x^5 + (a*(6*b^5*d^5 + 75*a*b^4*d^ 4*e + 200*a^2*b^3*d^3*e^2 + 150*a^3*b^2*d^2*e^3 + 30*a^4*b*d*e^4 + a^5*e^5 )*x^6)/6 + (b*(b^5*d^5 + 30*a*b^4*d^4*e + 150*a^2*b^3*d^3*e^2 + 200*a^3*b^ 2*d^2*e^3 + 75*a^4*b*d*e^4 + 6*a^5*e^5)*x^7)/7 + (5*b^2*e*(b^4*d^4 + 12*a* b^3*d^3*e + 30*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 + 3*a^4*e^4)*x^8)/8 + (5*b ^3*e^2*(2*b^3*d^3 + 12*a*b^2*d^2*e + 15*a^2*b*d*e^2 + 4*a^3*e^3)*x^9)/9 + (b^4*e^3*(2*b^2*d^2 + 6*a*b*d*e + 3*a^2*e^2)*x^10)/2 + (b^5*e^4*(5*b*d + 6 *a*e)*x^11)/11 + (b^6*e^5*x^12)/12
Time = 0.81 (sec) , antiderivative size = 143, 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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^3 (d+e x)^5 \, dx\) |
| \(\Big \downarrow \) 1098 |
| \(\displaystyle \frac {\int b^6 (a+b x)^6 (d+e x)^5dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^6 (d+e x)^5dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (\frac {5 e^4 (a+b x)^{10} (b d-a e)}{b^5}+\frac {10 e^3 (a+b x)^9 (b d-a e)^2}{b^5}+\frac {10 e^2 (a+b x)^8 (b d-a e)^3}{b^5}+\frac {5 e (a+b x)^7 (b d-a e)^4}{b^5}+\frac {(a+b x)^6 (b d-a e)^5}{b^5}+\frac {e^5 (a+b x)^{11}}{b^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5 e^4 (a+b x)^{11} (b d-a e)}{11 b^6}+\frac {e^3 (a+b x)^{10} (b d-a e)^2}{b^6}+\frac {10 e^2 (a+b x)^9 (b d-a e)^3}{9 b^6}+\frac {5 e (a+b x)^8 (b d-a e)^4}{8 b^6}+\frac {(a+b x)^7 (b d-a e)^5}{7 b^6}+\frac {e^5 (a+b x)^{12}}{12 b^6}\) |
Input:
Int[(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
Output:
((b*d - a*e)^5*(a + b*x)^7)/(7*b^6) + (5*e*(b*d - a*e)^4*(a + b*x)^8)/(8*b ^6) + (10*e^2*(b*d - a*e)^3*(a + b*x)^9)/(9*b^6) + (e^3*(b*d - a*e)^2*(a + b*x)^10)/b^6 + (5*e^4*(b*d - a*e)*(a + b*x)^11)/(11*b^6) + (e^5*(a + b*x) ^12)/(12*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_.)*((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. \(509\) vs. \(2(133)=266\).
Time = 1.05 (sec) , antiderivative size = 510, normalized size of antiderivative = 3.57
| method | result | size |
| norman | \(\frac {e^{5} b^{6} x^{12}}{12}+\left (\frac {6}{11} e^{5} a \,b^{5}+\frac {5}{11} d \,e^{4} b^{6}\right ) x^{11}+\left (\frac {3}{2} e^{5} a^{2} b^{4}+3 d \,e^{4} a \,b^{5}+d^{2} e^{3} b^{6}\right ) x^{10}+\left (\frac {20}{9} e^{5} a^{3} b^{3}+\frac {25}{3} a^{2} b^{4} d \,e^{4}+\frac {20}{3} d^{2} e^{3} a \,b^{5}+\frac {10}{9} d^{3} e^{2} b^{6}\right ) x^{9}+\left (\frac {15}{8} e^{5} a^{4} b^{2}+\frac {25}{2} d \,e^{4} a^{3} b^{3}+\frac {75}{4} d^{2} e^{3} a^{2} b^{4}+\frac {15}{2} d^{3} e^{2} a \,b^{5}+\frac {5}{8} d^{4} e \,b^{6}\right ) x^{8}+\left (\frac {6}{7} a^{5} b \,e^{5}+\frac {75}{7} d \,e^{4} a^{4} b^{2}+\frac {200}{7} d^{2} e^{3} a^{3} b^{3}+\frac {150}{7} d^{3} e^{2} a^{2} b^{4}+\frac {30}{7} d^{4} e a \,b^{5}+\frac {1}{7} d^{5} b^{6}\right ) x^{7}+\left (\frac {1}{6} e^{5} a^{6}+5 d \,e^{4} a^{5} b +25 d^{2} e^{3} a^{4} b^{2}+\frac {100}{3} d^{3} e^{2} a^{3} b^{3}+\frac {25}{2} d^{4} e \,a^{2} b^{4}+d^{5} a \,b^{5}\right ) x^{6}+\left (d \,e^{4} a^{6}+12 d^{2} e^{3} a^{5} b +30 d^{3} e^{2} a^{4} b^{2}+20 d^{4} e \,a^{3} b^{3}+3 d^{5} a^{2} b^{4}\right ) x^{5}+\left (\frac {5}{2} d^{2} e^{3} a^{6}+15 d^{3} e^{2} a^{5} b +\frac {75}{4} d^{4} e \,a^{4} b^{2}+5 d^{5} a^{3} b^{3}\right ) x^{4}+\left (\frac {10}{3} d^{3} e^{2} a^{6}+10 d^{4} e \,a^{5} b +5 d^{5} a^{4} b^{2}\right ) x^{3}+\left (\frac {5}{2} d^{4} e \,a^{6}+3 d^{5} a^{5} b \right ) x^{2}+d^{5} a^{6} x\) | \(510\) |
| default | \(\frac {e^{5} b^{6} x^{12}}{12}+\frac {\left (6 e^{5} a \,b^{5}+5 d \,e^{4} b^{6}\right ) x^{11}}{11}+\frac {\left (15 e^{5} a^{2} b^{4}+30 d \,e^{4} a \,b^{5}+10 d^{2} e^{3} b^{6}\right ) x^{10}}{10}+\frac {\left (20 e^{5} a^{3} b^{3}+75 a^{2} b^{4} d \,e^{4}+60 d^{2} e^{3} a \,b^{5}+10 d^{3} e^{2} b^{6}\right ) x^{9}}{9}+\frac {\left (15 e^{5} a^{4} b^{2}+100 d \,e^{4} a^{3} b^{3}+150 d^{2} e^{3} a^{2} b^{4}+60 d^{3} e^{2} a \,b^{5}+5 d^{4} e \,b^{6}\right ) x^{8}}{8}+\frac {\left (6 a^{5} b \,e^{5}+75 d \,e^{4} a^{4} b^{2}+200 d^{2} e^{3} a^{3} b^{3}+150 d^{3} e^{2} a^{2} b^{4}+30 d^{4} e a \,b^{5}+d^{5} b^{6}\right ) x^{7}}{7}+\frac {\left (e^{5} a^{6}+30 d \,e^{4} a^{5} b +150 d^{2} e^{3} a^{4} b^{2}+200 d^{3} e^{2} a^{3} b^{3}+75 d^{4} e \,a^{2} b^{4}+6 d^{5} a \,b^{5}\right ) x^{6}}{6}+\frac {\left (5 d \,e^{4} a^{6}+60 d^{2} e^{3} a^{5} b +150 d^{3} e^{2} a^{4} b^{2}+100 d^{4} e \,a^{3} b^{3}+15 d^{5} a^{2} b^{4}\right ) x^{5}}{5}+\frac {\left (10 d^{2} e^{3} a^{6}+60 d^{3} e^{2} a^{5} b +75 d^{4} e \,a^{4} b^{2}+20 d^{5} a^{3} b^{3}\right ) x^{4}}{4}+\frac {\left (10 d^{3} e^{2} a^{6}+30 d^{4} e \,a^{5} b +15 d^{5} a^{4} b^{2}\right ) x^{3}}{3}+\frac {\left (5 d^{4} e \,a^{6}+6 d^{5} a^{5} b \right ) x^{2}}{2}+d^{5} a^{6} x\) | \(521\) |
| risch | \(\frac {5}{11} x^{11} d \,e^{4} b^{6}+\frac {3}{2} x^{10} e^{5} a^{2} b^{4}+x^{10} d^{2} e^{3} b^{6}+\frac {20}{9} x^{9} e^{5} a^{3} b^{3}+\frac {10}{9} x^{9} d^{3} e^{2} b^{6}+\frac {15}{8} x^{8} e^{5} a^{4} b^{2}+\frac {5}{8} x^{8} d^{4} e \,b^{6}+\frac {6}{7} x^{7} a^{5} b \,e^{5}+x^{6} d^{5} a \,b^{5}+\frac {5}{2} x^{4} d^{2} e^{3} a^{6}+5 x^{4} d^{5} a^{3} b^{3}+\frac {10}{3} x^{3} d^{3} e^{2} a^{6}+5 x^{3} d^{5} a^{4} b^{2}+\frac {5}{2} x^{2} d^{4} e \,a^{6}+3 x^{2} d^{5} a^{5} b +a^{6} d \,e^{4} x^{5}+3 a^{2} b^{4} d^{5} x^{5}+3 x^{10} d \,e^{4} a \,b^{5}+\frac {25}{3} x^{9} a^{2} b^{4} d \,e^{4}+\frac {25}{2} x^{6} d^{4} e \,a^{2} b^{4}+15 x^{4} d^{3} e^{2} a^{5} b +\frac {75}{4} x^{4} d^{4} e \,a^{4} b^{2}+10 x^{3} d^{4} e \,a^{5} b +12 a^{5} b \,d^{2} e^{3} x^{5}+30 a^{4} b^{2} d^{3} e^{2} x^{5}+20 a^{3} b^{3} d^{4} e \,x^{5}+\frac {6}{11} x^{11} e^{5} a \,b^{5}+\frac {100}{3} x^{6} d^{3} e^{2} a^{3} b^{3}+\frac {20}{3} x^{9} d^{2} e^{3} a \,b^{5}+\frac {25}{2} x^{8} d \,e^{4} a^{3} b^{3}+\frac {75}{4} x^{8} d^{2} e^{3} a^{2} b^{4}+\frac {15}{2} x^{8} d^{3} e^{2} a \,b^{5}+\frac {75}{7} x^{7} d \,e^{4} a^{4} b^{2}+\frac {200}{7} x^{7} d^{2} e^{3} a^{3} b^{3}+\frac {150}{7} x^{7} d^{3} e^{2} a^{2} b^{4}+\frac {30}{7} x^{7} d^{4} e a \,b^{5}+5 x^{6} d \,e^{4} a^{5} b +25 x^{6} d^{2} e^{3} a^{4} b^{2}+d^{5} a^{6} x +\frac {1}{12} e^{5} b^{6} x^{12}+\frac {1}{7} x^{7} d^{5} b^{6}+\frac {1}{6} x^{6} e^{5} a^{6}\) | \(580\) |
| parallelrisch | \(\frac {5}{11} x^{11} d \,e^{4} b^{6}+\frac {3}{2} x^{10} e^{5} a^{2} b^{4}+x^{10} d^{2} e^{3} b^{6}+\frac {20}{9} x^{9} e^{5} a^{3} b^{3}+\frac {10}{9} x^{9} d^{3} e^{2} b^{6}+\frac {15}{8} x^{8} e^{5} a^{4} b^{2}+\frac {5}{8} x^{8} d^{4} e \,b^{6}+\frac {6}{7} x^{7} a^{5} b \,e^{5}+x^{6} d^{5} a \,b^{5}+\frac {5}{2} x^{4} d^{2} e^{3} a^{6}+5 x^{4} d^{5} a^{3} b^{3}+\frac {10}{3} x^{3} d^{3} e^{2} a^{6}+5 x^{3} d^{5} a^{4} b^{2}+\frac {5}{2} x^{2} d^{4} e \,a^{6}+3 x^{2} d^{5} a^{5} b +a^{6} d \,e^{4} x^{5}+3 a^{2} b^{4} d^{5} x^{5}+3 x^{10} d \,e^{4} a \,b^{5}+\frac {25}{3} x^{9} a^{2} b^{4} d \,e^{4}+\frac {25}{2} x^{6} d^{4} e \,a^{2} b^{4}+15 x^{4} d^{3} e^{2} a^{5} b +\frac {75}{4} x^{4} d^{4} e \,a^{4} b^{2}+10 x^{3} d^{4} e \,a^{5} b +12 a^{5} b \,d^{2} e^{3} x^{5}+30 a^{4} b^{2} d^{3} e^{2} x^{5}+20 a^{3} b^{3} d^{4} e \,x^{5}+\frac {6}{11} x^{11} e^{5} a \,b^{5}+\frac {100}{3} x^{6} d^{3} e^{2} a^{3} b^{3}+\frac {20}{3} x^{9} d^{2} e^{3} a \,b^{5}+\frac {25}{2} x^{8} d \,e^{4} a^{3} b^{3}+\frac {75}{4} x^{8} d^{2} e^{3} a^{2} b^{4}+\frac {15}{2} x^{8} d^{3} e^{2} a \,b^{5}+\frac {75}{7} x^{7} d \,e^{4} a^{4} b^{2}+\frac {200}{7} x^{7} d^{2} e^{3} a^{3} b^{3}+\frac {150}{7} x^{7} d^{3} e^{2} a^{2} b^{4}+\frac {30}{7} x^{7} d^{4} e a \,b^{5}+5 x^{6} d \,e^{4} a^{5} b +25 x^{6} d^{2} e^{3} a^{4} b^{2}+d^{5} a^{6} x +\frac {1}{12} e^{5} b^{6} x^{12}+\frac {1}{7} x^{7} d^{5} b^{6}+\frac {1}{6} x^{6} e^{5} a^{6}\) | \(580\) |
| gosper | \(\frac {x \left (462 e^{5} b^{6} x^{11}+3024 x^{10} e^{5} a \,b^{5}+2520 x^{10} d \,e^{4} b^{6}+8316 x^{9} e^{5} a^{2} b^{4}+16632 x^{9} d \,e^{4} a \,b^{5}+5544 x^{9} d^{2} e^{3} b^{6}+12320 x^{8} e^{5} a^{3} b^{3}+46200 x^{8} a^{2} b^{4} d \,e^{4}+36960 x^{8} d^{2} e^{3} a \,b^{5}+6160 x^{8} d^{3} e^{2} b^{6}+10395 x^{7} e^{5} a^{4} b^{2}+69300 x^{7} d \,e^{4} a^{3} b^{3}+103950 x^{7} d^{2} e^{3} a^{2} b^{4}+41580 x^{7} d^{3} e^{2} a \,b^{5}+3465 x^{7} d^{4} e \,b^{6}+4752 x^{6} a^{5} b \,e^{5}+59400 x^{6} d \,e^{4} a^{4} b^{2}+158400 x^{6} d^{2} e^{3} a^{3} b^{3}+118800 x^{6} d^{3} e^{2} a^{2} b^{4}+23760 x^{6} d^{4} e a \,b^{5}+792 x^{6} d^{5} b^{6}+924 x^{5} e^{5} a^{6}+27720 x^{5} d \,e^{4} a^{5} b +138600 x^{5} d^{2} e^{3} a^{4} b^{2}+184800 x^{5} d^{3} e^{2} a^{3} b^{3}+69300 x^{5} d^{4} e \,a^{2} b^{4}+5544 x^{5} d^{5} a \,b^{5}+5544 a^{6} d \,e^{4} x^{4}+66528 a^{5} b \,d^{2} e^{3} x^{4}+166320 a^{4} b^{2} d^{3} e^{2} x^{4}+110880 a^{3} b^{3} d^{4} e \,x^{4}+16632 a^{2} b^{4} d^{5} x^{4}+13860 x^{3} d^{2} e^{3} a^{6}+83160 x^{3} d^{3} e^{2} a^{5} b +103950 x^{3} d^{4} e \,a^{4} b^{2}+27720 x^{3} d^{5} a^{3} b^{3}+18480 x^{2} d^{3} e^{2} a^{6}+55440 x^{2} d^{4} e \,a^{5} b +27720 x^{2} d^{5} a^{4} b^{2}+13860 x \,d^{4} e \,a^{6}+16632 x \,d^{5} a^{5} b +5544 d^{5} a^{6}\right )}{5544}\) | \(582\) |
| orering | \(\frac {x \left (462 e^{5} b^{6} x^{11}+3024 x^{10} e^{5} a \,b^{5}+2520 x^{10} d \,e^{4} b^{6}+8316 x^{9} e^{5} a^{2} b^{4}+16632 x^{9} d \,e^{4} a \,b^{5}+5544 x^{9} d^{2} e^{3} b^{6}+12320 x^{8} e^{5} a^{3} b^{3}+46200 x^{8} a^{2} b^{4} d \,e^{4}+36960 x^{8} d^{2} e^{3} a \,b^{5}+6160 x^{8} d^{3} e^{2} b^{6}+10395 x^{7} e^{5} a^{4} b^{2}+69300 x^{7} d \,e^{4} a^{3} b^{3}+103950 x^{7} d^{2} e^{3} a^{2} b^{4}+41580 x^{7} d^{3} e^{2} a \,b^{5}+3465 x^{7} d^{4} e \,b^{6}+4752 x^{6} a^{5} b \,e^{5}+59400 x^{6} d \,e^{4} a^{4} b^{2}+158400 x^{6} d^{2} e^{3} a^{3} b^{3}+118800 x^{6} d^{3} e^{2} a^{2} b^{4}+23760 x^{6} d^{4} e a \,b^{5}+792 x^{6} d^{5} b^{6}+924 x^{5} e^{5} a^{6}+27720 x^{5} d \,e^{4} a^{5} b +138600 x^{5} d^{2} e^{3} a^{4} b^{2}+184800 x^{5} d^{3} e^{2} a^{3} b^{3}+69300 x^{5} d^{4} e \,a^{2} b^{4}+5544 x^{5} d^{5} a \,b^{5}+5544 a^{6} d \,e^{4} x^{4}+66528 a^{5} b \,d^{2} e^{3} x^{4}+166320 a^{4} b^{2} d^{3} e^{2} x^{4}+110880 a^{3} b^{3} d^{4} e \,x^{4}+16632 a^{2} b^{4} d^{5} x^{4}+13860 x^{3} d^{2} e^{3} a^{6}+83160 x^{3} d^{3} e^{2} a^{5} b +103950 x^{3} d^{4} e \,a^{4} b^{2}+27720 x^{3} d^{5} a^{3} b^{3}+18480 x^{2} d^{3} e^{2} a^{6}+55440 x^{2} d^{4} e \,a^{5} b +27720 x^{2} d^{5} a^{4} b^{2}+13860 x \,d^{4} e \,a^{6}+16632 x \,d^{5} a^{5} b +5544 d^{5} a^{6}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{3}}{5544 \left (b x +a \right )^{6}}\) | \(607\) |
Input:
int((e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)
Output:
1/12*e^5*b^6*x^12+(6/11*e^5*a*b^5+5/11*d*e^4*b^6)*x^11+(3/2*e^5*a^2*b^4+3* d*e^4*a*b^5+d^2*e^3*b^6)*x^10+(20/9*e^5*a^3*b^3+25/3*a^2*b^4*d*e^4+20/3*d^ 2*e^3*a*b^5+10/9*d^3*e^2*b^6)*x^9+(15/8*e^5*a^4*b^2+25/2*d*e^4*a^3*b^3+75/ 4*d^2*e^3*a^2*b^4+15/2*d^3*e^2*a*b^5+5/8*d^4*e*b^6)*x^8+(6/7*a^5*b*e^5+75/ 7*d*e^4*a^4*b^2+200/7*d^2*e^3*a^3*b^3+150/7*d^3*e^2*a^2*b^4+30/7*d^4*e*a*b ^5+1/7*d^5*b^6)*x^7+(1/6*e^5*a^6+5*d*e^4*a^5*b+25*d^2*e^3*a^4*b^2+100/3*d^ 3*e^2*a^3*b^3+25/2*d^4*e*a^2*b^4+d^5*a*b^5)*x^6+(a^6*d*e^4+12*a^5*b*d^2*e^ 3+30*a^4*b^2*d^3*e^2+20*a^3*b^3*d^4*e+3*a^2*b^4*d^5)*x^5+(5/2*d^2*e^3*a^6+ 15*d^3*e^2*a^5*b+75/4*d^4*e*a^4*b^2+5*d^5*a^3*b^3)*x^4+(10/3*d^3*e^2*a^6+1 0*d^4*e*a^5*b+5*d^5*a^4*b^2)*x^3+(5/2*d^4*e*a^6+3*d^5*a^5*b)*x^2+d^5*a^6*x
Leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (133) = 266\).
Time = 0.08 (sec) , antiderivative size = 517, normalized size of antiderivative = 3.62 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{12} \, b^{6} e^{5} x^{12} + a^{6} d^{5} x + \frac {1}{11} \, {\left (5 \, b^{6} d e^{4} + 6 \, a b^{5} e^{5}\right )} x^{11} + \frac {1}{2} \, {\left (2 \, b^{6} d^{2} e^{3} + 6 \, a b^{5} d e^{4} + 3 \, a^{2} b^{4} e^{5}\right )} x^{10} + \frac {5}{9} \, {\left (2 \, b^{6} d^{3} e^{2} + 12 \, a b^{5} d^{2} e^{3} + 15 \, a^{2} b^{4} d e^{4} + 4 \, a^{3} b^{3} e^{5}\right )} x^{9} + \frac {5}{8} \, {\left (b^{6} d^{4} e + 12 \, a b^{5} d^{3} e^{2} + 30 \, a^{2} b^{4} d^{2} e^{3} + 20 \, a^{3} b^{3} d e^{4} + 3 \, a^{4} b^{2} e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{5} + 30 \, a b^{5} d^{4} e + 150 \, a^{2} b^{4} d^{3} e^{2} + 200 \, a^{3} b^{3} d^{2} e^{3} + 75 \, a^{4} b^{2} d e^{4} + 6 \, a^{5} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (6 \, a b^{5} d^{5} + 75 \, a^{2} b^{4} d^{4} e + 200 \, a^{3} b^{3} d^{3} e^{2} + 150 \, a^{4} b^{2} d^{2} e^{3} + 30 \, a^{5} b d e^{4} + a^{6} e^{5}\right )} x^{6} + {\left (3 \, a^{2} b^{4} d^{5} + 20 \, a^{3} b^{3} d^{4} e + 30 \, a^{4} b^{2} d^{3} e^{2} + 12 \, a^{5} b d^{2} e^{3} + a^{6} d e^{4}\right )} x^{5} + \frac {5}{4} \, {\left (4 \, a^{3} b^{3} d^{5} + 15 \, a^{4} b^{2} d^{4} e + 12 \, a^{5} b d^{3} e^{2} + 2 \, a^{6} d^{2} e^{3}\right )} x^{4} + \frac {5}{3} \, {\left (3 \, a^{4} b^{2} d^{5} + 6 \, a^{5} b d^{4} e + 2 \, a^{6} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (6 \, a^{5} b d^{5} + 5 \, a^{6} d^{4} e\right )} x^{2} \] Input:
integrate((e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
Output:
1/12*b^6*e^5*x^12 + a^6*d^5*x + 1/11*(5*b^6*d*e^4 + 6*a*b^5*e^5)*x^11 + 1/ 2*(2*b^6*d^2*e^3 + 6*a*b^5*d*e^4 + 3*a^2*b^4*e^5)*x^10 + 5/9*(2*b^6*d^3*e^ 2 + 12*a*b^5*d^2*e^3 + 15*a^2*b^4*d*e^4 + 4*a^3*b^3*e^5)*x^9 + 5/8*(b^6*d^ 4*e + 12*a*b^5*d^3*e^2 + 30*a^2*b^4*d^2*e^3 + 20*a^3*b^3*d*e^4 + 3*a^4*b^2 *e^5)*x^8 + 1/7*(b^6*d^5 + 30*a*b^5*d^4*e + 150*a^2*b^4*d^3*e^2 + 200*a^3* b^3*d^2*e^3 + 75*a^4*b^2*d*e^4 + 6*a^5*b*e^5)*x^7 + 1/6*(6*a*b^5*d^5 + 75* a^2*b^4*d^4*e + 200*a^3*b^3*d^3*e^2 + 150*a^4*b^2*d^2*e^3 + 30*a^5*b*d*e^4 + a^6*e^5)*x^6 + (3*a^2*b^4*d^5 + 20*a^3*b^3*d^4*e + 30*a^4*b^2*d^3*e^2 + 12*a^5*b*d^2*e^3 + a^6*d*e^4)*x^5 + 5/4*(4*a^3*b^3*d^5 + 15*a^4*b^2*d^4*e + 12*a^5*b*d^3*e^2 + 2*a^6*d^2*e^3)*x^4 + 5/3*(3*a^4*b^2*d^5 + 6*a^5*b*d^ 4*e + 2*a^6*d^3*e^2)*x^3 + 1/2*(6*a^5*b*d^5 + 5*a^6*d^4*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 580 vs. \(2 (129) = 258\).
Time = 0.05 (sec) , antiderivative size = 580, normalized size of antiderivative = 4.06 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=a^{6} d^{5} x + \frac {b^{6} e^{5} x^{12}}{12} + x^{11} \cdot \left (\frac {6 a b^{5} e^{5}}{11} + \frac {5 b^{6} d e^{4}}{11}\right ) + x^{10} \cdot \left (\frac {3 a^{2} b^{4} e^{5}}{2} + 3 a b^{5} d e^{4} + b^{6} d^{2} e^{3}\right ) + x^{9} \cdot \left (\frac {20 a^{3} b^{3} e^{5}}{9} + \frac {25 a^{2} b^{4} d e^{4}}{3} + \frac {20 a b^{5} d^{2} e^{3}}{3} + \frac {10 b^{6} d^{3} e^{2}}{9}\right ) + x^{8} \cdot \left (\frac {15 a^{4} b^{2} e^{5}}{8} + \frac {25 a^{3} b^{3} d e^{4}}{2} + \frac {75 a^{2} b^{4} d^{2} e^{3}}{4} + \frac {15 a b^{5} d^{3} e^{2}}{2} + \frac {5 b^{6} d^{4} e}{8}\right ) + x^{7} \cdot \left (\frac {6 a^{5} b e^{5}}{7} + \frac {75 a^{4} b^{2} d e^{4}}{7} + \frac {200 a^{3} b^{3} d^{2} e^{3}}{7} + \frac {150 a^{2} b^{4} d^{3} e^{2}}{7} + \frac {30 a b^{5} d^{4} e}{7} + \frac {b^{6} d^{5}}{7}\right ) + x^{6} \left (\frac {a^{6} e^{5}}{6} + 5 a^{5} b d e^{4} + 25 a^{4} b^{2} d^{2} e^{3} + \frac {100 a^{3} b^{3} d^{3} e^{2}}{3} + \frac {25 a^{2} b^{4} d^{4} e}{2} + a b^{5} d^{5}\right ) + x^{5} \left (a^{6} d e^{4} + 12 a^{5} b d^{2} e^{3} + 30 a^{4} b^{2} d^{3} e^{2} + 20 a^{3} b^{3} d^{4} e + 3 a^{2} b^{4} d^{5}\right ) + x^{4} \cdot \left (\frac {5 a^{6} d^{2} e^{3}}{2} + 15 a^{5} b d^{3} e^{2} + \frac {75 a^{4} b^{2} d^{4} e}{4} + 5 a^{3} b^{3} d^{5}\right ) + x^{3} \cdot \left (\frac {10 a^{6} d^{3} e^{2}}{3} + 10 a^{5} b d^{4} e + 5 a^{4} b^{2} d^{5}\right ) + x^{2} \cdot \left (\frac {5 a^{6} d^{4} e}{2} + 3 a^{5} b d^{5}\right ) \] Input:
integrate((e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**3,x)
Output:
a**6*d**5*x + b**6*e**5*x**12/12 + x**11*(6*a*b**5*e**5/11 + 5*b**6*d*e**4 /11) + x**10*(3*a**2*b**4*e**5/2 + 3*a*b**5*d*e**4 + b**6*d**2*e**3) + x** 9*(20*a**3*b**3*e**5/9 + 25*a**2*b**4*d*e**4/3 + 20*a*b**5*d**2*e**3/3 + 1 0*b**6*d**3*e**2/9) + x**8*(15*a**4*b**2*e**5/8 + 25*a**3*b**3*d*e**4/2 + 75*a**2*b**4*d**2*e**3/4 + 15*a*b**5*d**3*e**2/2 + 5*b**6*d**4*e/8) + x**7 *(6*a**5*b*e**5/7 + 75*a**4*b**2*d*e**4/7 + 200*a**3*b**3*d**2*e**3/7 + 15 0*a**2*b**4*d**3*e**2/7 + 30*a*b**5*d**4*e/7 + b**6*d**5/7) + x**6*(a**6*e **5/6 + 5*a**5*b*d*e**4 + 25*a**4*b**2*d**2*e**3 + 100*a**3*b**3*d**3*e**2 /3 + 25*a**2*b**4*d**4*e/2 + a*b**5*d**5) + x**5*(a**6*d*e**4 + 12*a**5*b* d**2*e**3 + 30*a**4*b**2*d**3*e**2 + 20*a**3*b**3*d**4*e + 3*a**2*b**4*d** 5) + x**4*(5*a**6*d**2*e**3/2 + 15*a**5*b*d**3*e**2 + 75*a**4*b**2*d**4*e/ 4 + 5*a**3*b**3*d**5) + x**3*(10*a**6*d**3*e**2/3 + 10*a**5*b*d**4*e + 5*a **4*b**2*d**5) + x**2*(5*a**6*d**4*e/2 + 3*a**5*b*d**5)
Leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (133) = 266\).
Time = 0.05 (sec) , antiderivative size = 517, normalized size of antiderivative = 3.62 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{12} \, b^{6} e^{5} x^{12} + a^{6} d^{5} x + \frac {1}{11} \, {\left (5 \, b^{6} d e^{4} + 6 \, a b^{5} e^{5}\right )} x^{11} + \frac {1}{2} \, {\left (2 \, b^{6} d^{2} e^{3} + 6 \, a b^{5} d e^{4} + 3 \, a^{2} b^{4} e^{5}\right )} x^{10} + \frac {5}{9} \, {\left (2 \, b^{6} d^{3} e^{2} + 12 \, a b^{5} d^{2} e^{3} + 15 \, a^{2} b^{4} d e^{4} + 4 \, a^{3} b^{3} e^{5}\right )} x^{9} + \frac {5}{8} \, {\left (b^{6} d^{4} e + 12 \, a b^{5} d^{3} e^{2} + 30 \, a^{2} b^{4} d^{2} e^{3} + 20 \, a^{3} b^{3} d e^{4} + 3 \, a^{4} b^{2} e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{5} + 30 \, a b^{5} d^{4} e + 150 \, a^{2} b^{4} d^{3} e^{2} + 200 \, a^{3} b^{3} d^{2} e^{3} + 75 \, a^{4} b^{2} d e^{4} + 6 \, a^{5} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (6 \, a b^{5} d^{5} + 75 \, a^{2} b^{4} d^{4} e + 200 \, a^{3} b^{3} d^{3} e^{2} + 150 \, a^{4} b^{2} d^{2} e^{3} + 30 \, a^{5} b d e^{4} + a^{6} e^{5}\right )} x^{6} + {\left (3 \, a^{2} b^{4} d^{5} + 20 \, a^{3} b^{3} d^{4} e + 30 \, a^{4} b^{2} d^{3} e^{2} + 12 \, a^{5} b d^{2} e^{3} + a^{6} d e^{4}\right )} x^{5} + \frac {5}{4} \, {\left (4 \, a^{3} b^{3} d^{5} + 15 \, a^{4} b^{2} d^{4} e + 12 \, a^{5} b d^{3} e^{2} + 2 \, a^{6} d^{2} e^{3}\right )} x^{4} + \frac {5}{3} \, {\left (3 \, a^{4} b^{2} d^{5} + 6 \, a^{5} b d^{4} e + 2 \, a^{6} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (6 \, a^{5} b d^{5} + 5 \, a^{6} d^{4} e\right )} x^{2} \] Input:
integrate((e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")
Output:
1/12*b^6*e^5*x^12 + a^6*d^5*x + 1/11*(5*b^6*d*e^4 + 6*a*b^5*e^5)*x^11 + 1/ 2*(2*b^6*d^2*e^3 + 6*a*b^5*d*e^4 + 3*a^2*b^4*e^5)*x^10 + 5/9*(2*b^6*d^3*e^ 2 + 12*a*b^5*d^2*e^3 + 15*a^2*b^4*d*e^4 + 4*a^3*b^3*e^5)*x^9 + 5/8*(b^6*d^ 4*e + 12*a*b^5*d^3*e^2 + 30*a^2*b^4*d^2*e^3 + 20*a^3*b^3*d*e^4 + 3*a^4*b^2 *e^5)*x^8 + 1/7*(b^6*d^5 + 30*a*b^5*d^4*e + 150*a^2*b^4*d^3*e^2 + 200*a^3* b^3*d^2*e^3 + 75*a^4*b^2*d*e^4 + 6*a^5*b*e^5)*x^7 + 1/6*(6*a*b^5*d^5 + 75* a^2*b^4*d^4*e + 200*a^3*b^3*d^3*e^2 + 150*a^4*b^2*d^2*e^3 + 30*a^5*b*d*e^4 + a^6*e^5)*x^6 + (3*a^2*b^4*d^5 + 20*a^3*b^3*d^4*e + 30*a^4*b^2*d^3*e^2 + 12*a^5*b*d^2*e^3 + a^6*d*e^4)*x^5 + 5/4*(4*a^3*b^3*d^5 + 15*a^4*b^2*d^4*e + 12*a^5*b*d^3*e^2 + 2*a^6*d^2*e^3)*x^4 + 5/3*(3*a^4*b^2*d^5 + 6*a^5*b*d^ 4*e + 2*a^6*d^3*e^2)*x^3 + 1/2*(6*a^5*b*d^5 + 5*a^6*d^4*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 579 vs. \(2 (133) = 266\).
Time = 0.20 (sec) , antiderivative size = 579, normalized size of antiderivative = 4.05 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{12} \, b^{6} e^{5} x^{12} + \frac {5}{11} \, b^{6} d e^{4} x^{11} + \frac {6}{11} \, a b^{5} e^{5} x^{11} + b^{6} d^{2} e^{3} x^{10} + 3 \, a b^{5} d e^{4} x^{10} + \frac {3}{2} \, a^{2} b^{4} e^{5} x^{10} + \frac {10}{9} \, b^{6} d^{3} e^{2} x^{9} + \frac {20}{3} \, a b^{5} d^{2} e^{3} x^{9} + \frac {25}{3} \, a^{2} b^{4} d e^{4} x^{9} + \frac {20}{9} \, a^{3} b^{3} e^{5} x^{9} + \frac {5}{8} \, b^{6} d^{4} e x^{8} + \frac {15}{2} \, a b^{5} d^{3} e^{2} x^{8} + \frac {75}{4} \, a^{2} b^{4} d^{2} e^{3} x^{8} + \frac {25}{2} \, a^{3} b^{3} d e^{4} x^{8} + \frac {15}{8} \, a^{4} b^{2} e^{5} x^{8} + \frac {1}{7} \, b^{6} d^{5} x^{7} + \frac {30}{7} \, a b^{5} d^{4} e x^{7} + \frac {150}{7} \, a^{2} b^{4} d^{3} e^{2} x^{7} + \frac {200}{7} \, a^{3} b^{3} d^{2} e^{3} x^{7} + \frac {75}{7} \, a^{4} b^{2} d e^{4} x^{7} + \frac {6}{7} \, a^{5} b e^{5} x^{7} + a b^{5} d^{5} x^{6} + \frac {25}{2} \, a^{2} b^{4} d^{4} e x^{6} + \frac {100}{3} \, a^{3} b^{3} d^{3} e^{2} x^{6} + 25 \, a^{4} b^{2} d^{2} e^{3} x^{6} + 5 \, a^{5} b d e^{4} x^{6} + \frac {1}{6} \, a^{6} e^{5} x^{6} + 3 \, a^{2} b^{4} d^{5} x^{5} + 20 \, a^{3} b^{3} d^{4} e x^{5} + 30 \, a^{4} b^{2} d^{3} e^{2} x^{5} + 12 \, a^{5} b d^{2} e^{3} x^{5} + a^{6} d e^{4} x^{5} + 5 \, a^{3} b^{3} d^{5} x^{4} + \frac {75}{4} \, a^{4} b^{2} d^{4} e x^{4} + 15 \, a^{5} b d^{3} e^{2} x^{4} + \frac {5}{2} \, a^{6} d^{2} e^{3} x^{4} + 5 \, a^{4} b^{2} d^{5} x^{3} + 10 \, a^{5} b d^{4} e x^{3} + \frac {10}{3} \, a^{6} d^{3} e^{2} x^{3} + 3 \, a^{5} b d^{5} x^{2} + \frac {5}{2} \, a^{6} d^{4} e x^{2} + a^{6} d^{5} x \] Input:
integrate((e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
Output:
1/12*b^6*e^5*x^12 + 5/11*b^6*d*e^4*x^11 + 6/11*a*b^5*e^5*x^11 + b^6*d^2*e^ 3*x^10 + 3*a*b^5*d*e^4*x^10 + 3/2*a^2*b^4*e^5*x^10 + 10/9*b^6*d^3*e^2*x^9 + 20/3*a*b^5*d^2*e^3*x^9 + 25/3*a^2*b^4*d*e^4*x^9 + 20/9*a^3*b^3*e^5*x^9 + 5/8*b^6*d^4*e*x^8 + 15/2*a*b^5*d^3*e^2*x^8 + 75/4*a^2*b^4*d^2*e^3*x^8 + 2 5/2*a^3*b^3*d*e^4*x^8 + 15/8*a^4*b^2*e^5*x^8 + 1/7*b^6*d^5*x^7 + 30/7*a*b^ 5*d^4*e*x^7 + 150/7*a^2*b^4*d^3*e^2*x^7 + 200/7*a^3*b^3*d^2*e^3*x^7 + 75/7 *a^4*b^2*d*e^4*x^7 + 6/7*a^5*b*e^5*x^7 + a*b^5*d^5*x^6 + 25/2*a^2*b^4*d^4* e*x^6 + 100/3*a^3*b^3*d^3*e^2*x^6 + 25*a^4*b^2*d^2*e^3*x^6 + 5*a^5*b*d*e^4 *x^6 + 1/6*a^6*e^5*x^6 + 3*a^2*b^4*d^5*x^5 + 20*a^3*b^3*d^4*e*x^5 + 30*a^4 *b^2*d^3*e^2*x^5 + 12*a^5*b*d^2*e^3*x^5 + a^6*d*e^4*x^5 + 5*a^3*b^3*d^5*x^ 4 + 75/4*a^4*b^2*d^4*e*x^4 + 15*a^5*b*d^3*e^2*x^4 + 5/2*a^6*d^2*e^3*x^4 + 5*a^4*b^2*d^5*x^3 + 10*a^5*b*d^4*e*x^3 + 10/3*a^6*d^3*e^2*x^3 + 3*a^5*b*d^ 5*x^2 + 5/2*a^6*d^4*e*x^2 + a^6*d^5*x
Time = 8.94 (sec) , antiderivative size = 492, normalized size of antiderivative = 3.44 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=x^5\,\left (a^6\,d\,e^4+12\,a^5\,b\,d^2\,e^3+30\,a^4\,b^2\,d^3\,e^2+20\,a^3\,b^3\,d^4\,e+3\,a^2\,b^4\,d^5\right )+x^8\,\left (\frac {15\,a^4\,b^2\,e^5}{8}+\frac {25\,a^3\,b^3\,d\,e^4}{2}+\frac {75\,a^2\,b^4\,d^2\,e^3}{4}+\frac {15\,a\,b^5\,d^3\,e^2}{2}+\frac {5\,b^6\,d^4\,e}{8}\right )+x^6\,\left (\frac {a^6\,e^5}{6}+5\,a^5\,b\,d\,e^4+25\,a^4\,b^2\,d^2\,e^3+\frac {100\,a^3\,b^3\,d^3\,e^2}{3}+\frac {25\,a^2\,b^4\,d^4\,e}{2}+a\,b^5\,d^5\right )+x^7\,\left (\frac {6\,a^5\,b\,e^5}{7}+\frac {75\,a^4\,b^2\,d\,e^4}{7}+\frac {200\,a^3\,b^3\,d^2\,e^3}{7}+\frac {150\,a^2\,b^4\,d^3\,e^2}{7}+\frac {30\,a\,b^5\,d^4\,e}{7}+\frac {b^6\,d^5}{7}\right )+a^6\,d^5\,x+\frac {b^6\,e^5\,x^{12}}{12}+\frac {5\,a^3\,d^2\,x^4\,\left (2\,a^3\,e^3+12\,a^2\,b\,d\,e^2+15\,a\,b^2\,d^2\,e+4\,b^3\,d^3\right )}{4}+\frac {5\,b^3\,e^2\,x^9\,\left (4\,a^3\,e^3+15\,a^2\,b\,d\,e^2+12\,a\,b^2\,d^2\,e+2\,b^3\,d^3\right )}{9}+\frac {a^5\,d^4\,x^2\,\left (5\,a\,e+6\,b\,d\right )}{2}+\frac {b^5\,e^4\,x^{11}\,\left (6\,a\,e+5\,b\,d\right )}{11}+\frac {5\,a^4\,d^3\,x^3\,\left (2\,a^2\,e^2+6\,a\,b\,d\,e+3\,b^2\,d^2\right )}{3}+\frac {b^4\,e^3\,x^{10}\,\left (3\,a^2\,e^2+6\,a\,b\,d\,e+2\,b^2\,d^2\right )}{2} \] Input:
int((d + e*x)^5*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)
Output:
x^5*(a^6*d*e^4 + 3*a^2*b^4*d^5 + 20*a^3*b^3*d^4*e + 12*a^5*b*d^2*e^3 + 30* a^4*b^2*d^3*e^2) + x^8*((5*b^6*d^4*e)/8 + (15*a^4*b^2*e^5)/8 + (15*a*b^5*d ^3*e^2)/2 + (25*a^3*b^3*d*e^4)/2 + (75*a^2*b^4*d^2*e^3)/4) + x^6*((a^6*e^5 )/6 + a*b^5*d^5 + (25*a^2*b^4*d^4*e)/2 + (100*a^3*b^3*d^3*e^2)/3 + 25*a^4* b^2*d^2*e^3 + 5*a^5*b*d*e^4) + x^7*((b^6*d^5)/7 + (6*a^5*b*e^5)/7 + (75*a^ 4*b^2*d*e^4)/7 + (150*a^2*b^4*d^3*e^2)/7 + (200*a^3*b^3*d^2*e^3)/7 + (30*a *b^5*d^4*e)/7) + a^6*d^5*x + (b^6*e^5*x^12)/12 + (5*a^3*d^2*x^4*(2*a^3*e^3 + 4*b^3*d^3 + 15*a*b^2*d^2*e + 12*a^2*b*d*e^2))/4 + (5*b^3*e^2*x^9*(4*a^3 *e^3 + 2*b^3*d^3 + 12*a*b^2*d^2*e + 15*a^2*b*d*e^2))/9 + (a^5*d^4*x^2*(5*a *e + 6*b*d))/2 + (b^5*e^4*x^11*(6*a*e + 5*b*d))/11 + (5*a^4*d^3*x^3*(2*a^2 *e^2 + 3*b^2*d^2 + 6*a*b*d*e))/3 + (b^4*e^3*x^10*(3*a^2*e^2 + 2*b^2*d^2 + 6*a*b*d*e))/2
Time = 0.21 (sec) , antiderivative size = 581, normalized size of antiderivative = 4.06 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {x \left (462 b^{6} e^{5} x^{11}+3024 a \,b^{5} e^{5} x^{10}+2520 b^{6} d \,e^{4} x^{10}+8316 a^{2} b^{4} e^{5} x^{9}+16632 a \,b^{5} d \,e^{4} x^{9}+5544 b^{6} d^{2} e^{3} x^{9}+12320 a^{3} b^{3} e^{5} x^{8}+46200 a^{2} b^{4} d \,e^{4} x^{8}+36960 a \,b^{5} d^{2} e^{3} x^{8}+6160 b^{6} d^{3} e^{2} x^{8}+10395 a^{4} b^{2} e^{5} x^{7}+69300 a^{3} b^{3} d \,e^{4} x^{7}+103950 a^{2} b^{4} d^{2} e^{3} x^{7}+41580 a \,b^{5} d^{3} e^{2} x^{7}+3465 b^{6} d^{4} e \,x^{7}+4752 a^{5} b \,e^{5} x^{6}+59400 a^{4} b^{2} d \,e^{4} x^{6}+158400 a^{3} b^{3} d^{2} e^{3} x^{6}+118800 a^{2} b^{4} d^{3} e^{2} x^{6}+23760 a \,b^{5} d^{4} e \,x^{6}+792 b^{6} d^{5} x^{6}+924 a^{6} e^{5} x^{5}+27720 a^{5} b d \,e^{4} x^{5}+138600 a^{4} b^{2} d^{2} e^{3} x^{5}+184800 a^{3} b^{3} d^{3} e^{2} x^{5}+69300 a^{2} b^{4} d^{4} e \,x^{5}+5544 a \,b^{5} d^{5} x^{5}+5544 a^{6} d \,e^{4} x^{4}+66528 a^{5} b \,d^{2} e^{3} x^{4}+166320 a^{4} b^{2} d^{3} e^{2} x^{4}+110880 a^{3} b^{3} d^{4} e \,x^{4}+16632 a^{2} b^{4} d^{5} x^{4}+13860 a^{6} d^{2} e^{3} x^{3}+83160 a^{5} b \,d^{3} e^{2} x^{3}+103950 a^{4} b^{2} d^{4} e \,x^{3}+27720 a^{3} b^{3} d^{5} x^{3}+18480 a^{6} d^{3} e^{2} x^{2}+55440 a^{5} b \,d^{4} e \,x^{2}+27720 a^{4} b^{2} d^{5} x^{2}+13860 a^{6} d^{4} e x +16632 a^{5} b \,d^{5} x +5544 a^{6} d^{5}\right )}{5544} \] Input:
int((e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^3,x)
Output:
(x*(5544*a**6*d**5 + 13860*a**6*d**4*e*x + 18480*a**6*d**3*e**2*x**2 + 138 60*a**6*d**2*e**3*x**3 + 5544*a**6*d*e**4*x**4 + 924*a**6*e**5*x**5 + 1663 2*a**5*b*d**5*x + 55440*a**5*b*d**4*e*x**2 + 83160*a**5*b*d**3*e**2*x**3 + 66528*a**5*b*d**2*e**3*x**4 + 27720*a**5*b*d*e**4*x**5 + 4752*a**5*b*e**5 *x**6 + 27720*a**4*b**2*d**5*x**2 + 103950*a**4*b**2*d**4*e*x**3 + 166320* a**4*b**2*d**3*e**2*x**4 + 138600*a**4*b**2*d**2*e**3*x**5 + 59400*a**4*b* *2*d*e**4*x**6 + 10395*a**4*b**2*e**5*x**7 + 27720*a**3*b**3*d**5*x**3 + 1 10880*a**3*b**3*d**4*e*x**4 + 184800*a**3*b**3*d**3*e**2*x**5 + 158400*a** 3*b**3*d**2*e**3*x**6 + 69300*a**3*b**3*d*e**4*x**7 + 12320*a**3*b**3*e**5 *x**8 + 16632*a**2*b**4*d**5*x**4 + 69300*a**2*b**4*d**4*e*x**5 + 118800*a **2*b**4*d**3*e**2*x**6 + 103950*a**2*b**4*d**2*e**3*x**7 + 46200*a**2*b** 4*d*e**4*x**8 + 8316*a**2*b**4*e**5*x**9 + 5544*a*b**5*d**5*x**5 + 23760*a *b**5*d**4*e*x**6 + 41580*a*b**5*d**3*e**2*x**7 + 36960*a*b**5*d**2*e**3*x **8 + 16632*a*b**5*d*e**4*x**9 + 3024*a*b**5*e**5*x**10 + 792*b**6*d**5*x* *6 + 3465*b**6*d**4*e*x**7 + 6160*b**6*d**3*e**2*x**8 + 5544*b**6*d**2*e** 3*x**9 + 2520*b**6*d*e**4*x**10 + 462*b**6*e**5*x**11))/5544