Integrand size = 20, antiderivative size = 243 \[ \int (a+b x)^{10} (A+B x) (d+e x)^5 \, dx=\frac {(A b-a B) (b d-a e)^5 (a+b x)^{11}}{11 b^7}+\frac {(b d-a e)^4 (b B d+5 A b e-6 a B e) (a+b x)^{12}}{12 b^7}+\frac {5 e (b d-a e)^3 (b B d+2 A b e-3 a B e) (a+b x)^{13}}{13 b^7}+\frac {5 e^2 (b d-a e)^2 (b B d+A b e-2 a B e) (a+b x)^{14}}{7 b^7}+\frac {e^3 (b d-a e) (2 b B d+A b e-3 a B e) (a+b x)^{15}}{3 b^7}+\frac {e^4 (5 b B d+A b e-6 a B e) (a+b x)^{16}}{16 b^7}+\frac {B e^5 (a+b x)^{17}}{17 b^7} \]
1/11*(A*b-B*a)*(-a*e+b*d)^5*(b*x+a)^11/b^7+1/12*(-a*e+b*d)^4*(5*A*b*e-6*B* a*e+B*b*d)*(b*x+a)^12/b^7+5/13*e*(-a*e+b*d)^3*(2*A*b*e-3*B*a*e+B*b*d)*(b*x +a)^13/b^7+5/7*e^2*(-a*e+b*d)^2*(A*b*e-2*B*a*e+B*b*d)*(b*x+a)^14/b^7+1/3*e ^3*(-a*e+b*d)*(A*b*e-3*B*a*e+2*B*b*d)*(b*x+a)^15/b^7+1/16*e^4*(A*b*e-6*B*a *e+5*B*b*d)*(b*x+a)^16/b^7+1/17*B*e^5*(b*x+a)^17/b^7
Leaf count is larger than twice the leaf count of optimal. \(1509\) vs. \(2(243)=486\).
Time = 0.34 (sec) , antiderivative size = 1509, normalized size of antiderivative = 6.21 \[ \int (a+b x)^{10} (A+B x) (d+e x)^5 \, dx=a^{10} A d^5 x+\frac {1}{2} a^9 d^4 (a B d+5 A (2 b d+a e)) x^2+\frac {5}{3} a^8 d^3 \left (a B d (2 b d+a e)+A \left (9 b^2 d^2+10 a b d e+2 a^2 e^2\right )\right ) x^3+\frac {5}{4} a^7 d^2 \left (a B d \left (9 b^2 d^2+10 a b d e+2 a^2 e^2\right )+A \left (24 b^3 d^3+45 a b^2 d^2 e+20 a^2 b d e^2+2 a^3 e^3\right )\right ) x^4+a^6 d \left (a B d \left (24 b^3 d^3+45 a b^2 d^2 e+20 a^2 b d e^2+2 a^3 e^3\right )+A \left (42 b^4 d^4+120 a b^3 d^3 e+90 a^2 b^2 d^2 e^2+20 a^3 b d e^3+a^4 e^4\right )\right ) x^5+\frac {1}{6} a^5 \left (5 a B d \left (42 b^4 d^4+120 a b^3 d^3 e+90 a^2 b^2 d^2 e^2+20 a^3 b d e^3+a^4 e^4\right )+A \left (252 b^5 d^5+1050 a b^4 d^4 e+1200 a^2 b^3 d^3 e^2+450 a^3 b^2 d^2 e^3+50 a^4 b d e^4+a^5 e^5\right )\right ) x^6+\frac {1}{7} a^4 \left (a B \left (252 b^5 d^5+1050 a b^4 d^4 e+1200 a^2 b^3 d^3 e^2+450 a^3 b^2 d^2 e^3+50 a^4 b d e^4+a^5 e^5\right )+5 A b \left (42 b^5 d^5+252 a b^4 d^4 e+420 a^2 b^3 d^3 e^2+240 a^3 b^2 d^2 e^3+45 a^4 b d e^4+2 a^5 e^5\right )\right ) x^7+\frac {5}{8} a^3 b \left (a B \left (42 b^5 d^5+252 a b^4 d^4 e+420 a^2 b^3 d^3 e^2+240 a^3 b^2 d^2 e^3+45 a^4 b d e^4+2 a^5 e^5\right )+3 A b \left (8 b^5 d^5+70 a b^4 d^4 e+168 a^2 b^3 d^3 e^2+140 a^3 b^2 d^2 e^3+40 a^4 b d e^4+3 a^5 e^5\right )\right ) x^8+\frac {5}{3} a^2 b^2 \left (a B \left (8 b^5 d^5+70 a b^4 d^4 e+168 a^2 b^3 d^3 e^2+140 a^3 b^2 d^2 e^3+40 a^4 b d e^4+3 a^5 e^5\right )+A b \left (3 b^5 d^5+40 a b^4 d^4 e+140 a^2 b^3 d^3 e^2+168 a^3 b^2 d^2 e^3+70 a^4 b d e^4+8 a^5 e^5\right )\right ) x^9+\frac {1}{2} a b^3 \left (3 a B \left (3 b^5 d^5+40 a b^4 d^4 e+140 a^2 b^3 d^3 e^2+168 a^3 b^2 d^2 e^3+70 a^4 b d e^4+8 a^5 e^5\right )+A b \left (2 b^5 d^5+45 a b^4 d^4 e+240 a^2 b^3 d^3 e^2+420 a^3 b^2 d^2 e^3+252 a^4 b d e^4+42 a^5 e^5\right )\right ) x^{10}+\frac {1}{11} b^4 \left (5 a B \left (2 b^5 d^5+45 a b^4 d^4 e+240 a^2 b^3 d^3 e^2+420 a^3 b^2 d^2 e^3+252 a^4 b d e^4+42 a^5 e^5\right )+A b \left (b^5 d^5+50 a b^4 d^4 e+450 a^2 b^3 d^3 e^2+1200 a^3 b^2 d^2 e^3+1050 a^4 b d e^4+252 a^5 e^5\right )\right ) x^{11}+\frac {1}{12} b^5 \left (252 a^5 B e^5+450 a^2 b^3 d^2 e^2 (B d+A e)+600 a^3 b^2 d e^3 (2 B d+A e)+210 a^4 b e^4 (5 B d+A e)+50 a b^4 d^3 e (B d+2 A e)+b^5 d^4 (B d+5 A e)\right ) x^{12}+\frac {5}{13} b^6 e \left (42 a^4 B e^4+20 a b^3 d^2 e (B d+A e)+45 a^2 b^2 d e^2 (2 B d+A e)+24 a^3 b e^3 (5 B d+A e)+b^4 d^3 (B d+2 A e)\right ) x^{13}+\frac {5}{14} b^7 e^2 \left (24 a^3 B e^3+2 b^3 d^2 (B d+A e)+10 a b^2 d e (2 B d+A e)+9 a^2 b e^2 (5 B d+A e)\right ) x^{14}+\frac {1}{3} b^8 e^3 \left (9 a^2 B e^2+b^2 d (2 B d+A e)+2 a b e (5 B d+A e)\right ) x^{15}+\frac {1}{16} b^9 e^4 (5 b B d+A b e+10 a B e) x^{16}+\frac {1}{17} b^{10} B e^5 x^{17} \]
a^10*A*d^5*x + (a^9*d^4*(a*B*d + 5*A*(2*b*d + a*e))*x^2)/2 + (5*a^8*d^3*(a *B*d*(2*b*d + a*e) + A*(9*b^2*d^2 + 10*a*b*d*e + 2*a^2*e^2))*x^3)/3 + (5*a ^7*d^2*(a*B*d*(9*b^2*d^2 + 10*a*b*d*e + 2*a^2*e^2) + A*(24*b^3*d^3 + 45*a* b^2*d^2*e + 20*a^2*b*d*e^2 + 2*a^3*e^3))*x^4)/4 + a^6*d*(a*B*d*(24*b^3*d^3 + 45*a*b^2*d^2*e + 20*a^2*b*d*e^2 + 2*a^3*e^3) + A*(42*b^4*d^4 + 120*a*b^ 3*d^3*e + 90*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 + a^4*e^4))*x^5 + (a^5*(5*a* B*d*(42*b^4*d^4 + 120*a*b^3*d^3*e + 90*a^2*b^2*d^2*e^2 + 20*a^3*b*d*e^3 + a^4*e^4) + A*(252*b^5*d^5 + 1050*a*b^4*d^4*e + 1200*a^2*b^3*d^3*e^2 + 450* a^3*b^2*d^2*e^3 + 50*a^4*b*d*e^4 + a^5*e^5))*x^6)/6 + (a^4*(a*B*(252*b^5*d ^5 + 1050*a*b^4*d^4*e + 1200*a^2*b^3*d^3*e^2 + 450*a^3*b^2*d^2*e^3 + 50*a^ 4*b*d*e^4 + a^5*e^5) + 5*A*b*(42*b^5*d^5 + 252*a*b^4*d^4*e + 420*a^2*b^3*d ^3*e^2 + 240*a^3*b^2*d^2*e^3 + 45*a^4*b*d*e^4 + 2*a^5*e^5))*x^7)/7 + (5*a^ 3*b*(a*B*(42*b^5*d^5 + 252*a*b^4*d^4*e + 420*a^2*b^3*d^3*e^2 + 240*a^3*b^2 *d^2*e^3 + 45*a^4*b*d*e^4 + 2*a^5*e^5) + 3*A*b*(8*b^5*d^5 + 70*a*b^4*d^4*e + 168*a^2*b^3*d^3*e^2 + 140*a^3*b^2*d^2*e^3 + 40*a^4*b*d*e^4 + 3*a^5*e^5) )*x^8)/8 + (5*a^2*b^2*(a*B*(8*b^5*d^5 + 70*a*b^4*d^4*e + 168*a^2*b^3*d^3*e ^2 + 140*a^3*b^2*d^2*e^3 + 40*a^4*b*d*e^4 + 3*a^5*e^5) + A*b*(3*b^5*d^5 + 40*a*b^4*d^4*e + 140*a^2*b^3*d^3*e^2 + 168*a^3*b^2*d^2*e^3 + 70*a^4*b*d*e^ 4 + 8*a^5*e^5))*x^9)/3 + (a*b^3*(3*a*B*(3*b^5*d^5 + 40*a*b^4*d^4*e + 140*a ^2*b^3*d^3*e^2 + 168*a^3*b^2*d^2*e^3 + 70*a^4*b*d*e^4 + 8*a^5*e^5) + A*...
Time = 1.39 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {86, 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.
| \(\displaystyle \int (a+b x)^{10} (A+B x) (d+e x)^5 \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {e^4 (a+b x)^{15} (-6 a B e+A b e+5 b B d)}{b^6}+\frac {5 e^3 (a+b x)^{14} (b d-a e) (-3 a B e+A b e+2 b B d)}{b^6}+\frac {10 e^2 (a+b x)^{13} (b d-a e)^2 (-2 a B e+A b e+b B d)}{b^6}+\frac {5 e (a+b x)^{12} (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{b^6}+\frac {(a+b x)^{11} (b d-a e)^4 (-6 a B e+5 A b e+b B d)}{b^6}+\frac {(a+b x)^{10} (A b-a B) (b d-a e)^5}{b^6}+\frac {B e^5 (a+b x)^{16}}{b^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^4 (a+b x)^{16} (-6 a B e+A b e+5 b B d)}{16 b^7}+\frac {e^3 (a+b x)^{15} (b d-a e) (-3 a B e+A b e+2 b B d)}{3 b^7}+\frac {5 e^2 (a+b x)^{14} (b d-a e)^2 (-2 a B e+A b e+b B d)}{7 b^7}+\frac {5 e (a+b x)^{13} (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{13 b^7}+\frac {(a+b x)^{12} (b d-a e)^4 (-6 a B e+5 A b e+b B d)}{12 b^7}+\frac {(a+b x)^{11} (A b-a B) (b d-a e)^5}{11 b^7}+\frac {B e^5 (a+b x)^{17}}{17 b^7}\) |
((A*b - a*B)*(b*d - a*e)^5*(a + b*x)^11)/(11*b^7) + ((b*d - a*e)^4*(b*B*d + 5*A*b*e - 6*a*B*e)*(a + b*x)^12)/(12*b^7) + (5*e*(b*d - a*e)^3*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^13)/(13*b^7) + (5*e^2*(b*d - a*e)^2*(b*B*d + A*b*e - 2*a*B*e)*(a + b*x)^14)/(7*b^7) + (e^3*(b*d - a*e)*(2*b*B*d + A*b*e - 3*a*B*e)*(a + b*x)^15)/(3*b^7) + (e^4*(5*b*B*d + A*b*e - 6*a*B*e)*(a + b*x)^16)/(16*b^7) + (B*e^5*(a + b*x)^17)/(17*b^7)
3.11.83.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Leaf count of result is larger than twice the leaf count of optimal. \(1620\) vs. \(2(229)=458\).
Time = 0.70 (sec) , antiderivative size = 1621, normalized size of antiderivative = 6.67
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1621\) |
| norman | \(\text {Expression too large to display}\) | \(1718\) |
| gosper | \(\text {Expression too large to display}\) | \(2033\) |
| risch | \(\text {Expression too large to display}\) | \(2033\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2033\) |
1/17*b^10*B*e^5*x^17+1/16*((A*b^10+10*B*a*b^9)*e^5+5*b^10*B*d*e^4)*x^16+1/ 15*((10*A*a*b^9+45*B*a^2*b^8)*e^5+5*(A*b^10+10*B*a*b^9)*d*e^4+10*b^10*B*d^ 2*e^3)*x^15+1/14*((45*A*a^2*b^8+120*B*a^3*b^7)*e^5+5*(10*A*a*b^9+45*B*a^2* b^8)*d*e^4+10*(A*b^10+10*B*a*b^9)*d^2*e^3+10*b^10*B*d^3*e^2)*x^14+1/13*((1 20*A*a^3*b^7+210*B*a^4*b^6)*e^5+5*(45*A*a^2*b^8+120*B*a^3*b^7)*d*e^4+10*(1 0*A*a*b^9+45*B*a^2*b^8)*d^2*e^3+10*(A*b^10+10*B*a*b^9)*d^3*e^2+5*b^10*B*d^ 4*e)*x^13+1/12*((210*A*a^4*b^6+252*B*a^5*b^5)*e^5+5*(120*A*a^3*b^7+210*B*a ^4*b^6)*d*e^4+10*(45*A*a^2*b^8+120*B*a^3*b^7)*d^2*e^3+10*(10*A*a*b^9+45*B* a^2*b^8)*d^3*e^2+5*(A*b^10+10*B*a*b^9)*d^4*e+b^10*B*d^5)*x^12+1/11*((252*A *a^5*b^5+210*B*a^6*b^4)*e^5+5*(210*A*a^4*b^6+252*B*a^5*b^5)*d*e^4+10*(120* A*a^3*b^7+210*B*a^4*b^6)*d^2*e^3+10*(45*A*a^2*b^8+120*B*a^3*b^7)*d^3*e^2+5 *(10*A*a*b^9+45*B*a^2*b^8)*d^4*e+(A*b^10+10*B*a*b^9)*d^5)*x^11+1/10*((210* A*a^6*b^4+120*B*a^7*b^3)*e^5+5*(252*A*a^5*b^5+210*B*a^6*b^4)*d*e^4+10*(210 *A*a^4*b^6+252*B*a^5*b^5)*d^2*e^3+10*(120*A*a^3*b^7+210*B*a^4*b^6)*d^3*e^2 +5*(45*A*a^2*b^8+120*B*a^3*b^7)*d^4*e+(10*A*a*b^9+45*B*a^2*b^8)*d^5)*x^10+ 1/9*((120*A*a^7*b^3+45*B*a^8*b^2)*e^5+5*(210*A*a^6*b^4+120*B*a^7*b^3)*d*e^ 4+10*(252*A*a^5*b^5+210*B*a^6*b^4)*d^2*e^3+10*(210*A*a^4*b^6+252*B*a^5*b^5 )*d^3*e^2+5*(120*A*a^3*b^7+210*B*a^4*b^6)*d^4*e+(45*A*a^2*b^8+120*B*a^3*b^ 7)*d^5)*x^9+1/8*((45*A*a^8*b^2+10*B*a^9*b)*e^5+5*(120*A*a^7*b^3+45*B*a^8*b ^2)*d*e^4+10*(210*A*a^6*b^4+120*B*a^7*b^3)*d^2*e^3+10*(252*A*a^5*b^5+21...
Leaf count of result is larger than twice the leaf count of optimal. 1625 vs. \(2 (229) = 458\).
Time = 0.23 (sec) , antiderivative size = 1625, normalized size of antiderivative = 6.69 \[ \int (a+b x)^{10} (A+B x) (d+e x)^5 \, dx=\text {Too large to display} \]
1/17*B*b^10*e^5*x^17 + A*a^10*d^5*x + 1/16*(5*B*b^10*d*e^4 + (10*B*a*b^9 + A*b^10)*e^5)*x^16 + 1/3*(2*B*b^10*d^2*e^3 + (10*B*a*b^9 + A*b^10)*d*e^4 + (9*B*a^2*b^8 + 2*A*a*b^9)*e^5)*x^15 + 5/14*(2*B*b^10*d^3*e^2 + 2*(10*B*a* b^9 + A*b^10)*d^2*e^3 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^4 + 3*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^5)*x^14 + 5/13*(B*b^10*d^4*e + 2*(10*B*a*b^9 + A*b^10)*d ^3*e^2 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^3 + 15*(8*B*a^3*b^7 + 3*A*a^2* b^8)*d*e^4 + 6*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^5)*x^13 + 1/12*(B*b^10*d^5 + 5*(10*B*a*b^9 + A*b^10)*d^4*e + 50*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^2 + 150 *(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^3 + 150*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e ^4 + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^5)*x^12 + 1/11*((10*B*a*b^9 + A*b^10 )*d^5 + 25*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e + 150*(8*B*a^3*b^7 + 3*A*a^2*b^ 8)*d^3*e^2 + 300*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^3 + 210*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^4 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^5)*x^11 + 1/2*((9*B* a^2*b^8 + 2*A*a*b^9)*d^5 + 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e + 60*(7*B* a^4*b^6 + 4*A*a^3*b^7)*d^3*e^2 + 84*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^3 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^4 + 6*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^5)* x^10 + 5/3*((8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5 + 10*(7*B*a^4*b^6 + 4*A*a^3*b^ 7)*d^4*e + 28*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^2 + 28*(5*B*a^6*b^4 + 6*A* a^5*b^5)*d^2*e^3 + 10*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^4 + (3*B*a^8*b^2 + 8 *A*a^7*b^3)*e^5)*x^9 + 5/8*(6*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5 + 42*(6*B...
Leaf count of result is larger than twice the leaf count of optimal. 2076 vs. \(2 (243) = 486\).
Time = 0.14 (sec) , antiderivative size = 2076, normalized size of antiderivative = 8.54 \[ \int (a+b x)^{10} (A+B x) (d+e x)^5 \, dx=\text {Too large to display} \]
A*a**10*d**5*x + B*b**10*e**5*x**17/17 + x**16*(A*b**10*e**5/16 + 5*B*a*b* *9*e**5/8 + 5*B*b**10*d*e**4/16) + x**15*(2*A*a*b**9*e**5/3 + A*b**10*d*e* *4/3 + 3*B*a**2*b**8*e**5 + 10*B*a*b**9*d*e**4/3 + 2*B*b**10*d**2*e**3/3) + x**14*(45*A*a**2*b**8*e**5/14 + 25*A*a*b**9*d*e**4/7 + 5*A*b**10*d**2*e* *3/7 + 60*B*a**3*b**7*e**5/7 + 225*B*a**2*b**8*d*e**4/14 + 50*B*a*b**9*d** 2*e**3/7 + 5*B*b**10*d**3*e**2/7) + x**13*(120*A*a**3*b**7*e**5/13 + 225*A *a**2*b**8*d*e**4/13 + 100*A*a*b**9*d**2*e**3/13 + 10*A*b**10*d**3*e**2/13 + 210*B*a**4*b**6*e**5/13 + 600*B*a**3*b**7*d*e**4/13 + 450*B*a**2*b**8*d **2*e**3/13 + 100*B*a*b**9*d**3*e**2/13 + 5*B*b**10*d**4*e/13) + x**12*(35 *A*a**4*b**6*e**5/2 + 50*A*a**3*b**7*d*e**4 + 75*A*a**2*b**8*d**2*e**3/2 + 25*A*a*b**9*d**3*e**2/3 + 5*A*b**10*d**4*e/12 + 21*B*a**5*b**5*e**5 + 175 *B*a**4*b**6*d*e**4/2 + 100*B*a**3*b**7*d**2*e**3 + 75*B*a**2*b**8*d**3*e* *2/2 + 25*B*a*b**9*d**4*e/6 + B*b**10*d**5/12) + x**11*(252*A*a**5*b**5*e* *5/11 + 1050*A*a**4*b**6*d*e**4/11 + 1200*A*a**3*b**7*d**2*e**3/11 + 450*A *a**2*b**8*d**3*e**2/11 + 50*A*a*b**9*d**4*e/11 + A*b**10*d**5/11 + 210*B* a**6*b**4*e**5/11 + 1260*B*a**5*b**5*d*e**4/11 + 2100*B*a**4*b**6*d**2*e** 3/11 + 1200*B*a**3*b**7*d**3*e**2/11 + 225*B*a**2*b**8*d**4*e/11 + 10*B*a* b**9*d**5/11) + x**10*(21*A*a**6*b**4*e**5 + 126*A*a**5*b**5*d*e**4 + 210* A*a**4*b**6*d**2*e**3 + 120*A*a**3*b**7*d**3*e**2 + 45*A*a**2*b**8*d**4*e/ 2 + A*a*b**9*d**5 + 12*B*a**7*b**3*e**5 + 105*B*a**6*b**4*d*e**4 + 252*...
Leaf count of result is larger than twice the leaf count of optimal. 1625 vs. \(2 (229) = 458\).
Time = 0.22 (sec) , antiderivative size = 1625, normalized size of antiderivative = 6.69 \[ \int (a+b x)^{10} (A+B x) (d+e x)^5 \, dx=\text {Too large to display} \]
1/17*B*b^10*e^5*x^17 + A*a^10*d^5*x + 1/16*(5*B*b^10*d*e^4 + (10*B*a*b^9 + A*b^10)*e^5)*x^16 + 1/3*(2*B*b^10*d^2*e^3 + (10*B*a*b^9 + A*b^10)*d*e^4 + (9*B*a^2*b^8 + 2*A*a*b^9)*e^5)*x^15 + 5/14*(2*B*b^10*d^3*e^2 + 2*(10*B*a* b^9 + A*b^10)*d^2*e^3 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^4 + 3*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^5)*x^14 + 5/13*(B*b^10*d^4*e + 2*(10*B*a*b^9 + A*b^10)*d ^3*e^2 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^3 + 15*(8*B*a^3*b^7 + 3*A*a^2* b^8)*d*e^4 + 6*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^5)*x^13 + 1/12*(B*b^10*d^5 + 5*(10*B*a*b^9 + A*b^10)*d^4*e + 50*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^2 + 150 *(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^3 + 150*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e ^4 + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^5)*x^12 + 1/11*((10*B*a*b^9 + A*b^10 )*d^5 + 25*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e + 150*(8*B*a^3*b^7 + 3*A*a^2*b^ 8)*d^3*e^2 + 300*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^3 + 210*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^4 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^5)*x^11 + 1/2*((9*B* a^2*b^8 + 2*A*a*b^9)*d^5 + 15*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e + 60*(7*B* a^4*b^6 + 4*A*a^3*b^7)*d^3*e^2 + 84*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^3 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^4 + 6*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^5)* x^10 + 5/3*((8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5 + 10*(7*B*a^4*b^6 + 4*A*a^3*b^ 7)*d^4*e + 28*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^2 + 28*(5*B*a^6*b^4 + 6*A* a^5*b^5)*d^2*e^3 + 10*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^4 + (3*B*a^8*b^2 + 8 *A*a^7*b^3)*e^5)*x^9 + 5/8*(6*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5 + 42*(6*B...
Leaf count of result is larger than twice the leaf count of optimal. 2032 vs. \(2 (229) = 458\).
Time = 0.29 (sec) , antiderivative size = 2032, normalized size of antiderivative = 8.36 \[ \int (a+b x)^{10} (A+B x) (d+e x)^5 \, dx=\text {Too large to display} \]
1/17*B*b^10*e^5*x^17 + 5/16*B*b^10*d*e^4*x^16 + 5/8*B*a*b^9*e^5*x^16 + 1/1 6*A*b^10*e^5*x^16 + 2/3*B*b^10*d^2*e^3*x^15 + 10/3*B*a*b^9*d*e^4*x^15 + 1/ 3*A*b^10*d*e^4*x^15 + 3*B*a^2*b^8*e^5*x^15 + 2/3*A*a*b^9*e^5*x^15 + 5/7*B* b^10*d^3*e^2*x^14 + 50/7*B*a*b^9*d^2*e^3*x^14 + 5/7*A*b^10*d^2*e^3*x^14 + 225/14*B*a^2*b^8*d*e^4*x^14 + 25/7*A*a*b^9*d*e^4*x^14 + 60/7*B*a^3*b^7*e^5 *x^14 + 45/14*A*a^2*b^8*e^5*x^14 + 5/13*B*b^10*d^4*e*x^13 + 100/13*B*a*b^9 *d^3*e^2*x^13 + 10/13*A*b^10*d^3*e^2*x^13 + 450/13*B*a^2*b^8*d^2*e^3*x^13 + 100/13*A*a*b^9*d^2*e^3*x^13 + 600/13*B*a^3*b^7*d*e^4*x^13 + 225/13*A*a^2 *b^8*d*e^4*x^13 + 210/13*B*a^4*b^6*e^5*x^13 + 120/13*A*a^3*b^7*e^5*x^13 + 1/12*B*b^10*d^5*x^12 + 25/6*B*a*b^9*d^4*e*x^12 + 5/12*A*b^10*d^4*e*x^12 + 75/2*B*a^2*b^8*d^3*e^2*x^12 + 25/3*A*a*b^9*d^3*e^2*x^12 + 100*B*a^3*b^7*d^ 2*e^3*x^12 + 75/2*A*a^2*b^8*d^2*e^3*x^12 + 175/2*B*a^4*b^6*d*e^4*x^12 + 50 *A*a^3*b^7*d*e^4*x^12 + 21*B*a^5*b^5*e^5*x^12 + 35/2*A*a^4*b^6*e^5*x^12 + 10/11*B*a*b^9*d^5*x^11 + 1/11*A*b^10*d^5*x^11 + 225/11*B*a^2*b^8*d^4*e*x^1 1 + 50/11*A*a*b^9*d^4*e*x^11 + 1200/11*B*a^3*b^7*d^3*e^2*x^11 + 450/11*A*a ^2*b^8*d^3*e^2*x^11 + 2100/11*B*a^4*b^6*d^2*e^3*x^11 + 1200/11*A*a^3*b^7*d ^2*e^3*x^11 + 1260/11*B*a^5*b^5*d*e^4*x^11 + 1050/11*A*a^4*b^6*d*e^4*x^11 + 210/11*B*a^6*b^4*e^5*x^11 + 252/11*A*a^5*b^5*e^5*x^11 + 9/2*B*a^2*b^8*d^ 5*x^10 + A*a*b^9*d^5*x^10 + 60*B*a^3*b^7*d^4*e*x^10 + 45/2*A*a^2*b^8*d^4*e *x^10 + 210*B*a^4*b^6*d^3*e^2*x^10 + 120*A*a^3*b^7*d^3*e^2*x^10 + 252*B...
Time = 1.76 (sec) , antiderivative size = 1685, normalized size of antiderivative = 6.93 \[ \int (a+b x)^{10} (A+B x) (d+e x)^5 \, dx=\text {Too large to display} \]
x^9*(5*A*a^2*b^8*d^5 + (40*A*a^7*b^3*e^5)/3 + (40*B*a^3*b^7*d^5)/3 + 5*B*a ^8*b^2*e^5 + (200*A*a^3*b^7*d^4*e)/3 + (350*A*a^6*b^4*d*e^4)/3 + (350*B*a^ 4*b^6*d^4*e)/3 + (200*B*a^7*b^3*d*e^4)/3 + (700*A*a^4*b^6*d^3*e^2)/3 + 280 *A*a^5*b^5*d^2*e^3 + 280*B*a^5*b^5*d^3*e^2 + (700*B*a^6*b^4*d^2*e^3)/3) + x^7*((B*a^10*e^5)/7 + (10*A*a^9*b*e^5)/7 + 30*A*a^4*b^6*d^5 + 36*B*a^5*b^5 *d^5 + 180*A*a^5*b^5*d^4*e + (225*A*a^8*b^2*d*e^4)/7 + 150*B*a^6*b^4*d^4*e + 300*A*a^6*b^4*d^3*e^2 + (1200*A*a^7*b^3*d^2*e^3)/7 + (1200*B*a^7*b^3*d^ 3*e^2)/7 + (450*B*a^8*b^2*d^2*e^3)/7 + (50*B*a^9*b*d*e^4)/7) + x^11*((A*b^ 10*d^5)/11 + (10*B*a*b^9*d^5)/11 + (252*A*a^5*b^5*e^5)/11 + (210*B*a^6*b^4 *e^5)/11 + (1050*A*a^4*b^6*d*e^4)/11 + (225*B*a^2*b^8*d^4*e)/11 + (1260*B* a^5*b^5*d*e^4)/11 + (450*A*a^2*b^8*d^3*e^2)/11 + (1200*A*a^3*b^7*d^2*e^3)/ 11 + (1200*B*a^3*b^7*d^3*e^2)/11 + (2100*B*a^4*b^6*d^2*e^3)/11 + (50*A*a*b ^9*d^4*e)/11) + x^10*(A*a*b^9*d^5 + 21*A*a^6*b^4*e^5 + (9*B*a^2*b^8*d^5)/2 + 12*B*a^7*b^3*e^5 + (45*A*a^2*b^8*d^4*e)/2 + 126*A*a^5*b^5*d*e^4 + 60*B* a^3*b^7*d^4*e + 105*B*a^6*b^4*d*e^4 + 120*A*a^3*b^7*d^3*e^2 + 210*A*a^4*b^ 6*d^2*e^3 + 210*B*a^4*b^6*d^3*e^2 + 252*B*a^5*b^5*d^2*e^3) + x^8*((5*B*a^9 *b*e^5)/4 + 15*A*a^3*b^7*d^5 + (45*A*a^8*b^2*e^5)/8 + (105*B*a^4*b^6*d^5)/ 4 + (525*A*a^4*b^6*d^4*e)/4 + 75*A*a^7*b^3*d*e^4 + (315*B*a^5*b^5*d^4*e)/2 + (225*B*a^8*b^2*d*e^4)/8 + 315*A*a^5*b^5*d^3*e^2 + (525*A*a^6*b^4*d^2*e^ 3)/2 + (525*B*a^6*b^4*d^3*e^2)/2 + 150*B*a^7*b^3*d^2*e^3) + x^5*(A*a^10...