Integrand size = 20, antiderivative size = 372 \[ \int (a+b x)^{10} (A+B x) (d+e x)^8 \, dx=\frac {(A b-a B) (b d-a e)^8 (a+b x)^{11}}{11 b^{10}}+\frac {(b d-a e)^7 (b B d+8 A b e-9 a B e) (a+b x)^{12}}{12 b^{10}}+\frac {4 e (b d-a e)^6 (2 b B d+7 A b e-9 a B e) (a+b x)^{13}}{13 b^{10}}+\frac {2 e^2 (b d-a e)^5 (b B d+2 A b e-3 a B e) (a+b x)^{14}}{b^{10}}+\frac {14 e^3 (b d-a e)^4 (4 b B d+5 A b e-9 a B e) (a+b x)^{15}}{15 b^{10}}+\frac {7 e^4 (b d-a e)^3 (5 b B d+4 A b e-9 a B e) (a+b x)^{16}}{8 b^{10}}+\frac {28 e^5 (b d-a e)^2 (2 b B d+A b e-3 a B e) (a+b x)^{17}}{17 b^{10}}+\frac {2 e^6 (b d-a e) (7 b B d+2 A b e-9 a B e) (a+b x)^{18}}{9 b^{10}}+\frac {e^7 (8 b B d+A b e-9 a B e) (a+b x)^{19}}{19 b^{10}}+\frac {B e^8 (a+b x)^{20}}{20 b^{10}} \]
1/11*(A*b-B*a)*(-a*e+b*d)^8*(b*x+a)^11/b^10+1/12*(-a*e+b*d)^7*(8*A*b*e-9*B *a*e+B*b*d)*(b*x+a)^12/b^10+4/13*e*(-a*e+b*d)^6*(7*A*b*e-9*B*a*e+2*B*b*d)* (b*x+a)^13/b^10+2*e^2*(-a*e+b*d)^5*(2*A*b*e-3*B*a*e+B*b*d)*(b*x+a)^14/b^10 +14/15*e^3*(-a*e+b*d)^4*(5*A*b*e-9*B*a*e+4*B*b*d)*(b*x+a)^15/b^10+7/8*e^4* (-a*e+b*d)^3*(4*A*b*e-9*B*a*e+5*B*b*d)*(b*x+a)^16/b^10+28/17*e^5*(-a*e+b*d )^2*(A*b*e-3*B*a*e+2*B*b*d)*(b*x+a)^17/b^10+2/9*e^6*(-a*e+b*d)*(2*A*b*e-9* B*a*e+7*B*b*d)*(b*x+a)^18/b^10+1/19*e^7*(A*b*e-9*B*a*e+8*B*b*d)*(b*x+a)^19 /b^10+1/20*B*e^8*(b*x+a)^20/b^10
Leaf count is larger than twice the leaf count of optimal. \(2307\) vs. \(2(372)=744\).
Time = 0.55 (sec) , antiderivative size = 2307, normalized size of antiderivative = 6.20 \[ \int (a+b x)^{10} (A+B x) (d+e x)^8 \, dx=\text {Result too large to show} \]
a^10*A*d^8*x + (a^9*d^7*(10*A*b*d + a*B*d + 8*a*A*e)*x^2)/2 + (a^8*d^6*(2* a*B*d*(5*b*d + 4*a*e) + A*(45*b^2*d^2 + 80*a*b*d*e + 28*a^2*e^2))*x^3)/3 + (a^7*d^5*(a*B*d*(45*b^2*d^2 + 80*a*b*d*e + 28*a^2*e^2) + 8*A*(15*b^3*d^3 + 45*a*b^2*d^2*e + 35*a^2*b*d*e^2 + 7*a^3*e^3))*x^4)/4 + (2*a^6*d^4*(4*a*B *d*(15*b^3*d^3 + 45*a*b^2*d^2*e + 35*a^2*b*d*e^2 + 7*a^3*e^3) + 5*A*(21*b^ 4*d^4 + 96*a*b^3*d^3*e + 126*a^2*b^2*d^2*e^2 + 56*a^3*b*d*e^3 + 7*a^4*e^4) )*x^5)/5 + (a^5*d^3*(5*a*B*d*(21*b^4*d^4 + 96*a*b^3*d^3*e + 126*a^2*b^2*d^ 2*e^2 + 56*a^3*b*d*e^3 + 7*a^4*e^4) + 14*A*(9*b^5*d^5 + 60*a*b^4*d^4*e + 1 20*a^2*b^3*d^3*e^2 + 90*a^3*b^2*d^2*e^3 + 25*a^4*b*d*e^4 + 2*a^5*e^5))*x^6 )/3 + 2*a^4*d^2*(2*a*B*d*(9*b^5*d^5 + 60*a*b^4*d^4*e + 120*a^2*b^3*d^3*e^2 + 90*a^3*b^2*d^2*e^3 + 25*a^4*b*d*e^4 + 2*a^5*e^5) + A*(15*b^6*d^6 + 144* a*b^5*d^5*e + 420*a^2*b^4*d^4*e^2 + 480*a^3*b^3*d^3*e^3 + 225*a^4*b^2*d^2* e^4 + 40*a^5*b*d*e^5 + 2*a^6*e^6))*x^7 + (a^3*d*(7*a*B*d*(15*b^6*d^6 + 144 *a*b^5*d^5*e + 420*a^2*b^4*d^4*e^2 + 480*a^3*b^3*d^3*e^3 + 225*a^4*b^2*d^2 *e^4 + 40*a^5*b*d*e^5 + 2*a^6*e^6) + 4*A*(15*b^7*d^7 + 210*a*b^6*d^6*e + 8 82*a^2*b^5*d^5*e^2 + 1470*a^3*b^4*d^4*e^3 + 1050*a^4*b^3*d^3*e^4 + 315*a^5 *b^2*d^2*e^5 + 35*a^6*b*d*e^6 + a^7*e^7))*x^8)/4 + (a^2*(8*a*B*d*(15*b^7*d ^7 + 210*a*b^6*d^6*e + 882*a^2*b^5*d^5*e^2 + 1470*a^3*b^4*d^4*e^3 + 1050*a ^4*b^3*d^3*e^4 + 315*a^5*b^2*d^2*e^5 + 35*a^6*b*d*e^6 + a^7*e^7) + A*(45*b ^8*d^8 + 960*a*b^7*d^7*e + 5880*a^2*b^6*d^6*e^2 + 14112*a^3*b^5*d^5*e^3...
Time = 2.18 (sec) , antiderivative size = 372, 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.
\(\displaystyle \int (a+b x)^{10} (A+B x) (d+e x)^8 \, dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (\frac {e^7 (a+b x)^{18} (-9 a B e+A b e+8 b B d)}{b^9}+\frac {4 e^6 (a+b x)^{17} (b d-a e) (-9 a B e+2 A b e+7 b B d)}{b^9}+\frac {28 e^5 (a+b x)^{16} (b d-a e)^2 (-3 a B e+A b e+2 b B d)}{b^9}+\frac {14 e^4 (a+b x)^{15} (b d-a e)^3 (-9 a B e+4 A b e+5 b B d)}{b^9}+\frac {14 e^3 (a+b x)^{14} (b d-a e)^4 (-9 a B e+5 A b e+4 b B d)}{b^9}+\frac {28 e^2 (a+b x)^{13} (b d-a e)^5 (-3 a B e+2 A b e+b B d)}{b^9}+\frac {4 e (a+b x)^{12} (b d-a e)^6 (-9 a B e+7 A b e+2 b B d)}{b^9}+\frac {(a+b x)^{11} (b d-a e)^7 (-9 a B e+8 A b e+b B d)}{b^9}+\frac {(a+b x)^{10} (A b-a B) (b d-a e)^8}{b^9}+\frac {B e^8 (a+b x)^{19}}{b^9}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e^7 (a+b x)^{19} (-9 a B e+A b e+8 b B d)}{19 b^{10}}+\frac {2 e^6 (a+b x)^{18} (b d-a e) (-9 a B e+2 A b e+7 b B d)}{9 b^{10}}+\frac {28 e^5 (a+b x)^{17} (b d-a e)^2 (-3 a B e+A b e+2 b B d)}{17 b^{10}}+\frac {7 e^4 (a+b x)^{16} (b d-a e)^3 (-9 a B e+4 A b e+5 b B d)}{8 b^{10}}+\frac {14 e^3 (a+b x)^{15} (b d-a e)^4 (-9 a B e+5 A b e+4 b B d)}{15 b^{10}}+\frac {2 e^2 (a+b x)^{14} (b d-a e)^5 (-3 a B e+2 A b e+b B d)}{b^{10}}+\frac {4 e (a+b x)^{13} (b d-a e)^6 (-9 a B e+7 A b e+2 b B d)}{13 b^{10}}+\frac {(a+b x)^{12} (b d-a e)^7 (-9 a B e+8 A b e+b B d)}{12 b^{10}}+\frac {(a+b x)^{11} (A b-a B) (b d-a e)^8}{11 b^{10}}+\frac {B e^8 (a+b x)^{20}}{20 b^{10}}\) |
((A*b - a*B)*(b*d - a*e)^8*(a + b*x)^11)/(11*b^10) + ((b*d - a*e)^7*(b*B*d + 8*A*b*e - 9*a*B*e)*(a + b*x)^12)/(12*b^10) + (4*e*(b*d - a*e)^6*(2*b*B* d + 7*A*b*e - 9*a*B*e)*(a + b*x)^13)/(13*b^10) + (2*e^2*(b*d - a*e)^5*(b*B *d + 2*A*b*e - 3*a*B*e)*(a + b*x)^14)/b^10 + (14*e^3*(b*d - a*e)^4*(4*b*B* d + 5*A*b*e - 9*a*B*e)*(a + b*x)^15)/(15*b^10) + (7*e^4*(b*d - a*e)^3*(5*b *B*d + 4*A*b*e - 9*a*B*e)*(a + b*x)^16)/(8*b^10) + (28*e^5*(b*d - a*e)^2*( 2*b*B*d + A*b*e - 3*a*B*e)*(a + b*x)^17)/(17*b^10) + (2*e^6*(b*d - a*e)*(7 *b*B*d + 2*A*b*e - 9*a*B*e)*(a + b*x)^18)/(9*b^10) + (e^7*(8*b*B*d + A*b*e - 9*a*B*e)*(a + b*x)^19)/(19*b^10) + (B*e^8*(a + b*x)^20)/(20*b^10)
3.11.80.3.1 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. \(2472\) vs. \(2(354)=708\).
Time = 2.08 (sec) , antiderivative size = 2473, normalized size of antiderivative = 6.65
method | result | size |
default | \(\text {Expression too large to display}\) | \(2473\) |
norman | \(\text {Expression too large to display}\) | \(2662\) |
gosper | \(\text {Expression too large to display}\) | \(3160\) |
risch | \(\text {Expression too large to display}\) | \(3160\) |
parallelrisch | \(\text {Expression too large to display}\) | \(3160\) |
1/20*b^10*B*e^8*x^20+1/19*((A*b^10+10*B*a*b^9)*e^8+8*b^10*B*d*e^7)*x^19+1/ 18*((10*A*a*b^9+45*B*a^2*b^8)*e^8+8*(A*b^10+10*B*a*b^9)*d*e^7+28*b^10*B*d^ 2*e^6)*x^18+1/17*((45*A*a^2*b^8+120*B*a^3*b^7)*e^8+8*(10*A*a*b^9+45*B*a^2* b^8)*d*e^7+28*(A*b^10+10*B*a*b^9)*d^2*e^6+56*b^10*B*d^3*e^5)*x^17+1/16*((1 20*A*a^3*b^7+210*B*a^4*b^6)*e^8+8*(45*A*a^2*b^8+120*B*a^3*b^7)*d*e^7+28*(1 0*A*a*b^9+45*B*a^2*b^8)*d^2*e^6+56*(A*b^10+10*B*a*b^9)*d^3*e^5+70*b^10*B*d ^4*e^4)*x^16+1/15*((210*A*a^4*b^6+252*B*a^5*b^5)*e^8+8*(120*A*a^3*b^7+210* B*a^4*b^6)*d*e^7+28*(45*A*a^2*b^8+120*B*a^3*b^7)*d^2*e^6+56*(10*A*a*b^9+45 *B*a^2*b^8)*d^3*e^5+70*(A*b^10+10*B*a*b^9)*d^4*e^4+56*b^10*B*d^5*e^3)*x^15 +1/14*((252*A*a^5*b^5+210*B*a^6*b^4)*e^8+8*(210*A*a^4*b^6+252*B*a^5*b^5)*d *e^7+28*(120*A*a^3*b^7+210*B*a^4*b^6)*d^2*e^6+56*(45*A*a^2*b^8+120*B*a^3*b ^7)*d^3*e^5+70*(10*A*a*b^9+45*B*a^2*b^8)*d^4*e^4+56*(A*b^10+10*B*a*b^9)*d^ 5*e^3+28*b^10*B*d^6*e^2)*x^14+1/13*((210*A*a^6*b^4+120*B*a^7*b^3)*e^8+8*(2 52*A*a^5*b^5+210*B*a^6*b^4)*d*e^7+28*(210*A*a^4*b^6+252*B*a^5*b^5)*d^2*e^6 +56*(120*A*a^3*b^7+210*B*a^4*b^6)*d^3*e^5+70*(45*A*a^2*b^8+120*B*a^3*b^7)* d^4*e^4+56*(10*A*a*b^9+45*B*a^2*b^8)*d^5*e^3+28*(A*b^10+10*B*a*b^9)*d^6*e^ 2+8*b^10*B*d^7*e)*x^13+1/12*((120*A*a^7*b^3+45*B*a^8*b^2)*e^8+8*(210*A*a^6 *b^4+120*B*a^7*b^3)*d*e^7+28*(252*A*a^5*b^5+210*B*a^6*b^4)*d^2*e^6+56*(210 *A*a^4*b^6+252*B*a^5*b^5)*d^3*e^5+70*(120*A*a^3*b^7+210*B*a^4*b^6)*d^4*e^4 +56*(45*A*a^2*b^8+120*B*a^3*b^7)*d^5*e^3+28*(10*A*a*b^9+45*B*a^2*b^8)*d...
Leaf count of result is larger than twice the leaf count of optimal. 2487 vs. \(2 (354) = 708\).
Time = 0.24 (sec) , antiderivative size = 2487, normalized size of antiderivative = 6.69 \[ \int (a+b x)^{10} (A+B x) (d+e x)^8 \, dx=\text {Too large to display} \]
1/20*B*b^10*e^8*x^20 + A*a^10*d^8*x + 1/19*(8*B*b^10*d*e^7 + (10*B*a*b^9 + A*b^10)*e^8)*x^19 + 1/18*(28*B*b^10*d^2*e^6 + 8*(10*B*a*b^9 + A*b^10)*d*e ^7 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^8)*x^18 + 1/17*(56*B*b^10*d^3*e^5 + 28* (10*B*a*b^9 + A*b^10)*d^2*e^6 + 40*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^7 + 15*(8 *B*a^3*b^7 + 3*A*a^2*b^8)*e^8)*x^17 + 1/8*(35*B*b^10*d^4*e^4 + 28*(10*B*a* b^9 + A*b^10)*d^3*e^5 + 70*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^6 + 60*(8*B*a^3 *b^7 + 3*A*a^2*b^8)*d*e^7 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^8)*x^16 + 2/1 5*(28*B*b^10*d^5*e^3 + 35*(10*B*a*b^9 + A*b^10)*d^4*e^4 + 140*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^5 + 210*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^6 + 120*(7*B *a^4*b^6 + 4*A*a^3*b^7)*d*e^7 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^8)*x^15 + (2*B*b^10*d^6*e^2 + 4*(10*B*a*b^9 + A*b^10)*d^5*e^3 + 25*(9*B*a^2*b^8 + 2 *A*a*b^9)*d^4*e^4 + 60*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^5 + 60*(7*B*a^4*b ^6 + 4*A*a^3*b^7)*d^2*e^6 + 24*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^7 + 3*(5*B* a^6*b^4 + 6*A*a^5*b^5)*e^8)*x^14 + 2/13*(4*B*b^10*d^7*e + 14*(10*B*a*b^9 + A*b^10)*d^6*e^2 + 140*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^3 + 525*(8*B*a^3*b^ 7 + 3*A*a^2*b^8)*d^4*e^4 + 840*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^5 + 588*( 6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^6 + 168*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^7 + 15*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^8)*x^13 + 1/12*(B*b^10*d^8 + 8*(10*B*a *b^9 + A*b^10)*d^7*e + 140*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^2 + 840*(8*B*a^ 3*b^7 + 3*A*a^2*b^8)*d^5*e^3 + 2100*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^4...
Leaf count of result is larger than twice the leaf count of optimal. 3165 vs. \(2 (384) = 768\).
Time = 0.18 (sec) , antiderivative size = 3165, normalized size of antiderivative = 8.51 \[ \int (a+b x)^{10} (A+B x) (d+e x)^8 \, dx=\text {Too large to display} \]
A*a**10*d**8*x + B*b**10*e**8*x**20/20 + x**19*(A*b**10*e**8/19 + 10*B*a*b **9*e**8/19 + 8*B*b**10*d*e**7/19) + x**18*(5*A*a*b**9*e**8/9 + 4*A*b**10* d*e**7/9 + 5*B*a**2*b**8*e**8/2 + 40*B*a*b**9*d*e**7/9 + 14*B*b**10*d**2*e **6/9) + x**17*(45*A*a**2*b**8*e**8/17 + 80*A*a*b**9*d*e**7/17 + 28*A*b**1 0*d**2*e**6/17 + 120*B*a**3*b**7*e**8/17 + 360*B*a**2*b**8*d*e**7/17 + 280 *B*a*b**9*d**2*e**6/17 + 56*B*b**10*d**3*e**5/17) + x**16*(15*A*a**3*b**7* e**8/2 + 45*A*a**2*b**8*d*e**7/2 + 35*A*a*b**9*d**2*e**6/2 + 7*A*b**10*d** 3*e**5/2 + 105*B*a**4*b**6*e**8/8 + 60*B*a**3*b**7*d*e**7 + 315*B*a**2*b** 8*d**2*e**6/4 + 35*B*a*b**9*d**3*e**5 + 35*B*b**10*d**4*e**4/8) + x**15*(1 4*A*a**4*b**6*e**8 + 64*A*a**3*b**7*d*e**7 + 84*A*a**2*b**8*d**2*e**6 + 11 2*A*a*b**9*d**3*e**5/3 + 14*A*b**10*d**4*e**4/3 + 84*B*a**5*b**5*e**8/5 + 112*B*a**4*b**6*d*e**7 + 224*B*a**3*b**7*d**2*e**6 + 168*B*a**2*b**8*d**3* e**5 + 140*B*a*b**9*d**4*e**4/3 + 56*B*b**10*d**5*e**3/15) + x**14*(18*A*a **5*b**5*e**8 + 120*A*a**4*b**6*d*e**7 + 240*A*a**3*b**7*d**2*e**6 + 180*A *a**2*b**8*d**3*e**5 + 50*A*a*b**9*d**4*e**4 + 4*A*b**10*d**5*e**3 + 15*B* a**6*b**4*e**8 + 144*B*a**5*b**5*d*e**7 + 420*B*a**4*b**6*d**2*e**6 + 480* B*a**3*b**7*d**3*e**5 + 225*B*a**2*b**8*d**4*e**4 + 40*B*a*b**9*d**5*e**3 + 2*B*b**10*d**6*e**2) + x**13*(210*A*a**6*b**4*e**8/13 + 2016*A*a**5*b**5 *d*e**7/13 + 5880*A*a**4*b**6*d**2*e**6/13 + 6720*A*a**3*b**7*d**3*e**5/13 + 3150*A*a**2*b**8*d**4*e**4/13 + 560*A*a*b**9*d**5*e**3/13 + 28*A*b**...
Leaf count of result is larger than twice the leaf count of optimal. 2487 vs. \(2 (354) = 708\).
Time = 0.20 (sec) , antiderivative size = 2487, normalized size of antiderivative = 6.69 \[ \int (a+b x)^{10} (A+B x) (d+e x)^8 \, dx=\text {Too large to display} \]
1/20*B*b^10*e^8*x^20 + A*a^10*d^8*x + 1/19*(8*B*b^10*d*e^7 + (10*B*a*b^9 + A*b^10)*e^8)*x^19 + 1/18*(28*B*b^10*d^2*e^6 + 8*(10*B*a*b^9 + A*b^10)*d*e ^7 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^8)*x^18 + 1/17*(56*B*b^10*d^3*e^5 + 28* (10*B*a*b^9 + A*b^10)*d^2*e^6 + 40*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^7 + 15*(8 *B*a^3*b^7 + 3*A*a^2*b^8)*e^8)*x^17 + 1/8*(35*B*b^10*d^4*e^4 + 28*(10*B*a* b^9 + A*b^10)*d^3*e^5 + 70*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^6 + 60*(8*B*a^3 *b^7 + 3*A*a^2*b^8)*d*e^7 + 15*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^8)*x^16 + 2/1 5*(28*B*b^10*d^5*e^3 + 35*(10*B*a*b^9 + A*b^10)*d^4*e^4 + 140*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^5 + 210*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^6 + 120*(7*B *a^4*b^6 + 4*A*a^3*b^7)*d*e^7 + 21*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^8)*x^15 + (2*B*b^10*d^6*e^2 + 4*(10*B*a*b^9 + A*b^10)*d^5*e^3 + 25*(9*B*a^2*b^8 + 2 *A*a*b^9)*d^4*e^4 + 60*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^5 + 60*(7*B*a^4*b ^6 + 4*A*a^3*b^7)*d^2*e^6 + 24*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^7 + 3*(5*B* a^6*b^4 + 6*A*a^5*b^5)*e^8)*x^14 + 2/13*(4*B*b^10*d^7*e + 14*(10*B*a*b^9 + A*b^10)*d^6*e^2 + 140*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^3 + 525*(8*B*a^3*b^ 7 + 3*A*a^2*b^8)*d^4*e^4 + 840*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^5 + 588*( 6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^6 + 168*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^7 + 15*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^8)*x^13 + 1/12*(B*b^10*d^8 + 8*(10*B*a *b^9 + A*b^10)*d^7*e + 140*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^2 + 840*(8*B*a^ 3*b^7 + 3*A*a^2*b^8)*d^5*e^3 + 2100*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^4...
Leaf count of result is larger than twice the leaf count of optimal. 3159 vs. \(2 (354) = 708\).
Time = 0.33 (sec) , antiderivative size = 3159, normalized size of antiderivative = 8.49 \[ \int (a+b x)^{10} (A+B x) (d+e x)^8 \, dx=\text {Too large to display} \]
1/20*B*b^10*e^8*x^20 + 8/19*B*b^10*d*e^7*x^19 + 10/19*B*a*b^9*e^8*x^19 + 1 /19*A*b^10*e^8*x^19 + 14/9*B*b^10*d^2*e^6*x^18 + 40/9*B*a*b^9*d*e^7*x^18 + 4/9*A*b^10*d*e^7*x^18 + 5/2*B*a^2*b^8*e^8*x^18 + 5/9*A*a*b^9*e^8*x^18 + 5 6/17*B*b^10*d^3*e^5*x^17 + 280/17*B*a*b^9*d^2*e^6*x^17 + 28/17*A*b^10*d^2* e^6*x^17 + 360/17*B*a^2*b^8*d*e^7*x^17 + 80/17*A*a*b^9*d*e^7*x^17 + 120/17 *B*a^3*b^7*e^8*x^17 + 45/17*A*a^2*b^8*e^8*x^17 + 35/8*B*b^10*d^4*e^4*x^16 + 35*B*a*b^9*d^3*e^5*x^16 + 7/2*A*b^10*d^3*e^5*x^16 + 315/4*B*a^2*b^8*d^2* e^6*x^16 + 35/2*A*a*b^9*d^2*e^6*x^16 + 60*B*a^3*b^7*d*e^7*x^16 + 45/2*A*a^ 2*b^8*d*e^7*x^16 + 105/8*B*a^4*b^6*e^8*x^16 + 15/2*A*a^3*b^7*e^8*x^16 + 56 /15*B*b^10*d^5*e^3*x^15 + 140/3*B*a*b^9*d^4*e^4*x^15 + 14/3*A*b^10*d^4*e^4 *x^15 + 168*B*a^2*b^8*d^3*e^5*x^15 + 112/3*A*a*b^9*d^3*e^5*x^15 + 224*B*a^ 3*b^7*d^2*e^6*x^15 + 84*A*a^2*b^8*d^2*e^6*x^15 + 112*B*a^4*b^6*d*e^7*x^15 + 64*A*a^3*b^7*d*e^7*x^15 + 84/5*B*a^5*b^5*e^8*x^15 + 14*A*a^4*b^6*e^8*x^1 5 + 2*B*b^10*d^6*e^2*x^14 + 40*B*a*b^9*d^5*e^3*x^14 + 4*A*b^10*d^5*e^3*x^1 4 + 225*B*a^2*b^8*d^4*e^4*x^14 + 50*A*a*b^9*d^4*e^4*x^14 + 480*B*a^3*b^7*d ^3*e^5*x^14 + 180*A*a^2*b^8*d^3*e^5*x^14 + 420*B*a^4*b^6*d^2*e^6*x^14 + 24 0*A*a^3*b^7*d^2*e^6*x^14 + 144*B*a^5*b^5*d*e^7*x^14 + 120*A*a^4*b^6*d*e^7* x^14 + 15*B*a^6*b^4*e^8*x^14 + 18*A*a^5*b^5*e^8*x^14 + 8/13*B*b^10*d^7*e*x ^13 + 280/13*B*a*b^9*d^6*e^2*x^13 + 28/13*A*b^10*d^6*e^2*x^13 + 2520/13*B* a^2*b^8*d^5*e^3*x^13 + 560/13*A*a*b^9*d^5*e^3*x^13 + 8400/13*B*a^3*b^7*...
Time = 2.17 (sec) , antiderivative size = 2631, normalized size of antiderivative = 7.07 \[ \int (a+b x)^{10} (A+B x) (d+e x)^8 \, dx=\text {Too large to display} \]
x^5*(42*A*a^6*b^4*d^8 + 24*B*a^7*b^3*d^8 + 14*A*a^10*d^4*e^4 + (56*B*a^10* d^5*e^3)/5 + 192*A*a^7*b^3*d^7*e + 112*A*a^9*b*d^5*e^3 + 72*B*a^8*b^2*d^7* e + 56*B*a^9*b*d^6*e^2 + 252*A*a^8*b^2*d^6*e^2) + x^16*((15*A*a^3*b^7*e^8) /2 + (105*B*a^4*b^6*e^8)/8 + (7*A*b^10*d^3*e^5)/2 + (35*B*b^10*d^4*e^4)/8 + (35*A*a*b^9*d^2*e^6)/2 + (45*A*a^2*b^8*d*e^7)/2 + 35*B*a*b^9*d^3*e^5 + 6 0*B*a^3*b^7*d*e^7 + (315*B*a^2*b^8*d^2*e^6)/4) + x^10*((B*a^10*e^8)/10 + A *a*b^9*d^8 + A*a^9*b*e^8 + (9*B*a^2*b^8*d^8)/2 + 36*A*a^2*b^8*d^7*e + 36*A *a^8*b^2*d*e^7 + 96*B*a^3*b^7*d^7*e + 336*A*a^3*b^7*d^6*e^2 + 1176*A*a^4*b ^6*d^5*e^3 + 1764*A*a^5*b^5*d^4*e^4 + 1176*A*a^6*b^4*d^3*e^5 + 336*A*a^7*b ^3*d^2*e^6 + 588*B*a^4*b^6*d^6*e^2 + (7056*B*a^5*b^5*d^5*e^3)/5 + 1470*B*a ^6*b^4*d^4*e^4 + 672*B*a^7*b^3*d^3*e^5 + 126*B*a^8*b^2*d^2*e^6 + 8*B*a^9*b *d*e^7) + x^11*((A*b^10*d^8)/11 + (10*B*a*b^9*d^8)/11 + (10*B*a^9*b*e^8)/1 1 + (45*A*a^8*b^2*e^8)/11 + (960*A*a^7*b^3*d*e^7)/11 + (360*B*a^2*b^8*d^7* e)/11 + (360*B*a^8*b^2*d*e^7)/11 + (1260*A*a^2*b^8*d^6*e^2)/11 + (6720*A*a ^3*b^7*d^5*e^3)/11 + (14700*A*a^4*b^6*d^4*e^4)/11 + (14112*A*a^5*b^5*d^3*e ^5)/11 + (5880*A*a^6*b^4*d^2*e^6)/11 + (3360*B*a^3*b^7*d^6*e^2)/11 + (1176 0*B*a^4*b^6*d^5*e^3)/11 + (17640*B*a^5*b^5*d^4*e^4)/11 + (11760*B*a^6*b^4* d^3*e^5)/11 + (3360*B*a^7*b^3*d^2*e^6)/11 + (80*A*a*b^9*d^7*e)/11) + x^6*( 42*A*a^5*b^5*d^8 + 35*B*a^6*b^4*d^8 + (28*A*a^10*d^3*e^5)/3 + (35*B*a^10*d ^4*e^4)/3 + 280*A*a^6*b^4*d^7*e + (350*A*a^9*b*d^4*e^4)/3 + 160*B*a^7*b...