Integrand size = 20, antiderivative size = 415 \[ \int (a+b x)^{10} (A+B x) (d+e x)^9 \, dx=\frac {(A b-a B) (b d-a e)^9 (a+b x)^{11}}{11 b^{11}}+\frac {(b d-a e)^8 (b B d+9 A b e-10 a B e) (a+b x)^{12}}{12 b^{11}}+\frac {9 e (b d-a e)^7 (b B d+4 A b e-5 a B e) (a+b x)^{13}}{13 b^{11}}+\frac {6 e^2 (b d-a e)^6 (3 b B d+7 A b e-10 a B e) (a+b x)^{14}}{7 b^{11}}+\frac {14 e^3 (b d-a e)^5 (2 b B d+3 A b e-5 a B e) (a+b x)^{15}}{5 b^{11}}+\frac {63 e^4 (b d-a e)^4 (b B d+A b e-2 a B e) (a+b x)^{16}}{8 b^{11}}+\frac {42 e^5 (b d-a e)^3 (3 b B d+2 A b e-5 a B e) (a+b x)^{17}}{17 b^{11}}+\frac {2 e^6 (b d-a e)^2 (7 b B d+3 A b e-10 a B e) (a+b x)^{18}}{3 b^{11}}+\frac {9 e^7 (b d-a e) (4 b B d+A b e-5 a B e) (a+b x)^{19}}{19 b^{11}}+\frac {e^8 (9 b B d+A b e-10 a B e) (a+b x)^{20}}{20 b^{11}}+\frac {B e^9 (a+b x)^{21}}{21 b^{11}} \] Output:
1/11*(A*b-B*a)*(-a*e+b*d)^9*(b*x+a)^11/b^11+1/12*(-a*e+b*d)^8*(9*A*b*e-10* B*a*e+B*b*d)*(b*x+a)^12/b^11+9/13*e*(-a*e+b*d)^7*(4*A*b*e-5*B*a*e+B*b*d)*( b*x+a)^13/b^11+6/7*e^2*(-a*e+b*d)^6*(7*A*b*e-10*B*a*e+3*B*b*d)*(b*x+a)^14/ b^11+14/5*e^3*(-a*e+b*d)^5*(3*A*b*e-5*B*a*e+2*B*b*d)*(b*x+a)^15/b^11+63/8* e^4*(-a*e+b*d)^4*(A*b*e-2*B*a*e+B*b*d)*(b*x+a)^16/b^11+42/17*e^5*(-a*e+b*d )^3*(2*A*b*e-5*B*a*e+3*B*b*d)*(b*x+a)^17/b^11+2/3*e^6*(-a*e+b*d)^2*(3*A*b* e-10*B*a*e+7*B*b*d)*(b*x+a)^18/b^11+9/19*e^7*(-a*e+b*d)*(A*b*e-5*B*a*e+4*B *b*d)*(b*x+a)^19/b^11+1/20*e^8*(A*b*e-10*B*a*e+9*B*b*d)*(b*x+a)^20/b^11+1/ 21*B*e^9*(b*x+a)^21/b^11
Leaf count is larger than twice the leaf count of optimal. \(2553\) vs. \(2(415)=830\).
Time = 0.62 (sec) , antiderivative size = 2553, normalized size of antiderivative = 6.15 \[ \int (a+b x)^{10} (A+B x) (d+e x)^9 \, dx=\text {Result too large to show} \] Input:
Integrate[(a + b*x)^10*(A + B*x)*(d + e*x)^9,x]
Output:
a^10*A*d^9*x + (a^9*d^8*(10*A*b*d + a*B*d + 9*a*A*e)*x^2)/2 + (a^8*d^7*(a* B*d*(10*b*d + 9*a*e) + 9*A*(5*b^2*d^2 + 10*a*b*d*e + 4*a^2*e^2))*x^3)/3 + (3*a^7*d^6*(3*a*B*d*(5*b^2*d^2 + 10*a*b*d*e + 4*a^2*e^2) + A*(40*b^3*d^3 + 135*a*b^2*d^2*e + 120*a^2*b*d*e^2 + 28*a^3*e^3))*x^4)/4 + (3*a^6*d^5*(a*B *d*(40*b^3*d^3 + 135*a*b^2*d^2*e + 120*a^2*b*d*e^2 + 28*a^3*e^3) + A*(70*b ^4*d^4 + 360*a*b^3*d^3*e + 540*a^2*b^2*d^2*e^2 + 280*a^3*b*d*e^3 + 42*a^4* e^4))*x^5)/5 + a^5*d^4*(a*B*d*(35*b^4*d^4 + 180*a*b^3*d^3*e + 270*a^2*b^2* d^2*e^2 + 140*a^3*b*d*e^3 + 21*a^4*e^4) + 3*A*(14*b^5*d^5 + 105*a*b^4*d^4* e + 240*a^2*b^3*d^3*e^2 + 210*a^3*b^2*d^2*e^3 + 70*a^4*b*d*e^4 + 7*a^5*e^5 ))*x^6 + (6*a^4*d^3*(3*a*B*d*(14*b^5*d^5 + 105*a*b^4*d^4*e + 240*a^2*b^3*d ^3*e^2 + 210*a^3*b^2*d^2*e^3 + 70*a^4*b*d*e^4 + 7*a^5*e^5) + 7*A*(5*b^6*d^ 6 + 54*a*b^5*d^5*e + 180*a^2*b^4*d^4*e^2 + 240*a^3*b^3*d^3*e^3 + 135*a^4*b ^2*d^2*e^4 + 30*a^5*b*d*e^5 + 2*a^6*e^6))*x^7)/7 + (3*a^3*d^2*(7*a*B*d*(5* b^6*d^6 + 54*a*b^5*d^5*e + 180*a^2*b^4*d^4*e^2 + 240*a^3*b^3*d^3*e^3 + 135 *a^4*b^2*d^2*e^4 + 30*a^5*b*d*e^5 + 2*a^6*e^6) + A*(20*b^7*d^7 + 315*a*b^6 *d^6*e + 1512*a^2*b^5*d^5*e^2 + 2940*a^3*b^4*d^4*e^3 + 2520*a^4*b^3*d^3*e^ 4 + 945*a^5*b^2*d^2*e^5 + 140*a^6*b*d*e^6 + 6*a^7*e^7))*x^8)/4 + (a^2*d*(2 *a*B*d*(20*b^7*d^7 + 315*a*b^6*d^6*e + 1512*a^2*b^5*d^5*e^2 + 2940*a^3*b^4 *d^4*e^3 + 2520*a^4*b^3*d^3*e^4 + 945*a^5*b^2*d^2*e^5 + 140*a^6*b*d*e^6 + 6*a^7*e^7) + 3*A*(5*b^8*d^8 + 120*a*b^7*d^7*e + 840*a^2*b^6*d^6*e^2 + 2...
Time = 3.40 (sec) , antiderivative size = 415, 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.
| \(\displaystyle \int (a+b x)^{10} (A+B x) (d+e x)^9 \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {e^8 (a+b x)^{19} (-10 a B e+A b e+9 b B d)}{b^{10}}+\frac {9 e^7 (a+b x)^{18} (b d-a e) (-5 a B e+A b e+4 b B d)}{b^{10}}+\frac {12 e^6 (a+b x)^{17} (b d-a e)^2 (-10 a B e+3 A b e+7 b B d)}{b^{10}}+\frac {42 e^5 (a+b x)^{16} (b d-a e)^3 (-5 a B e+2 A b e+3 b B d)}{b^{10}}+\frac {126 e^4 (a+b x)^{15} (b d-a e)^4 (-2 a B e+A b e+b B d)}{b^{10}}+\frac {42 e^3 (a+b x)^{14} (b d-a e)^5 (-5 a B e+3 A b e+2 b B d)}{b^{10}}+\frac {12 e^2 (a+b x)^{13} (b d-a e)^6 (-10 a B e+7 A b e+3 b B d)}{b^{10}}+\frac {9 e (a+b x)^{12} (b d-a e)^7 (-5 a B e+4 A b e+b B d)}{b^{10}}+\frac {(a+b x)^{11} (b d-a e)^8 (-10 a B e+9 A b e+b B d)}{b^{10}}+\frac {(a+b x)^{10} (A b-a B) (b d-a e)^9}{b^{10}}+\frac {B e^9 (a+b x)^{20}}{b^{10}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^8 (a+b x)^{20} (-10 a B e+A b e+9 b B d)}{20 b^{11}}+\frac {9 e^7 (a+b x)^{19} (b d-a e) (-5 a B e+A b e+4 b B d)}{19 b^{11}}+\frac {2 e^6 (a+b x)^{18} (b d-a e)^2 (-10 a B e+3 A b e+7 b B d)}{3 b^{11}}+\frac {42 e^5 (a+b x)^{17} (b d-a e)^3 (-5 a B e+2 A b e+3 b B d)}{17 b^{11}}+\frac {63 e^4 (a+b x)^{16} (b d-a e)^4 (-2 a B e+A b e+b B d)}{8 b^{11}}+\frac {14 e^3 (a+b x)^{15} (b d-a e)^5 (-5 a B e+3 A b e+2 b B d)}{5 b^{11}}+\frac {6 e^2 (a+b x)^{14} (b d-a e)^6 (-10 a B e+7 A b e+3 b B d)}{7 b^{11}}+\frac {9 e (a+b x)^{13} (b d-a e)^7 (-5 a B e+4 A b e+b B d)}{13 b^{11}}+\frac {(a+b x)^{12} (b d-a e)^8 (-10 a B e+9 A b e+b B d)}{12 b^{11}}+\frac {(a+b x)^{11} (A b-a B) (b d-a e)^9}{11 b^{11}}+\frac {B e^9 (a+b x)^{21}}{21 b^{11}}\) |
Input:
Int[(a + b*x)^10*(A + B*x)*(d + e*x)^9,x]
Output:
((A*b - a*B)*(b*d - a*e)^9*(a + b*x)^11)/(11*b^11) + ((b*d - a*e)^8*(b*B*d + 9*A*b*e - 10*a*B*e)*(a + b*x)^12)/(12*b^11) + (9*e*(b*d - a*e)^7*(b*B*d + 4*A*b*e - 5*a*B*e)*(a + b*x)^13)/(13*b^11) + (6*e^2*(b*d - a*e)^6*(3*b* B*d + 7*A*b*e - 10*a*B*e)*(a + b*x)^14)/(7*b^11) + (14*e^3*(b*d - a*e)^5*( 2*b*B*d + 3*A*b*e - 5*a*B*e)*(a + b*x)^15)/(5*b^11) + (63*e^4*(b*d - a*e)^ 4*(b*B*d + A*b*e - 2*a*B*e)*(a + b*x)^16)/(8*b^11) + (42*e^5*(b*d - a*e)^3 *(3*b*B*d + 2*A*b*e - 5*a*B*e)*(a + b*x)^17)/(17*b^11) + (2*e^6*(b*d - a*e )^2*(7*b*B*d + 3*A*b*e - 10*a*B*e)*(a + b*x)^18)/(3*b^11) + (9*e^7*(b*d - a*e)*(4*b*B*d + A*b*e - 5*a*B*e)*(a + b*x)^19)/(19*b^11) + (e^8*(9*b*B*d + A*b*e - 10*a*B*e)*(a + b*x)^20)/(20*b^11) + (B*e^9*(a + b*x)^21)/(21*b^11 )
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. \(2756\) vs. \(2(393)=786\).
Time = 0.27 (sec) , antiderivative size = 2757, normalized size of antiderivative = 6.64
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2757\) |
| norman | \(\text {Expression too large to display}\) | \(2978\) |
| orering | \(\text {Expression too large to display}\) | \(3536\) |
| gosper | \(\text {Expression too large to display}\) | \(3537\) |
| risch | \(\text {Expression too large to display}\) | \(3537\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(3537\) |
Input:
int((b*x+a)^10*(B*x+A)*(e*x+d)^9,x,method=_RETURNVERBOSE)
Output:
1/21*b^10*B*e^9*x^21+1/20*((A*b^10+10*B*a*b^9)*e^9+9*b^10*B*d*e^8)*x^20+1/ 19*((10*A*a*b^9+45*B*a^2*b^8)*e^9+9*(A*b^10+10*B*a*b^9)*d*e^8+36*b^10*B*d^ 2*e^7)*x^19+1/18*((45*A*a^2*b^8+120*B*a^3*b^7)*e^9+9*(10*A*a*b^9+45*B*a^2* b^8)*d*e^8+36*(A*b^10+10*B*a*b^9)*d^2*e^7+84*b^10*B*d^3*e^6)*x^18+1/17*((1 20*A*a^3*b^7+210*B*a^4*b^6)*e^9+9*(45*A*a^2*b^8+120*B*a^3*b^7)*d*e^8+36*(1 0*A*a*b^9+45*B*a^2*b^8)*d^2*e^7+84*(A*b^10+10*B*a*b^9)*d^3*e^6+126*b^10*B* d^4*e^5)*x^17+1/16*((210*A*a^4*b^6+252*B*a^5*b^5)*e^9+9*(120*A*a^3*b^7+210 *B*a^4*b^6)*d*e^8+36*(45*A*a^2*b^8+120*B*a^3*b^7)*d^2*e^7+84*(10*A*a*b^9+4 5*B*a^2*b^8)*d^3*e^6+126*(A*b^10+10*B*a*b^9)*d^4*e^5+126*b^10*B*d^5*e^4)*x ^16+1/15*((252*A*a^5*b^5+210*B*a^6*b^4)*e^9+9*(210*A*a^4*b^6+252*B*a^5*b^5 )*d*e^8+36*(120*A*a^3*b^7+210*B*a^4*b^6)*d^2*e^7+84*(45*A*a^2*b^8+120*B*a^ 3*b^7)*d^3*e^6+126*(10*A*a*b^9+45*B*a^2*b^8)*d^4*e^5+126*(A*b^10+10*B*a*b^ 9)*d^5*e^4+84*b^10*B*d^6*e^3)*x^15+1/14*((210*A*a^6*b^4+120*B*a^7*b^3)*e^9 +9*(252*A*a^5*b^5+210*B*a^6*b^4)*d*e^8+36*(210*A*a^4*b^6+252*B*a^5*b^5)*d^ 2*e^7+84*(120*A*a^3*b^7+210*B*a^4*b^6)*d^3*e^6+126*(45*A*a^2*b^8+120*B*a^3 *b^7)*d^4*e^5+126*(10*A*a*b^9+45*B*a^2*b^8)*d^5*e^4+84*(A*b^10+10*B*a*b^9) *d^6*e^3+36*b^10*B*d^7*e^2)*x^14+1/13*((120*A*a^7*b^3+45*B*a^8*b^2)*e^9+9* (210*A*a^6*b^4+120*B*a^7*b^3)*d*e^8+36*(252*A*a^5*b^5+210*B*a^6*b^4)*d^2*e ^7+84*(210*A*a^4*b^6+252*B*a^5*b^5)*d^3*e^6+126*(120*A*a^3*b^7+210*B*a^4*b ^6)*d^4*e^5+126*(45*A*a^2*b^8+120*B*a^3*b^7)*d^5*e^4+84*(10*A*a*b^9+45*...
Leaf count of result is larger than twice the leaf count of optimal. 2771 vs. \(2 (393) = 786\).
Time = 0.10 (sec) , antiderivative size = 2771, normalized size of antiderivative = 6.68 \[ \int (a+b x)^{10} (A+B x) (d+e x)^9 \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^10*(B*x+A)*(e*x+d)^9,x, algorithm="fricas")
Output:
1/21*B*b^10*e^9*x^21 + A*a^10*d^9*x + 1/20*(9*B*b^10*d*e^8 + (10*B*a*b^9 + A*b^10)*e^9)*x^20 + 1/19*(36*B*b^10*d^2*e^7 + 9*(10*B*a*b^9 + A*b^10)*d*e ^8 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^9)*x^19 + 1/6*(28*B*b^10*d^3*e^6 + 12*( 10*B*a*b^9 + A*b^10)*d^2*e^7 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^8 + 5*(8*B *a^3*b^7 + 3*A*a^2*b^8)*e^9)*x^18 + 3/17*(42*B*b^10*d^4*e^5 + 28*(10*B*a*b ^9 + A*b^10)*d^3*e^6 + 60*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^7 + 45*(8*B*a^3* b^7 + 3*A*a^2*b^8)*d*e^8 + 10*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^9)*x^17 + 3/8* (21*B*b^10*d^5*e^4 + 21*(10*B*a*b^9 + A*b^10)*d^4*e^5 + 70*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^6 + 90*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^7 + 45*(7*B*a^4* b^6 + 4*A*a^3*b^7)*d*e^8 + 7*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^9)*x^16 + 2/5*( 14*B*b^10*d^6*e^3 + 21*(10*B*a*b^9 + A*b^10)*d^5*e^4 + 105*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^5 + 210*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^6 + 180*(7*B*a^ 4*b^6 + 4*A*a^3*b^7)*d^2*e^7 + 63*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^8 + 7*(5 *B*a^6*b^4 + 6*A*a^5*b^5)*e^9)*x^15 + 3/7*(6*B*b^10*d^7*e^2 + 14*(10*B*a*b ^9 + A*b^10)*d^6*e^3 + 105*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^4 + 315*(8*B*a^ 3*b^7 + 3*A*a^2*b^8)*d^4*e^5 + 420*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^6 + 2 52*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^7 + 63*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d* e^8 + 5*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^9)*x^14 + 3/13*(3*B*b^10*d^8*e + 12* (10*B*a*b^9 + A*b^10)*d^7*e^2 + 140*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^3 + 63 0*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^4 + 1260*(7*B*a^4*b^6 + 4*A*a^3*b^7...
Leaf count of result is larger than twice the leaf count of optimal. 3541 vs. \(2 (428) = 856\).
Time = 0.25 (sec) , antiderivative size = 3541, normalized size of antiderivative = 8.53 \[ \int (a+b x)^{10} (A+B x) (d+e x)^9 \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)**10*(B*x+A)*(e*x+d)**9,x)
Output:
A*a**10*d**9*x + B*b**10*e**9*x**21/21 + x**20*(A*b**10*e**9/20 + B*a*b**9 *e**9/2 + 9*B*b**10*d*e**8/20) + x**19*(10*A*a*b**9*e**9/19 + 9*A*b**10*d* e**8/19 + 45*B*a**2*b**8*e**9/19 + 90*B*a*b**9*d*e**8/19 + 36*B*b**10*d**2 *e**7/19) + x**18*(5*A*a**2*b**8*e**9/2 + 5*A*a*b**9*d*e**8 + 2*A*b**10*d* *2*e**7 + 20*B*a**3*b**7*e**9/3 + 45*B*a**2*b**8*d*e**8/2 + 20*B*a*b**9*d* *2*e**7 + 14*B*b**10*d**3*e**6/3) + x**17*(120*A*a**3*b**7*e**9/17 + 405*A *a**2*b**8*d*e**8/17 + 360*A*a*b**9*d**2*e**7/17 + 84*A*b**10*d**3*e**6/17 + 210*B*a**4*b**6*e**9/17 + 1080*B*a**3*b**7*d*e**8/17 + 1620*B*a**2*b**8 *d**2*e**7/17 + 840*B*a*b**9*d**3*e**6/17 + 126*B*b**10*d**4*e**5/17) + x* *16*(105*A*a**4*b**6*e**9/8 + 135*A*a**3*b**7*d*e**8/2 + 405*A*a**2*b**8*d **2*e**7/4 + 105*A*a*b**9*d**3*e**6/2 + 63*A*b**10*d**4*e**5/8 + 63*B*a**5 *b**5*e**9/4 + 945*B*a**4*b**6*d*e**8/8 + 270*B*a**3*b**7*d**2*e**7 + 945* B*a**2*b**8*d**3*e**6/4 + 315*B*a*b**9*d**4*e**5/4 + 63*B*b**10*d**5*e**4/ 8) + x**15*(84*A*a**5*b**5*e**9/5 + 126*A*a**4*b**6*d*e**8 + 288*A*a**3*b* *7*d**2*e**7 + 252*A*a**2*b**8*d**3*e**6 + 84*A*a*b**9*d**4*e**5 + 42*A*b* *10*d**5*e**4/5 + 14*B*a**6*b**4*e**9 + 756*B*a**5*b**5*d*e**8/5 + 504*B*a **4*b**6*d**2*e**7 + 672*B*a**3*b**7*d**3*e**6 + 378*B*a**2*b**8*d**4*e**5 + 84*B*a*b**9*d**5*e**4 + 28*B*b**10*d**6*e**3/5) + x**14*(15*A*a**6*b**4 *e**9 + 162*A*a**5*b**5*d*e**8 + 540*A*a**4*b**6*d**2*e**7 + 720*A*a**3*b* *7*d**3*e**6 + 405*A*a**2*b**8*d**4*e**5 + 90*A*a*b**9*d**5*e**4 + 6*A*...
Leaf count of result is larger than twice the leaf count of optimal. 2771 vs. \(2 (393) = 786\).
Time = 0.05 (sec) , antiderivative size = 2771, normalized size of antiderivative = 6.68 \[ \int (a+b x)^{10} (A+B x) (d+e x)^9 \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^10*(B*x+A)*(e*x+d)^9,x, algorithm="maxima")
Output:
1/21*B*b^10*e^9*x^21 + A*a^10*d^9*x + 1/20*(9*B*b^10*d*e^8 + (10*B*a*b^9 + A*b^10)*e^9)*x^20 + 1/19*(36*B*b^10*d^2*e^7 + 9*(10*B*a*b^9 + A*b^10)*d*e ^8 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^9)*x^19 + 1/6*(28*B*b^10*d^3*e^6 + 12*( 10*B*a*b^9 + A*b^10)*d^2*e^7 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^8 + 5*(8*B *a^3*b^7 + 3*A*a^2*b^8)*e^9)*x^18 + 3/17*(42*B*b^10*d^4*e^5 + 28*(10*B*a*b ^9 + A*b^10)*d^3*e^6 + 60*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^7 + 45*(8*B*a^3* b^7 + 3*A*a^2*b^8)*d*e^8 + 10*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^9)*x^17 + 3/8* (21*B*b^10*d^5*e^4 + 21*(10*B*a*b^9 + A*b^10)*d^4*e^5 + 70*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^6 + 90*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^7 + 45*(7*B*a^4* b^6 + 4*A*a^3*b^7)*d*e^8 + 7*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^9)*x^16 + 2/5*( 14*B*b^10*d^6*e^3 + 21*(10*B*a*b^9 + A*b^10)*d^5*e^4 + 105*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^5 + 210*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^6 + 180*(7*B*a^ 4*b^6 + 4*A*a^3*b^7)*d^2*e^7 + 63*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^8 + 7*(5 *B*a^6*b^4 + 6*A*a^5*b^5)*e^9)*x^15 + 3/7*(6*B*b^10*d^7*e^2 + 14*(10*B*a*b ^9 + A*b^10)*d^6*e^3 + 105*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^4 + 315*(8*B*a^ 3*b^7 + 3*A*a^2*b^8)*d^4*e^5 + 420*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^6 + 2 52*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^7 + 63*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d* e^8 + 5*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^9)*x^14 + 3/13*(3*B*b^10*d^8*e + 12* (10*B*a*b^9 + A*b^10)*d^7*e^2 + 140*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^3 + 63 0*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^4 + 1260*(7*B*a^4*b^6 + 4*A*a^3*b^7...
Leaf count of result is larger than twice the leaf count of optimal. 3536 vs. \(2 (393) = 786\).
Time = 0.12 (sec) , antiderivative size = 3536, normalized size of antiderivative = 8.52 \[ \int (a+b x)^{10} (A+B x) (d+e x)^9 \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^10*(B*x+A)*(e*x+d)^9,x, algorithm="giac")
Output:
1/21*B*b^10*e^9*x^21 + 9/20*B*b^10*d*e^8*x^20 + 1/2*B*a*b^9*e^9*x^20 + 1/2 0*A*b^10*e^9*x^20 + 36/19*B*b^10*d^2*e^7*x^19 + 90/19*B*a*b^9*d*e^8*x^19 + 9/19*A*b^10*d*e^8*x^19 + 45/19*B*a^2*b^8*e^9*x^19 + 10/19*A*a*b^9*e^9*x^1 9 + 14/3*B*b^10*d^3*e^6*x^18 + 20*B*a*b^9*d^2*e^7*x^18 + 2*A*b^10*d^2*e^7* x^18 + 45/2*B*a^2*b^8*d*e^8*x^18 + 5*A*a*b^9*d*e^8*x^18 + 20/3*B*a^3*b^7*e ^9*x^18 + 5/2*A*a^2*b^8*e^9*x^18 + 126/17*B*b^10*d^4*e^5*x^17 + 840/17*B*a *b^9*d^3*e^6*x^17 + 84/17*A*b^10*d^3*e^6*x^17 + 1620/17*B*a^2*b^8*d^2*e^7* x^17 + 360/17*A*a*b^9*d^2*e^7*x^17 + 1080/17*B*a^3*b^7*d*e^8*x^17 + 405/17 *A*a^2*b^8*d*e^8*x^17 + 210/17*B*a^4*b^6*e^9*x^17 + 120/17*A*a^3*b^7*e^9*x ^17 + 63/8*B*b^10*d^5*e^4*x^16 + 315/4*B*a*b^9*d^4*e^5*x^16 + 63/8*A*b^10* d^4*e^5*x^16 + 945/4*B*a^2*b^8*d^3*e^6*x^16 + 105/2*A*a*b^9*d^3*e^6*x^16 + 270*B*a^3*b^7*d^2*e^7*x^16 + 405/4*A*a^2*b^8*d^2*e^7*x^16 + 945/8*B*a^4*b ^6*d*e^8*x^16 + 135/2*A*a^3*b^7*d*e^8*x^16 + 63/4*B*a^5*b^5*e^9*x^16 + 105 /8*A*a^4*b^6*e^9*x^16 + 28/5*B*b^10*d^6*e^3*x^15 + 84*B*a*b^9*d^5*e^4*x^15 + 42/5*A*b^10*d^5*e^4*x^15 + 378*B*a^2*b^8*d^4*e^5*x^15 + 84*A*a*b^9*d^4* e^5*x^15 + 672*B*a^3*b^7*d^3*e^6*x^15 + 252*A*a^2*b^8*d^3*e^6*x^15 + 504*B *a^4*b^6*d^2*e^7*x^15 + 288*A*a^3*b^7*d^2*e^7*x^15 + 756/5*B*a^5*b^5*d*e^8 *x^15 + 126*A*a^4*b^6*d*e^8*x^15 + 14*B*a^6*b^4*e^9*x^15 + 84/5*A*a^5*b^5* e^9*x^15 + 18/7*B*b^10*d^7*e^2*x^14 + 60*B*a*b^9*d^6*e^3*x^14 + 6*A*b^10*d ^6*e^3*x^14 + 405*B*a^2*b^8*d^5*e^4*x^14 + 90*A*a*b^9*d^5*e^4*x^14 + 10...
Time = 1.58 (sec) , antiderivative size = 2947, normalized size of antiderivative = 7.10 \[ \int (a+b x)^{10} (A+B x) (d+e x)^9 \, dx=\text {Too large to display} \] Input:
int((A + B*x)*(a + b*x)^10*(d + e*x)^9,x)
Output:
x^9*(A*a^10*d*e^8 + 5*A*a^2*b^8*d^9 + (40*B*a^3*b^7*d^9)/3 + 4*B*a^10*d^2* e^7 + 120*A*a^3*b^7*d^8*e + 40*A*a^9*b*d^2*e^7 + 210*B*a^4*b^6*d^8*e + (28 0*B*a^9*b*d^3*e^6)/3 + 840*A*a^4*b^6*d^7*e^2 + 2352*A*a^5*b^5*d^6*e^3 + 29 40*A*a^6*b^4*d^5*e^4 + 1680*A*a^7*b^3*d^4*e^5 + 420*A*a^8*b^2*d^3*e^6 + 10 08*B*a^5*b^5*d^7*e^2 + 1960*B*a^6*b^4*d^6*e^3 + 1680*B*a^7*b^3*d^5*e^4 + 6 30*B*a^8*b^2*d^4*e^5) + x^13*((9*B*b^10*d^8*e)/13 + (120*A*a^7*b^3*e^9)/13 + (45*B*a^8*b^2*e^9)/13 + (36*A*b^10*d^7*e^2)/13 + (840*A*a*b^9*d^6*e^3)/ 13 + (1890*A*a^6*b^4*d*e^8)/13 + (360*B*a*b^9*d^7*e^2)/13 + (1080*B*a^7*b^ 3*d*e^8)/13 + (5670*A*a^2*b^8*d^5*e^4)/13 + (15120*A*a^3*b^7*d^4*e^5)/13 + (17640*A*a^4*b^6*d^3*e^6)/13 + (9072*A*a^5*b^5*d^2*e^7)/13 + (3780*B*a^2* b^8*d^6*e^3)/13 + (15120*B*a^3*b^7*d^5*e^4)/13 + (26460*B*a^4*b^6*d^4*e^5) /13 + (21168*B*a^5*b^5*d^3*e^6)/13 + (7560*B*a^6*b^4*d^2*e^7)/13) + x^5*(4 2*A*a^6*b^4*d^9 + 24*B*a^7*b^3*d^9 + (126*A*a^10*d^5*e^4)/5 + (84*B*a^10*d ^6*e^3)/5 + 216*A*a^7*b^3*d^8*e + 168*A*a^9*b*d^6*e^3 + 81*B*a^8*b^2*d^8*e + 72*B*a^9*b*d^7*e^2 + 324*A*a^8*b^2*d^7*e^2) + x^8*(15*A*a^3*b^7*d^9 + ( 105*B*a^4*b^6*d^9)/4 + (9*A*a^10*d^2*e^7)/2 + (21*B*a^10*d^3*e^6)/2 + (945 *A*a^4*b^6*d^8*e)/4 + 105*A*a^9*b*d^3*e^6 + (567*B*a^5*b^5*d^8*e)/2 + (315 *B*a^9*b*d^4*e^5)/2 + 1134*A*a^5*b^5*d^7*e^2 + 2205*A*a^6*b^4*d^6*e^3 + 18 90*A*a^7*b^3*d^5*e^4 + (2835*A*a^8*b^2*d^4*e^5)/4 + 945*B*a^6*b^4*d^7*e^2 + 1260*B*a^7*b^3*d^6*e^3 + (2835*B*a^8*b^2*d^5*e^4)/4) + x^17*((120*A*a...
Time = 0.17 (sec) , antiderivative size = 1817, normalized size of antiderivative = 4.38 \[ \int (a+b x)^{10} (A+B x) (d+e x)^9 \, dx =\text {Too large to display} \] Input:
int((b*x+a)^10*(B*x+A)*(e*x+d)^9,x)
Output:
(x*(3527160*a**11*d**9 + 15872220*a**11*d**8*e*x + 42325920*a**11*d**7*e** 2*x**2 + 74070360*a**11*d**6*e**3*x**3 + 88884432*a**11*d**5*e**4*x**4 + 7 4070360*a**11*d**4*e**5*x**5 + 42325920*a**11*d**3*e**6*x**6 + 15872220*a* *11*d**2*e**7*x**7 + 3527160*a**11*d*e**8*x**8 + 352716*a**11*e**9*x**9 + 19399380*a**10*b*d**9*x + 116396280*a**10*b*d**8*e*x**2 + 349188840*a**10* b*d**7*e**2*x**3 + 651819168*a**10*b*d**6*e**3*x**4 + 814773960*a**10*b*d* *5*e**4*x**5 + 698377680*a**10*b*d**4*e**5*x**6 + 407386980*a**10*b*d**3*e **6*x**7 + 155195040*a**10*b*d**2*e**7*x**8 + 34918884*a**10*b*d*e**8*x**9 + 3527160*a**10*b*e**9*x**10 + 64664600*a**9*b**2*d**9*x**2 + 436486050*a **9*b**2*d**8*e*x**3 + 1396755360*a**9*b**2*d**7*e**2*x**4 + 2715913200*a* *9*b**2*d**6*e**3*x**5 + 3491888400*a**9*b**2*d**5*e**4*x**6 + 3055402350* a**9*b**2*d**4*e**5*x**7 + 1810608800*a**9*b**2*d**3*e**6*x**8 + 698377680 *a**9*b**2*d**2*e**7*x**9 + 158722200*a**9*b**2*d*e**8*x**10 + 16166150*a* *9*b**2*e**9*x**11 + 145495350*a**8*b**3*d**9*x**3 + 1047566520*a**8*b**3* d**8*e*x**4 + 3491888400*a**8*b**3*d**7*e**2*x**5 + 6983776800*a**8*b**3*d **6*e**3*x**6 + 9166207050*a**8*b**3*d**5*e**4*x**7 + 8147739600*a**8*b**3 *d**4*e**5*x**8 + 4888643760*a**8*b**3*d**3*e**6*x**9 + 1904666400*a**8*b* *3*d**2*e**7*x**10 + 436486050*a**8*b**3*d*e**8*x**11 + 44767800*a**8*b**3 *e**9*x**12 + 232792560*a**7*b**4*d**9*x**4 + 1745944200*a**7*b**4*d**8*e* x**5 + 5986094400*a**7*b**4*d**7*e**2*x**6 + 12221609400*a**7*b**4*d**6...