Integrand size = 20, antiderivative size = 460 \[ \int (a+b x)^{10} (A+B x) (d+e x)^{10} \, dx=\frac {(A b-a B) (b d-a e)^{10} (a+b x)^{11}}{11 b^{12}}+\frac {(b d-a e)^9 (b B d+10 A b e-11 a B e) (a+b x)^{12}}{12 b^{12}}+\frac {5 e (b d-a e)^8 (2 b B d+9 A b e-11 a B e) (a+b x)^{13}}{13 b^{12}}+\frac {15 e^2 (b d-a e)^7 (3 b B d+8 A b e-11 a B e) (a+b x)^{14}}{14 b^{12}}+\frac {2 e^3 (b d-a e)^6 (4 b B d+7 A b e-11 a B e) (a+b x)^{15}}{b^{12}}+\frac {21 e^4 (b d-a e)^5 (5 b B d+6 A b e-11 a B e) (a+b x)^{16}}{8 b^{12}}+\frac {42 e^5 (b d-a e)^4 (6 b B d+5 A b e-11 a B e) (a+b x)^{17}}{17 b^{12}}+\frac {5 e^6 (b d-a e)^3 (7 b B d+4 A b e-11 a B e) (a+b x)^{18}}{3 b^{12}}+\frac {15 e^7 (b d-a e)^2 (8 b B d+3 A b e-11 a B e) (a+b x)^{19}}{19 b^{12}}+\frac {e^8 (b d-a e) (9 b B d+2 A b e-11 a B e) (a+b x)^{20}}{4 b^{12}}+\frac {e^9 (10 b B d+A b e-11 a B e) (a+b x)^{21}}{21 b^{12}}+\frac {B e^{10} (a+b x)^{22}}{22 b^{12}} \] Output:
1/11*(A*b-B*a)*(-a*e+b*d)^10*(b*x+a)^11/b^12+1/12*(-a*e+b*d)^9*(10*A*b*e-1 1*B*a*e+B*b*d)*(b*x+a)^12/b^12+5/13*e*(-a*e+b*d)^8*(9*A*b*e-11*B*a*e+2*B*b *d)*(b*x+a)^13/b^12+15/14*e^2*(-a*e+b*d)^7*(8*A*b*e-11*B*a*e+3*B*b*d)*(b*x +a)^14/b^12+2*e^3*(-a*e+b*d)^6*(7*A*b*e-11*B*a*e+4*B*b*d)*(b*x+a)^15/b^12+ 21/8*e^4*(-a*e+b*d)^5*(6*A*b*e-11*B*a*e+5*B*b*d)*(b*x+a)^16/b^12+42/17*e^5 *(-a*e+b*d)^4*(5*A*b*e-11*B*a*e+6*B*b*d)*(b*x+a)^17/b^12+5/3*e^6*(-a*e+b*d )^3*(4*A*b*e-11*B*a*e+7*B*b*d)*(b*x+a)^18/b^12+15/19*e^7*(-a*e+b*d)^2*(3*A *b*e-11*B*a*e+8*B*b*d)*(b*x+a)^19/b^12+1/4*e^8*(-a*e+b*d)*(2*A*b*e-11*B*a* e+9*B*b*d)*(b*x+a)^20/b^12+1/21*e^9*(A*b*e-11*B*a*e+10*B*b*d)*(b*x+a)^21/b ^12+1/22*B*e^10*(b*x+a)^22/b^12
Leaf count is larger than twice the leaf count of optimal. \(2815\) vs. \(2(460)=920\).
Time = 0.71 (sec) , antiderivative size = 2815, normalized size of antiderivative = 6.12 \[ \int (a+b x)^{10} (A+B x) (d+e x)^{10} \, dx=\text {Result too large to show} \] Input:
Integrate[(a + b*x)^10*(A + B*x)*(d + e*x)^10,x]
Output:
a^10*A*d^10*x + (a^9*d^9*(a*B*d + 10*A*(b*d + a*e))*x^2)/2 + (5*a^8*d^8*(2 *a*B*d*(b*d + a*e) + A*(9*b^2*d^2 + 20*a*b*d*e + 9*a^2*e^2))*x^3)/3 + (5*a ^7*d^7*(a*B*d*(9*b^2*d^2 + 20*a*b*d*e + 9*a^2*e^2) + 6*A*(4*b^3*d^3 + 15*a *b^2*d^2*e + 15*a^2*b*d*e^2 + 4*a^3*e^3))*x^4)/4 + 3*a^6*d^6*(2*a*B*d*(4*b ^3*d^3 + 15*a*b^2*d^2*e + 15*a^2*b*d*e^2 + 4*a^3*e^3) + A*(14*b^4*d^4 + 80 *a*b^3*d^3*e + 135*a^2*b^2*d^2*e^2 + 80*a^3*b*d*e^3 + 14*a^4*e^4))*x^5 + ( a^5*d^5*(5*a*B*d*(14*b^4*d^4 + 80*a*b^3*d^3*e + 135*a^2*b^2*d^2*e^2 + 80*a ^3*b*d*e^3 + 14*a^4*e^4) + 4*A*(21*b^5*d^5 + 175*a*b^4*d^4*e + 450*a^2*b^3 *d^3*e^2 + 450*a^3*b^2*d^2*e^3 + 175*a^4*b*d*e^4 + 21*a^5*e^5))*x^6)/2 + ( 6*a^4*d^4*(2*a*B*d*(21*b^5*d^5 + 175*a*b^4*d^4*e + 450*a^2*b^3*d^3*e^2 + 4 50*a^3*b^2*d^2*e^3 + 175*a^4*b*d*e^4 + 21*a^5*e^5) + 5*A*(7*b^6*d^6 + 84*a *b^5*d^5*e + 315*a^2*b^4*d^4*e^2 + 480*a^3*b^3*d^3*e^3 + 315*a^4*b^2*d^2*e ^4 + 84*a^5*b*d*e^5 + 7*a^6*e^6))*x^7)/7 + (15*a^3*d^3*(a*B*d*(7*b^6*d^6 + 84*a*b^5*d^5*e + 315*a^2*b^4*d^4*e^2 + 480*a^3*b^3*d^3*e^3 + 315*a^4*b^2* d^2*e^4 + 84*a^5*b*d*e^5 + 7*a^6*e^6) + A*(4*b^7*d^7 + 70*a*b^6*d^6*e + 37 8*a^2*b^5*d^5*e^2 + 840*a^3*b^4*d^4*e^3 + 840*a^4*b^3*d^3*e^4 + 378*a^5*b^ 2*d^2*e^5 + 70*a^6*b*d*e^6 + 4*a^7*e^7))*x^8)/4 + (5*a^2*d^2*(4*a*B*d*(2*b ^7*d^7 + 35*a*b^6*d^6*e + 189*a^2*b^5*d^5*e^2 + 420*a^3*b^4*d^4*e^3 + 420* a^4*b^3*d^3*e^4 + 189*a^5*b^2*d^2*e^5 + 35*a^6*b*d*e^6 + 2*a^7*e^7) + A*(3 *b^8*d^8 + 80*a*b^7*d^7*e + 630*a^2*b^6*d^6*e^2 + 2016*a^3*b^5*d^5*e^3 ...
Time = 4.13 (sec) , antiderivative size = 460, 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.
| \(\displaystyle \int (a+b x)^{10} (A+B x) (d+e x)^{10} \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {e^9 (a+b x)^{20} (-11 a B e+A b e+10 b B d)}{b^{11}}+\frac {5 e^8 (a+b x)^{19} (b d-a e) (-11 a B e+2 A b e+9 b B d)}{b^{11}}+\frac {15 e^7 (a+b x)^{18} (b d-a e)^2 (-11 a B e+3 A b e+8 b B d)}{b^{11}}+\frac {30 e^6 (a+b x)^{17} (b d-a e)^3 (-11 a B e+4 A b e+7 b B d)}{b^{11}}+\frac {42 e^5 (a+b x)^{16} (b d-a e)^4 (-11 a B e+5 A b e+6 b B d)}{b^{11}}+\frac {42 e^4 (a+b x)^{15} (b d-a e)^5 (-11 a B e+6 A b e+5 b B d)}{b^{11}}+\frac {30 e^3 (a+b x)^{14} (b d-a e)^6 (-11 a B e+7 A b e+4 b B d)}{b^{11}}+\frac {15 e^2 (a+b x)^{13} (b d-a e)^7 (-11 a B e+8 A b e+3 b B d)}{b^{11}}+\frac {5 e (a+b x)^{12} (b d-a e)^8 (-11 a B e+9 A b e+2 b B d)}{b^{11}}+\frac {(a+b x)^{11} (b d-a e)^9 (-11 a B e+10 A b e+b B d)}{b^{11}}+\frac {(a+b x)^{10} (A b-a B) (b d-a e)^{10}}{b^{11}}+\frac {B e^{10} (a+b x)^{21}}{b^{11}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^9 (a+b x)^{21} (-11 a B e+A b e+10 b B d)}{21 b^{12}}+\frac {e^8 (a+b x)^{20} (b d-a e) (-11 a B e+2 A b e+9 b B d)}{4 b^{12}}+\frac {15 e^7 (a+b x)^{19} (b d-a e)^2 (-11 a B e+3 A b e+8 b B d)}{19 b^{12}}+\frac {5 e^6 (a+b x)^{18} (b d-a e)^3 (-11 a B e+4 A b e+7 b B d)}{3 b^{12}}+\frac {42 e^5 (a+b x)^{17} (b d-a e)^4 (-11 a B e+5 A b e+6 b B d)}{17 b^{12}}+\frac {21 e^4 (a+b x)^{16} (b d-a e)^5 (-11 a B e+6 A b e+5 b B d)}{8 b^{12}}+\frac {2 e^3 (a+b x)^{15} (b d-a e)^6 (-11 a B e+7 A b e+4 b B d)}{b^{12}}+\frac {15 e^2 (a+b x)^{14} (b d-a e)^7 (-11 a B e+8 A b e+3 b B d)}{14 b^{12}}+\frac {5 e (a+b x)^{13} (b d-a e)^8 (-11 a B e+9 A b e+2 b B d)}{13 b^{12}}+\frac {(a+b x)^{12} (b d-a e)^9 (-11 a B e+10 A b e+b B d)}{12 b^{12}}+\frac {(a+b x)^{11} (A b-a B) (b d-a e)^{10}}{11 b^{12}}+\frac {B e^{10} (a+b x)^{22}}{22 b^{12}}\) |
Input:
Int[(a + b*x)^10*(A + B*x)*(d + e*x)^10,x]
Output:
((A*b - a*B)*(b*d - a*e)^10*(a + b*x)^11)/(11*b^12) + ((b*d - a*e)^9*(b*B* d + 10*A*b*e - 11*a*B*e)*(a + b*x)^12)/(12*b^12) + (5*e*(b*d - a*e)^8*(2*b *B*d + 9*A*b*e - 11*a*B*e)*(a + b*x)^13)/(13*b^12) + (15*e^2*(b*d - a*e)^7 *(3*b*B*d + 8*A*b*e - 11*a*B*e)*(a + b*x)^14)/(14*b^12) + (2*e^3*(b*d - a* e)^6*(4*b*B*d + 7*A*b*e - 11*a*B*e)*(a + b*x)^15)/b^12 + (21*e^4*(b*d - a* e)^5*(5*b*B*d + 6*A*b*e - 11*a*B*e)*(a + b*x)^16)/(8*b^12) + (42*e^5*(b*d - a*e)^4*(6*b*B*d + 5*A*b*e - 11*a*B*e)*(a + b*x)^17)/(17*b^12) + (5*e^6*( b*d - a*e)^3*(7*b*B*d + 4*A*b*e - 11*a*B*e)*(a + b*x)^18)/(3*b^12) + (15*e ^7*(b*d - a*e)^2*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(a + b*x)^19)/(19*b^12) + (e^8*(b*d - a*e)*(9*b*B*d + 2*A*b*e - 11*a*B*e)*(a + b*x)^20)/(4*b^12) + ( e^9*(10*b*B*d + A*b*e - 11*a*B*e)*(a + b*x)^21)/(21*b^12) + (B*e^10*(a + b *x)^22)/(22*b^12)
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. \(3040\) vs. \(2(438)=876\).
Time = 0.29 (sec) , antiderivative size = 3041, normalized size of antiderivative = 6.61
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3041\) |
| norman | \(\text {Expression too large to display}\) | \(3293\) |
| orering | \(\text {Expression too large to display}\) | \(3912\) |
| gosper | \(\text {Expression too large to display}\) | \(3913\) |
| risch | \(\text {Expression too large to display}\) | \(3913\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(3913\) |
Input:
int((b*x+a)^10*(B*x+A)*(e*x+d)^10,x,method=_RETURNVERBOSE)
Output:
1/22*b^10*B*e^10*x^22+1/21*((A*b^10+10*B*a*b^9)*e^10+10*b^10*B*d*e^9)*x^21 +1/20*((10*A*a*b^9+45*B*a^2*b^8)*e^10+10*(A*b^10+10*B*a*b^9)*d*e^9+45*b^10 *B*d^2*e^8)*x^20+1/19*((45*A*a^2*b^8+120*B*a^3*b^7)*e^10+10*(10*A*a*b^9+45 *B*a^2*b^8)*d*e^9+45*(A*b^10+10*B*a*b^9)*d^2*e^8+120*b^10*B*d^3*e^7)*x^19+ 1/18*((120*A*a^3*b^7+210*B*a^4*b^6)*e^10+10*(45*A*a^2*b^8+120*B*a^3*b^7)*d *e^9+45*(10*A*a*b^9+45*B*a^2*b^8)*d^2*e^8+120*(A*b^10+10*B*a*b^9)*d^3*e^7+ 210*b^10*B*d^4*e^6)*x^18+1/17*((210*A*a^4*b^6+252*B*a^5*b^5)*e^10+10*(120* A*a^3*b^7+210*B*a^4*b^6)*d*e^9+45*(45*A*a^2*b^8+120*B*a^3*b^7)*d^2*e^8+120 *(10*A*a*b^9+45*B*a^2*b^8)*d^3*e^7+210*(A*b^10+10*B*a*b^9)*d^4*e^6+252*b^1 0*B*d^5*e^5)*x^17+1/16*((252*A*a^5*b^5+210*B*a^6*b^4)*e^10+10*(210*A*a^4*b ^6+252*B*a^5*b^5)*d*e^9+45*(120*A*a^3*b^7+210*B*a^4*b^6)*d^2*e^8+120*(45*A *a^2*b^8+120*B*a^3*b^7)*d^3*e^7+210*(10*A*a*b^9+45*B*a^2*b^8)*d^4*e^6+252* (A*b^10+10*B*a*b^9)*d^5*e^5+210*b^10*B*d^6*e^4)*x^16+1/15*((210*A*a^6*b^4+ 120*B*a^7*b^3)*e^10+10*(252*A*a^5*b^5+210*B*a^6*b^4)*d*e^9+45*(210*A*a^4*b ^6+252*B*a^5*b^5)*d^2*e^8+120*(120*A*a^3*b^7+210*B*a^4*b^6)*d^3*e^7+210*(4 5*A*a^2*b^8+120*B*a^3*b^7)*d^4*e^6+252*(10*A*a*b^9+45*B*a^2*b^8)*d^5*e^5+2 10*(A*b^10+10*B*a*b^9)*d^6*e^4+120*b^10*B*d^7*e^3)*x^15+1/14*((120*A*a^7*b ^3+45*B*a^8*b^2)*e^10+10*(210*A*a^6*b^4+120*B*a^7*b^3)*d*e^9+45*(252*A*a^5 *b^5+210*B*a^6*b^4)*d^2*e^8+120*(210*A*a^4*b^6+252*B*a^5*b^5)*d^3*e^7+210* (120*A*a^3*b^7+210*B*a^4*b^6)*d^4*e^6+252*(45*A*a^2*b^8+120*B*a^3*b^7)*...
Leaf count of result is larger than twice the leaf count of optimal. 3048 vs. \(2 (438) = 876\).
Time = 0.13 (sec) , antiderivative size = 3048, normalized size of antiderivative = 6.63 \[ \int (a+b x)^{10} (A+B x) (d+e x)^{10} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^10*(B*x+A)*(e*x+d)^10,x, algorithm="fricas")
Output:
Too large to include
Leaf count of result is larger than twice the leaf count of optimal. 3936 vs. \(2 (478) = 956\).
Time = 0.25 (sec) , antiderivative size = 3936, normalized size of antiderivative = 8.56 \[ \int (a+b x)^{10} (A+B x) (d+e x)^{10} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)**10*(B*x+A)*(e*x+d)**10,x)
Output:
A*a**10*d**10*x + B*b**10*e**10*x**22/22 + x**21*(A*b**10*e**10/21 + 10*B* a*b**9*e**10/21 + 10*B*b**10*d*e**9/21) + x**20*(A*a*b**9*e**10/2 + A*b**1 0*d*e**9/2 + 9*B*a**2*b**8*e**10/4 + 5*B*a*b**9*d*e**9 + 9*B*b**10*d**2*e* *8/4) + x**19*(45*A*a**2*b**8*e**10/19 + 100*A*a*b**9*d*e**9/19 + 45*A*b** 10*d**2*e**8/19 + 120*B*a**3*b**7*e**10/19 + 450*B*a**2*b**8*d*e**9/19 + 4 50*B*a*b**9*d**2*e**8/19 + 120*B*b**10*d**3*e**7/19) + x**18*(20*A*a**3*b* *7*e**10/3 + 25*A*a**2*b**8*d*e**9 + 25*A*a*b**9*d**2*e**8 + 20*A*b**10*d* *3*e**7/3 + 35*B*a**4*b**6*e**10/3 + 200*B*a**3*b**7*d*e**9/3 + 225*B*a**2 *b**8*d**2*e**8/2 + 200*B*a*b**9*d**3*e**7/3 + 35*B*b**10*d**4*e**6/3) + x **17*(210*A*a**4*b**6*e**10/17 + 1200*A*a**3*b**7*d*e**9/17 + 2025*A*a**2* b**8*d**2*e**8/17 + 1200*A*a*b**9*d**3*e**7/17 + 210*A*b**10*d**4*e**6/17 + 252*B*a**5*b**5*e**10/17 + 2100*B*a**4*b**6*d*e**9/17 + 5400*B*a**3*b**7 *d**2*e**8/17 + 5400*B*a**2*b**8*d**3*e**7/17 + 2100*B*a*b**9*d**4*e**6/17 + 252*B*b**10*d**5*e**5/17) + x**16*(63*A*a**5*b**5*e**10/4 + 525*A*a**4* b**6*d*e**9/4 + 675*A*a**3*b**7*d**2*e**8/2 + 675*A*a**2*b**8*d**3*e**7/2 + 525*A*a*b**9*d**4*e**6/4 + 63*A*b**10*d**5*e**5/4 + 105*B*a**6*b**4*e**1 0/8 + 315*B*a**5*b**5*d*e**9/2 + 4725*B*a**4*b**6*d**2*e**8/8 + 900*B*a**3 *b**7*d**3*e**7 + 4725*B*a**2*b**8*d**4*e**6/8 + 315*B*a*b**9*d**5*e**5/2 + 105*B*b**10*d**6*e**4/8) + x**15*(14*A*a**6*b**4*e**10 + 168*A*a**5*b**5 *d*e**9 + 630*A*a**4*b**6*d**2*e**8 + 960*A*a**3*b**7*d**3*e**7 + 630*A...
Leaf count of result is larger than twice the leaf count of optimal. 3048 vs. \(2 (438) = 876\).
Time = 0.06 (sec) , antiderivative size = 3048, normalized size of antiderivative = 6.63 \[ \int (a+b x)^{10} (A+B x) (d+e x)^{10} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^10*(B*x+A)*(e*x+d)^10,x, algorithm="maxima")
Output:
1/22*B*b^10*e^10*x^22 + A*a^10*d^10*x + 1/21*(10*B*b^10*d*e^9 + (10*B*a*b^ 9 + A*b^10)*e^10)*x^21 + 1/4*(9*B*b^10*d^2*e^8 + 2*(10*B*a*b^9 + A*b^10)*d *e^9 + (9*B*a^2*b^8 + 2*A*a*b^9)*e^10)*x^20 + 5/19*(24*B*b^10*d^3*e^7 + 9* (10*B*a*b^9 + A*b^10)*d^2*e^8 + 10*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^9 + 3*(8* B*a^3*b^7 + 3*A*a^2*b^8)*e^10)*x^19 + 5/6*(14*B*b^10*d^4*e^6 + 8*(10*B*a*b ^9 + A*b^10)*d^3*e^7 + 15*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^8 + 10*(8*B*a^3* b^7 + 3*A*a^2*b^8)*d*e^9 + 2*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^10)*x^18 + 3/17 *(84*B*b^10*d^5*e^5 + 70*(10*B*a*b^9 + A*b^10)*d^4*e^6 + 200*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^7 + 225*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^8 + 100*(7*B* a^4*b^6 + 4*A*a^3*b^7)*d*e^9 + 14*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^10)*x^17 + 3/8*(35*B*b^10*d^6*e^4 + 42*(10*B*a*b^9 + A*b^10)*d^5*e^5 + 175*(9*B*a^2* b^8 + 2*A*a*b^9)*d^4*e^6 + 300*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^7 + 225*( 7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^8 + 70*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^9 + 7*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^10)*x^16 + 2*(4*B*b^10*d^7*e^3 + 7*(10*B *a*b^9 + A*b^10)*d^6*e^4 + 42*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^5 + 105*(8*B *a^3*b^7 + 3*A*a^2*b^8)*d^4*e^6 + 120*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^7 + 63*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^2*e^8 + 14*(5*B*a^6*b^4 + 6*A*a^5*b^5)* d*e^9 + (4*B*a^7*b^3 + 7*A*a^6*b^4)*e^10)*x^15 + 15/14*(3*B*b^10*d^8*e^2 + 8*(10*B*a*b^9 + A*b^10)*d^7*e^3 + 70*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^4 + 252*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^5 + 420*(7*B*a^4*b^6 + 4*A*a^3*b^...
Leaf count of result is larger than twice the leaf count of optimal. 3912 vs. \(2 (438) = 876\).
Time = 0.13 (sec) , antiderivative size = 3912, normalized size of antiderivative = 8.50 \[ \int (a+b x)^{10} (A+B x) (d+e x)^{10} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^10*(B*x+A)*(e*x+d)^10,x, algorithm="giac")
Output:
1/22*B*b^10*e^10*x^22 + 10/21*B*b^10*d*e^9*x^21 + 10/21*B*a*b^9*e^10*x^21 + 1/21*A*b^10*e^10*x^21 + 9/4*B*b^10*d^2*e^8*x^20 + 5*B*a*b^9*d*e^9*x^20 + 1/2*A*b^10*d*e^9*x^20 + 9/4*B*a^2*b^8*e^10*x^20 + 1/2*A*a*b^9*e^10*x^20 + 120/19*B*b^10*d^3*e^7*x^19 + 450/19*B*a*b^9*d^2*e^8*x^19 + 45/19*A*b^10*d ^2*e^8*x^19 + 450/19*B*a^2*b^8*d*e^9*x^19 + 100/19*A*a*b^9*d*e^9*x^19 + 12 0/19*B*a^3*b^7*e^10*x^19 + 45/19*A*a^2*b^8*e^10*x^19 + 35/3*B*b^10*d^4*e^6 *x^18 + 200/3*B*a*b^9*d^3*e^7*x^18 + 20/3*A*b^10*d^3*e^7*x^18 + 225/2*B*a^ 2*b^8*d^2*e^8*x^18 + 25*A*a*b^9*d^2*e^8*x^18 + 200/3*B*a^3*b^7*d*e^9*x^18 + 25*A*a^2*b^8*d*e^9*x^18 + 35/3*B*a^4*b^6*e^10*x^18 + 20/3*A*a^3*b^7*e^10 *x^18 + 252/17*B*b^10*d^5*e^5*x^17 + 2100/17*B*a*b^9*d^4*e^6*x^17 + 210/17 *A*b^10*d^4*e^6*x^17 + 5400/17*B*a^2*b^8*d^3*e^7*x^17 + 1200/17*A*a*b^9*d^ 3*e^7*x^17 + 5400/17*B*a^3*b^7*d^2*e^8*x^17 + 2025/17*A*a^2*b^8*d^2*e^8*x^ 17 + 2100/17*B*a^4*b^6*d*e^9*x^17 + 1200/17*A*a^3*b^7*d*e^9*x^17 + 252/17* B*a^5*b^5*e^10*x^17 + 210/17*A*a^4*b^6*e^10*x^17 + 105/8*B*b^10*d^6*e^4*x^ 16 + 315/2*B*a*b^9*d^5*e^5*x^16 + 63/4*A*b^10*d^5*e^5*x^16 + 4725/8*B*a^2* b^8*d^4*e^6*x^16 + 525/4*A*a*b^9*d^4*e^6*x^16 + 900*B*a^3*b^7*d^3*e^7*x^16 + 675/2*A*a^2*b^8*d^3*e^7*x^16 + 4725/8*B*a^4*b^6*d^2*e^8*x^16 + 675/2*A* a^3*b^7*d^2*e^8*x^16 + 315/2*B*a^5*b^5*d*e^9*x^16 + 525/4*A*a^4*b^6*d*e^9* x^16 + 105/8*B*a^6*b^4*e^10*x^16 + 63/4*A*a^5*b^5*e^10*x^16 + 8*B*b^10*d^7 *e^3*x^15 + 140*B*a*b^9*d^6*e^4*x^15 + 14*A*b^10*d^6*e^4*x^15 + 756*B*a...
Time = 1.64 (sec) , antiderivative size = 3262, normalized size of antiderivative = 7.09 \[ \int (a+b x)^{10} (A+B x) (d+e x)^{10} \, dx=\text {Too large to display} \] Input:
int((A + B*x)*(a + b*x)^10*(d + e*x)^10,x)
Output:
x^11*((A*a^10*e^10)/11 + (A*b^10*d^10)/11 + (10*B*a*b^9*d^10)/11 + (10*B*a ^10*d*e^9)/11 + (450*B*a^2*b^8*d^9*e)/11 + (450*B*a^9*b*d^2*e^8)/11 + (202 5*A*a^2*b^8*d^8*e^2)/11 + (14400*A*a^3*b^7*d^7*e^3)/11 + (44100*A*a^4*b^6* d^6*e^4)/11 + (63504*A*a^5*b^5*d^5*e^5)/11 + (44100*A*a^6*b^4*d^4*e^6)/11 + (14400*A*a^7*b^3*d^3*e^7)/11 + (2025*A*a^8*b^2*d^2*e^8)/11 + (5400*B*a^3 *b^7*d^8*e^2)/11 + (25200*B*a^4*b^6*d^7*e^3)/11 + (52920*B*a^5*b^5*d^6*e^4 )/11 + (52920*B*a^6*b^4*d^5*e^5)/11 + (25200*B*a^7*b^3*d^4*e^6)/11 + (5400 *B*a^8*b^2*d^3*e^7)/11 + (100*A*a*b^9*d^9*e)/11 + (100*A*a^9*b*d*e^9)/11) + x^5*(42*A*a^6*b^4*d^10 + 24*B*a^7*b^3*d^10 + 42*A*a^10*d^6*e^4 + 24*B*a^ 10*d^7*e^3 + 240*A*a^7*b^3*d^9*e + 240*A*a^9*b*d^7*e^3 + 90*B*a^8*b^2*d^9* e + 90*B*a^9*b*d^8*e^2 + 405*A*a^8*b^2*d^8*e^2) + x^8*(15*A*a^3*b^7*d^10 + (105*B*a^4*b^6*d^10)/4 + 15*A*a^10*d^3*e^7 + (105*B*a^10*d^4*e^6)/4 + (52 5*A*a^4*b^6*d^9*e)/2 + (525*A*a^9*b*d^4*e^6)/2 + 315*B*a^5*b^5*d^9*e + 315 *B*a^9*b*d^5*e^5 + (2835*A*a^5*b^5*d^8*e^2)/2 + 3150*A*a^6*b^4*d^7*e^3 + 3 150*A*a^7*b^3*d^6*e^4 + (2835*A*a^8*b^2*d^5*e^5)/2 + (4725*B*a^6*b^4*d^8*e ^2)/4 + 1800*B*a^7*b^3*d^7*e^3 + (4725*B*a^8*b^2*d^6*e^4)/4) + x^12*((B*a^ 10*e^10)/12 + (B*b^10*d^10)/12 + (5*A*a^9*b*e^10)/6 + (5*A*b^10*d^9*e)/6 + (75*A*a*b^9*d^8*e^2)/2 + (75*A*a^8*b^2*d*e^9)/2 + 450*A*a^2*b^8*d^7*e^3 + 2100*A*a^3*b^7*d^6*e^4 + 4410*A*a^4*b^6*d^5*e^5 + 4410*A*a^5*b^5*d^4*e^6 + 2100*A*a^6*b^4*d^3*e^7 + 450*A*a^7*b^3*d^2*e^8 + (675*B*a^2*b^8*d^8*e...
Time = 0.17 (sec) , antiderivative size = 2011, normalized size of antiderivative = 4.37 \[ \int (a+b x)^{10} (A+B x) (d+e x)^{10} \, dx =\text {Too large to display} \] Input:
int((b*x+a)^10*(B*x+A)*(e*x+d)^10,x)
Output:
(x*(7759752*a**11*d**10 + 38798760*a**11*d**9*e*x + 116396280*a**11*d**8*e **2*x**2 + 232792560*a**11*d**7*e**3*x**3 + 325909584*a**11*d**6*e**4*x**4 + 325909584*a**11*d**5*e**5*x**5 + 232792560*a**11*d**4*e**6*x**6 + 11639 6280*a**11*d**3*e**7*x**7 + 38798760*a**11*d**2*e**8*x**8 + 7759752*a**11* d*e**9*x**9 + 705432*a**11*e**10*x**10 + 42678636*a**10*b*d**10*x + 284524 240*a**10*b*d**9*e*x**2 + 960269310*a**10*b*d**8*e**2*x**3 + 2048574528*a* *10*b*d**7*e**3*x**4 + 2987504520*a**10*b*d**6*e**4*x**5 + 3072861792*a**1 0*b*d**5*e**5*x**6 + 2240628390*a**10*b*d**4*e**6*x**7 + 1138096960*a**10* b*d**3*e**7*x**8 + 384107724*a**10*b*d**2*e**8*x**9 + 77597520*a**10*b*d*e **9*x**10 + 7113106*a**10*b*e**10*x**11 + 142262120*a**9*b**2*d**10*x**2 + 1066965900*a**9*b**2*d**9*e*x**3 + 3841077240*a**9*b**2*d**8*e**2*x**4 + 8535727200*a**9*b**2*d**7*e**3*x**5 + 12803590800*a**9*b**2*d**6*e**4*x**6 + 13443770340*a**9*b**2*d**5*e**5*x**7 + 9958348400*a**9*b**2*d**4*e**6*x **8 + 5121436320*a**9*b**2*d**3*e**7*x**9 + 1745944200*a**9*b**2*d**2*e**8 *x**10 + 355655300*a**9*b**2*d*e**9*x**11 + 32829720*a**9*b**2*e**10*x**12 + 320089770*a**8*b**3*d**10*x**3 + 2560718160*a**8*b**3*d**9*e*x**4 + 960 2693100*a**8*b**3*d**8*e**2*x**5 + 21949012800*a**8*b**3*d**7*e**3*x**6 + 33609425850*a**8*b**3*d**6*e**4*x**7 + 35850054240*a**8*b**3*d**5*e**5*x** 8 + 26887540680*a**8*b**3*d**4*e**6*x**9 + 13967553600*a**8*b**3*d**3*e**7 *x**10 + 4801346550*a**8*b**3*d**2*e**8*x**11 + 984891600*a**8*b**3*d*e...