Integrand size = 15, antiderivative size = 275 \[ \int (a+b x)^{12} (c+d x)^{10} \, dx=\frac {(b c-a d)^{10} (a+b x)^{13}}{13 b^{11}}+\frac {5 d (b c-a d)^9 (a+b x)^{14}}{7 b^{11}}+\frac {3 d^2 (b c-a d)^8 (a+b x)^{15}}{b^{11}}+\frac {15 d^3 (b c-a d)^7 (a+b x)^{16}}{2 b^{11}}+\frac {210 d^4 (b c-a d)^6 (a+b x)^{17}}{17 b^{11}}+\frac {14 d^5 (b c-a d)^5 (a+b x)^{18}}{b^{11}}+\frac {210 d^6 (b c-a d)^4 (a+b x)^{19}}{19 b^{11}}+\frac {6 d^7 (b c-a d)^3 (a+b x)^{20}}{b^{11}}+\frac {15 d^8 (b c-a d)^2 (a+b x)^{21}}{7 b^{11}}+\frac {5 d^9 (b c-a d) (a+b x)^{22}}{11 b^{11}}+\frac {d^{10} (a+b x)^{23}}{23 b^{11}} \]
1/13*(-a*d+b*c)^10*(b*x+a)^13/b^11+5/7*d*(-a*d+b*c)^9*(b*x+a)^14/b^11+3*d^ 2*(-a*d+b*c)^8*(b*x+a)^15/b^11+15/2*d^3*(-a*d+b*c)^7*(b*x+a)^16/b^11+210/1 7*d^4*(-a*d+b*c)^6*(b*x+a)^17/b^11+14*d^5*(-a*d+b*c)^5*(b*x+a)^18/b^11+210 /19*d^6*(-a*d+b*c)^4*(b*x+a)^19/b^11+6*d^7*(-a*d+b*c)^3*(b*x+a)^20/b^11+15 /7*d^8*(-a*d+b*c)^2*(b*x+a)^21/b^11+5/11*d^9*(-a*d+b*c)*(b*x+a)^22/b^11+1/ 23*d^10*(b*x+a)^23/b^11
Leaf count is larger than twice the leaf count of optimal. \(1817\) vs. \(2(275)=550\).
Time = 0.17 (sec) , antiderivative size = 1817, normalized size of antiderivative = 6.61 \[ \int (a+b x)^{12} (c+d x)^{10} \, dx =\text {Too large to display} \]
a^12*c^10*x + a^11*c^9*(6*b*c + 5*a*d)*x^2 + a^10*c^8*(22*b^2*c^2 + 40*a*b *c*d + 15*a^2*d^2)*x^3 + 5*a^9*c^7*(11*b^3*c^3 + 33*a*b^2*c^2*d + 27*a^2*b *c*d^2 + 6*a^3*d^3)*x^4 + a^8*c^6*(99*b^4*c^4 + 440*a*b^3*c^3*d + 594*a^2* b^2*c^2*d^2 + 288*a^3*b*c*d^3 + 42*a^4*d^4)*x^5 + 3*a^7*c^5*(44*b^5*c^5 + 275*a*b^4*c^4*d + 550*a^2*b^3*c^3*d^2 + 440*a^3*b^2*c^2*d^3 + 140*a^4*b*c* d^4 + 14*a^5*d^5)*x^6 + (3*a^6*c^4*(308*b^6*c^6 + 2640*a*b^5*c^5*d + 7425* a^2*b^4*c^4*d^2 + 8800*a^3*b^3*c^3*d^3 + 4620*a^4*b^2*c^2*d^4 + 1008*a^5*b *c*d^5 + 70*a^6*d^6)*x^7)/7 + 3*a^5*c^3*(33*b^7*c^7 + 385*a*b^6*c^6*d + 14 85*a^2*b^5*c^5*d^2 + 2475*a^3*b^4*c^4*d^3 + 1925*a^4*b^3*c^3*d^4 + 693*a^5 *b^2*c^2*d^5 + 105*a^6*b*c*d^6 + 5*a^7*d^7)*x^8 + 5*a^4*c^2*(11*b^8*c^8 + 176*a*b^7*c^7*d + 924*a^2*b^6*c^6*d^2 + 2112*a^3*b^5*c^5*d^3 + 2310*a^4*b^ 4*c^4*d^4 + 1232*a^5*b^3*c^3*d^5 + 308*a^6*b^2*c^2*d^6 + 32*a^7*b*c*d^7 + a^8*d^8)*x^9 + a^3*c*(22*b^9*c^9 + 495*a*b^8*c^8*d + 3564*a^2*b^7*c^7*d^2 + 11088*a^3*b^6*c^6*d^3 + 16632*a^4*b^5*c^5*d^4 + 12474*a^5*b^4*c^4*d^5 + 4620*a^6*b^3*c^3*d^6 + 792*a^7*b^2*c^2*d^7 + 54*a^8*b*c*d^8 + a^9*d^9)*x^1 0 + (a^2*(66*b^10*c^10 + 2200*a*b^9*c^9*d + 22275*a^2*b^8*c^8*d^2 + 95040* a^3*b^7*c^7*d^3 + 194040*a^4*b^6*c^6*d^4 + 199584*a^5*b^5*c^5*d^5 + 103950 *a^6*b^4*c^4*d^6 + 26400*a^7*b^3*c^3*d^7 + 2970*a^8*b^2*c^2*d^8 + 120*a^9* b*c*d^9 + a^10*d^10)*x^11)/11 + a*b*(b^10*c^10 + 55*a*b^9*c^9*d + 825*a^2* b^8*c^8*d^2 + 4950*a^3*b^7*c^7*d^3 + 13860*a^4*b^6*c^6*d^4 + 19404*a^5*...
Time = 1.35 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {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.
| \(\displaystyle \int (a+b x)^{12} (c+d x)^{10} \, dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (\frac {10 d^9 (a+b x)^{21} (b c-a d)}{b^{10}}+\frac {45 d^8 (a+b x)^{20} (b c-a d)^2}{b^{10}}+\frac {120 d^7 (a+b x)^{19} (b c-a d)^3}{b^{10}}+\frac {210 d^6 (a+b x)^{18} (b c-a d)^4}{b^{10}}+\frac {252 d^5 (a+b x)^{17} (b c-a d)^5}{b^{10}}+\frac {210 d^4 (a+b x)^{16} (b c-a d)^6}{b^{10}}+\frac {120 d^3 (a+b x)^{15} (b c-a d)^7}{b^{10}}+\frac {45 d^2 (a+b x)^{14} (b c-a d)^8}{b^{10}}+\frac {10 d (a+b x)^{13} (b c-a d)^9}{b^{10}}+\frac {(a+b x)^{12} (b c-a d)^{10}}{b^{10}}+\frac {d^{10} (a+b x)^{22}}{b^{10}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5 d^9 (a+b x)^{22} (b c-a d)}{11 b^{11}}+\frac {15 d^8 (a+b x)^{21} (b c-a d)^2}{7 b^{11}}+\frac {6 d^7 (a+b x)^{20} (b c-a d)^3}{b^{11}}+\frac {210 d^6 (a+b x)^{19} (b c-a d)^4}{19 b^{11}}+\frac {14 d^5 (a+b x)^{18} (b c-a d)^5}{b^{11}}+\frac {210 d^4 (a+b x)^{17} (b c-a d)^6}{17 b^{11}}+\frac {15 d^3 (a+b x)^{16} (b c-a d)^7}{2 b^{11}}+\frac {3 d^2 (a+b x)^{15} (b c-a d)^8}{b^{11}}+\frac {5 d (a+b x)^{14} (b c-a d)^9}{7 b^{11}}+\frac {(a+b x)^{13} (b c-a d)^{10}}{13 b^{11}}+\frac {d^{10} (a+b x)^{23}}{23 b^{11}}\) |
((b*c - a*d)^10*(a + b*x)^13)/(13*b^11) + (5*d*(b*c - a*d)^9*(a + b*x)^14) /(7*b^11) + (3*d^2*(b*c - a*d)^8*(a + b*x)^15)/b^11 + (15*d^3*(b*c - a*d)^ 7*(a + b*x)^16)/(2*b^11) + (210*d^4*(b*c - a*d)^6*(a + b*x)^17)/(17*b^11) + (14*d^5*(b*c - a*d)^5*(a + b*x)^18)/b^11 + (210*d^6*(b*c - a*d)^4*(a + b *x)^19)/(19*b^11) + (6*d^7*(b*c - a*d)^3*(a + b*x)^20)/b^11 + (15*d^8*(b*c - a*d)^2*(a + b*x)^21)/(7*b^11) + (5*d^9*(b*c - a*d)*(a + b*x)^22)/(11*b^ 11) + (d^10*(a + b*x)^23)/(23*b^11)
3.13.99.3.1 Defintions of rubi rules used
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1868\) vs. \(2(259)=518\).
Time = 0.21 (sec) , antiderivative size = 1869, normalized size of antiderivative = 6.80
| method | result | size |
| norman | \(\text {Expression too large to display}\) | \(1869\) |
| default | \(\text {Expression too large to display}\) | \(1891\) |
| gosper | \(\text {Expression too large to display}\) | \(2187\) |
| risch | \(\text {Expression too large to display}\) | \(2187\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2187\) |
a^12*c^10*x+(5*a^12*c^9*d+6*a^11*b*c^10)*x^2+(15*a^12*c^8*d^2+40*a^11*b*c^ 9*d+22*a^10*b^2*c^10)*x^3+(30*a^12*c^7*d^3+135*a^11*b*c^8*d^2+165*a^10*b^2 *c^9*d+55*a^9*b^3*c^10)*x^4+(42*a^12*c^6*d^4+288*a^11*b*c^7*d^3+594*a^10*b ^2*c^8*d^2+440*a^9*b^3*c^9*d+99*a^8*b^4*c^10)*x^5+(42*a^12*c^5*d^5+420*a^1 1*b*c^6*d^4+1320*a^10*b^2*c^7*d^3+1650*a^9*b^3*c^8*d^2+825*a^8*b^4*c^9*d+1 32*a^7*b^5*c^10)*x^6+(30*a^12*c^4*d^6+432*a^11*b*c^5*d^5+1980*a^10*b^2*c^6 *d^4+26400/7*a^9*b^3*c^7*d^3+22275/7*a^8*b^4*c^8*d^2+7920/7*a^7*b^5*c^9*d+ 132*a^6*b^6*c^10)*x^7+(15*a^12*c^3*d^7+315*a^11*b*c^4*d^6+2079*a^10*b^2*c^ 5*d^5+5775*a^9*b^3*c^6*d^4+7425*a^8*b^4*c^7*d^3+4455*a^7*b^5*c^8*d^2+1155* a^6*b^6*c^9*d+99*a^5*b^7*c^10)*x^8+(5*a^12*c^2*d^8+160*a^11*b*c^3*d^7+1540 *a^10*b^2*c^4*d^6+6160*a^9*b^3*c^5*d^5+11550*a^8*b^4*c^6*d^4+10560*a^7*b^5 *c^7*d^3+4620*a^6*b^6*c^8*d^2+880*a^5*b^7*c^9*d+55*a^4*b^8*c^10)*x^9+(a^12 *c*d^9+54*a^11*b*c^2*d^8+792*a^10*b^2*c^3*d^7+4620*a^9*b^3*c^4*d^6+12474*a ^8*b^4*c^5*d^5+16632*a^7*b^5*c^6*d^4+11088*a^6*b^6*c^7*d^3+3564*a^5*b^7*c^ 8*d^2+495*a^4*b^8*c^9*d+22*a^3*b^9*c^10)*x^10+(1/11*a^12*d^10+120/11*a^11* b*c*d^9+270*a^10*b^2*c^2*d^8+2400*a^9*b^3*c^3*d^7+9450*a^8*b^4*c^4*d^6+181 44*a^7*b^5*c^5*d^5+17640*a^6*b^6*c^6*d^4+8640*a^5*b^7*c^7*d^3+2025*a^4*b^8 *c^8*d^2+200*a^3*b^9*c^9*d+6*a^2*b^10*c^10)*x^11+(a^11*b*d^10+55*a^10*b^2* c*d^9+825*a^9*b^3*c^2*d^8+4950*a^8*b^4*c^3*d^7+13860*a^7*b^5*c^4*d^6+19404 *a^6*b^6*c^5*d^5+13860*a^5*b^7*c^6*d^4+4950*a^4*b^8*c^7*d^3+825*a^3*b^9...
Leaf count of result is larger than twice the leaf count of optimal. 1877 vs. \(2 (259) = 518\).
Time = 0.24 (sec) , antiderivative size = 1877, normalized size of antiderivative = 6.83 \[ \int (a+b x)^{12} (c+d x)^{10} \, dx=\text {Too large to display} \]
1/23*b^12*d^10*x^23 + a^12*c^10*x + 1/11*(5*b^12*c*d^9 + 6*a*b^11*d^10)*x^ 22 + 1/7*(15*b^12*c^2*d^8 + 40*a*b^11*c*d^9 + 22*a^2*b^10*d^10)*x^21 + (6* b^12*c^3*d^7 + 27*a*b^11*c^2*d^8 + 33*a^2*b^10*c*d^9 + 11*a^3*b^9*d^10)*x^ 20 + 5/19*(42*b^12*c^4*d^6 + 288*a*b^11*c^3*d^7 + 594*a^2*b^10*c^2*d^8 + 4 40*a^3*b^9*c*d^9 + 99*a^4*b^8*d^10)*x^19 + (14*b^12*c^5*d^5 + 140*a*b^11*c ^4*d^6 + 440*a^2*b^10*c^3*d^7 + 550*a^3*b^9*c^2*d^8 + 275*a^4*b^8*c*d^9 + 44*a^5*b^7*d^10)*x^18 + 3/17*(70*b^12*c^6*d^4 + 1008*a*b^11*c^5*d^5 + 4620 *a^2*b^10*c^4*d^6 + 8800*a^3*b^9*c^3*d^7 + 7425*a^4*b^8*c^2*d^8 + 2640*a^5 *b^7*c*d^9 + 308*a^6*b^6*d^10)*x^17 + 3/2*(5*b^12*c^7*d^3 + 105*a*b^11*c^6 *d^4 + 693*a^2*b^10*c^5*d^5 + 1925*a^3*b^9*c^4*d^6 + 2475*a^4*b^8*c^3*d^7 + 1485*a^5*b^7*c^2*d^8 + 385*a^6*b^6*c*d^9 + 33*a^7*b^5*d^10)*x^16 + 3*(b^ 12*c^8*d^2 + 32*a*b^11*c^7*d^3 + 308*a^2*b^10*c^6*d^4 + 1232*a^3*b^9*c^5*d ^5 + 2310*a^4*b^8*c^4*d^6 + 2112*a^5*b^7*c^3*d^7 + 924*a^6*b^6*c^2*d^8 + 1 76*a^7*b^5*c*d^9 + 11*a^8*b^4*d^10)*x^15 + 5/7*(b^12*c^9*d + 54*a*b^11*c^8 *d^2 + 792*a^2*b^10*c^7*d^3 + 4620*a^3*b^9*c^6*d^4 + 12474*a^4*b^8*c^5*d^5 + 16632*a^5*b^7*c^4*d^6 + 11088*a^6*b^6*c^3*d^7 + 3564*a^7*b^5*c^2*d^8 + 495*a^8*b^4*c*d^9 + 22*a^9*b^3*d^10)*x^14 + 1/13*(b^12*c^10 + 120*a*b^11*c ^9*d + 2970*a^2*b^10*c^8*d^2 + 26400*a^3*b^9*c^7*d^3 + 103950*a^4*b^8*c^6* d^4 + 199584*a^5*b^7*c^5*d^5 + 194040*a^6*b^6*c^4*d^6 + 95040*a^7*b^5*c^3* d^7 + 22275*a^8*b^4*c^2*d^8 + 2200*a^9*b^3*c*d^9 + 66*a^10*b^2*d^10)*x^...
Leaf count of result is larger than twice the leaf count of optimal. 2088 vs. \(2 (255) = 510\).
Time = 0.17 (sec) , antiderivative size = 2088, normalized size of antiderivative = 7.59 \[ \int (a+b x)^{12} (c+d x)^{10} \, dx=\text {Too large to display} \]
a**12*c**10*x + b**12*d**10*x**23/23 + x**22*(6*a*b**11*d**10/11 + 5*b**12 *c*d**9/11) + x**21*(22*a**2*b**10*d**10/7 + 40*a*b**11*c*d**9/7 + 15*b**1 2*c**2*d**8/7) + x**20*(11*a**3*b**9*d**10 + 33*a**2*b**10*c*d**9 + 27*a*b **11*c**2*d**8 + 6*b**12*c**3*d**7) + x**19*(495*a**4*b**8*d**10/19 + 2200 *a**3*b**9*c*d**9/19 + 2970*a**2*b**10*c**2*d**8/19 + 1440*a*b**11*c**3*d* *7/19 + 210*b**12*c**4*d**6/19) + x**18*(44*a**5*b**7*d**10 + 275*a**4*b** 8*c*d**9 + 550*a**3*b**9*c**2*d**8 + 440*a**2*b**10*c**3*d**7 + 140*a*b**1 1*c**4*d**6 + 14*b**12*c**5*d**5) + x**17*(924*a**6*b**6*d**10/17 + 7920*a **5*b**7*c*d**9/17 + 22275*a**4*b**8*c**2*d**8/17 + 26400*a**3*b**9*c**3*d **7/17 + 13860*a**2*b**10*c**4*d**6/17 + 3024*a*b**11*c**5*d**5/17 + 210*b **12*c**6*d**4/17) + x**16*(99*a**7*b**5*d**10/2 + 1155*a**6*b**6*c*d**9/2 + 4455*a**5*b**7*c**2*d**8/2 + 7425*a**4*b**8*c**3*d**7/2 + 5775*a**3*b** 9*c**4*d**6/2 + 2079*a**2*b**10*c**5*d**5/2 + 315*a*b**11*c**6*d**4/2 + 15 *b**12*c**7*d**3/2) + x**15*(33*a**8*b**4*d**10 + 528*a**7*b**5*c*d**9 + 2 772*a**6*b**6*c**2*d**8 + 6336*a**5*b**7*c**3*d**7 + 6930*a**4*b**8*c**4*d **6 + 3696*a**3*b**9*c**5*d**5 + 924*a**2*b**10*c**6*d**4 + 96*a*b**11*c** 7*d**3 + 3*b**12*c**8*d**2) + x**14*(110*a**9*b**3*d**10/7 + 2475*a**8*b** 4*c*d**9/7 + 17820*a**7*b**5*c**2*d**8/7 + 7920*a**6*b**6*c**3*d**7 + 1188 0*a**5*b**7*c**4*d**6 + 8910*a**4*b**8*c**5*d**5 + 3300*a**3*b**9*c**6*d** 4 + 3960*a**2*b**10*c**7*d**3/7 + 270*a*b**11*c**8*d**2/7 + 5*b**12*c**...
Leaf count of result is larger than twice the leaf count of optimal. 1877 vs. \(2 (259) = 518\).
Time = 0.22 (sec) , antiderivative size = 1877, normalized size of antiderivative = 6.83 \[ \int (a+b x)^{12} (c+d x)^{10} \, dx=\text {Too large to display} \]
1/23*b^12*d^10*x^23 + a^12*c^10*x + 1/11*(5*b^12*c*d^9 + 6*a*b^11*d^10)*x^ 22 + 1/7*(15*b^12*c^2*d^8 + 40*a*b^11*c*d^9 + 22*a^2*b^10*d^10)*x^21 + (6* b^12*c^3*d^7 + 27*a*b^11*c^2*d^8 + 33*a^2*b^10*c*d^9 + 11*a^3*b^9*d^10)*x^ 20 + 5/19*(42*b^12*c^4*d^6 + 288*a*b^11*c^3*d^7 + 594*a^2*b^10*c^2*d^8 + 4 40*a^3*b^9*c*d^9 + 99*a^4*b^8*d^10)*x^19 + (14*b^12*c^5*d^5 + 140*a*b^11*c ^4*d^6 + 440*a^2*b^10*c^3*d^7 + 550*a^3*b^9*c^2*d^8 + 275*a^4*b^8*c*d^9 + 44*a^5*b^7*d^10)*x^18 + 3/17*(70*b^12*c^6*d^4 + 1008*a*b^11*c^5*d^5 + 4620 *a^2*b^10*c^4*d^6 + 8800*a^3*b^9*c^3*d^7 + 7425*a^4*b^8*c^2*d^8 + 2640*a^5 *b^7*c*d^9 + 308*a^6*b^6*d^10)*x^17 + 3/2*(5*b^12*c^7*d^3 + 105*a*b^11*c^6 *d^4 + 693*a^2*b^10*c^5*d^5 + 1925*a^3*b^9*c^4*d^6 + 2475*a^4*b^8*c^3*d^7 + 1485*a^5*b^7*c^2*d^8 + 385*a^6*b^6*c*d^9 + 33*a^7*b^5*d^10)*x^16 + 3*(b^ 12*c^8*d^2 + 32*a*b^11*c^7*d^3 + 308*a^2*b^10*c^6*d^4 + 1232*a^3*b^9*c^5*d ^5 + 2310*a^4*b^8*c^4*d^6 + 2112*a^5*b^7*c^3*d^7 + 924*a^6*b^6*c^2*d^8 + 1 76*a^7*b^5*c*d^9 + 11*a^8*b^4*d^10)*x^15 + 5/7*(b^12*c^9*d + 54*a*b^11*c^8 *d^2 + 792*a^2*b^10*c^7*d^3 + 4620*a^3*b^9*c^6*d^4 + 12474*a^4*b^8*c^5*d^5 + 16632*a^5*b^7*c^4*d^6 + 11088*a^6*b^6*c^3*d^7 + 3564*a^7*b^5*c^2*d^8 + 495*a^8*b^4*c*d^9 + 22*a^9*b^3*d^10)*x^14 + 1/13*(b^12*c^10 + 120*a*b^11*c ^9*d + 2970*a^2*b^10*c^8*d^2 + 26400*a^3*b^9*c^7*d^3 + 103950*a^4*b^8*c^6* d^4 + 199584*a^5*b^7*c^5*d^5 + 194040*a^6*b^6*c^4*d^6 + 95040*a^7*b^5*c^3* d^7 + 22275*a^8*b^4*c^2*d^8 + 2200*a^9*b^3*c*d^9 + 66*a^10*b^2*d^10)*x^...
Leaf count of result is larger than twice the leaf count of optimal. 2186 vs. \(2 (259) = 518\).
Time = 0.30 (sec) , antiderivative size = 2186, normalized size of antiderivative = 7.95 \[ \int (a+b x)^{12} (c+d x)^{10} \, dx=\text {Too large to display} \]
1/23*b^12*d^10*x^23 + 5/11*b^12*c*d^9*x^22 + 6/11*a*b^11*d^10*x^22 + 15/7* b^12*c^2*d^8*x^21 + 40/7*a*b^11*c*d^9*x^21 + 22/7*a^2*b^10*d^10*x^21 + 6*b ^12*c^3*d^7*x^20 + 27*a*b^11*c^2*d^8*x^20 + 33*a^2*b^10*c*d^9*x^20 + 11*a^ 3*b^9*d^10*x^20 + 210/19*b^12*c^4*d^6*x^19 + 1440/19*a*b^11*c^3*d^7*x^19 + 2970/19*a^2*b^10*c^2*d^8*x^19 + 2200/19*a^3*b^9*c*d^9*x^19 + 495/19*a^4*b ^8*d^10*x^19 + 14*b^12*c^5*d^5*x^18 + 140*a*b^11*c^4*d^6*x^18 + 440*a^2*b^ 10*c^3*d^7*x^18 + 550*a^3*b^9*c^2*d^8*x^18 + 275*a^4*b^8*c*d^9*x^18 + 44*a ^5*b^7*d^10*x^18 + 210/17*b^12*c^6*d^4*x^17 + 3024/17*a*b^11*c^5*d^5*x^17 + 13860/17*a^2*b^10*c^4*d^6*x^17 + 26400/17*a^3*b^9*c^3*d^7*x^17 + 22275/1 7*a^4*b^8*c^2*d^8*x^17 + 7920/17*a^5*b^7*c*d^9*x^17 + 924/17*a^6*b^6*d^10* x^17 + 15/2*b^12*c^7*d^3*x^16 + 315/2*a*b^11*c^6*d^4*x^16 + 2079/2*a^2*b^1 0*c^5*d^5*x^16 + 5775/2*a^3*b^9*c^4*d^6*x^16 + 7425/2*a^4*b^8*c^3*d^7*x^16 + 4455/2*a^5*b^7*c^2*d^8*x^16 + 1155/2*a^6*b^6*c*d^9*x^16 + 99/2*a^7*b^5* d^10*x^16 + 3*b^12*c^8*d^2*x^15 + 96*a*b^11*c^7*d^3*x^15 + 924*a^2*b^10*c^ 6*d^4*x^15 + 3696*a^3*b^9*c^5*d^5*x^15 + 6930*a^4*b^8*c^4*d^6*x^15 + 6336* a^5*b^7*c^3*d^7*x^15 + 2772*a^6*b^6*c^2*d^8*x^15 + 528*a^7*b^5*c*d^9*x^15 + 33*a^8*b^4*d^10*x^15 + 5/7*b^12*c^9*d*x^14 + 270/7*a*b^11*c^8*d^2*x^14 + 3960/7*a^2*b^10*c^7*d^3*x^14 + 3300*a^3*b^9*c^6*d^4*x^14 + 8910*a^4*b^8*c ^5*d^5*x^14 + 11880*a^5*b^7*c^4*d^6*x^14 + 7920*a^6*b^6*c^3*d^7*x^14 + 178 20/7*a^7*b^5*c^2*d^8*x^14 + 2475/7*a^8*b^4*c*d^9*x^14 + 110/7*a^9*b^3*d...
Time = 1.30 (sec) , antiderivative size = 1847, normalized size of antiderivative = 6.72 \[ \int (a+b x)^{12} (c+d x)^{10} \, dx=\text {Too large to display} \]
x^12*(a*b^11*c^10 + a^11*b*d^10 + 55*a^2*b^10*c^9*d + 55*a^10*b^2*c*d^9 + 825*a^3*b^9*c^8*d^2 + 4950*a^4*b^8*c^7*d^3 + 13860*a^5*b^7*c^6*d^4 + 19404 *a^6*b^6*c^5*d^5 + 13860*a^7*b^5*c^4*d^6 + 4950*a^8*b^4*c^3*d^7 + 825*a^9* b^3*c^2*d^8) + x^7*(132*a^6*b^6*c^10 + 30*a^12*c^4*d^6 + (7920*a^7*b^5*c^9 *d)/7 + 432*a^11*b*c^5*d^5 + (22275*a^8*b^4*c^8*d^2)/7 + (26400*a^9*b^3*c^ 7*d^3)/7 + 1980*a^10*b^2*c^6*d^4) + x^17*((924*a^6*b^6*d^10)/17 + (210*b^1 2*c^6*d^4)/17 + (3024*a*b^11*c^5*d^5)/17 + (7920*a^5*b^7*c*d^9)/17 + (1386 0*a^2*b^10*c^4*d^6)/17 + (26400*a^3*b^9*c^3*d^7)/17 + (22275*a^4*b^8*c^2*d ^8)/17) + x^5*(99*a^8*b^4*c^10 + 42*a^12*c^6*d^4 + 440*a^9*b^3*c^9*d + 288 *a^11*b*c^7*d^3 + 594*a^10*b^2*c^8*d^2) + x^19*((495*a^4*b^8*d^10)/19 + (2 10*b^12*c^4*d^6)/19 + (1440*a*b^11*c^3*d^7)/19 + (2200*a^3*b^9*c*d^9)/19 + (2970*a^2*b^10*c^2*d^8)/19) + x^8*(99*a^5*b^7*c^10 + 15*a^12*c^3*d^7 + 11 55*a^6*b^6*c^9*d + 315*a^11*b*c^4*d^6 + 4455*a^7*b^5*c^8*d^2 + 7425*a^8*b^ 4*c^7*d^3 + 5775*a^9*b^3*c^6*d^4 + 2079*a^10*b^2*c^5*d^5) + x^16*((99*a^7* b^5*d^10)/2 + (15*b^12*c^7*d^3)/2 + (315*a*b^11*c^6*d^4)/2 + (1155*a^6*b^6 *c*d^9)/2 + (2079*a^2*b^10*c^5*d^5)/2 + (5775*a^3*b^9*c^4*d^6)/2 + (7425*a ^4*b^8*c^3*d^7)/2 + (4455*a^5*b^7*c^2*d^8)/2) + x^11*((a^12*d^10)/11 + 6*a ^2*b^10*c^10 + 200*a^3*b^9*c^9*d + 2025*a^4*b^8*c^8*d^2 + 8640*a^5*b^7*c^7 *d^3 + 17640*a^6*b^6*c^6*d^4 + 18144*a^7*b^5*c^5*d^5 + 9450*a^8*b^4*c^4*d^ 6 + 2400*a^9*b^3*c^3*d^7 + 270*a^10*b^2*c^2*d^8 + (120*a^11*b*c*d^9)/11...
Time = 0.00 (sec) , antiderivative size = 2188, normalized size of antiderivative = 7.96 \[ \int (a+b x)^{12} (c+d x)^{10} \, dx =\text {Too large to display} \]
int(a**12*c**10 + 10*a**12*c**9*d*x + 45*a**12*c**8*d**2*x**2 + 120*a**12* c**7*d**3*x**3 + 210*a**12*c**6*d**4*x**4 + 252*a**12*c**5*d**5*x**5 + 210 *a**12*c**4*d**6*x**6 + 120*a**12*c**3*d**7*x**7 + 45*a**12*c**2*d**8*x**8 + 10*a**12*c*d**9*x**9 + a**12*d**10*x**10 + 12*a**11*b*c**10*x + 120*a** 11*b*c**9*d*x**2 + 540*a**11*b*c**8*d**2*x**3 + 1440*a**11*b*c**7*d**3*x** 4 + 2520*a**11*b*c**6*d**4*x**5 + 3024*a**11*b*c**5*d**5*x**6 + 2520*a**11 *b*c**4*d**6*x**7 + 1440*a**11*b*c**3*d**7*x**8 + 540*a**11*b*c**2*d**8*x* *9 + 120*a**11*b*c*d**9*x**10 + 12*a**11*b*d**10*x**11 + 66*a**10*b**2*c** 10*x**2 + 660*a**10*b**2*c**9*d*x**3 + 2970*a**10*b**2*c**8*d**2*x**4 + 79 20*a**10*b**2*c**7*d**3*x**5 + 13860*a**10*b**2*c**6*d**4*x**6 + 16632*a** 10*b**2*c**5*d**5*x**7 + 13860*a**10*b**2*c**4*d**6*x**8 + 7920*a**10*b**2 *c**3*d**7*x**9 + 2970*a**10*b**2*c**2*d**8*x**10 + 660*a**10*b**2*c*d**9* x**11 + 66*a**10*b**2*d**10*x**12 + 220*a**9*b**3*c**10*x**3 + 2200*a**9*b **3*c**9*d*x**4 + 9900*a**9*b**3*c**8*d**2*x**5 + 26400*a**9*b**3*c**7*d** 3*x**6 + 46200*a**9*b**3*c**6*d**4*x**7 + 55440*a**9*b**3*c**5*d**5*x**8 + 46200*a**9*b**3*c**4*d**6*x**9 + 26400*a**9*b**3*c**3*d**7*x**10 + 9900*a **9*b**3*c**2*d**8*x**11 + 2200*a**9*b**3*c*d**9*x**12 + 220*a**9*b**3*d** 10*x**13 + 495*a**8*b**4*c**10*x**4 + 4950*a**8*b**4*c**9*d*x**5 + 22275*a **8*b**4*c**8*d**2*x**6 + 59400*a**8*b**4*c**7*d**3*x**7 + 103950*a**8*b** 4*c**6*d**4*x**8 + 124740*a**8*b**4*c**5*d**5*x**9 + 103950*a**8*b**4*c**4 *d**6*x**10 + 59400*a**8*b**4*c**3*d**7*x**11 + 22275*a**8*b**4*c**2*d**8* x**12 + 4950*a**8*b**4*c*d**9*x**13 + 495*a**8*b**4*d**10*x**14 + 792*a**7 *b**5*c**10*x**5 + 7920*a**7*b**5*c**9*d*x**6 + 35640*a**7*b**5*c**8*d**2* x**7 + 95040*a**7*b**5*c**7*d**3*x**8 + 166320*a**7*b**5*c**6*d**4*x**9 + 199584*a**7*b**5*c**5*d**5*x**10 + 166320*a**7*b**5*c**4*d**6*x**11 + 9504 0*a**7*b**5*c**3*d**7*x**12 + 35640*a**7*b**5*c**2*d**8*x**13 + 7920*a**7* b**5*c*d**9*x**14 + 792*a**7*b**5*d**10*x**15 + 924*a**6*b**6*c**10*x**6 + 9240*a**6*b**6*c**9*d*x**7 + 41580*a**6*b**6*c**8*d**2*x**8 + 110880*a**6 *b**6*c**7*d**3*x**9 + 194040*a**6*b**6*c**6*d**4*x**10 + 232848*a**6*b**6 *c**5*d**5*x**11 + 194040*a**6*b**6*c**4*d**6*x**12 + 110880*a**6*b**6*c** 3*d**7*x**13 + 41580*a**6*b**6*c**2*d**8*x**14 + 9240*a**6*b**6*c*d**9*x** 15 + 924*a**6*b**6*d**10*x**16 + 792*a**5*b**7*c**10*x**7 + 7920*a**5*b**7 *c**9*d*x**8 + 35640*a**5*b**7*c**8*d**2*x**9 + 95040*a**5*b**7*c**7*d**3* x**10 + 166320*a**5*b**7*c**6*d**4*x**11 + 199584*a**5*b**7*c**5*d**5*x**1 2 + 166320*a**5*b**7*c**4*d**6*x**13 + 95040*a**5*b**7*c**3*d**7*x**14 + 3 5640*a**5*b**7*c**2*d**8*x**15 + 7920*a**5*b**7*c*d**9*x**16 + 792*a**5*b* *7*d**10*x**17 + 495*a**4*b**8*c**10*x**8 + 4950*a**4*b**8*c**9*d*x**9 + 2 2275*a**4*b**8*c**8*d**2*x**10 + 59400*a**4*b**8*c**7*d**3*x**11 + 103950* a**4*b**8*c**6*d**4*x**12 + 124740*a**4*b**8*c**5*d**5*x**13 + 103950*a**4 *b**8*c**4*d**6*x**14 + 59400*a**4*b**8*c**3*d**7*x**15 + 22275*a**4*b**8* c**2*d**8*x**16 + 4950*a**4*b**8*c*d**9*x**17 + 495*a**4*b**8*d**10*x**18 + 220*a**3*b**9*c**10*x**9 + 2200*a**3*b**9*c**9*d*x**10 + 9900*a**3*b**9* c**8*d**2*x**11 + 26400*a**3*b**9*c**7*d**3*x**12 + 46200*a**3*b**9*c**6*d **4*x**13 + 55440*a**3*b**9*c**5*d**5*x**14 + 46200*a**3*b**9*c**4*d**6*x* *15 + 26400*a**3*b**9*c**3*d**7*x**16 + 9900*a**3*b**9*c**2*d**8*x**17 + 2 200*a**3*b**9*c*d**9*x**18 + 220*a**3*b**9*d**10*x**19 + 66*a**2*b**10*c** 10*x**10 + 660*a**2*b**10*c**9*d*x**11 + 2970*a**2*b**10*c**8*d**2*x**12 + 7920*a**2*b**10*c**7*d**3*x**13 + 13860*a**2*b**10*c**6*d**4*x**14 + 1663 2*a**2*b**10*c**5*d**5*x**15 + 13860*a**2*b**10*c**4*d**6*x**16 + 7920*a** 2*b**10*c**3*d**7*x**17 + 2970*a**2*b**10*c**2*d**8*x**18 + 660*a**2*b**10 *c*d**9*x**19 + 66*a**2*b**10*d**10*x**20 + 12*a*b**11*c**10*x**11 + 120*a *b**11*c**9*d*x**12 + 540*a*b**11*c**8*d**2*x**13 + 1440*a*b**11*c**7*d**3 *x**14 + 2520*a*b**11*c**6*d**4*x**15 + 3024*a*b**11*c**5*d**5*x**16 + 252 0*a*b**11*c**4*d**6*x**17 + 1440*a*b**11*c**3*d**7*x**18 + 540*a*b**11*c** 2*d**8*x**19 + 120*a*b**11*c*d**9*x**20 + 12*a*b**11*d**10*x**21 + b**12*c **10*x**12 + 10*b**12*c**9*d*x**13 + 45*b**12*c**8*d**2*x**14 + 120*b**12* c**7*d**3*x**15 + 210*b**12*c**6*d**4*x**16 + 252*b**12*c**5*d**5*x**17 + 210*b**12*c**4*d**6*x**18 + 120*b**12*c**3*d**7*x**19 + 45*b**12*c**2*d**8 *x**20 + 10*b**12*c*d**9*x**21 + b**12*d**10*x**22,x)
(x*(14872858*a**12*c**10 + 74364290*a**12*c**9*d*x + 223092870*a**12*c**8* d**2*x**2 + 446185740*a**12*c**7*d**3*x**3 + 624660036*a**12*c**6*d**4*x** 4 + 624660036*a**12*c**5*d**5*x**5 + 446185740*a**12*c**4*d**6*x**6 + 2230 92870*a**12*c**3*d**7*x**7 + 74364290*a**12*c**2*d**8*x**8 + 14872858*a**1 2*c*d**9*x**9 + 1352078*a**12*d**10*x**10 + 89237148*a**11*b*c**10*x + 594 914320*a**11*b*c**9*d*x**2 + 2007835830*a**11*b*c**8*d**2*x**3 + 428338310 4*a**11*b*c**7*d**3*x**4 + 6246600360*a**11*b*c**6*d**4*x**5 + 6425074656* a**11*b*c**5*d**5*x**6 + 4684950270*a**11*b*c**4*d**6*x**7 + 2379657280*a* *11*b*c**3*d**7*x**8 + 803134332*a**11*b*c**2*d**8*x**9 + 162249360*a**11* b*c*d**9*x**10 + 14872858*a**11*b*d**10*x**11 + 327202876*a**10*b**2*c**10 *x**2 + 2454021570*a**10*b**2*c**9*d*x**3 + 8834477652*a**10*b**2*c**8*d** 2*x**4 + 19632172560*a**10*b**2*c**7*d**3*x**5 + 29448258840*a**10*b**2*c* *6*d**4*x**6 + 30920671782*a**10*b**2*c**5*d**5*x**7 + 22904201320*a**10*b **2*c**4*d**6*x**8 + 11779303536*a**10*b**2*c**3*d**7*x**9 + 4015671660*a* *10*b**2*c**2*d**8*x**10 + 818007190*a**10*b**2*c*d**9*x**11 + 75508356*a* *10*b**2*d**10*x**12 + 818007190*a**9*b**3*c**10*x**3 + 6544057520*a**9*b* *3*c**9*d*x**4 + 24540215700*a**9*b**3*c**8*d**2*x**5 + 56091921600*a**9*b **3*c**7*d**3*x**6 + 85890754950*a**9*b**3*c**6*d**4*x**7 + 91616805280*a* *9*b**3*c**5*d**5*x**8 + 68712603960*a**9*b**3*c**4*d**6*x**9 + 3569485920 0*a**9*b**3*c**3*d**7*x**10 + 12270107850*a**9*b**3*c**2*d**8*x**11 + 2...