Integrand size = 15, antiderivative size = 279 \[ \int (a+b x)^{10} (c+d x)^{10} \, dx=\frac {(b c-a d)^{10} (a+b x)^{11}}{11 b^{11}}+\frac {5 d (b c-a d)^9 (a+b x)^{12}}{6 b^{11}}+\frac {45 d^2 (b c-a d)^8 (a+b x)^{13}}{13 b^{11}}+\frac {60 d^3 (b c-a d)^7 (a+b x)^{14}}{7 b^{11}}+\frac {14 d^4 (b c-a d)^6 (a+b x)^{15}}{b^{11}}+\frac {63 d^5 (b c-a d)^5 (a+b x)^{16}}{4 b^{11}}+\frac {210 d^6 (b c-a d)^4 (a+b x)^{17}}{17 b^{11}}+\frac {20 d^7 (b c-a d)^3 (a+b x)^{18}}{3 b^{11}}+\frac {45 d^8 (b c-a d)^2 (a+b x)^{19}}{19 b^{11}}+\frac {d^9 (b c-a d) (a+b x)^{20}}{2 b^{11}}+\frac {d^{10} (a+b x)^{21}}{21 b^{11}} \]
1/11*(-a*d+b*c)^10*(b*x+a)^11/b^11+5/6*d*(-a*d+b*c)^9*(b*x+a)^12/b^11+45/1 3*d^2*(-a*d+b*c)^8*(b*x+a)^13/b^11+60/7*d^3*(-a*d+b*c)^7*(b*x+a)^14/b^11+1 4*d^4*(-a*d+b*c)^6*(b*x+a)^15/b^11+63/4*d^5*(-a*d+b*c)^5*(b*x+a)^16/b^11+2 10/17*d^6*(-a*d+b*c)^4*(b*x+a)^17/b^11+20/3*d^7*(-a*d+b*c)^3*(b*x+a)^18/b^ 11+45/19*d^8*(-a*d+b*c)^2*(b*x+a)^19/b^11+1/2*d^9*(-a*d+b*c)*(b*x+a)^20/b^ 11+1/21*d^10*(b*x+a)^21/b^11
Leaf count is larger than twice the leaf count of optimal. \(1539\) vs. \(2(279)=558\).
Time = 0.10 (sec) , antiderivative size = 1539, normalized size of antiderivative = 5.52 \[ \int (a+b x)^{10} (c+d x)^{10} \, dx =\text {Too large to display} \]
a^10*c^10*x + 5*a^9*c^9*(b*c + a*d)*x^2 + (5*a^8*c^8*(9*b^2*c^2 + 20*a*b*c *d + 9*a^2*d^2)*x^3)/3 + (15*a^7*c^7*(4*b^3*c^3 + 15*a*b^2*c^2*d + 15*a^2* b*c*d^2 + 4*a^3*d^3)*x^4)/2 + 3*a^6*c^6*(14*b^4*c^4 + 80*a*b^3*c^3*d + 135 *a^2*b^2*c^2*d^2 + 80*a^3*b*c*d^3 + 14*a^4*d^4)*x^5 + 2*a^5*c^5*(21*b^5*c^ 5 + 175*a*b^4*c^4*d + 450*a^2*b^3*c^3*d^2 + 450*a^3*b^2*c^2*d^3 + 175*a^4* b*c*d^4 + 21*a^5*d^5)*x^6 + (30*a^4*c^4*(7*b^6*c^6 + 84*a*b^5*c^5*d + 315* a^2*b^4*c^4*d^2 + 480*a^3*b^3*c^3*d^3 + 315*a^4*b^2*c^2*d^4 + 84*a^5*b*c*d ^5 + 7*a^6*d^6)*x^7)/7 + (15*a^3*c^3*(2*b^7*c^7 + 35*a*b^6*c^6*d + 189*a^2 *b^5*c^5*d^2 + 420*a^3*b^4*c^4*d^3 + 420*a^4*b^3*c^3*d^4 + 189*a^5*b^2*c^2 *d^5 + 35*a^6*b*c*d^6 + 2*a^7*d^7)*x^8)/2 + (5*a^2*c^2*(3*b^8*c^8 + 80*a*b ^7*c^7*d + 630*a^2*b^6*c^6*d^2 + 2016*a^3*b^5*c^5*d^3 + 2940*a^4*b^4*c^4*d ^4 + 2016*a^5*b^3*c^3*d^5 + 630*a^6*b^2*c^2*d^6 + 80*a^7*b*c*d^7 + 3*a^8*d ^8)*x^9)/3 + a*c*(b^9*c^9 + 45*a*b^8*c^8*d + 540*a^2*b^7*c^7*d^2 + 2520*a^ 3*b^6*c^6*d^3 + 5292*a^4*b^5*c^5*d^4 + 5292*a^5*b^4*c^4*d^5 + 2520*a^6*b^3 *c^3*d^6 + 540*a^7*b^2*c^2*d^7 + 45*a^8*b*c*d^8 + a^9*d^9)*x^10 + ((b^10*c ^10 + 100*a*b^9*c^9*d + 2025*a^2*b^8*c^8*d^2 + 14400*a^3*b^7*c^7*d^3 + 441 00*a^4*b^6*c^6*d^4 + 63504*a^5*b^5*c^5*d^5 + 44100*a^6*b^4*c^4*d^6 + 14400 *a^7*b^3*c^3*d^7 + 2025*a^8*b^2*c^2*d^8 + 100*a^9*b*c*d^9 + a^10*d^10)*x^1 1)/11 + (5*b*d*(b^9*c^9 + 45*a*b^8*c^8*d + 540*a^2*b^7*c^7*d^2 + 2520*a^3* b^6*c^6*d^3 + 5292*a^4*b^5*c^5*d^4 + 5292*a^5*b^4*c^4*d^5 + 2520*a^6*b^...
Time = 1.15 (sec) , antiderivative size = 279, 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.
| \(\displaystyle \int (a+b x)^{10} (c+d x)^{10} \, dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (\frac {10 d^9 (a+b x)^{19} (b c-a d)}{b^{10}}+\frac {45 d^8 (a+b x)^{18} (b c-a d)^2}{b^{10}}+\frac {120 d^7 (a+b x)^{17} (b c-a d)^3}{b^{10}}+\frac {210 d^6 (a+b x)^{16} (b c-a d)^4}{b^{10}}+\frac {252 d^5 (a+b x)^{15} (b c-a d)^5}{b^{10}}+\frac {210 d^4 (a+b x)^{14} (b c-a d)^6}{b^{10}}+\frac {120 d^3 (a+b x)^{13} (b c-a d)^7}{b^{10}}+\frac {45 d^2 (a+b x)^{12} (b c-a d)^8}{b^{10}}+\frac {10 d (a+b x)^{11} (b c-a d)^9}{b^{10}}+\frac {(a+b x)^{10} (b c-a d)^{10}}{b^{10}}+\frac {d^{10} (a+b x)^{20}}{b^{10}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^9 (a+b x)^{20} (b c-a d)}{2 b^{11}}+\frac {45 d^8 (a+b x)^{19} (b c-a d)^2}{19 b^{11}}+\frac {20 d^7 (a+b x)^{18} (b c-a d)^3}{3 b^{11}}+\frac {210 d^6 (a+b x)^{17} (b c-a d)^4}{17 b^{11}}+\frac {63 d^5 (a+b x)^{16} (b c-a d)^5}{4 b^{11}}+\frac {14 d^4 (a+b x)^{15} (b c-a d)^6}{b^{11}}+\frac {60 d^3 (a+b x)^{14} (b c-a d)^7}{7 b^{11}}+\frac {45 d^2 (a+b x)^{13} (b c-a d)^8}{13 b^{11}}+\frac {5 d (a+b x)^{12} (b c-a d)^9}{6 b^{11}}+\frac {(a+b x)^{11} (b c-a d)^{10}}{11 b^{11}}+\frac {d^{10} (a+b x)^{21}}{21 b^{11}}\) |
((b*c - a*d)^10*(a + b*x)^11)/(11*b^11) + (5*d*(b*c - a*d)^9*(a + b*x)^12) /(6*b^11) + (45*d^2*(b*c - a*d)^8*(a + b*x)^13)/(13*b^11) + (60*d^3*(b*c - a*d)^7*(a + b*x)^14)/(7*b^11) + (14*d^4*(b*c - a*d)^6*(a + b*x)^15)/b^11 + (63*d^5*(b*c - a*d)^5*(a + b*x)^16)/(4*b^11) + (210*d^6*(b*c - a*d)^4*(a + b*x)^17)/(17*b^11) + (20*d^7*(b*c - a*d)^3*(a + b*x)^18)/(3*b^11) + (45 *d^8*(b*c - a*d)^2*(a + b*x)^19)/(19*b^11) + (d^9*(b*c - a*d)*(a + b*x)^20 )/(2*b^11) + (d^10*(a + b*x)^21)/(21*b^11)
3.14.1.3.1 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. \(1571\) vs. \(2(259)=518\).
Time = 0.22 (sec) , antiderivative size = 1572, normalized size of antiderivative = 5.63
| method | result | size |
| norman | \(\text {Expression too large to display}\) | \(1572\) |
| default | \(\text {Expression too large to display}\) | \(1591\) |
| gosper | \(\text {Expression too large to display}\) | \(1834\) |
| risch | \(\text {Expression too large to display}\) | \(1834\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1834\) |
a^10*c^10*x+(5*a^10*c^9*d+5*a^9*b*c^10)*x^2+(15*a^10*c^8*d^2+100/3*a^9*b*c ^9*d+15*a^8*b^2*c^10)*x^3+(30*a^10*c^7*d^3+225/2*a^9*b*c^8*d^2+225/2*a^8*b ^2*c^9*d+30*a^7*b^3*c^10)*x^4+(42*a^10*c^6*d^4+240*a^9*b*c^7*d^3+405*a^8*b ^2*c^8*d^2+240*a^7*b^3*c^9*d+42*a^6*b^4*c^10)*x^5+(42*a^10*c^5*d^5+350*a^9 *b*c^6*d^4+900*a^8*b^2*c^7*d^3+900*a^7*b^3*c^8*d^2+350*a^6*b^4*c^9*d+42*a^ 5*b^5*c^10)*x^6+(30*a^10*c^4*d^6+360*a^9*b*c^5*d^5+1350*a^8*b^2*c^6*d^4+14 400/7*a^7*b^3*c^7*d^3+1350*a^6*b^4*c^8*d^2+360*a^5*b^5*c^9*d+30*a^4*b^6*c^ 10)*x^7+(15*a^10*c^3*d^7+525/2*a^9*b*c^4*d^6+2835/2*a^8*b^2*c^5*d^5+3150*a ^7*b^3*c^6*d^4+3150*a^6*b^4*c^7*d^3+2835/2*a^5*b^5*c^8*d^2+525/2*a^4*b^6*c ^9*d+15*a^3*b^7*c^10)*x^8+(5*a^10*c^2*d^8+400/3*a^9*b*c^3*d^7+1050*a^8*b^2 *c^4*d^6+3360*a^7*b^3*c^5*d^5+4900*a^6*b^4*c^6*d^4+3360*a^5*b^5*c^7*d^3+10 50*a^4*b^6*c^8*d^2+400/3*a^3*b^7*c^9*d+5*a^2*b^8*c^10)*x^9+(a^10*c*d^9+45* a^9*b*c^2*d^8+540*a^8*b^2*c^3*d^7+2520*a^7*b^3*c^4*d^6+5292*a^6*b^4*c^5*d^ 5+5292*a^5*b^5*c^6*d^4+2520*a^4*b^6*c^7*d^3+540*a^3*b^7*c^8*d^2+45*a^2*b^8 *c^9*d+a*b^9*c^10)*x^10+(1/11*a^10*d^10+100/11*a^9*b*c*d^9+2025/11*a^8*b^2 *c^2*d^8+14400/11*a^7*b^3*c^3*d^7+44100/11*a^6*b^4*c^4*d^6+63504/11*a^5*b^ 5*c^5*d^5+44100/11*a^4*b^6*c^6*d^4+14400/11*a^3*b^7*c^7*d^3+2025/11*a^2*b^ 8*c^8*d^2+100/11*a*b^9*c^9*d+1/11*b^10*c^10)*x^11+(5/6*a^9*b*d^10+75/2*a^8 *b^2*c*d^9+450*a^7*b^3*c^2*d^8+2100*a^6*b^4*c^3*d^7+4410*a^5*b^5*c^4*d^6+4 410*a^4*b^6*c^5*d^5+2100*a^3*b^7*c^6*d^4+450*a^2*b^8*c^7*d^3+75/2*a*b^9...
Leaf count of result is larger than twice the leaf count of optimal. 1581 vs. \(2 (259) = 518\).
Time = 0.23 (sec) , antiderivative size = 1581, normalized size of antiderivative = 5.67 \[ \int (a+b x)^{10} (c+d x)^{10} \, dx=\text {Too large to display} \]
1/21*b^10*d^10*x^21 + a^10*c^10*x + 1/2*(b^10*c*d^9 + a*b^9*d^10)*x^20 + 5 /19*(9*b^10*c^2*d^8 + 20*a*b^9*c*d^9 + 9*a^2*b^8*d^10)*x^19 + 5/3*(4*b^10* c^3*d^7 + 15*a*b^9*c^2*d^8 + 15*a^2*b^8*c*d^9 + 4*a^3*b^7*d^10)*x^18 + 15/ 17*(14*b^10*c^4*d^6 + 80*a*b^9*c^3*d^7 + 135*a^2*b^8*c^2*d^8 + 80*a^3*b^7* c*d^9 + 14*a^4*b^6*d^10)*x^17 + 3/4*(21*b^10*c^5*d^5 + 175*a*b^9*c^4*d^6 + 450*a^2*b^8*c^3*d^7 + 450*a^3*b^7*c^2*d^8 + 175*a^4*b^6*c*d^9 + 21*a^5*b^ 5*d^10)*x^16 + 2*(7*b^10*c^6*d^4 + 84*a*b^9*c^5*d^5 + 315*a^2*b^8*c^4*d^6 + 480*a^3*b^7*c^3*d^7 + 315*a^4*b^6*c^2*d^8 + 84*a^5*b^5*c*d^9 + 7*a^6*b^4 *d^10)*x^15 + 30/7*(2*b^10*c^7*d^3 + 35*a*b^9*c^6*d^4 + 189*a^2*b^8*c^5*d^ 5 + 420*a^3*b^7*c^4*d^6 + 420*a^4*b^6*c^3*d^7 + 189*a^5*b^5*c^2*d^8 + 35*a ^6*b^4*c*d^9 + 2*a^7*b^3*d^10)*x^14 + 15/13*(3*b^10*c^8*d^2 + 80*a*b^9*c^7 *d^3 + 630*a^2*b^8*c^6*d^4 + 2016*a^3*b^7*c^5*d^5 + 2940*a^4*b^6*c^4*d^6 + 2016*a^5*b^5*c^3*d^7 + 630*a^6*b^4*c^2*d^8 + 80*a^7*b^3*c*d^9 + 3*a^8*b^2 *d^10)*x^13 + 5/6*(b^10*c^9*d + 45*a*b^9*c^8*d^2 + 540*a^2*b^8*c^7*d^3 + 2 520*a^3*b^7*c^6*d^4 + 5292*a^4*b^6*c^5*d^5 + 5292*a^5*b^5*c^4*d^6 + 2520*a ^6*b^4*c^3*d^7 + 540*a^7*b^3*c^2*d^8 + 45*a^8*b^2*c*d^9 + a^9*b*d^10)*x^12 + 1/11*(b^10*c^10 + 100*a*b^9*c^9*d + 2025*a^2*b^8*c^8*d^2 + 14400*a^3*b^ 7*c^7*d^3 + 44100*a^4*b^6*c^6*d^4 + 63504*a^5*b^5*c^5*d^5 + 44100*a^6*b^4* c^4*d^6 + 14400*a^7*b^3*c^3*d^7 + 2025*a^8*b^2*c^2*d^8 + 100*a^9*b*c*d^9 + a^10*d^10)*x^11 + (a*b^9*c^10 + 45*a^2*b^8*c^9*d + 540*a^3*b^7*c^8*d^2...
Leaf count of result is larger than twice the leaf count of optimal. 1775 vs. \(2 (257) = 514\).
Time = 0.12 (sec) , antiderivative size = 1775, normalized size of antiderivative = 6.36 \[ \int (a+b x)^{10} (c+d x)^{10} \, dx=\text {Too large to display} \]
a**10*c**10*x + b**10*d**10*x**21/21 + x**20*(a*b**9*d**10/2 + b**10*c*d** 9/2) + x**19*(45*a**2*b**8*d**10/19 + 100*a*b**9*c*d**9/19 + 45*b**10*c**2 *d**8/19) + x**18*(20*a**3*b**7*d**10/3 + 25*a**2*b**8*c*d**9 + 25*a*b**9* c**2*d**8 + 20*b**10*c**3*d**7/3) + x**17*(210*a**4*b**6*d**10/17 + 1200*a **3*b**7*c*d**9/17 + 2025*a**2*b**8*c**2*d**8/17 + 1200*a*b**9*c**3*d**7/1 7 + 210*b**10*c**4*d**6/17) + x**16*(63*a**5*b**5*d**10/4 + 525*a**4*b**6* c*d**9/4 + 675*a**3*b**7*c**2*d**8/2 + 675*a**2*b**8*c**3*d**7/2 + 525*a*b **9*c**4*d**6/4 + 63*b**10*c**5*d**5/4) + x**15*(14*a**6*b**4*d**10 + 168* a**5*b**5*c*d**9 + 630*a**4*b**6*c**2*d**8 + 960*a**3*b**7*c**3*d**7 + 630 *a**2*b**8*c**4*d**6 + 168*a*b**9*c**5*d**5 + 14*b**10*c**6*d**4) + x**14* (60*a**7*b**3*d**10/7 + 150*a**6*b**4*c*d**9 + 810*a**5*b**5*c**2*d**8 + 1 800*a**4*b**6*c**3*d**7 + 1800*a**3*b**7*c**4*d**6 + 810*a**2*b**8*c**5*d* *5 + 150*a*b**9*c**6*d**4 + 60*b**10*c**7*d**3/7) + x**13*(45*a**8*b**2*d* *10/13 + 1200*a**7*b**3*c*d**9/13 + 9450*a**6*b**4*c**2*d**8/13 + 30240*a* *5*b**5*c**3*d**7/13 + 44100*a**4*b**6*c**4*d**6/13 + 30240*a**3*b**7*c**5 *d**5/13 + 9450*a**2*b**8*c**6*d**4/13 + 1200*a*b**9*c**7*d**3/13 + 45*b** 10*c**8*d**2/13) + x**12*(5*a**9*b*d**10/6 + 75*a**8*b**2*c*d**9/2 + 450*a **7*b**3*c**2*d**8 + 2100*a**6*b**4*c**3*d**7 + 4410*a**5*b**5*c**4*d**6 + 4410*a**4*b**6*c**5*d**5 + 2100*a**3*b**7*c**6*d**4 + 450*a**2*b**8*c**7* d**3 + 75*a*b**9*c**8*d**2/2 + 5*b**10*c**9*d/6) + x**11*(a**10*d**10/1...
Leaf count of result is larger than twice the leaf count of optimal. 1581 vs. \(2 (259) = 518\).
Time = 0.22 (sec) , antiderivative size = 1581, normalized size of antiderivative = 5.67 \[ \int (a+b x)^{10} (c+d x)^{10} \, dx=\text {Too large to display} \]
1/21*b^10*d^10*x^21 + a^10*c^10*x + 1/2*(b^10*c*d^9 + a*b^9*d^10)*x^20 + 5 /19*(9*b^10*c^2*d^8 + 20*a*b^9*c*d^9 + 9*a^2*b^8*d^10)*x^19 + 5/3*(4*b^10* c^3*d^7 + 15*a*b^9*c^2*d^8 + 15*a^2*b^8*c*d^9 + 4*a^3*b^7*d^10)*x^18 + 15/ 17*(14*b^10*c^4*d^6 + 80*a*b^9*c^3*d^7 + 135*a^2*b^8*c^2*d^8 + 80*a^3*b^7* c*d^9 + 14*a^4*b^6*d^10)*x^17 + 3/4*(21*b^10*c^5*d^5 + 175*a*b^9*c^4*d^6 + 450*a^2*b^8*c^3*d^7 + 450*a^3*b^7*c^2*d^8 + 175*a^4*b^6*c*d^9 + 21*a^5*b^ 5*d^10)*x^16 + 2*(7*b^10*c^6*d^4 + 84*a*b^9*c^5*d^5 + 315*a^2*b^8*c^4*d^6 + 480*a^3*b^7*c^3*d^7 + 315*a^4*b^6*c^2*d^8 + 84*a^5*b^5*c*d^9 + 7*a^6*b^4 *d^10)*x^15 + 30/7*(2*b^10*c^7*d^3 + 35*a*b^9*c^6*d^4 + 189*a^2*b^8*c^5*d^ 5 + 420*a^3*b^7*c^4*d^6 + 420*a^4*b^6*c^3*d^7 + 189*a^5*b^5*c^2*d^8 + 35*a ^6*b^4*c*d^9 + 2*a^7*b^3*d^10)*x^14 + 15/13*(3*b^10*c^8*d^2 + 80*a*b^9*c^7 *d^3 + 630*a^2*b^8*c^6*d^4 + 2016*a^3*b^7*c^5*d^5 + 2940*a^4*b^6*c^4*d^6 + 2016*a^5*b^5*c^3*d^7 + 630*a^6*b^4*c^2*d^8 + 80*a^7*b^3*c*d^9 + 3*a^8*b^2 *d^10)*x^13 + 5/6*(b^10*c^9*d + 45*a*b^9*c^8*d^2 + 540*a^2*b^8*c^7*d^3 + 2 520*a^3*b^7*c^6*d^4 + 5292*a^4*b^6*c^5*d^5 + 5292*a^5*b^5*c^4*d^6 + 2520*a ^6*b^4*c^3*d^7 + 540*a^7*b^3*c^2*d^8 + 45*a^8*b^2*c*d^9 + a^9*b*d^10)*x^12 + 1/11*(b^10*c^10 + 100*a*b^9*c^9*d + 2025*a^2*b^8*c^8*d^2 + 14400*a^3*b^ 7*c^7*d^3 + 44100*a^4*b^6*c^6*d^4 + 63504*a^5*b^5*c^5*d^5 + 44100*a^6*b^4* c^4*d^6 + 14400*a^7*b^3*c^3*d^7 + 2025*a^8*b^2*c^2*d^8 + 100*a^9*b*c*d^9 + a^10*d^10)*x^11 + (a*b^9*c^10 + 45*a^2*b^8*c^9*d + 540*a^3*b^7*c^8*d^2...
Leaf count of result is larger than twice the leaf count of optimal. 1833 vs. \(2 (259) = 518\).
Time = 0.32 (sec) , antiderivative size = 1833, normalized size of antiderivative = 6.57 \[ \int (a+b x)^{10} (c+d x)^{10} \, dx=\text {Too large to display} \]
1/21*b^10*d^10*x^21 + 1/2*b^10*c*d^9*x^20 + 1/2*a*b^9*d^10*x^20 + 45/19*b^ 10*c^2*d^8*x^19 + 100/19*a*b^9*c*d^9*x^19 + 45/19*a^2*b^8*d^10*x^19 + 20/3 *b^10*c^3*d^7*x^18 + 25*a*b^9*c^2*d^8*x^18 + 25*a^2*b^8*c*d^9*x^18 + 20/3* a^3*b^7*d^10*x^18 + 210/17*b^10*c^4*d^6*x^17 + 1200/17*a*b^9*c^3*d^7*x^17 + 2025/17*a^2*b^8*c^2*d^8*x^17 + 1200/17*a^3*b^7*c*d^9*x^17 + 210/17*a^4*b ^6*d^10*x^17 + 63/4*b^10*c^5*d^5*x^16 + 525/4*a*b^9*c^4*d^6*x^16 + 675/2*a ^2*b^8*c^3*d^7*x^16 + 675/2*a^3*b^7*c^2*d^8*x^16 + 525/4*a^4*b^6*c*d^9*x^1 6 + 63/4*a^5*b^5*d^10*x^16 + 14*b^10*c^6*d^4*x^15 + 168*a*b^9*c^5*d^5*x^15 + 630*a^2*b^8*c^4*d^6*x^15 + 960*a^3*b^7*c^3*d^7*x^15 + 630*a^4*b^6*c^2*d ^8*x^15 + 168*a^5*b^5*c*d^9*x^15 + 14*a^6*b^4*d^10*x^15 + 60/7*b^10*c^7*d^ 3*x^14 + 150*a*b^9*c^6*d^4*x^14 + 810*a^2*b^8*c^5*d^5*x^14 + 1800*a^3*b^7* c^4*d^6*x^14 + 1800*a^4*b^6*c^3*d^7*x^14 + 810*a^5*b^5*c^2*d^8*x^14 + 150* a^6*b^4*c*d^9*x^14 + 60/7*a^7*b^3*d^10*x^14 + 45/13*b^10*c^8*d^2*x^13 + 12 00/13*a*b^9*c^7*d^3*x^13 + 9450/13*a^2*b^8*c^6*d^4*x^13 + 30240/13*a^3*b^7 *c^5*d^5*x^13 + 44100/13*a^4*b^6*c^4*d^6*x^13 + 30240/13*a^5*b^5*c^3*d^7*x ^13 + 9450/13*a^6*b^4*c^2*d^8*x^13 + 1200/13*a^7*b^3*c*d^9*x^13 + 45/13*a^ 8*b^2*d^10*x^13 + 5/6*b^10*c^9*d*x^12 + 75/2*a*b^9*c^8*d^2*x^12 + 450*a^2* b^8*c^7*d^3*x^12 + 2100*a^3*b^7*c^6*d^4*x^12 + 4410*a^4*b^6*c^5*d^5*x^12 + 4410*a^5*b^5*c^4*d^6*x^12 + 2100*a^6*b^4*c^3*d^7*x^12 + 450*a^7*b^3*c^2*d ^8*x^12 + 75/2*a^8*b^2*c*d^9*x^12 + 5/6*a^9*b*d^10*x^12 + 1/11*b^10*c^1...
Time = 0.73 (sec) , antiderivative size = 1549, normalized size of antiderivative = 5.55 \[ \int (a+b x)^{10} (c+d x)^{10} \, dx=\text {Too large to display} \]
x^7*(30*a^4*b^6*c^10 + 30*a^10*c^4*d^6 + 360*a^5*b^5*c^9*d + 360*a^9*b*c^5 *d^5 + 1350*a^6*b^4*c^8*d^2 + (14400*a^7*b^3*c^7*d^3)/7 + 1350*a^8*b^2*c^6 *d^4) + x^15*(14*a^6*b^4*d^10 + 14*b^10*c^6*d^4 + 168*a*b^9*c^5*d^5 + 168* a^5*b^5*c*d^9 + 630*a^2*b^8*c^4*d^6 + 960*a^3*b^7*c^3*d^7 + 630*a^4*b^6*c^ 2*d^8) + x^5*(42*a^6*b^4*c^10 + 42*a^10*c^6*d^4 + 240*a^7*b^3*c^9*d + 240* a^9*b*c^7*d^3 + 405*a^8*b^2*c^8*d^2) + x^17*((210*a^4*b^6*d^10)/17 + (210* b^10*c^4*d^6)/17 + (1200*a*b^9*c^3*d^7)/17 + (1200*a^3*b^7*c*d^9)/17 + (20 25*a^2*b^8*c^2*d^8)/17) + x^11*((a^10*d^10)/11 + (b^10*c^10)/11 + (2025*a^ 2*b^8*c^8*d^2)/11 + (14400*a^3*b^7*c^7*d^3)/11 + (44100*a^4*b^6*c^6*d^4)/1 1 + (63504*a^5*b^5*c^5*d^5)/11 + (44100*a^6*b^4*c^4*d^6)/11 + (14400*a^7*b ^3*c^3*d^7)/11 + (2025*a^8*b^2*c^2*d^8)/11 + (100*a*b^9*c^9*d)/11 + (100*a ^9*b*c*d^9)/11) + x^8*(15*a^3*b^7*c^10 + 15*a^10*c^3*d^7 + (525*a^4*b^6*c^ 9*d)/2 + (525*a^9*b*c^4*d^6)/2 + (2835*a^5*b^5*c^8*d^2)/2 + 3150*a^6*b^4*c ^7*d^3 + 3150*a^7*b^3*c^6*d^4 + (2835*a^8*b^2*c^5*d^5)/2) + x^14*((60*a^7* b^3*d^10)/7 + (60*b^10*c^7*d^3)/7 + 150*a*b^9*c^6*d^4 + 150*a^6*b^4*c*d^9 + 810*a^2*b^8*c^5*d^5 + 1800*a^3*b^7*c^4*d^6 + 1800*a^4*b^6*c^3*d^7 + 810* a^5*b^5*c^2*d^8) + x^10*(a*b^9*c^10 + a^10*c*d^9 + 45*a^2*b^8*c^9*d + 45*a ^9*b*c^2*d^8 + 540*a^3*b^7*c^8*d^2 + 2520*a^4*b^6*c^7*d^3 + 5292*a^5*b^5*c ^6*d^4 + 5292*a^6*b^4*c^5*d^5 + 2520*a^7*b^3*c^4*d^6 + 540*a^8*b^2*c^3*d^7 ) + x^12*((5*a^9*b*d^10)/6 + (5*b^10*c^9*d)/6 + (75*a*b^9*c^8*d^2)/2 + ...
Time = 0.00 (sec) , antiderivative size = 1834, normalized size of antiderivative = 6.57 \[ \int (a+b x)^{10} (c+d x)^{10} \, dx =\text {Too large to display} \]
int(a**10*c**10 + 10*a**10*c**9*d*x + 45*a**10*c**8*d**2*x**2 + 120*a**10* c**7*d**3*x**3 + 210*a**10*c**6*d**4*x**4 + 252*a**10*c**5*d**5*x**5 + 210 *a**10*c**4*d**6*x**6 + 120*a**10*c**3*d**7*x**7 + 45*a**10*c**2*d**8*x**8 + 10*a**10*c*d**9*x**9 + a**10*d**10*x**10 + 10*a**9*b*c**10*x + 100*a**9 *b*c**9*d*x**2 + 450*a**9*b*c**8*d**2*x**3 + 1200*a**9*b*c**7*d**3*x**4 + 2100*a**9*b*c**6*d**4*x**5 + 2520*a**9*b*c**5*d**5*x**6 + 2100*a**9*b*c**4 *d**6*x**7 + 1200*a**9*b*c**3*d**7*x**8 + 450*a**9*b*c**2*d**8*x**9 + 100* a**9*b*c*d**9*x**10 + 10*a**9*b*d**10*x**11 + 45*a**8*b**2*c**10*x**2 + 45 0*a**8*b**2*c**9*d*x**3 + 2025*a**8*b**2*c**8*d**2*x**4 + 5400*a**8*b**2*c **7*d**3*x**5 + 9450*a**8*b**2*c**6*d**4*x**6 + 11340*a**8*b**2*c**5*d**5* x**7 + 9450*a**8*b**2*c**4*d**6*x**8 + 5400*a**8*b**2*c**3*d**7*x**9 + 202 5*a**8*b**2*c**2*d**8*x**10 + 450*a**8*b**2*c*d**9*x**11 + 45*a**8*b**2*d* *10*x**12 + 120*a**7*b**3*c**10*x**3 + 1200*a**7*b**3*c**9*d*x**4 + 5400*a **7*b**3*c**8*d**2*x**5 + 14400*a**7*b**3*c**7*d**3*x**6 + 25200*a**7*b**3 *c**6*d**4*x**7 + 30240*a**7*b**3*c**5*d**5*x**8 + 25200*a**7*b**3*c**4*d* *6*x**9 + 14400*a**7*b**3*c**3*d**7*x**10 + 5400*a**7*b**3*c**2*d**8*x**11 + 1200*a**7*b**3*c*d**9*x**12 + 120*a**7*b**3*d**10*x**13 + 210*a**6*b**4 *c**10*x**4 + 2100*a**6*b**4*c**9*d*x**5 + 9450*a**6*b**4*c**8*d**2*x**6 + 25200*a**6*b**4*c**7*d**3*x**7 + 44100*a**6*b**4*c**6*d**4*x**8 + 52920*a **6*b**4*c**5*d**5*x**9 + 44100*a**6*b**4*c**4*d**6*x**10 + 25200*a**6*b** 4*c**3*d**7*x**11 + 9450*a**6*b**4*c**2*d**8*x**12 + 2100*a**6*b**4*c*d**9 *x**13 + 210*a**6*b**4*d**10*x**14 + 252*a**5*b**5*c**10*x**5 + 2520*a**5* b**5*c**9*d*x**6 + 11340*a**5*b**5*c**8*d**2*x**7 + 30240*a**5*b**5*c**7*d **3*x**8 + 52920*a**5*b**5*c**6*d**4*x**9 + 63504*a**5*b**5*c**5*d**5*x**1 0 + 52920*a**5*b**5*c**4*d**6*x**11 + 30240*a**5*b**5*c**3*d**7*x**12 + 11 340*a**5*b**5*c**2*d**8*x**13 + 2520*a**5*b**5*c*d**9*x**14 + 252*a**5*b** 5*d**10*x**15 + 210*a**4*b**6*c**10*x**6 + 2100*a**4*b**6*c**9*d*x**7 + 94 50*a**4*b**6*c**8*d**2*x**8 + 25200*a**4*b**6*c**7*d**3*x**9 + 44100*a**4* b**6*c**6*d**4*x**10 + 52920*a**4*b**6*c**5*d**5*x**11 + 44100*a**4*b**6*c **4*d**6*x**12 + 25200*a**4*b**6*c**3*d**7*x**13 + 9450*a**4*b**6*c**2*d** 8*x**14 + 2100*a**4*b**6*c*d**9*x**15 + 210*a**4*b**6*d**10*x**16 + 120*a* *3*b**7*c**10*x**7 + 1200*a**3*b**7*c**9*d*x**8 + 5400*a**3*b**7*c**8*d**2 *x**9 + 14400*a**3*b**7*c**7*d**3*x**10 + 25200*a**3*b**7*c**6*d**4*x**11 + 30240*a**3*b**7*c**5*d**5*x**12 + 25200*a**3*b**7*c**4*d**6*x**13 + 1440 0*a**3*b**7*c**3*d**7*x**14 + 5400*a**3*b**7*c**2*d**8*x**15 + 1200*a**3*b **7*c*d**9*x**16 + 120*a**3*b**7*d**10*x**17 + 45*a**2*b**8*c**10*x**8 + 4 50*a**2*b**8*c**9*d*x**9 + 2025*a**2*b**8*c**8*d**2*x**10 + 5400*a**2*b**8 *c**7*d**3*x**11 + 9450*a**2*b**8*c**6*d**4*x**12 + 11340*a**2*b**8*c**5*d **5*x**13 + 9450*a**2*b**8*c**4*d**6*x**14 + 5400*a**2*b**8*c**3*d**7*x**1 5 + 2025*a**2*b**8*c**2*d**8*x**16 + 450*a**2*b**8*c*d**9*x**17 + 45*a**2* b**8*d**10*x**18 + 10*a*b**9*c**10*x**9 + 100*a*b**9*c**9*d*x**10 + 450*a* b**9*c**8*d**2*x**11 + 1200*a*b**9*c**7*d**3*x**12 + 2100*a*b**9*c**6*d**4 *x**13 + 2520*a*b**9*c**5*d**5*x**14 + 2100*a*b**9*c**4*d**6*x**15 + 1200* a*b**9*c**3*d**7*x**16 + 450*a*b**9*c**2*d**8*x**17 + 100*a*b**9*c*d**9*x* *18 + 10*a*b**9*d**10*x**19 + b**10*c**10*x**10 + 10*b**10*c**9*d*x**11 + 45*b**10*c**8*d**2*x**12 + 120*b**10*c**7*d**3*x**13 + 210*b**10*c**6*d**4 *x**14 + 252*b**10*c**5*d**5*x**15 + 210*b**10*c**4*d**6*x**16 + 120*b**10 *c**3*d**7*x**17 + 45*b**10*c**2*d**8*x**18 + 10*b**10*c*d**9*x**19 + b**1 0*d**10*x**20,x)
(x*(3879876*a**10*c**10 + 19399380*a**10*c**9*d*x + 58198140*a**10*c**8*d* *2*x**2 + 116396280*a**10*c**7*d**3*x**3 + 162954792*a**10*c**6*d**4*x**4 + 162954792*a**10*c**5*d**5*x**5 + 116396280*a**10*c**4*d**6*x**6 + 581981 40*a**10*c**3*d**7*x**7 + 19399380*a**10*c**2*d**8*x**8 + 3879876*a**10*c* d**9*x**9 + 352716*a**10*d**10*x**10 + 19399380*a**9*b*c**10*x + 129329200 *a**9*b*c**9*d*x**2 + 436486050*a**9*b*c**8*d**2*x**3 + 931170240*a**9*b*c **7*d**3*x**4 + 1357956600*a**9*b*c**6*d**4*x**5 + 1396755360*a**9*b*c**5* d**5*x**6 + 1018467450*a**9*b*c**4*d**6*x**7 + 517316800*a**9*b*c**3*d**7* x**8 + 174594420*a**9*b*c**2*d**8*x**9 + 35271600*a**9*b*c*d**9*x**10 + 32 33230*a**9*b*d**10*x**11 + 58198140*a**8*b**2*c**10*x**2 + 436486050*a**8* b**2*c**9*d*x**3 + 1571349780*a**8*b**2*c**8*d**2*x**4 + 3491888400*a**8*b **2*c**7*d**3*x**5 + 5237832600*a**8*b**2*c**6*d**4*x**6 + 5499724230*a**8 *b**2*c**5*d**5*x**7 + 4073869800*a**8*b**2*c**4*d**6*x**8 + 2095133040*a* *8*b**2*c**3*d**7*x**9 + 714249900*a**8*b**2*c**2*d**8*x**10 + 145495350*a **8*b**2*c*d**9*x**11 + 13430340*a**8*b**2*d**10*x**12 + 116396280*a**7*b* *3*c**10*x**3 + 931170240*a**7*b**3*c**9*d*x**4 + 3491888400*a**7*b**3*c** 8*d**2*x**5 + 7981459200*a**7*b**3*c**7*d**3*x**6 + 12221609400*a**7*b**3* c**6*d**4*x**7 + 13036383360*a**7*b**3*c**5*d**5*x**8 + 9777287520*a**7*b* *3*c**4*d**6*x**9 + 5079110400*a**7*b**3*c**3*d**7*x**10 + 1745944200*a**7 *b**3*c**2*d**8*x**11 + 358142400*a**7*b**3*c*d**9*x**12 + 33256080*a**...