Integrand size = 15, antiderivative size = 225 \[ \int (a+b x)^8 (c+d x)^{10} \, dx=\frac {(b c-a d)^8 (c+d x)^{11}}{11 d^9}-\frac {2 b (b c-a d)^7 (c+d x)^{12}}{3 d^9}+\frac {28 b^2 (b c-a d)^6 (c+d x)^{13}}{13 d^9}-\frac {4 b^3 (b c-a d)^5 (c+d x)^{14}}{d^9}+\frac {14 b^4 (b c-a d)^4 (c+d x)^{15}}{3 d^9}-\frac {7 b^5 (b c-a d)^3 (c+d x)^{16}}{2 d^9}+\frac {28 b^6 (b c-a d)^2 (c+d x)^{17}}{17 d^9}-\frac {4 b^7 (b c-a d) (c+d x)^{18}}{9 d^9}+\frac {b^8 (c+d x)^{19}}{19 d^9} \]
1/11*(-a*d+b*c)^8*(d*x+c)^11/d^9-2/3*b*(-a*d+b*c)^7*(d*x+c)^12/d^9+28/13*b ^2*(-a*d+b*c)^6*(d*x+c)^13/d^9-4*b^3*(-a*d+b*c)^5*(d*x+c)^14/d^9+14/3*b^4* (-a*d+b*c)^4*(d*x+c)^15/d^9-7/2*b^5*(-a*d+b*c)^3*(d*x+c)^16/d^9+28/17*b^6* (-a*d+b*c)^2*(d*x+c)^17/d^9-4/9*b^7*(-a*d+b*c)*(d*x+c)^18/d^9+1/19*b^8*(d* x+c)^19/d^9
Leaf count is larger than twice the leaf count of optimal. \(1241\) vs. \(2(225)=450\).
Time = 0.09 (sec) , antiderivative size = 1241, normalized size of antiderivative = 5.52 \[ \int (a+b x)^8 (c+d x)^{10} \, dx=a^8 c^{10} x+a^7 c^9 (4 b c+5 a d) x^2+\frac {1}{3} a^6 c^8 \left (28 b^2 c^2+80 a b c d+45 a^2 d^2\right ) x^3+2 a^5 c^7 \left (7 b^3 c^3+35 a b^2 c^2 d+45 a^2 b c d^2+15 a^3 d^3\right ) x^4+2 a^4 c^6 \left (7 b^4 c^4+56 a b^3 c^3 d+126 a^2 b^2 c^2 d^2+96 a^3 b c d^3+21 a^4 d^4\right ) x^5+\frac {14}{3} a^3 c^5 \left (2 b^5 c^5+25 a b^4 c^4 d+90 a^2 b^3 c^3 d^2+120 a^3 b^2 c^2 d^3+60 a^4 b c d^4+9 a^5 d^5\right ) x^6+2 a^2 c^4 \left (2 b^6 c^6+40 a b^5 c^5 d+225 a^2 b^4 c^4 d^2+480 a^3 b^3 c^3 d^3+420 a^4 b^2 c^2 d^4+144 a^5 b c d^5+15 a^6 d^6\right ) x^7+a c^3 \left (b^7 c^7+35 a b^6 c^6 d+315 a^2 b^5 c^5 d^2+1050 a^3 b^4 c^4 d^3+1470 a^4 b^3 c^3 d^4+882 a^5 b^2 c^2 d^5+210 a^6 b c d^6+15 a^7 d^7\right ) x^8+\frac {1}{9} c^2 \left (b^8 c^8+80 a b^7 c^7 d+1260 a^2 b^6 c^6 d^2+6720 a^3 b^5 c^5 d^3+14700 a^4 b^4 c^4 d^4+14112 a^5 b^3 c^3 d^5+5880 a^6 b^2 c^2 d^6+960 a^7 b c d^7+45 a^8 d^8\right ) x^9+c d \left (b^8 c^8+36 a b^7 c^7 d+336 a^2 b^6 c^6 d^2+1176 a^3 b^5 c^5 d^3+1764 a^4 b^4 c^4 d^4+1176 a^5 b^3 c^3 d^5+336 a^6 b^2 c^2 d^6+36 a^7 b c d^7+a^8 d^8\right ) x^{10}+\frac {1}{11} d^2 \left (45 b^8 c^8+960 a b^7 c^7 d+5880 a^2 b^6 c^6 d^2+14112 a^3 b^5 c^5 d^3+14700 a^4 b^4 c^4 d^4+6720 a^5 b^3 c^3 d^5+1260 a^6 b^2 c^2 d^6+80 a^7 b c d^7+a^8 d^8\right ) x^{11}+\frac {2}{3} b d^3 \left (15 b^7 c^7+210 a b^6 c^6 d+882 a^2 b^5 c^5 d^2+1470 a^3 b^4 c^4 d^3+1050 a^4 b^3 c^3 d^4+315 a^5 b^2 c^2 d^5+35 a^6 b c d^6+a^7 d^7\right ) x^{12}+\frac {14}{13} b^2 d^4 \left (15 b^6 c^6+144 a b^5 c^5 d+420 a^2 b^4 c^4 d^2+480 a^3 b^3 c^3 d^3+225 a^4 b^2 c^2 d^4+40 a^5 b c d^5+2 a^6 d^6\right ) x^{13}+2 b^3 d^5 \left (9 b^5 c^5+60 a b^4 c^4 d+120 a^2 b^3 c^3 d^2+90 a^3 b^2 c^2 d^3+25 a^4 b c d^4+2 a^5 d^5\right ) x^{14}+\frac {2}{3} b^4 d^6 \left (21 b^4 c^4+96 a b^3 c^3 d+126 a^2 b^2 c^2 d^2+56 a^3 b c d^3+7 a^4 d^4\right ) x^{15}+\frac {1}{2} b^5 d^7 \left (15 b^3 c^3+45 a b^2 c^2 d+35 a^2 b c d^2+7 a^3 d^3\right ) x^{16}+\frac {1}{17} b^6 d^8 \left (45 b^2 c^2+80 a b c d+28 a^2 d^2\right ) x^{17}+\frac {1}{9} b^7 d^9 (5 b c+4 a d) x^{18}+\frac {1}{19} b^8 d^{10} x^{19} \]
a^8*c^10*x + a^7*c^9*(4*b*c + 5*a*d)*x^2 + (a^6*c^8*(28*b^2*c^2 + 80*a*b*c *d + 45*a^2*d^2)*x^3)/3 + 2*a^5*c^7*(7*b^3*c^3 + 35*a*b^2*c^2*d + 45*a^2*b *c*d^2 + 15*a^3*d^3)*x^4 + 2*a^4*c^6*(7*b^4*c^4 + 56*a*b^3*c^3*d + 126*a^2 *b^2*c^2*d^2 + 96*a^3*b*c*d^3 + 21*a^4*d^4)*x^5 + (14*a^3*c^5*(2*b^5*c^5 + 25*a*b^4*c^4*d + 90*a^2*b^3*c^3*d^2 + 120*a^3*b^2*c^2*d^3 + 60*a^4*b*c*d^ 4 + 9*a^5*d^5)*x^6)/3 + 2*a^2*c^4*(2*b^6*c^6 + 40*a*b^5*c^5*d + 225*a^2*b^ 4*c^4*d^2 + 480*a^3*b^3*c^3*d^3 + 420*a^4*b^2*c^2*d^4 + 144*a^5*b*c*d^5 + 15*a^6*d^6)*x^7 + a*c^3*(b^7*c^7 + 35*a*b^6*c^6*d + 315*a^2*b^5*c^5*d^2 + 1050*a^3*b^4*c^4*d^3 + 1470*a^4*b^3*c^3*d^4 + 882*a^5*b^2*c^2*d^5 + 210*a^ 6*b*c*d^6 + 15*a^7*d^7)*x^8 + (c^2*(b^8*c^8 + 80*a*b^7*c^7*d + 1260*a^2*b^ 6*c^6*d^2 + 6720*a^3*b^5*c^5*d^3 + 14700*a^4*b^4*c^4*d^4 + 14112*a^5*b^3*c ^3*d^5 + 5880*a^6*b^2*c^2*d^6 + 960*a^7*b*c*d^7 + 45*a^8*d^8)*x^9)/9 + c*d *(b^8*c^8 + 36*a*b^7*c^7*d + 336*a^2*b^6*c^6*d^2 + 1176*a^3*b^5*c^5*d^3 + 1764*a^4*b^4*c^4*d^4 + 1176*a^5*b^3*c^3*d^5 + 336*a^6*b^2*c^2*d^6 + 36*a^7 *b*c*d^7 + a^8*d^8)*x^10 + (d^2*(45*b^8*c^8 + 960*a*b^7*c^7*d + 5880*a^2*b ^6*c^6*d^2 + 14112*a^3*b^5*c^5*d^3 + 14700*a^4*b^4*c^4*d^4 + 6720*a^5*b^3* c^3*d^5 + 1260*a^6*b^2*c^2*d^6 + 80*a^7*b*c*d^7 + a^8*d^8)*x^11)/11 + (2*b *d^3*(15*b^7*c^7 + 210*a*b^6*c^6*d + 882*a^2*b^5*c^5*d^2 + 1470*a^3*b^4*c^ 4*d^3 + 1050*a^4*b^3*c^3*d^4 + 315*a^5*b^2*c^2*d^5 + 35*a^6*b*c*d^6 + a^7* d^7)*x^12)/3 + (14*b^2*d^4*(15*b^6*c^6 + 144*a*b^5*c^5*d + 420*a^2*b^4*...
Time = 0.94 (sec) , antiderivative size = 225, 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.
| \(\displaystyle \int (a+b x)^8 (c+d x)^{10} \, dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (-\frac {8 b^7 (c+d x)^{17} (b c-a d)}{d^8}+\frac {28 b^6 (c+d x)^{16} (b c-a d)^2}{d^8}-\frac {56 b^5 (c+d x)^{15} (b c-a d)^3}{d^8}+\frac {70 b^4 (c+d x)^{14} (b c-a d)^4}{d^8}-\frac {56 b^3 (c+d x)^{13} (b c-a d)^5}{d^8}+\frac {28 b^2 (c+d x)^{12} (b c-a d)^6}{d^8}-\frac {8 b (c+d x)^{11} (b c-a d)^7}{d^8}+\frac {(c+d x)^{10} (a d-b c)^8}{d^8}+\frac {b^8 (c+d x)^{18}}{d^8}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 b^7 (c+d x)^{18} (b c-a d)}{9 d^9}+\frac {28 b^6 (c+d x)^{17} (b c-a d)^2}{17 d^9}-\frac {7 b^5 (c+d x)^{16} (b c-a d)^3}{2 d^9}+\frac {14 b^4 (c+d x)^{15} (b c-a d)^4}{3 d^9}-\frac {4 b^3 (c+d x)^{14} (b c-a d)^5}{d^9}+\frac {28 b^2 (c+d x)^{13} (b c-a d)^6}{13 d^9}-\frac {2 b (c+d x)^{12} (b c-a d)^7}{3 d^9}+\frac {(c+d x)^{11} (b c-a d)^8}{11 d^9}+\frac {b^8 (c+d x)^{19}}{19 d^9}\) |
((b*c - a*d)^8*(c + d*x)^11)/(11*d^9) - (2*b*(b*c - a*d)^7*(c + d*x)^12)/( 3*d^9) + (28*b^2*(b*c - a*d)^6*(c + d*x)^13)/(13*d^9) - (4*b^3*(b*c - a*d) ^5*(c + d*x)^14)/d^9 + (14*b^4*(b*c - a*d)^4*(c + d*x)^15)/(3*d^9) - (7*b^ 5*(b*c - a*d)^3*(c + d*x)^16)/(2*d^9) + (28*b^6*(b*c - a*d)^2*(c + d*x)^17 )/(17*d^9) - (4*b^7*(b*c - a*d)*(c + d*x)^18)/(9*d^9) + (b^8*(c + d*x)^19) /(19*d^9)
3.14.3.3.1 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. \(1272\) vs. \(2(209)=418\).
Time = 0.21 (sec) , antiderivative size = 1273, normalized size of antiderivative = 5.66
| method | result | size |
| norman | \(\text {Expression too large to display}\) | \(1273\) |
| default | \(\text {Expression too large to display}\) | \(1291\) |
| gosper | \(\text {Expression too large to display}\) | \(1479\) |
| risch | \(\text {Expression too large to display}\) | \(1479\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1479\) |
a^8*c^10*x+(5*a^8*c^9*d+4*a^7*b*c^10)*x^2+(15*a^8*c^8*d^2+80/3*a^7*b*c^9*d +28/3*a^6*b^2*c^10)*x^3+(30*a^8*c^7*d^3+90*a^7*b*c^8*d^2+70*a^6*b^2*c^9*d+ 14*a^5*b^3*c^10)*x^4+(42*a^8*c^6*d^4+192*a^7*b*c^7*d^3+252*a^6*b^2*c^8*d^2 +112*a^5*b^3*c^9*d+14*a^4*b^4*c^10)*x^5+(42*a^8*c^5*d^5+280*a^7*b*c^6*d^4+ 560*a^6*b^2*c^7*d^3+420*a^5*b^3*c^8*d^2+350/3*a^4*b^4*c^9*d+28/3*a^3*b^5*c ^10)*x^6+(30*a^8*c^4*d^6+288*a^7*b*c^5*d^5+840*a^6*b^2*c^6*d^4+960*a^5*b^3 *c^7*d^3+450*a^4*b^4*c^8*d^2+80*a^3*b^5*c^9*d+4*a^2*b^6*c^10)*x^7+(15*a^8* c^3*d^7+210*a^7*b*c^4*d^6+882*a^6*b^2*c^5*d^5+1470*a^5*b^3*c^6*d^4+1050*a^ 4*b^4*c^7*d^3+315*a^3*b^5*c^8*d^2+35*a^2*b^6*c^9*d+a*b^7*c^10)*x^8+(5*a^8* c^2*d^8+320/3*a^7*b*c^3*d^7+1960/3*a^6*b^2*c^4*d^6+1568*a^5*b^3*c^5*d^5+49 00/3*a^4*b^4*c^6*d^4+2240/3*a^3*b^5*c^7*d^3+140*a^2*b^6*c^8*d^2+80/9*a*b^7 *c^9*d+1/9*b^8*c^10)*x^9+(a^8*c*d^9+36*a^7*b*c^2*d^8+336*a^6*b^2*c^3*d^7+1 176*a^5*b^3*c^4*d^6+1764*a^4*b^4*c^5*d^5+1176*a^3*b^5*c^6*d^4+336*a^2*b^6* c^7*d^3+36*a*b^7*c^8*d^2+b^8*c^9*d)*x^10+(1/11*a^8*d^10+80/11*a^7*b*c*d^9+ 1260/11*a^6*b^2*c^2*d^8+6720/11*a^5*b^3*c^3*d^7+14700/11*a^4*b^4*c^4*d^6+1 4112/11*a^3*b^5*c^5*d^5+5880/11*a^2*b^6*c^6*d^4+960/11*a*b^7*c^7*d^3+45/11 *b^8*c^8*d^2)*x^11+(2/3*a^7*b*d^10+70/3*a^6*b^2*c*d^9+210*a^5*b^3*c^2*d^8+ 700*a^4*b^4*c^3*d^7+980*a^3*b^5*c^4*d^6+588*a^2*b^6*c^5*d^5+140*a*b^7*c^6* d^4+10*b^8*c^7*d^3)*x^12+(28/13*a^6*b^2*d^10+560/13*a^5*b^3*c*d^9+3150/13* a^4*b^4*c^2*d^8+6720/13*a^3*b^5*c^3*d^7+5880/13*a^2*b^6*c^4*d^6+2016/13...
Leaf count of result is larger than twice the leaf count of optimal. 1283 vs. \(2 (209) = 418\).
Time = 0.23 (sec) , antiderivative size = 1283, normalized size of antiderivative = 5.70 \[ \int (a+b x)^8 (c+d x)^{10} \, dx=\text {Too large to display} \]
1/19*b^8*d^10*x^19 + a^8*c^10*x + 1/9*(5*b^8*c*d^9 + 4*a*b^7*d^10)*x^18 + 1/17*(45*b^8*c^2*d^8 + 80*a*b^7*c*d^9 + 28*a^2*b^6*d^10)*x^17 + 1/2*(15*b^ 8*c^3*d^7 + 45*a*b^7*c^2*d^8 + 35*a^2*b^6*c*d^9 + 7*a^3*b^5*d^10)*x^16 + 2 /3*(21*b^8*c^4*d^6 + 96*a*b^7*c^3*d^7 + 126*a^2*b^6*c^2*d^8 + 56*a^3*b^5*c *d^9 + 7*a^4*b^4*d^10)*x^15 + 2*(9*b^8*c^5*d^5 + 60*a*b^7*c^4*d^6 + 120*a^ 2*b^6*c^3*d^7 + 90*a^3*b^5*c^2*d^8 + 25*a^4*b^4*c*d^9 + 2*a^5*b^3*d^10)*x^ 14 + 14/13*(15*b^8*c^6*d^4 + 144*a*b^7*c^5*d^5 + 420*a^2*b^6*c^4*d^6 + 480 *a^3*b^5*c^3*d^7 + 225*a^4*b^4*c^2*d^8 + 40*a^5*b^3*c*d^9 + 2*a^6*b^2*d^10 )*x^13 + 2/3*(15*b^8*c^7*d^3 + 210*a*b^7*c^6*d^4 + 882*a^2*b^6*c^5*d^5 + 1 470*a^3*b^5*c^4*d^6 + 1050*a^4*b^4*c^3*d^7 + 315*a^5*b^3*c^2*d^8 + 35*a^6* b^2*c*d^9 + a^7*b*d^10)*x^12 + 1/11*(45*b^8*c^8*d^2 + 960*a*b^7*c^7*d^3 + 5880*a^2*b^6*c^6*d^4 + 14112*a^3*b^5*c^5*d^5 + 14700*a^4*b^4*c^4*d^6 + 672 0*a^5*b^3*c^3*d^7 + 1260*a^6*b^2*c^2*d^8 + 80*a^7*b*c*d^9 + a^8*d^10)*x^11 + (b^8*c^9*d + 36*a*b^7*c^8*d^2 + 336*a^2*b^6*c^7*d^3 + 1176*a^3*b^5*c^6* d^4 + 1764*a^4*b^4*c^5*d^5 + 1176*a^5*b^3*c^4*d^6 + 336*a^6*b^2*c^3*d^7 + 36*a^7*b*c^2*d^8 + a^8*c*d^9)*x^10 + 1/9*(b^8*c^10 + 80*a*b^7*c^9*d + 1260 *a^2*b^6*c^8*d^2 + 6720*a^3*b^5*c^7*d^3 + 14700*a^4*b^4*c^6*d^4 + 14112*a^ 5*b^3*c^5*d^5 + 5880*a^6*b^2*c^4*d^6 + 960*a^7*b*c^3*d^7 + 45*a^8*c^2*d^8) *x^9 + (a*b^7*c^10 + 35*a^2*b^6*c^9*d + 315*a^3*b^5*c^8*d^2 + 1050*a^4*b^4 *c^7*d^3 + 1470*a^5*b^3*c^6*d^4 + 882*a^6*b^2*c^5*d^5 + 210*a^7*b*c^4*d...
Leaf count of result is larger than twice the leaf count of optimal. 1428 vs. \(2 (207) = 414\).
Time = 0.11 (sec) , antiderivative size = 1428, normalized size of antiderivative = 6.35 \[ \int (a+b x)^8 (c+d x)^{10} \, dx=\text {Too large to display} \]
a**8*c**10*x + b**8*d**10*x**19/19 + x**18*(4*a*b**7*d**10/9 + 5*b**8*c*d* *9/9) + x**17*(28*a**2*b**6*d**10/17 + 80*a*b**7*c*d**9/17 + 45*b**8*c**2* d**8/17) + x**16*(7*a**3*b**5*d**10/2 + 35*a**2*b**6*c*d**9/2 + 45*a*b**7* c**2*d**8/2 + 15*b**8*c**3*d**7/2) + x**15*(14*a**4*b**4*d**10/3 + 112*a** 3*b**5*c*d**9/3 + 84*a**2*b**6*c**2*d**8 + 64*a*b**7*c**3*d**7 + 14*b**8*c **4*d**6) + x**14*(4*a**5*b**3*d**10 + 50*a**4*b**4*c*d**9 + 180*a**3*b**5 *c**2*d**8 + 240*a**2*b**6*c**3*d**7 + 120*a*b**7*c**4*d**6 + 18*b**8*c**5 *d**5) + x**13*(28*a**6*b**2*d**10/13 + 560*a**5*b**3*c*d**9/13 + 3150*a** 4*b**4*c**2*d**8/13 + 6720*a**3*b**5*c**3*d**7/13 + 5880*a**2*b**6*c**4*d* *6/13 + 2016*a*b**7*c**5*d**5/13 + 210*b**8*c**6*d**4/13) + x**12*(2*a**7* b*d**10/3 + 70*a**6*b**2*c*d**9/3 + 210*a**5*b**3*c**2*d**8 + 700*a**4*b** 4*c**3*d**7 + 980*a**3*b**5*c**4*d**6 + 588*a**2*b**6*c**5*d**5 + 140*a*b* *7*c**6*d**4 + 10*b**8*c**7*d**3) + x**11*(a**8*d**10/11 + 80*a**7*b*c*d** 9/11 + 1260*a**6*b**2*c**2*d**8/11 + 6720*a**5*b**3*c**3*d**7/11 + 14700*a **4*b**4*c**4*d**6/11 + 14112*a**3*b**5*c**5*d**5/11 + 5880*a**2*b**6*c**6 *d**4/11 + 960*a*b**7*c**7*d**3/11 + 45*b**8*c**8*d**2/11) + x**10*(a**8*c *d**9 + 36*a**7*b*c**2*d**8 + 336*a**6*b**2*c**3*d**7 + 1176*a**5*b**3*c** 4*d**6 + 1764*a**4*b**4*c**5*d**5 + 1176*a**3*b**5*c**6*d**4 + 336*a**2*b* *6*c**7*d**3 + 36*a*b**7*c**8*d**2 + b**8*c**9*d) + x**9*(5*a**8*c**2*d**8 + 320*a**7*b*c**3*d**7/3 + 1960*a**6*b**2*c**4*d**6/3 + 1568*a**5*b**3...
Leaf count of result is larger than twice the leaf count of optimal. 1283 vs. \(2 (209) = 418\).
Time = 0.22 (sec) , antiderivative size = 1283, normalized size of antiderivative = 5.70 \[ \int (a+b x)^8 (c+d x)^{10} \, dx=\text {Too large to display} \]
1/19*b^8*d^10*x^19 + a^8*c^10*x + 1/9*(5*b^8*c*d^9 + 4*a*b^7*d^10)*x^18 + 1/17*(45*b^8*c^2*d^8 + 80*a*b^7*c*d^9 + 28*a^2*b^6*d^10)*x^17 + 1/2*(15*b^ 8*c^3*d^7 + 45*a*b^7*c^2*d^8 + 35*a^2*b^6*c*d^9 + 7*a^3*b^5*d^10)*x^16 + 2 /3*(21*b^8*c^4*d^6 + 96*a*b^7*c^3*d^7 + 126*a^2*b^6*c^2*d^8 + 56*a^3*b^5*c *d^9 + 7*a^4*b^4*d^10)*x^15 + 2*(9*b^8*c^5*d^5 + 60*a*b^7*c^4*d^6 + 120*a^ 2*b^6*c^3*d^7 + 90*a^3*b^5*c^2*d^8 + 25*a^4*b^4*c*d^9 + 2*a^5*b^3*d^10)*x^ 14 + 14/13*(15*b^8*c^6*d^4 + 144*a*b^7*c^5*d^5 + 420*a^2*b^6*c^4*d^6 + 480 *a^3*b^5*c^3*d^7 + 225*a^4*b^4*c^2*d^8 + 40*a^5*b^3*c*d^9 + 2*a^6*b^2*d^10 )*x^13 + 2/3*(15*b^8*c^7*d^3 + 210*a*b^7*c^6*d^4 + 882*a^2*b^6*c^5*d^5 + 1 470*a^3*b^5*c^4*d^6 + 1050*a^4*b^4*c^3*d^7 + 315*a^5*b^3*c^2*d^8 + 35*a^6* b^2*c*d^9 + a^7*b*d^10)*x^12 + 1/11*(45*b^8*c^8*d^2 + 960*a*b^7*c^7*d^3 + 5880*a^2*b^6*c^6*d^4 + 14112*a^3*b^5*c^5*d^5 + 14700*a^4*b^4*c^4*d^6 + 672 0*a^5*b^3*c^3*d^7 + 1260*a^6*b^2*c^2*d^8 + 80*a^7*b*c*d^9 + a^8*d^10)*x^11 + (b^8*c^9*d + 36*a*b^7*c^8*d^2 + 336*a^2*b^6*c^7*d^3 + 1176*a^3*b^5*c^6* d^4 + 1764*a^4*b^4*c^5*d^5 + 1176*a^5*b^3*c^4*d^6 + 336*a^6*b^2*c^3*d^7 + 36*a^7*b*c^2*d^8 + a^8*c*d^9)*x^10 + 1/9*(b^8*c^10 + 80*a*b^7*c^9*d + 1260 *a^2*b^6*c^8*d^2 + 6720*a^3*b^5*c^7*d^3 + 14700*a^4*b^4*c^6*d^4 + 14112*a^ 5*b^3*c^5*d^5 + 5880*a^6*b^2*c^4*d^6 + 960*a^7*b*c^3*d^7 + 45*a^8*c^2*d^8) *x^9 + (a*b^7*c^10 + 35*a^2*b^6*c^9*d + 315*a^3*b^5*c^8*d^2 + 1050*a^4*b^4 *c^7*d^3 + 1470*a^5*b^3*c^6*d^4 + 882*a^6*b^2*c^5*d^5 + 210*a^7*b*c^4*d...
Leaf count of result is larger than twice the leaf count of optimal. 1478 vs. \(2 (209) = 418\).
Time = 0.28 (sec) , antiderivative size = 1478, normalized size of antiderivative = 6.57 \[ \int (a+b x)^8 (c+d x)^{10} \, dx=\text {Too large to display} \]
1/19*b^8*d^10*x^19 + 5/9*b^8*c*d^9*x^18 + 4/9*a*b^7*d^10*x^18 + 45/17*b^8* c^2*d^8*x^17 + 80/17*a*b^7*c*d^9*x^17 + 28/17*a^2*b^6*d^10*x^17 + 15/2*b^8 *c^3*d^7*x^16 + 45/2*a*b^7*c^2*d^8*x^16 + 35/2*a^2*b^6*c*d^9*x^16 + 7/2*a^ 3*b^5*d^10*x^16 + 14*b^8*c^4*d^6*x^15 + 64*a*b^7*c^3*d^7*x^15 + 84*a^2*b^6 *c^2*d^8*x^15 + 112/3*a^3*b^5*c*d^9*x^15 + 14/3*a^4*b^4*d^10*x^15 + 18*b^8 *c^5*d^5*x^14 + 120*a*b^7*c^4*d^6*x^14 + 240*a^2*b^6*c^3*d^7*x^14 + 180*a^ 3*b^5*c^2*d^8*x^14 + 50*a^4*b^4*c*d^9*x^14 + 4*a^5*b^3*d^10*x^14 + 210/13* b^8*c^6*d^4*x^13 + 2016/13*a*b^7*c^5*d^5*x^13 + 5880/13*a^2*b^6*c^4*d^6*x^ 13 + 6720/13*a^3*b^5*c^3*d^7*x^13 + 3150/13*a^4*b^4*c^2*d^8*x^13 + 560/13* a^5*b^3*c*d^9*x^13 + 28/13*a^6*b^2*d^10*x^13 + 10*b^8*c^7*d^3*x^12 + 140*a *b^7*c^6*d^4*x^12 + 588*a^2*b^6*c^5*d^5*x^12 + 980*a^3*b^5*c^4*d^6*x^12 + 700*a^4*b^4*c^3*d^7*x^12 + 210*a^5*b^3*c^2*d^8*x^12 + 70/3*a^6*b^2*c*d^9*x ^12 + 2/3*a^7*b*d^10*x^12 + 45/11*b^8*c^8*d^2*x^11 + 960/11*a*b^7*c^7*d^3* x^11 + 5880/11*a^2*b^6*c^6*d^4*x^11 + 14112/11*a^3*b^5*c^5*d^5*x^11 + 1470 0/11*a^4*b^4*c^4*d^6*x^11 + 6720/11*a^5*b^3*c^3*d^7*x^11 + 1260/11*a^6*b^2 *c^2*d^8*x^11 + 80/11*a^7*b*c*d^9*x^11 + 1/11*a^8*d^10*x^11 + b^8*c^9*d*x^ 10 + 36*a*b^7*c^8*d^2*x^10 + 336*a^2*b^6*c^7*d^3*x^10 + 1176*a^3*b^5*c^6*d ^4*x^10 + 1764*a^4*b^4*c^5*d^5*x^10 + 1176*a^5*b^3*c^4*d^6*x^10 + 336*a^6* b^2*c^3*d^7*x^10 + 36*a^7*b*c^2*d^8*x^10 + a^8*c*d^9*x^10 + 1/9*b^8*c^10*x ^9 + 80/9*a*b^7*c^9*d*x^9 + 140*a^2*b^6*c^8*d^2*x^9 + 2240/3*a^3*b^5*c^...
Time = 0.78 (sec) , antiderivative size = 1253, normalized size of antiderivative = 5.57 \[ \int (a+b x)^8 (c+d x)^{10} \, dx=x^7\,\left (30\,a^8\,c^4\,d^6+288\,a^7\,b\,c^5\,d^5+840\,a^6\,b^2\,c^6\,d^4+960\,a^5\,b^3\,c^7\,d^3+450\,a^4\,b^4\,c^8\,d^2+80\,a^3\,b^5\,c^9\,d+4\,a^2\,b^6\,c^{10}\right )+x^{13}\,\left (\frac {28\,a^6\,b^2\,d^{10}}{13}+\frac {560\,a^5\,b^3\,c\,d^9}{13}+\frac {3150\,a^4\,b^4\,c^2\,d^8}{13}+\frac {6720\,a^3\,b^5\,c^3\,d^7}{13}+\frac {5880\,a^2\,b^6\,c^4\,d^6}{13}+\frac {2016\,a\,b^7\,c^5\,d^5}{13}+\frac {210\,b^8\,c^6\,d^4}{13}\right )+x^8\,\left (15\,a^8\,c^3\,d^7+210\,a^7\,b\,c^4\,d^6+882\,a^6\,b^2\,c^5\,d^5+1470\,a^5\,b^3\,c^6\,d^4+1050\,a^4\,b^4\,c^7\,d^3+315\,a^3\,b^5\,c^8\,d^2+35\,a^2\,b^6\,c^9\,d+a\,b^7\,c^{10}\right )+x^{12}\,\left (\frac {2\,a^7\,b\,d^{10}}{3}+\frac {70\,a^6\,b^2\,c\,d^9}{3}+210\,a^5\,b^3\,c^2\,d^8+700\,a^4\,b^4\,c^3\,d^7+980\,a^3\,b^5\,c^4\,d^6+588\,a^2\,b^6\,c^5\,d^5+140\,a\,b^7\,c^6\,d^4+10\,b^8\,c^7\,d^3\right )+x^{10}\,\left (a^8\,c\,d^9+36\,a^7\,b\,c^2\,d^8+336\,a^6\,b^2\,c^3\,d^7+1176\,a^5\,b^3\,c^4\,d^6+1764\,a^4\,b^4\,c^5\,d^5+1176\,a^3\,b^5\,c^6\,d^4+336\,a^2\,b^6\,c^7\,d^3+36\,a\,b^7\,c^8\,d^2+b^8\,c^9\,d\right )+x^5\,\left (42\,a^8\,c^6\,d^4+192\,a^7\,b\,c^7\,d^3+252\,a^6\,b^2\,c^8\,d^2+112\,a^5\,b^3\,c^9\,d+14\,a^4\,b^4\,c^{10}\right )+x^{15}\,\left (\frac {14\,a^4\,b^4\,d^{10}}{3}+\frac {112\,a^3\,b^5\,c\,d^9}{3}+84\,a^2\,b^6\,c^2\,d^8+64\,a\,b^7\,c^3\,d^7+14\,b^8\,c^4\,d^6\right )+x^6\,\left (42\,a^8\,c^5\,d^5+280\,a^7\,b\,c^6\,d^4+560\,a^6\,b^2\,c^7\,d^3+420\,a^5\,b^3\,c^8\,d^2+\frac {350\,a^4\,b^4\,c^9\,d}{3}+\frac {28\,a^3\,b^5\,c^{10}}{3}\right )+x^{14}\,\left (4\,a^5\,b^3\,d^{10}+50\,a^4\,b^4\,c\,d^9+180\,a^3\,b^5\,c^2\,d^8+240\,a^2\,b^6\,c^3\,d^7+120\,a\,b^7\,c^4\,d^6+18\,b^8\,c^5\,d^5\right )+x^9\,\left (5\,a^8\,c^2\,d^8+\frac {320\,a^7\,b\,c^3\,d^7}{3}+\frac {1960\,a^6\,b^2\,c^4\,d^6}{3}+1568\,a^5\,b^3\,c^5\,d^5+\frac {4900\,a^4\,b^4\,c^6\,d^4}{3}+\frac {2240\,a^3\,b^5\,c^7\,d^3}{3}+140\,a^2\,b^6\,c^8\,d^2+\frac {80\,a\,b^7\,c^9\,d}{9}+\frac {b^8\,c^{10}}{9}\right )+x^{11}\,\left (\frac {a^8\,d^{10}}{11}+\frac {80\,a^7\,b\,c\,d^9}{11}+\frac {1260\,a^6\,b^2\,c^2\,d^8}{11}+\frac {6720\,a^5\,b^3\,c^3\,d^7}{11}+\frac {14700\,a^4\,b^4\,c^4\,d^6}{11}+\frac {14112\,a^3\,b^5\,c^5\,d^5}{11}+\frac {5880\,a^2\,b^6\,c^6\,d^4}{11}+\frac {960\,a\,b^7\,c^7\,d^3}{11}+\frac {45\,b^8\,c^8\,d^2}{11}\right )+a^8\,c^{10}\,x+\frac {b^8\,d^{10}\,x^{19}}{19}+2\,a^5\,c^7\,x^4\,\left (15\,a^3\,d^3+45\,a^2\,b\,c\,d^2+35\,a\,b^2\,c^2\,d+7\,b^3\,c^3\right )+\frac {b^5\,d^7\,x^{16}\,\left (7\,a^3\,d^3+35\,a^2\,b\,c\,d^2+45\,a\,b^2\,c^2\,d+15\,b^3\,c^3\right )}{2}+a^7\,c^9\,x^2\,\left (5\,a\,d+4\,b\,c\right )+\frac {b^7\,d^9\,x^{18}\,\left (4\,a\,d+5\,b\,c\right )}{9}+\frac {a^6\,c^8\,x^3\,\left (45\,a^2\,d^2+80\,a\,b\,c\,d+28\,b^2\,c^2\right )}{3}+\frac {b^6\,d^8\,x^{17}\,\left (28\,a^2\,d^2+80\,a\,b\,c\,d+45\,b^2\,c^2\right )}{17} \]
x^7*(4*a^2*b^6*c^10 + 30*a^8*c^4*d^6 + 80*a^3*b^5*c^9*d + 288*a^7*b*c^5*d^ 5 + 450*a^4*b^4*c^8*d^2 + 960*a^5*b^3*c^7*d^3 + 840*a^6*b^2*c^6*d^4) + x^1 3*((28*a^6*b^2*d^10)/13 + (210*b^8*c^6*d^4)/13 + (2016*a*b^7*c^5*d^5)/13 + (560*a^5*b^3*c*d^9)/13 + (5880*a^2*b^6*c^4*d^6)/13 + (6720*a^3*b^5*c^3*d^ 7)/13 + (3150*a^4*b^4*c^2*d^8)/13) + x^8*(a*b^7*c^10 + 15*a^8*c^3*d^7 + 35 *a^2*b^6*c^9*d + 210*a^7*b*c^4*d^6 + 315*a^3*b^5*c^8*d^2 + 1050*a^4*b^4*c^ 7*d^3 + 1470*a^5*b^3*c^6*d^4 + 882*a^6*b^2*c^5*d^5) + x^12*((2*a^7*b*d^10) /3 + 10*b^8*c^7*d^3 + 140*a*b^7*c^6*d^4 + (70*a^6*b^2*c*d^9)/3 + 588*a^2*b ^6*c^5*d^5 + 980*a^3*b^5*c^4*d^6 + 700*a^4*b^4*c^3*d^7 + 210*a^5*b^3*c^2*d ^8) + x^10*(a^8*c*d^9 + b^8*c^9*d + 36*a*b^7*c^8*d^2 + 36*a^7*b*c^2*d^8 + 336*a^2*b^6*c^7*d^3 + 1176*a^3*b^5*c^6*d^4 + 1764*a^4*b^4*c^5*d^5 + 1176*a ^5*b^3*c^4*d^6 + 336*a^6*b^2*c^3*d^7) + x^5*(14*a^4*b^4*c^10 + 42*a^8*c^6* d^4 + 112*a^5*b^3*c^9*d + 192*a^7*b*c^7*d^3 + 252*a^6*b^2*c^8*d^2) + x^15* ((14*a^4*b^4*d^10)/3 + 14*b^8*c^4*d^6 + 64*a*b^7*c^3*d^7 + (112*a^3*b^5*c* d^9)/3 + 84*a^2*b^6*c^2*d^8) + x^6*((28*a^3*b^5*c^10)/3 + 42*a^8*c^5*d^5 + (350*a^4*b^4*c^9*d)/3 + 280*a^7*b*c^6*d^4 + 420*a^5*b^3*c^8*d^2 + 560*a^6 *b^2*c^7*d^3) + x^14*(4*a^5*b^3*d^10 + 18*b^8*c^5*d^5 + 120*a*b^7*c^4*d^6 + 50*a^4*b^4*c*d^9 + 240*a^2*b^6*c^3*d^7 + 180*a^3*b^5*c^2*d^8) + x^9*((b^ 8*c^10)/9 + 5*a^8*c^2*d^8 + (320*a^7*b*c^3*d^7)/3 + 140*a^2*b^6*c^8*d^2 + (2240*a^3*b^5*c^7*d^3)/3 + (4900*a^4*b^4*c^6*d^4)/3 + 1568*a^5*b^3*c^5*...