Integrand size = 13, antiderivative size = 38 \[ \int (a+b x) (c+d x)^{16} \, dx=-\frac {(b c-a d) (c+d x)^{17}}{17 d^2}+\frac {b (c+d x)^{18}}{18 d^2} \]
Leaf count is larger than twice the leaf count of optimal. \(342\) vs. \(2(38)=76\).
Time = 0.03 (sec) , antiderivative size = 342, normalized size of antiderivative = 9.00 \[ \int (a+b x) (c+d x)^{16} \, dx=a c^{16} x+\frac {1}{2} c^{15} (b c+16 a d) x^2+\frac {8}{3} c^{14} d (2 b c+15 a d) x^3+10 c^{13} d^2 (3 b c+14 a d) x^4+28 c^{12} d^3 (4 b c+13 a d) x^5+\frac {182}{3} c^{11} d^4 (5 b c+12 a d) x^6+104 c^{10} d^5 (6 b c+11 a d) x^7+143 c^9 d^6 (7 b c+10 a d) x^8+\frac {1430}{9} c^8 d^7 (8 b c+9 a d) x^9+143 c^7 d^8 (9 b c+8 a d) x^{10}+104 c^6 d^9 (10 b c+7 a d) x^{11}+\frac {182}{3} c^5 d^{10} (11 b c+6 a d) x^{12}+28 c^4 d^{11} (12 b c+5 a d) x^{13}+10 c^3 d^{12} (13 b c+4 a d) x^{14}+\frac {8}{3} c^2 d^{13} (14 b c+3 a d) x^{15}+\frac {1}{2} c d^{14} (15 b c+2 a d) x^{16}+\frac {1}{17} d^{15} (16 b c+a d) x^{17}+\frac {1}{18} b d^{16} x^{18} \]
a*c^16*x + (c^15*(b*c + 16*a*d)*x^2)/2 + (8*c^14*d*(2*b*c + 15*a*d)*x^3)/3 + 10*c^13*d^2*(3*b*c + 14*a*d)*x^4 + 28*c^12*d^3*(4*b*c + 13*a*d)*x^5 + ( 182*c^11*d^4*(5*b*c + 12*a*d)*x^6)/3 + 104*c^10*d^5*(6*b*c + 11*a*d)*x^7 + 143*c^9*d^6*(7*b*c + 10*a*d)*x^8 + (1430*c^8*d^7*(8*b*c + 9*a*d)*x^9)/9 + 143*c^7*d^8*(9*b*c + 8*a*d)*x^10 + 104*c^6*d^9*(10*b*c + 7*a*d)*x^11 + (1 82*c^5*d^10*(11*b*c + 6*a*d)*x^12)/3 + 28*c^4*d^11*(12*b*c + 5*a*d)*x^13 + 10*c^3*d^12*(13*b*c + 4*a*d)*x^14 + (8*c^2*d^13*(14*b*c + 3*a*d)*x^15)/3 + (c*d^14*(15*b*c + 2*a*d)*x^16)/2 + (d^15*(16*b*c + a*d)*x^17)/17 + (b*d^ 16*x^18)/18
Time = 0.19 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, 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.
| \(\displaystyle \int (a+b x) (c+d x)^{16} \, dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (\frac {(c+d x)^{16} (a d-b c)}{d}+\frac {b (c+d x)^{17}}{d}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b (c+d x)^{18}}{18 d^2}-\frac {(c+d x)^{17} (b c-a d)}{17 d^2}\) |
3.2.72.3.1 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. \(369\) vs. \(2(34)=68\).
Time = 0.41 (sec) , antiderivative size = 370, normalized size of antiderivative = 9.74
| method | result | size |
| norman | \(a \,c^{16} x +\left (8 a \,c^{15} d +\frac {1}{2} b \,c^{16}\right ) x^{2}+\left (40 a \,c^{14} d^{2}+\frac {16}{3} b \,c^{15} d \right ) x^{3}+\left (140 a \,c^{13} d^{3}+30 b \,c^{14} d^{2}\right ) x^{4}+\left (364 a \,c^{12} d^{4}+112 b \,c^{13} d^{3}\right ) x^{5}+\left (728 a \,c^{11} d^{5}+\frac {910}{3} b \,c^{12} d^{4}\right ) x^{6}+\left (1144 a \,c^{10} d^{6}+624 b \,c^{11} d^{5}\right ) x^{7}+\left (1430 a \,c^{9} d^{7}+1001 b \,c^{10} d^{6}\right ) x^{8}+\left (1430 a \,c^{8} d^{8}+\frac {11440}{9} b \,c^{9} d^{7}\right ) x^{9}+\left (1144 a \,c^{7} d^{9}+1287 b \,c^{8} d^{8}\right ) x^{10}+\left (728 a \,c^{6} d^{10}+1040 b \,c^{7} d^{9}\right ) x^{11}+\left (364 a \,c^{5} d^{11}+\frac {2002}{3} b \,c^{6} d^{10}\right ) x^{12}+\left (140 a \,c^{4} d^{12}+336 b \,c^{5} d^{11}\right ) x^{13}+\left (40 a \,c^{3} d^{13}+130 b \,c^{4} d^{12}\right ) x^{14}+\left (8 a \,c^{2} d^{14}+\frac {112}{3} b \,c^{3} d^{13}\right ) x^{15}+\left (a c \,d^{15}+\frac {15}{2} b \,c^{2} d^{14}\right ) x^{16}+\left (\frac {1}{17} a \,d^{16}+\frac {16}{17} b c \,d^{15}\right ) x^{17}+\frac {b \,d^{16} x^{18}}{18}\) | \(370\) |
| default | \(\frac {b \,d^{16} x^{18}}{18}+\frac {\left (a \,d^{16}+16 b c \,d^{15}\right ) x^{17}}{17}+\frac {\left (16 a c \,d^{15}+120 b \,c^{2} d^{14}\right ) x^{16}}{16}+\frac {\left (120 a \,c^{2} d^{14}+560 b \,c^{3} d^{13}\right ) x^{15}}{15}+\frac {\left (560 a \,c^{3} d^{13}+1820 b \,c^{4} d^{12}\right ) x^{14}}{14}+\frac {\left (1820 a \,c^{4} d^{12}+4368 b \,c^{5} d^{11}\right ) x^{13}}{13}+\frac {\left (4368 a \,c^{5} d^{11}+8008 b \,c^{6} d^{10}\right ) x^{12}}{12}+\frac {\left (8008 a \,c^{6} d^{10}+11440 b \,c^{7} d^{9}\right ) x^{11}}{11}+\frac {\left (11440 a \,c^{7} d^{9}+12870 b \,c^{8} d^{8}\right ) x^{10}}{10}+\frac {\left (12870 a \,c^{8} d^{8}+11440 b \,c^{9} d^{7}\right ) x^{9}}{9}+\frac {\left (11440 a \,c^{9} d^{7}+8008 b \,c^{10} d^{6}\right ) x^{8}}{8}+\frac {\left (8008 a \,c^{10} d^{6}+4368 b \,c^{11} d^{5}\right ) x^{7}}{7}+\frac {\left (4368 a \,c^{11} d^{5}+1820 b \,c^{12} d^{4}\right ) x^{6}}{6}+\frac {\left (1820 a \,c^{12} d^{4}+560 b \,c^{13} d^{3}\right ) x^{5}}{5}+\frac {\left (560 a \,c^{13} d^{3}+120 b \,c^{14} d^{2}\right ) x^{4}}{4}+\frac {\left (120 a \,c^{14} d^{2}+16 b \,c^{15} d \right ) x^{3}}{3}+\frac {\left (16 a \,c^{15} d +b \,c^{16}\right ) x^{2}}{2}+a \,c^{16} x\) | \(385\) |
| gosper | \(\frac {1}{18} b \,d^{16} x^{18}+a \,c^{16} x +\frac {1}{2} x^{2} b \,c^{16}+\frac {1}{17} x^{17} a \,d^{16}+336 b \,c^{5} d^{11} x^{13}+40 a \,c^{3} d^{13} x^{14}+130 b \,c^{4} d^{12} x^{14}+1287 b \,c^{8} d^{8} x^{10}+728 a \,c^{6} d^{10} x^{11}+1040 b \,c^{7} d^{9} x^{11}+140 a \,c^{4} d^{12} x^{13}+624 b \,c^{11} d^{5} x^{7}+1430 a \,c^{9} d^{7} x^{8}+1001 b \,c^{10} d^{6} x^{8}+1144 a \,c^{7} d^{9} x^{10}+30 b \,c^{14} d^{2} x^{4}+364 a \,c^{12} d^{4} x^{5}+112 b \,c^{13} d^{3} x^{5}+1144 a \,c^{10} d^{6} x^{7}+\frac {15}{2} x^{16} b \,c^{2} d^{14}+\frac {16}{17} x^{17} b c \,d^{15}+140 a \,c^{13} d^{3} x^{4}+\frac {910}{3} x^{6} b \,c^{12} d^{4}+1430 x^{9} a \,c^{8} d^{8}+\frac {11440}{9} x^{9} b \,c^{9} d^{7}+364 x^{12} a \,c^{5} d^{11}+\frac {2002}{3} x^{12} b \,c^{6} d^{10}+8 x^{15} a \,c^{2} d^{14}+\frac {112}{3} x^{15} b \,c^{3} d^{13}+x^{16} a c \,d^{15}+8 x^{2} a \,c^{15} d +40 x^{3} a \,c^{14} d^{2}+\frac {16}{3} x^{3} b \,c^{15} d +728 x^{6} a \,c^{11} d^{5}\) | \(386\) |
| risch | \(\frac {1}{18} b \,d^{16} x^{18}+a \,c^{16} x +\frac {1}{2} x^{2} b \,c^{16}+\frac {1}{17} x^{17} a \,d^{16}+336 b \,c^{5} d^{11} x^{13}+40 a \,c^{3} d^{13} x^{14}+130 b \,c^{4} d^{12} x^{14}+1287 b \,c^{8} d^{8} x^{10}+728 a \,c^{6} d^{10} x^{11}+1040 b \,c^{7} d^{9} x^{11}+140 a \,c^{4} d^{12} x^{13}+624 b \,c^{11} d^{5} x^{7}+1430 a \,c^{9} d^{7} x^{8}+1001 b \,c^{10} d^{6} x^{8}+1144 a \,c^{7} d^{9} x^{10}+30 b \,c^{14} d^{2} x^{4}+364 a \,c^{12} d^{4} x^{5}+112 b \,c^{13} d^{3} x^{5}+1144 a \,c^{10} d^{6} x^{7}+\frac {15}{2} x^{16} b \,c^{2} d^{14}+\frac {16}{17} x^{17} b c \,d^{15}+140 a \,c^{13} d^{3} x^{4}+\frac {910}{3} x^{6} b \,c^{12} d^{4}+1430 x^{9} a \,c^{8} d^{8}+\frac {11440}{9} x^{9} b \,c^{9} d^{7}+364 x^{12} a \,c^{5} d^{11}+\frac {2002}{3} x^{12} b \,c^{6} d^{10}+8 x^{15} a \,c^{2} d^{14}+\frac {112}{3} x^{15} b \,c^{3} d^{13}+x^{16} a c \,d^{15}+8 x^{2} a \,c^{15} d +40 x^{3} a \,c^{14} d^{2}+\frac {16}{3} x^{3} b \,c^{15} d +728 x^{6} a \,c^{11} d^{5}\) | \(386\) |
| parallelrisch | \(\frac {1}{18} b \,d^{16} x^{18}+a \,c^{16} x +\frac {1}{2} x^{2} b \,c^{16}+\frac {1}{17} x^{17} a \,d^{16}+336 b \,c^{5} d^{11} x^{13}+40 a \,c^{3} d^{13} x^{14}+130 b \,c^{4} d^{12} x^{14}+1287 b \,c^{8} d^{8} x^{10}+728 a \,c^{6} d^{10} x^{11}+1040 b \,c^{7} d^{9} x^{11}+140 a \,c^{4} d^{12} x^{13}+624 b \,c^{11} d^{5} x^{7}+1430 a \,c^{9} d^{7} x^{8}+1001 b \,c^{10} d^{6} x^{8}+1144 a \,c^{7} d^{9} x^{10}+30 b \,c^{14} d^{2} x^{4}+364 a \,c^{12} d^{4} x^{5}+112 b \,c^{13} d^{3} x^{5}+1144 a \,c^{10} d^{6} x^{7}+\frac {15}{2} x^{16} b \,c^{2} d^{14}+\frac {16}{17} x^{17} b c \,d^{15}+140 a \,c^{13} d^{3} x^{4}+\frac {910}{3} x^{6} b \,c^{12} d^{4}+1430 x^{9} a \,c^{8} d^{8}+\frac {11440}{9} x^{9} b \,c^{9} d^{7}+364 x^{12} a \,c^{5} d^{11}+\frac {2002}{3} x^{12} b \,c^{6} d^{10}+8 x^{15} a \,c^{2} d^{14}+\frac {112}{3} x^{15} b \,c^{3} d^{13}+x^{16} a c \,d^{15}+8 x^{2} a \,c^{15} d +40 x^{3} a \,c^{14} d^{2}+\frac {16}{3} x^{3} b \,c^{15} d +728 x^{6} a \,c^{11} d^{5}\) | \(386\) |
a*c^16*x+(8*a*c^15*d+1/2*b*c^16)*x^2+(40*a*c^14*d^2+16/3*b*c^15*d)*x^3+(14 0*a*c^13*d^3+30*b*c^14*d^2)*x^4+(364*a*c^12*d^4+112*b*c^13*d^3)*x^5+(728*a *c^11*d^5+910/3*b*c^12*d^4)*x^6+(1144*a*c^10*d^6+624*b*c^11*d^5)*x^7+(1430 *a*c^9*d^7+1001*b*c^10*d^6)*x^8+(1430*a*c^8*d^8+11440/9*b*c^9*d^7)*x^9+(11 44*a*c^7*d^9+1287*b*c^8*d^8)*x^10+(728*a*c^6*d^10+1040*b*c^7*d^9)*x^11+(36 4*a*c^5*d^11+2002/3*b*c^6*d^10)*x^12+(140*a*c^4*d^12+336*b*c^5*d^11)*x^13+ (40*a*c^3*d^13+130*b*c^4*d^12)*x^14+(8*a*c^2*d^14+112/3*b*c^3*d^13)*x^15+( a*c*d^15+15/2*b*c^2*d^14)*x^16+(1/17*a*d^16+16/17*b*c*d^15)*x^17+1/18*b*d^ 16*x^18
Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (34) = 68\).
Time = 0.22 (sec) , antiderivative size = 384, normalized size of antiderivative = 10.11 \[ \int (a+b x) (c+d x)^{16} \, dx=\frac {1}{18} \, b d^{16} x^{18} + a c^{16} x + \frac {1}{17} \, {\left (16 \, b c d^{15} + a d^{16}\right )} x^{17} + \frac {1}{2} \, {\left (15 \, b c^{2} d^{14} + 2 \, a c d^{15}\right )} x^{16} + \frac {8}{3} \, {\left (14 \, b c^{3} d^{13} + 3 \, a c^{2} d^{14}\right )} x^{15} + 10 \, {\left (13 \, b c^{4} d^{12} + 4 \, a c^{3} d^{13}\right )} x^{14} + 28 \, {\left (12 \, b c^{5} d^{11} + 5 \, a c^{4} d^{12}\right )} x^{13} + \frac {182}{3} \, {\left (11 \, b c^{6} d^{10} + 6 \, a c^{5} d^{11}\right )} x^{12} + 104 \, {\left (10 \, b c^{7} d^{9} + 7 \, a c^{6} d^{10}\right )} x^{11} + 143 \, {\left (9 \, b c^{8} d^{8} + 8 \, a c^{7} d^{9}\right )} x^{10} + \frac {1430}{9} \, {\left (8 \, b c^{9} d^{7} + 9 \, a c^{8} d^{8}\right )} x^{9} + 143 \, {\left (7 \, b c^{10} d^{6} + 10 \, a c^{9} d^{7}\right )} x^{8} + 104 \, {\left (6 \, b c^{11} d^{5} + 11 \, a c^{10} d^{6}\right )} x^{7} + \frac {182}{3} \, {\left (5 \, b c^{12} d^{4} + 12 \, a c^{11} d^{5}\right )} x^{6} + 28 \, {\left (4 \, b c^{13} d^{3} + 13 \, a c^{12} d^{4}\right )} x^{5} + 10 \, {\left (3 \, b c^{14} d^{2} + 14 \, a c^{13} d^{3}\right )} x^{4} + \frac {8}{3} \, {\left (2 \, b c^{15} d + 15 \, a c^{14} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b c^{16} + 16 \, a c^{15} d\right )} x^{2} \]
1/18*b*d^16*x^18 + a*c^16*x + 1/17*(16*b*c*d^15 + a*d^16)*x^17 + 1/2*(15*b *c^2*d^14 + 2*a*c*d^15)*x^16 + 8/3*(14*b*c^3*d^13 + 3*a*c^2*d^14)*x^15 + 1 0*(13*b*c^4*d^12 + 4*a*c^3*d^13)*x^14 + 28*(12*b*c^5*d^11 + 5*a*c^4*d^12)* x^13 + 182/3*(11*b*c^6*d^10 + 6*a*c^5*d^11)*x^12 + 104*(10*b*c^7*d^9 + 7*a *c^6*d^10)*x^11 + 143*(9*b*c^8*d^8 + 8*a*c^7*d^9)*x^10 + 1430/9*(8*b*c^9*d ^7 + 9*a*c^8*d^8)*x^9 + 143*(7*b*c^10*d^6 + 10*a*c^9*d^7)*x^8 + 104*(6*b*c ^11*d^5 + 11*a*c^10*d^6)*x^7 + 182/3*(5*b*c^12*d^4 + 12*a*c^11*d^5)*x^6 + 28*(4*b*c^13*d^3 + 13*a*c^12*d^4)*x^5 + 10*(3*b*c^14*d^2 + 14*a*c^13*d^3)* x^4 + 8/3*(2*b*c^15*d + 15*a*c^14*d^2)*x^3 + 1/2*(b*c^16 + 16*a*c^15*d)*x^ 2
Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (32) = 64\).
Time = 0.07 (sec) , antiderivative size = 393, normalized size of antiderivative = 10.34 \[ \int (a+b x) (c+d x)^{16} \, dx=a c^{16} x + \frac {b d^{16} x^{18}}{18} + x^{17} \left (\frac {a d^{16}}{17} + \frac {16 b c d^{15}}{17}\right ) + x^{16} \left (a c d^{15} + \frac {15 b c^{2} d^{14}}{2}\right ) + x^{15} \cdot \left (8 a c^{2} d^{14} + \frac {112 b c^{3} d^{13}}{3}\right ) + x^{14} \cdot \left (40 a c^{3} d^{13} + 130 b c^{4} d^{12}\right ) + x^{13} \cdot \left (140 a c^{4} d^{12} + 336 b c^{5} d^{11}\right ) + x^{12} \cdot \left (364 a c^{5} d^{11} + \frac {2002 b c^{6} d^{10}}{3}\right ) + x^{11} \cdot \left (728 a c^{6} d^{10} + 1040 b c^{7} d^{9}\right ) + x^{10} \cdot \left (1144 a c^{7} d^{9} + 1287 b c^{8} d^{8}\right ) + x^{9} \cdot \left (1430 a c^{8} d^{8} + \frac {11440 b c^{9} d^{7}}{9}\right ) + x^{8} \cdot \left (1430 a c^{9} d^{7} + 1001 b c^{10} d^{6}\right ) + x^{7} \cdot \left (1144 a c^{10} d^{6} + 624 b c^{11} d^{5}\right ) + x^{6} \cdot \left (728 a c^{11} d^{5} + \frac {910 b c^{12} d^{4}}{3}\right ) + x^{5} \cdot \left (364 a c^{12} d^{4} + 112 b c^{13} d^{3}\right ) + x^{4} \cdot \left (140 a c^{13} d^{3} + 30 b c^{14} d^{2}\right ) + x^{3} \cdot \left (40 a c^{14} d^{2} + \frac {16 b c^{15} d}{3}\right ) + x^{2} \cdot \left (8 a c^{15} d + \frac {b c^{16}}{2}\right ) \]
a*c**16*x + b*d**16*x**18/18 + x**17*(a*d**16/17 + 16*b*c*d**15/17) + x**1 6*(a*c*d**15 + 15*b*c**2*d**14/2) + x**15*(8*a*c**2*d**14 + 112*b*c**3*d** 13/3) + x**14*(40*a*c**3*d**13 + 130*b*c**4*d**12) + x**13*(140*a*c**4*d** 12 + 336*b*c**5*d**11) + x**12*(364*a*c**5*d**11 + 2002*b*c**6*d**10/3) + x**11*(728*a*c**6*d**10 + 1040*b*c**7*d**9) + x**10*(1144*a*c**7*d**9 + 12 87*b*c**8*d**8) + x**9*(1430*a*c**8*d**8 + 11440*b*c**9*d**7/9) + x**8*(14 30*a*c**9*d**7 + 1001*b*c**10*d**6) + x**7*(1144*a*c**10*d**6 + 624*b*c**1 1*d**5) + x**6*(728*a*c**11*d**5 + 910*b*c**12*d**4/3) + x**5*(364*a*c**12 *d**4 + 112*b*c**13*d**3) + x**4*(140*a*c**13*d**3 + 30*b*c**14*d**2) + x* *3*(40*a*c**14*d**2 + 16*b*c**15*d/3) + x**2*(8*a*c**15*d + b*c**16/2)
Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (34) = 68\).
Time = 0.20 (sec) , antiderivative size = 384, normalized size of antiderivative = 10.11 \[ \int (a+b x) (c+d x)^{16} \, dx=\frac {1}{18} \, b d^{16} x^{18} + a c^{16} x + \frac {1}{17} \, {\left (16 \, b c d^{15} + a d^{16}\right )} x^{17} + \frac {1}{2} \, {\left (15 \, b c^{2} d^{14} + 2 \, a c d^{15}\right )} x^{16} + \frac {8}{3} \, {\left (14 \, b c^{3} d^{13} + 3 \, a c^{2} d^{14}\right )} x^{15} + 10 \, {\left (13 \, b c^{4} d^{12} + 4 \, a c^{3} d^{13}\right )} x^{14} + 28 \, {\left (12 \, b c^{5} d^{11} + 5 \, a c^{4} d^{12}\right )} x^{13} + \frac {182}{3} \, {\left (11 \, b c^{6} d^{10} + 6 \, a c^{5} d^{11}\right )} x^{12} + 104 \, {\left (10 \, b c^{7} d^{9} + 7 \, a c^{6} d^{10}\right )} x^{11} + 143 \, {\left (9 \, b c^{8} d^{8} + 8 \, a c^{7} d^{9}\right )} x^{10} + \frac {1430}{9} \, {\left (8 \, b c^{9} d^{7} + 9 \, a c^{8} d^{8}\right )} x^{9} + 143 \, {\left (7 \, b c^{10} d^{6} + 10 \, a c^{9} d^{7}\right )} x^{8} + 104 \, {\left (6 \, b c^{11} d^{5} + 11 \, a c^{10} d^{6}\right )} x^{7} + \frac {182}{3} \, {\left (5 \, b c^{12} d^{4} + 12 \, a c^{11} d^{5}\right )} x^{6} + 28 \, {\left (4 \, b c^{13} d^{3} + 13 \, a c^{12} d^{4}\right )} x^{5} + 10 \, {\left (3 \, b c^{14} d^{2} + 14 \, a c^{13} d^{3}\right )} x^{4} + \frac {8}{3} \, {\left (2 \, b c^{15} d + 15 \, a c^{14} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b c^{16} + 16 \, a c^{15} d\right )} x^{2} \]
1/18*b*d^16*x^18 + a*c^16*x + 1/17*(16*b*c*d^15 + a*d^16)*x^17 + 1/2*(15*b *c^2*d^14 + 2*a*c*d^15)*x^16 + 8/3*(14*b*c^3*d^13 + 3*a*c^2*d^14)*x^15 + 1 0*(13*b*c^4*d^12 + 4*a*c^3*d^13)*x^14 + 28*(12*b*c^5*d^11 + 5*a*c^4*d^12)* x^13 + 182/3*(11*b*c^6*d^10 + 6*a*c^5*d^11)*x^12 + 104*(10*b*c^7*d^9 + 7*a *c^6*d^10)*x^11 + 143*(9*b*c^8*d^8 + 8*a*c^7*d^9)*x^10 + 1430/9*(8*b*c^9*d ^7 + 9*a*c^8*d^8)*x^9 + 143*(7*b*c^10*d^6 + 10*a*c^9*d^7)*x^8 + 104*(6*b*c ^11*d^5 + 11*a*c^10*d^6)*x^7 + 182/3*(5*b*c^12*d^4 + 12*a*c^11*d^5)*x^6 + 28*(4*b*c^13*d^3 + 13*a*c^12*d^4)*x^5 + 10*(3*b*c^14*d^2 + 14*a*c^13*d^3)* x^4 + 8/3*(2*b*c^15*d + 15*a*c^14*d^2)*x^3 + 1/2*(b*c^16 + 16*a*c^15*d)*x^ 2
Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (34) = 68\).
Time = 0.30 (sec) , antiderivative size = 385, normalized size of antiderivative = 10.13 \[ \int (a+b x) (c+d x)^{16} \, dx=\frac {1}{18} \, b d^{16} x^{18} + \frac {16}{17} \, b c d^{15} x^{17} + \frac {1}{17} \, a d^{16} x^{17} + \frac {15}{2} \, b c^{2} d^{14} x^{16} + a c d^{15} x^{16} + \frac {112}{3} \, b c^{3} d^{13} x^{15} + 8 \, a c^{2} d^{14} x^{15} + 130 \, b c^{4} d^{12} x^{14} + 40 \, a c^{3} d^{13} x^{14} + 336 \, b c^{5} d^{11} x^{13} + 140 \, a c^{4} d^{12} x^{13} + \frac {2002}{3} \, b c^{6} d^{10} x^{12} + 364 \, a c^{5} d^{11} x^{12} + 1040 \, b c^{7} d^{9} x^{11} + 728 \, a c^{6} d^{10} x^{11} + 1287 \, b c^{8} d^{8} x^{10} + 1144 \, a c^{7} d^{9} x^{10} + \frac {11440}{9} \, b c^{9} d^{7} x^{9} + 1430 \, a c^{8} d^{8} x^{9} + 1001 \, b c^{10} d^{6} x^{8} + 1430 \, a c^{9} d^{7} x^{8} + 624 \, b c^{11} d^{5} x^{7} + 1144 \, a c^{10} d^{6} x^{7} + \frac {910}{3} \, b c^{12} d^{4} x^{6} + 728 \, a c^{11} d^{5} x^{6} + 112 \, b c^{13} d^{3} x^{5} + 364 \, a c^{12} d^{4} x^{5} + 30 \, b c^{14} d^{2} x^{4} + 140 \, a c^{13} d^{3} x^{4} + \frac {16}{3} \, b c^{15} d x^{3} + 40 \, a c^{14} d^{2} x^{3} + \frac {1}{2} \, b c^{16} x^{2} + 8 \, a c^{15} d x^{2} + a c^{16} x \]
1/18*b*d^16*x^18 + 16/17*b*c*d^15*x^17 + 1/17*a*d^16*x^17 + 15/2*b*c^2*d^1 4*x^16 + a*c*d^15*x^16 + 112/3*b*c^3*d^13*x^15 + 8*a*c^2*d^14*x^15 + 130*b *c^4*d^12*x^14 + 40*a*c^3*d^13*x^14 + 336*b*c^5*d^11*x^13 + 140*a*c^4*d^12 *x^13 + 2002/3*b*c^6*d^10*x^12 + 364*a*c^5*d^11*x^12 + 1040*b*c^7*d^9*x^11 + 728*a*c^6*d^10*x^11 + 1287*b*c^8*d^8*x^10 + 1144*a*c^7*d^9*x^10 + 11440 /9*b*c^9*d^7*x^9 + 1430*a*c^8*d^8*x^9 + 1001*b*c^10*d^6*x^8 + 1430*a*c^9*d ^7*x^8 + 624*b*c^11*d^5*x^7 + 1144*a*c^10*d^6*x^7 + 910/3*b*c^12*d^4*x^6 + 728*a*c^11*d^5*x^6 + 112*b*c^13*d^3*x^5 + 364*a*c^12*d^4*x^5 + 30*b*c^14* d^2*x^4 + 140*a*c^13*d^3*x^4 + 16/3*b*c^15*d*x^3 + 40*a*c^14*d^2*x^3 + 1/2 *b*c^16*x^2 + 8*a*c^15*d*x^2 + a*c^16*x
Time = 0.56 (sec) , antiderivative size = 328, normalized size of antiderivative = 8.63 \[ \int (a+b x) (c+d x)^{16} \, dx=x^2\,\left (\frac {b\,c^{16}}{2}+8\,a\,d\,c^{15}\right )+x^{17}\,\left (\frac {a\,d^{16}}{17}+\frac {16\,b\,c\,d^{15}}{17}\right )+\frac {b\,d^{16}\,x^{18}}{18}+a\,c^{16}\,x+\frac {8\,c^{14}\,d\,x^3\,\left (15\,a\,d+2\,b\,c\right )}{3}+\frac {c\,d^{14}\,x^{16}\,\left (2\,a\,d+15\,b\,c\right )}{2}+10\,c^{13}\,d^2\,x^4\,\left (14\,a\,d+3\,b\,c\right )+28\,c^{12}\,d^3\,x^5\,\left (13\,a\,d+4\,b\,c\right )+\frac {182\,c^{11}\,d^4\,x^6\,\left (12\,a\,d+5\,b\,c\right )}{3}+104\,c^{10}\,d^5\,x^7\,\left (11\,a\,d+6\,b\,c\right )+143\,c^9\,d^6\,x^8\,\left (10\,a\,d+7\,b\,c\right )+\frac {1430\,c^8\,d^7\,x^9\,\left (9\,a\,d+8\,b\,c\right )}{9}+143\,c^7\,d^8\,x^{10}\,\left (8\,a\,d+9\,b\,c\right )+104\,c^6\,d^9\,x^{11}\,\left (7\,a\,d+10\,b\,c\right )+\frac {182\,c^5\,d^{10}\,x^{12}\,\left (6\,a\,d+11\,b\,c\right )}{3}+28\,c^4\,d^{11}\,x^{13}\,\left (5\,a\,d+12\,b\,c\right )+10\,c^3\,d^{12}\,x^{14}\,\left (4\,a\,d+13\,b\,c\right )+\frac {8\,c^2\,d^{13}\,x^{15}\,\left (3\,a\,d+14\,b\,c\right )}{3} \]
x^2*((b*c^16)/2 + 8*a*c^15*d) + x^17*((a*d^16)/17 + (16*b*c*d^15)/17) + (b *d^16*x^18)/18 + a*c^16*x + (8*c^14*d*x^3*(15*a*d + 2*b*c))/3 + (c*d^14*x^ 16*(2*a*d + 15*b*c))/2 + 10*c^13*d^2*x^4*(14*a*d + 3*b*c) + 28*c^12*d^3*x^ 5*(13*a*d + 4*b*c) + (182*c^11*d^4*x^6*(12*a*d + 5*b*c))/3 + 104*c^10*d^5* x^7*(11*a*d + 6*b*c) + 143*c^9*d^6*x^8*(10*a*d + 7*b*c) + (1430*c^8*d^7*x^ 9*(9*a*d + 8*b*c))/9 + 143*c^7*d^8*x^10*(8*a*d + 9*b*c) + 104*c^6*d^9*x^11 *(7*a*d + 10*b*c) + (182*c^5*d^10*x^12*(6*a*d + 11*b*c))/3 + 28*c^4*d^11*x ^13*(5*a*d + 12*b*c) + 10*c^3*d^12*x^14*(4*a*d + 13*b*c) + (8*c^2*d^13*x^1 5*(3*a*d + 14*b*c))/3