Integrand size = 16, antiderivative size = 88 \[ \int x^2 (a+b x) (c+d x)^{16} \, dx=-\frac {c^2 (b c-a d) (c+d x)^{17}}{17 d^4}+\frac {c (3 b c-2 a d) (c+d x)^{18}}{18 d^4}-\frac {(3 b c-a d) (c+d x)^{19}}{19 d^4}+\frac {b (c+d x)^{20}}{20 d^4} \]
-1/17*c^2*(-a*d+b*c)*(d*x+c)^17/d^4+1/18*c*(-2*a*d+3*b*c)*(d*x+c)^18/d^4-1 /19*(-a*d+3*b*c)*(d*x+c)^19/d^4+1/20*b*(d*x+c)^20/d^4
Leaf count is larger than twice the leaf count of optimal. \(355\) vs. \(2(88)=176\).
Time = 0.04 (sec) , antiderivative size = 355, normalized size of antiderivative = 4.03 \[ \int x^2 (a+b x) (c+d x)^{16} \, dx=\frac {1}{3} a c^{16} x^3+\frac {1}{4} c^{15} (b c+16 a d) x^4+\frac {8}{5} c^{14} d (2 b c+15 a d) x^5+\frac {20}{3} c^{13} d^2 (3 b c+14 a d) x^6+20 c^{12} d^3 (4 b c+13 a d) x^7+\frac {91}{2} c^{11} d^4 (5 b c+12 a d) x^8+\frac {728}{9} c^{10} d^5 (6 b c+11 a d) x^9+\frac {572}{5} c^9 d^6 (7 b c+10 a d) x^{10}+130 c^8 d^7 (8 b c+9 a d) x^{11}+\frac {715}{6} c^7 d^8 (9 b c+8 a d) x^{12}+88 c^6 d^9 (10 b c+7 a d) x^{13}+52 c^5 d^{10} (11 b c+6 a d) x^{14}+\frac {364}{15} c^4 d^{11} (12 b c+5 a d) x^{15}+\frac {35}{4} c^3 d^{12} (13 b c+4 a d) x^{16}+\frac {40}{17} c^2 d^{13} (14 b c+3 a d) x^{17}+\frac {4}{9} c d^{14} (15 b c+2 a d) x^{18}+\frac {1}{19} d^{15} (16 b c+a d) x^{19}+\frac {1}{20} b d^{16} x^{20} \]
(a*c^16*x^3)/3 + (c^15*(b*c + 16*a*d)*x^4)/4 + (8*c^14*d*(2*b*c + 15*a*d)* x^5)/5 + (20*c^13*d^2*(3*b*c + 14*a*d)*x^6)/3 + 20*c^12*d^3*(4*b*c + 13*a* d)*x^7 + (91*c^11*d^4*(5*b*c + 12*a*d)*x^8)/2 + (728*c^10*d^5*(6*b*c + 11* a*d)*x^9)/9 + (572*c^9*d^6*(7*b*c + 10*a*d)*x^10)/5 + 130*c^8*d^7*(8*b*c + 9*a*d)*x^11 + (715*c^7*d^8*(9*b*c + 8*a*d)*x^12)/6 + 88*c^6*d^9*(10*b*c + 7*a*d)*x^13 + 52*c^5*d^10*(11*b*c + 6*a*d)*x^14 + (364*c^4*d^11*(12*b*c + 5*a*d)*x^15)/15 + (35*c^3*d^12*(13*b*c + 4*a*d)*x^16)/4 + (40*c^2*d^13*(1 4*b*c + 3*a*d)*x^17)/17 + (4*c*d^14*(15*b*c + 2*a*d)*x^18)/9 + (d^15*(16*b *c + a*d)*x^19)/19 + (b*d^16*x^20)/20
Time = 0.37 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {85, 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.
| \(\displaystyle \int x^2 (a+b x) (c+d x)^{16} \, dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left (-\frac {c^2 (c+d x)^{16} (b c-a d)}{d^3}+\frac {(c+d x)^{18} (a d-3 b c)}{d^3}+\frac {c (c+d x)^{17} (3 b c-2 a d)}{d^3}+\frac {b (c+d x)^{19}}{d^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {c^2 (c+d x)^{17} (b c-a d)}{17 d^4}-\frac {(c+d x)^{19} (3 b c-a d)}{19 d^4}+\frac {c (c+d x)^{18} (3 b c-2 a d)}{18 d^4}+\frac {b (c+d x)^{20}}{20 d^4}\) |
-1/17*(c^2*(b*c - a*d)*(c + d*x)^17)/d^4 + (c*(3*b*c - 2*a*d)*(c + d*x)^18 )/(18*d^4) - ((3*b*c - a*d)*(c + d*x)^19)/(19*d^4) + (b*(c + d*x)^20)/(20* d^4)
3.2.70.3.1 Defintions of rubi rules used
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a* f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])
Leaf count of result is larger than twice the leaf count of optimal. \(373\) vs. \(2(80)=160\).
Time = 0.41 (sec) , antiderivative size = 374, normalized size of antiderivative = 4.25
| method | result | size |
| norman | \(\frac {a \,c^{16} x^{3}}{3}+\left (4 a \,c^{15} d +\frac {1}{4} b \,c^{16}\right ) x^{4}+\left (24 a \,c^{14} d^{2}+\frac {16}{5} b \,c^{15} d \right ) x^{5}+\left (\frac {280}{3} a \,c^{13} d^{3}+20 b \,c^{14} d^{2}\right ) x^{6}+\left (260 a \,c^{12} d^{4}+80 b \,c^{13} d^{3}\right ) x^{7}+\left (546 a \,c^{11} d^{5}+\frac {455}{2} b \,c^{12} d^{4}\right ) x^{8}+\left (\frac {8008}{9} a \,c^{10} d^{6}+\frac {1456}{3} b \,c^{11} d^{5}\right ) x^{9}+\left (1144 a \,c^{9} d^{7}+\frac {4004}{5} b \,c^{10} d^{6}\right ) x^{10}+\left (1170 a \,c^{8} d^{8}+1040 b \,c^{9} d^{7}\right ) x^{11}+\left (\frac {2860}{3} a \,c^{7} d^{9}+\frac {2145}{2} b \,c^{8} d^{8}\right ) x^{12}+\left (616 a \,c^{6} d^{10}+880 b \,c^{7} d^{9}\right ) x^{13}+\left (312 a \,c^{5} d^{11}+572 b \,c^{6} d^{10}\right ) x^{14}+\left (\frac {364}{3} a \,c^{4} d^{12}+\frac {1456}{5} b \,c^{5} d^{11}\right ) x^{15}+\left (35 a \,c^{3} d^{13}+\frac {455}{4} b \,c^{4} d^{12}\right ) x^{16}+\left (\frac {120}{17} a \,c^{2} d^{14}+\frac {560}{17} b \,c^{3} d^{13}\right ) x^{17}+\left (\frac {8}{9} a c \,d^{15}+\frac {20}{3} b \,c^{2} d^{14}\right ) x^{18}+\left (\frac {1}{19} a \,d^{16}+\frac {16}{19} b c \,d^{15}\right ) x^{19}+\frac {b \,d^{16} x^{20}}{20}\) | \(374\) |
| default | \(\frac {b \,d^{16} x^{20}}{20}+\frac {\left (a \,d^{16}+16 b c \,d^{15}\right ) x^{19}}{19}+\frac {\left (16 a c \,d^{15}+120 b \,c^{2} d^{14}\right ) x^{18}}{18}+\frac {\left (120 a \,c^{2} d^{14}+560 b \,c^{3} d^{13}\right ) x^{17}}{17}+\frac {\left (560 a \,c^{3} d^{13}+1820 b \,c^{4} d^{12}\right ) x^{16}}{16}+\frac {\left (1820 a \,c^{4} d^{12}+4368 b \,c^{5} d^{11}\right ) x^{15}}{15}+\frac {\left (4368 a \,c^{5} d^{11}+8008 b \,c^{6} d^{10}\right ) x^{14}}{14}+\frac {\left (8008 a \,c^{6} d^{10}+11440 b \,c^{7} d^{9}\right ) x^{13}}{13}+\frac {\left (11440 a \,c^{7} d^{9}+12870 b \,c^{8} d^{8}\right ) x^{12}}{12}+\frac {\left (12870 a \,c^{8} d^{8}+11440 b \,c^{9} d^{7}\right ) x^{11}}{11}+\frac {\left (11440 a \,c^{9} d^{7}+8008 b \,c^{10} d^{6}\right ) x^{10}}{10}+\frac {\left (8008 a \,c^{10} d^{6}+4368 b \,c^{11} d^{5}\right ) x^{9}}{9}+\frac {\left (4368 a \,c^{11} d^{5}+1820 b \,c^{12} d^{4}\right ) x^{8}}{8}+\frac {\left (1820 a \,c^{12} d^{4}+560 b \,c^{13} d^{3}\right ) x^{7}}{7}+\frac {\left (560 a \,c^{13} d^{3}+120 b \,c^{14} d^{2}\right ) x^{6}}{6}+\frac {\left (120 a \,c^{14} d^{2}+16 b \,c^{15} d \right ) x^{5}}{5}+\frac {\left (16 a \,c^{15} d +b \,c^{16}\right ) x^{4}}{4}+\frac {a \,c^{16} x^{3}}{3}\) | \(388\) |
| gosper | \(\frac {1}{20} b \,d^{16} x^{20}+\frac {1}{3} a \,c^{16} x^{3}+\frac {1}{4} x^{4} b \,c^{16}+\frac {1}{19} x^{19} a \,d^{16}+\frac {20}{3} x^{18} b \,c^{2} d^{14}+\frac {16}{19} x^{19} b c \,d^{15}+260 a \,c^{12} d^{4} x^{7}+80 b \,c^{13} d^{3} x^{7}+1170 a \,c^{8} d^{8} x^{11}+1040 b \,c^{9} d^{7} x^{11}+616 a \,c^{6} d^{10} x^{13}+880 b \,c^{7} d^{9} x^{13}+312 a \,c^{5} d^{11} x^{14}+572 b \,c^{6} d^{10} x^{14}+\frac {2860}{3} x^{12} a \,c^{7} d^{9}+\frac {2145}{2} x^{12} b \,c^{8} d^{8}+\frac {364}{3} x^{15} a \,c^{4} d^{12}+\frac {1456}{5} x^{15} b \,c^{5} d^{11}+35 x^{16} a \,c^{3} d^{13}+\frac {455}{4} x^{16} b \,c^{4} d^{12}+\frac {120}{17} x^{17} a \,c^{2} d^{14}+\frac {560}{17} x^{17} b \,c^{3} d^{13}+\frac {8}{9} x^{18} a c \,d^{15}+\frac {280}{3} x^{6} a \,c^{13} d^{3}+20 x^{6} b \,c^{14} d^{2}+546 x^{8} a \,c^{11} d^{5}+\frac {455}{2} x^{8} b \,c^{12} d^{4}+\frac {8008}{9} x^{9} a \,c^{10} d^{6}+\frac {1456}{3} x^{9} b \,c^{11} d^{5}+1144 x^{10} a \,c^{9} d^{7}+\frac {4004}{5} x^{10} b \,c^{10} d^{6}+4 x^{4} a \,c^{15} d +24 x^{5} a \,c^{14} d^{2}+\frac {16}{5} x^{5} b \,c^{15} d\) | \(390\) |
| risch | \(\frac {1}{20} b \,d^{16} x^{20}+\frac {1}{3} a \,c^{16} x^{3}+\frac {1}{4} x^{4} b \,c^{16}+\frac {1}{19} x^{19} a \,d^{16}+\frac {20}{3} x^{18} b \,c^{2} d^{14}+\frac {16}{19} x^{19} b c \,d^{15}+260 a \,c^{12} d^{4} x^{7}+80 b \,c^{13} d^{3} x^{7}+1170 a \,c^{8} d^{8} x^{11}+1040 b \,c^{9} d^{7} x^{11}+616 a \,c^{6} d^{10} x^{13}+880 b \,c^{7} d^{9} x^{13}+312 a \,c^{5} d^{11} x^{14}+572 b \,c^{6} d^{10} x^{14}+\frac {2860}{3} x^{12} a \,c^{7} d^{9}+\frac {2145}{2} x^{12} b \,c^{8} d^{8}+\frac {364}{3} x^{15} a \,c^{4} d^{12}+\frac {1456}{5} x^{15} b \,c^{5} d^{11}+35 x^{16} a \,c^{3} d^{13}+\frac {455}{4} x^{16} b \,c^{4} d^{12}+\frac {120}{17} x^{17} a \,c^{2} d^{14}+\frac {560}{17} x^{17} b \,c^{3} d^{13}+\frac {8}{9} x^{18} a c \,d^{15}+\frac {280}{3} x^{6} a \,c^{13} d^{3}+20 x^{6} b \,c^{14} d^{2}+546 x^{8} a \,c^{11} d^{5}+\frac {455}{2} x^{8} b \,c^{12} d^{4}+\frac {8008}{9} x^{9} a \,c^{10} d^{6}+\frac {1456}{3} x^{9} b \,c^{11} d^{5}+1144 x^{10} a \,c^{9} d^{7}+\frac {4004}{5} x^{10} b \,c^{10} d^{6}+4 x^{4} a \,c^{15} d +24 x^{5} a \,c^{14} d^{2}+\frac {16}{5} x^{5} b \,c^{15} d\) | \(390\) |
| parallelrisch | \(\frac {1}{20} b \,d^{16} x^{20}+\frac {1}{3} a \,c^{16} x^{3}+\frac {1}{4} x^{4} b \,c^{16}+\frac {1}{19} x^{19} a \,d^{16}+\frac {20}{3} x^{18} b \,c^{2} d^{14}+\frac {16}{19} x^{19} b c \,d^{15}+260 a \,c^{12} d^{4} x^{7}+80 b \,c^{13} d^{3} x^{7}+1170 a \,c^{8} d^{8} x^{11}+1040 b \,c^{9} d^{7} x^{11}+616 a \,c^{6} d^{10} x^{13}+880 b \,c^{7} d^{9} x^{13}+312 a \,c^{5} d^{11} x^{14}+572 b \,c^{6} d^{10} x^{14}+\frac {2860}{3} x^{12} a \,c^{7} d^{9}+\frac {2145}{2} x^{12} b \,c^{8} d^{8}+\frac {364}{3} x^{15} a \,c^{4} d^{12}+\frac {1456}{5} x^{15} b \,c^{5} d^{11}+35 x^{16} a \,c^{3} d^{13}+\frac {455}{4} x^{16} b \,c^{4} d^{12}+\frac {120}{17} x^{17} a \,c^{2} d^{14}+\frac {560}{17} x^{17} b \,c^{3} d^{13}+\frac {8}{9} x^{18} a c \,d^{15}+\frac {280}{3} x^{6} a \,c^{13} d^{3}+20 x^{6} b \,c^{14} d^{2}+546 x^{8} a \,c^{11} d^{5}+\frac {455}{2} x^{8} b \,c^{12} d^{4}+\frac {8008}{9} x^{9} a \,c^{10} d^{6}+\frac {1456}{3} x^{9} b \,c^{11} d^{5}+1144 x^{10} a \,c^{9} d^{7}+\frac {4004}{5} x^{10} b \,c^{10} d^{6}+4 x^{4} a \,c^{15} d +24 x^{5} a \,c^{14} d^{2}+\frac {16}{5} x^{5} b \,c^{15} d\) | \(390\) |
1/3*a*c^16*x^3+(4*a*c^15*d+1/4*b*c^16)*x^4+(24*a*c^14*d^2+16/5*b*c^15*d)*x ^5+(280/3*a*c^13*d^3+20*b*c^14*d^2)*x^6+(260*a*c^12*d^4+80*b*c^13*d^3)*x^7 +(546*a*c^11*d^5+455/2*b*c^12*d^4)*x^8+(8008/9*a*c^10*d^6+1456/3*b*c^11*d^ 5)*x^9+(1144*a*c^9*d^7+4004/5*b*c^10*d^6)*x^10+(1170*a*c^8*d^8+1040*b*c^9* d^7)*x^11+(2860/3*a*c^7*d^9+2145/2*b*c^8*d^8)*x^12+(616*a*c^6*d^10+880*b*c ^7*d^9)*x^13+(312*a*c^5*d^11+572*b*c^6*d^10)*x^14+(364/3*a*c^4*d^12+1456/5 *b*c^5*d^11)*x^15+(35*a*c^3*d^13+455/4*b*c^4*d^12)*x^16+(120/17*a*c^2*d^14 +560/17*b*c^3*d^13)*x^17+(8/9*a*c*d^15+20/3*b*c^2*d^14)*x^18+(1/19*a*d^16+ 16/19*b*c*d^15)*x^19+1/20*b*d^16*x^20
Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (80) = 160\).
Time = 0.22 (sec) , antiderivative size = 387, normalized size of antiderivative = 4.40 \[ \int x^2 (a+b x) (c+d x)^{16} \, dx=\frac {1}{20} \, b d^{16} x^{20} + \frac {1}{3} \, a c^{16} x^{3} + \frac {1}{19} \, {\left (16 \, b c d^{15} + a d^{16}\right )} x^{19} + \frac {4}{9} \, {\left (15 \, b c^{2} d^{14} + 2 \, a c d^{15}\right )} x^{18} + \frac {40}{17} \, {\left (14 \, b c^{3} d^{13} + 3 \, a c^{2} d^{14}\right )} x^{17} + \frac {35}{4} \, {\left (13 \, b c^{4} d^{12} + 4 \, a c^{3} d^{13}\right )} x^{16} + \frac {364}{15} \, {\left (12 \, b c^{5} d^{11} + 5 \, a c^{4} d^{12}\right )} x^{15} + 52 \, {\left (11 \, b c^{6} d^{10} + 6 \, a c^{5} d^{11}\right )} x^{14} + 88 \, {\left (10 \, b c^{7} d^{9} + 7 \, a c^{6} d^{10}\right )} x^{13} + \frac {715}{6} \, {\left (9 \, b c^{8} d^{8} + 8 \, a c^{7} d^{9}\right )} x^{12} + 130 \, {\left (8 \, b c^{9} d^{7} + 9 \, a c^{8} d^{8}\right )} x^{11} + \frac {572}{5} \, {\left (7 \, b c^{10} d^{6} + 10 \, a c^{9} d^{7}\right )} x^{10} + \frac {728}{9} \, {\left (6 \, b c^{11} d^{5} + 11 \, a c^{10} d^{6}\right )} x^{9} + \frac {91}{2} \, {\left (5 \, b c^{12} d^{4} + 12 \, a c^{11} d^{5}\right )} x^{8} + 20 \, {\left (4 \, b c^{13} d^{3} + 13 \, a c^{12} d^{4}\right )} x^{7} + \frac {20}{3} \, {\left (3 \, b c^{14} d^{2} + 14 \, a c^{13} d^{3}\right )} x^{6} + \frac {8}{5} \, {\left (2 \, b c^{15} d + 15 \, a c^{14} d^{2}\right )} x^{5} + \frac {1}{4} \, {\left (b c^{16} + 16 \, a c^{15} d\right )} x^{4} \]
1/20*b*d^16*x^20 + 1/3*a*c^16*x^3 + 1/19*(16*b*c*d^15 + a*d^16)*x^19 + 4/9 *(15*b*c^2*d^14 + 2*a*c*d^15)*x^18 + 40/17*(14*b*c^3*d^13 + 3*a*c^2*d^14)* x^17 + 35/4*(13*b*c^4*d^12 + 4*a*c^3*d^13)*x^16 + 364/15*(12*b*c^5*d^11 + 5*a*c^4*d^12)*x^15 + 52*(11*b*c^6*d^10 + 6*a*c^5*d^11)*x^14 + 88*(10*b*c^7 *d^9 + 7*a*c^6*d^10)*x^13 + 715/6*(9*b*c^8*d^8 + 8*a*c^7*d^9)*x^12 + 130*( 8*b*c^9*d^7 + 9*a*c^8*d^8)*x^11 + 572/5*(7*b*c^10*d^6 + 10*a*c^9*d^7)*x^10 + 728/9*(6*b*c^11*d^5 + 11*a*c^10*d^6)*x^9 + 91/2*(5*b*c^12*d^4 + 12*a*c^ 11*d^5)*x^8 + 20*(4*b*c^13*d^3 + 13*a*c^12*d^4)*x^7 + 20/3*(3*b*c^14*d^2 + 14*a*c^13*d^3)*x^6 + 8/5*(2*b*c^15*d + 15*a*c^14*d^2)*x^5 + 1/4*(b*c^16 + 16*a*c^15*d)*x^4
Leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (80) = 160\).
Time = 0.06 (sec) , antiderivative size = 413, normalized size of antiderivative = 4.69 \[ \int x^2 (a+b x) (c+d x)^{16} \, dx=\frac {a c^{16} x^{3}}{3} + \frac {b d^{16} x^{20}}{20} + x^{19} \left (\frac {a d^{16}}{19} + \frac {16 b c d^{15}}{19}\right ) + x^{18} \cdot \left (\frac {8 a c d^{15}}{9} + \frac {20 b c^{2} d^{14}}{3}\right ) + x^{17} \cdot \left (\frac {120 a c^{2} d^{14}}{17} + \frac {560 b c^{3} d^{13}}{17}\right ) + x^{16} \cdot \left (35 a c^{3} d^{13} + \frac {455 b c^{4} d^{12}}{4}\right ) + x^{15} \cdot \left (\frac {364 a c^{4} d^{12}}{3} + \frac {1456 b c^{5} d^{11}}{5}\right ) + x^{14} \cdot \left (312 a c^{5} d^{11} + 572 b c^{6} d^{10}\right ) + x^{13} \cdot \left (616 a c^{6} d^{10} + 880 b c^{7} d^{9}\right ) + x^{12} \cdot \left (\frac {2860 a c^{7} d^{9}}{3} + \frac {2145 b c^{8} d^{8}}{2}\right ) + x^{11} \cdot \left (1170 a c^{8} d^{8} + 1040 b c^{9} d^{7}\right ) + x^{10} \cdot \left (1144 a c^{9} d^{7} + \frac {4004 b c^{10} d^{6}}{5}\right ) + x^{9} \cdot \left (\frac {8008 a c^{10} d^{6}}{9} + \frac {1456 b c^{11} d^{5}}{3}\right ) + x^{8} \cdot \left (546 a c^{11} d^{5} + \frac {455 b c^{12} d^{4}}{2}\right ) + x^{7} \cdot \left (260 a c^{12} d^{4} + 80 b c^{13} d^{3}\right ) + x^{6} \cdot \left (\frac {280 a c^{13} d^{3}}{3} + 20 b c^{14} d^{2}\right ) + x^{5} \cdot \left (24 a c^{14} d^{2} + \frac {16 b c^{15} d}{5}\right ) + x^{4} \cdot \left (4 a c^{15} d + \frac {b c^{16}}{4}\right ) \]
a*c**16*x**3/3 + b*d**16*x**20/20 + x**19*(a*d**16/19 + 16*b*c*d**15/19) + x**18*(8*a*c*d**15/9 + 20*b*c**2*d**14/3) + x**17*(120*a*c**2*d**14/17 + 560*b*c**3*d**13/17) + x**16*(35*a*c**3*d**13 + 455*b*c**4*d**12/4) + x**1 5*(364*a*c**4*d**12/3 + 1456*b*c**5*d**11/5) + x**14*(312*a*c**5*d**11 + 5 72*b*c**6*d**10) + x**13*(616*a*c**6*d**10 + 880*b*c**7*d**9) + x**12*(286 0*a*c**7*d**9/3 + 2145*b*c**8*d**8/2) + x**11*(1170*a*c**8*d**8 + 1040*b*c **9*d**7) + x**10*(1144*a*c**9*d**7 + 4004*b*c**10*d**6/5) + x**9*(8008*a* c**10*d**6/9 + 1456*b*c**11*d**5/3) + x**8*(546*a*c**11*d**5 + 455*b*c**12 *d**4/2) + x**7*(260*a*c**12*d**4 + 80*b*c**13*d**3) + x**6*(280*a*c**13*d **3/3 + 20*b*c**14*d**2) + x**5*(24*a*c**14*d**2 + 16*b*c**15*d/5) + x**4* (4*a*c**15*d + b*c**16/4)
Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (80) = 160\).
Time = 0.21 (sec) , antiderivative size = 387, normalized size of antiderivative = 4.40 \[ \int x^2 (a+b x) (c+d x)^{16} \, dx=\frac {1}{20} \, b d^{16} x^{20} + \frac {1}{3} \, a c^{16} x^{3} + \frac {1}{19} \, {\left (16 \, b c d^{15} + a d^{16}\right )} x^{19} + \frac {4}{9} \, {\left (15 \, b c^{2} d^{14} + 2 \, a c d^{15}\right )} x^{18} + \frac {40}{17} \, {\left (14 \, b c^{3} d^{13} + 3 \, a c^{2} d^{14}\right )} x^{17} + \frac {35}{4} \, {\left (13 \, b c^{4} d^{12} + 4 \, a c^{3} d^{13}\right )} x^{16} + \frac {364}{15} \, {\left (12 \, b c^{5} d^{11} + 5 \, a c^{4} d^{12}\right )} x^{15} + 52 \, {\left (11 \, b c^{6} d^{10} + 6 \, a c^{5} d^{11}\right )} x^{14} + 88 \, {\left (10 \, b c^{7} d^{9} + 7 \, a c^{6} d^{10}\right )} x^{13} + \frac {715}{6} \, {\left (9 \, b c^{8} d^{8} + 8 \, a c^{7} d^{9}\right )} x^{12} + 130 \, {\left (8 \, b c^{9} d^{7} + 9 \, a c^{8} d^{8}\right )} x^{11} + \frac {572}{5} \, {\left (7 \, b c^{10} d^{6} + 10 \, a c^{9} d^{7}\right )} x^{10} + \frac {728}{9} \, {\left (6 \, b c^{11} d^{5} + 11 \, a c^{10} d^{6}\right )} x^{9} + \frac {91}{2} \, {\left (5 \, b c^{12} d^{4} + 12 \, a c^{11} d^{5}\right )} x^{8} + 20 \, {\left (4 \, b c^{13} d^{3} + 13 \, a c^{12} d^{4}\right )} x^{7} + \frac {20}{3} \, {\left (3 \, b c^{14} d^{2} + 14 \, a c^{13} d^{3}\right )} x^{6} + \frac {8}{5} \, {\left (2 \, b c^{15} d + 15 \, a c^{14} d^{2}\right )} x^{5} + \frac {1}{4} \, {\left (b c^{16} + 16 \, a c^{15} d\right )} x^{4} \]
1/20*b*d^16*x^20 + 1/3*a*c^16*x^3 + 1/19*(16*b*c*d^15 + a*d^16)*x^19 + 4/9 *(15*b*c^2*d^14 + 2*a*c*d^15)*x^18 + 40/17*(14*b*c^3*d^13 + 3*a*c^2*d^14)* x^17 + 35/4*(13*b*c^4*d^12 + 4*a*c^3*d^13)*x^16 + 364/15*(12*b*c^5*d^11 + 5*a*c^4*d^12)*x^15 + 52*(11*b*c^6*d^10 + 6*a*c^5*d^11)*x^14 + 88*(10*b*c^7 *d^9 + 7*a*c^6*d^10)*x^13 + 715/6*(9*b*c^8*d^8 + 8*a*c^7*d^9)*x^12 + 130*( 8*b*c^9*d^7 + 9*a*c^8*d^8)*x^11 + 572/5*(7*b*c^10*d^6 + 10*a*c^9*d^7)*x^10 + 728/9*(6*b*c^11*d^5 + 11*a*c^10*d^6)*x^9 + 91/2*(5*b*c^12*d^4 + 12*a*c^ 11*d^5)*x^8 + 20*(4*b*c^13*d^3 + 13*a*c^12*d^4)*x^7 + 20/3*(3*b*c^14*d^2 + 14*a*c^13*d^3)*x^6 + 8/5*(2*b*c^15*d + 15*a*c^14*d^2)*x^5 + 1/4*(b*c^16 + 16*a*c^15*d)*x^4
Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (80) = 160\).
Time = 0.29 (sec) , antiderivative size = 389, normalized size of antiderivative = 4.42 \[ \int x^2 (a+b x) (c+d x)^{16} \, dx=\frac {1}{20} \, b d^{16} x^{20} + \frac {16}{19} \, b c d^{15} x^{19} + \frac {1}{19} \, a d^{16} x^{19} + \frac {20}{3} \, b c^{2} d^{14} x^{18} + \frac {8}{9} \, a c d^{15} x^{18} + \frac {560}{17} \, b c^{3} d^{13} x^{17} + \frac {120}{17} \, a c^{2} d^{14} x^{17} + \frac {455}{4} \, b c^{4} d^{12} x^{16} + 35 \, a c^{3} d^{13} x^{16} + \frac {1456}{5} \, b c^{5} d^{11} x^{15} + \frac {364}{3} \, a c^{4} d^{12} x^{15} + 572 \, b c^{6} d^{10} x^{14} + 312 \, a c^{5} d^{11} x^{14} + 880 \, b c^{7} d^{9} x^{13} + 616 \, a c^{6} d^{10} x^{13} + \frac {2145}{2} \, b c^{8} d^{8} x^{12} + \frac {2860}{3} \, a c^{7} d^{9} x^{12} + 1040 \, b c^{9} d^{7} x^{11} + 1170 \, a c^{8} d^{8} x^{11} + \frac {4004}{5} \, b c^{10} d^{6} x^{10} + 1144 \, a c^{9} d^{7} x^{10} + \frac {1456}{3} \, b c^{11} d^{5} x^{9} + \frac {8008}{9} \, a c^{10} d^{6} x^{9} + \frac {455}{2} \, b c^{12} d^{4} x^{8} + 546 \, a c^{11} d^{5} x^{8} + 80 \, b c^{13} d^{3} x^{7} + 260 \, a c^{12} d^{4} x^{7} + 20 \, b c^{14} d^{2} x^{6} + \frac {280}{3} \, a c^{13} d^{3} x^{6} + \frac {16}{5} \, b c^{15} d x^{5} + 24 \, a c^{14} d^{2} x^{5} + \frac {1}{4} \, b c^{16} x^{4} + 4 \, a c^{15} d x^{4} + \frac {1}{3} \, a c^{16} x^{3} \]
1/20*b*d^16*x^20 + 16/19*b*c*d^15*x^19 + 1/19*a*d^16*x^19 + 20/3*b*c^2*d^1 4*x^18 + 8/9*a*c*d^15*x^18 + 560/17*b*c^3*d^13*x^17 + 120/17*a*c^2*d^14*x^ 17 + 455/4*b*c^4*d^12*x^16 + 35*a*c^3*d^13*x^16 + 1456/5*b*c^5*d^11*x^15 + 364/3*a*c^4*d^12*x^15 + 572*b*c^6*d^10*x^14 + 312*a*c^5*d^11*x^14 + 880*b *c^7*d^9*x^13 + 616*a*c^6*d^10*x^13 + 2145/2*b*c^8*d^8*x^12 + 2860/3*a*c^7 *d^9*x^12 + 1040*b*c^9*d^7*x^11 + 1170*a*c^8*d^8*x^11 + 4004/5*b*c^10*d^6* x^10 + 1144*a*c^9*d^7*x^10 + 1456/3*b*c^11*d^5*x^9 + 8008/9*a*c^10*d^6*x^9 + 455/2*b*c^12*d^4*x^8 + 546*a*c^11*d^5*x^8 + 80*b*c^13*d^3*x^7 + 260*a*c ^12*d^4*x^7 + 20*b*c^14*d^2*x^6 + 280/3*a*c^13*d^3*x^6 + 16/5*b*c^15*d*x^5 + 24*a*c^14*d^2*x^5 + 1/4*b*c^16*x^4 + 4*a*c^15*d*x^4 + 1/3*a*c^16*x^3
Time = 0.54 (sec) , antiderivative size = 331, normalized size of antiderivative = 3.76 \[ \int x^2 (a+b x) (c+d x)^{16} \, dx=x^4\,\left (\frac {b\,c^{16}}{4}+4\,a\,d\,c^{15}\right )+x^{19}\,\left (\frac {a\,d^{16}}{19}+\frac {16\,b\,c\,d^{15}}{19}\right )+\frac {a\,c^{16}\,x^3}{3}+\frac {b\,d^{16}\,x^{20}}{20}+\frac {8\,c^{14}\,d\,x^5\,\left (15\,a\,d+2\,b\,c\right )}{5}+\frac {4\,c\,d^{14}\,x^{18}\,\left (2\,a\,d+15\,b\,c\right )}{9}+\frac {20\,c^{13}\,d^2\,x^6\,\left (14\,a\,d+3\,b\,c\right )}{3}+20\,c^{12}\,d^3\,x^7\,\left (13\,a\,d+4\,b\,c\right )+\frac {91\,c^{11}\,d^4\,x^8\,\left (12\,a\,d+5\,b\,c\right )}{2}+\frac {728\,c^{10}\,d^5\,x^9\,\left (11\,a\,d+6\,b\,c\right )}{9}+\frac {572\,c^9\,d^6\,x^{10}\,\left (10\,a\,d+7\,b\,c\right )}{5}+130\,c^8\,d^7\,x^{11}\,\left (9\,a\,d+8\,b\,c\right )+\frac {715\,c^7\,d^8\,x^{12}\,\left (8\,a\,d+9\,b\,c\right )}{6}+88\,c^6\,d^9\,x^{13}\,\left (7\,a\,d+10\,b\,c\right )+52\,c^5\,d^{10}\,x^{14}\,\left (6\,a\,d+11\,b\,c\right )+\frac {364\,c^4\,d^{11}\,x^{15}\,\left (5\,a\,d+12\,b\,c\right )}{15}+\frac {35\,c^3\,d^{12}\,x^{16}\,\left (4\,a\,d+13\,b\,c\right )}{4}+\frac {40\,c^2\,d^{13}\,x^{17}\,\left (3\,a\,d+14\,b\,c\right )}{17} \]
x^4*((b*c^16)/4 + 4*a*c^15*d) + x^19*((a*d^16)/19 + (16*b*c*d^15)/19) + (a *c^16*x^3)/3 + (b*d^16*x^20)/20 + (8*c^14*d*x^5*(15*a*d + 2*b*c))/5 + (4*c *d^14*x^18*(2*a*d + 15*b*c))/9 + (20*c^13*d^2*x^6*(14*a*d + 3*b*c))/3 + 20 *c^12*d^3*x^7*(13*a*d + 4*b*c) + (91*c^11*d^4*x^8*(12*a*d + 5*b*c))/2 + (7 28*c^10*d^5*x^9*(11*a*d + 6*b*c))/9 + (572*c^9*d^6*x^10*(10*a*d + 7*b*c))/ 5 + 130*c^8*d^7*x^11*(9*a*d + 8*b*c) + (715*c^7*d^8*x^12*(8*a*d + 9*b*c))/ 6 + 88*c^6*d^9*x^13*(7*a*d + 10*b*c) + 52*c^5*d^10*x^14*(6*a*d + 11*b*c) + (364*c^4*d^11*x^15*(5*a*d + 12*b*c))/15 + (35*c^3*d^12*x^16*(4*a*d + 13*b *c))/4 + (40*c^2*d^13*x^17*(3*a*d + 14*b*c))/17