Integrand size = 18, antiderivative size = 177 \[ \int x^3 (a+b x)^2 (c+d x)^{16} \, dx=-\frac {c^3 (b c-a d)^2 (c+d x)^{17}}{17 d^6}+\frac {c^2 (5 b c-3 a d) (b c-a d) (c+d x)^{18}}{18 d^6}-\frac {c \left (10 b^2 c^2-12 a b c d+3 a^2 d^2\right ) (c+d x)^{19}}{19 d^6}+\frac {\left (10 b^2 c^2-8 a b c d+a^2 d^2\right ) (c+d x)^{20}}{20 d^6}-\frac {b (5 b c-2 a d) (c+d x)^{21}}{21 d^6}+\frac {b^2 (c+d x)^{22}}{22 d^6} \]
-1/17*c^3*(-a*d+b*c)^2*(d*x+c)^17/d^6+1/18*c^2*(-3*a*d+5*b*c)*(-a*d+b*c)*( d*x+c)^18/d^6-1/19*c*(3*a^2*d^2-12*a*b*c*d+10*b^2*c^2)*(d*x+c)^19/d^6+1/20 *(a^2*d^2-8*a*b*c*d+10*b^2*c^2)*(d*x+c)^20/d^6-1/21*b*(-2*a*d+5*b*c)*(d*x+ c)^21/d^6+1/22*b^2*(d*x+c)^22/d^6
Leaf count is larger than twice the leaf count of optimal. \(589\) vs. \(2(177)=354\).
Time = 0.08 (sec) , antiderivative size = 589, normalized size of antiderivative = 3.33 \[ \int x^3 (a+b x)^2 (c+d x)^{16} \, dx=\frac {1}{4} a^2 c^{16} x^4+\frac {2}{5} a c^{15} (b c+8 a d) x^5+\frac {1}{6} c^{14} \left (b^2 c^2+32 a b c d+120 a^2 d^2\right ) x^6+\frac {16}{7} c^{13} d \left (b^2 c^2+15 a b c d+35 a^2 d^2\right ) x^7+\frac {5}{2} c^{12} d^2 \left (6 b^2 c^2+56 a b c d+91 a^2 d^2\right ) x^8+\frac {56}{9} c^{11} d^3 \left (10 b^2 c^2+65 a b c d+78 a^2 d^2\right ) x^9+\frac {182}{5} c^{10} d^4 \left (5 b^2 c^2+24 a b c d+22 a^2 d^2\right ) x^{10}+\frac {208}{11} c^9 d^5 \left (21 b^2 c^2+77 a b c d+55 a^2 d^2\right ) x^{11}+\frac {143}{6} c^8 d^6 \left (28 b^2 c^2+80 a b c d+45 a^2 d^2\right ) x^{12}+220 c^7 d^7 \left (4 b^2 c^2+9 a b c d+4 a^2 d^2\right ) x^{13}+\frac {143}{7} c^6 d^8 \left (45 b^2 c^2+80 a b c d+28 a^2 d^2\right ) x^{14}+\frac {208}{15} c^5 d^9 \left (55 b^2 c^2+77 a b c d+21 a^2 d^2\right ) x^{15}+\frac {91}{4} c^4 d^{10} \left (22 b^2 c^2+24 a b c d+5 a^2 d^2\right ) x^{16}+\frac {56}{17} c^3 d^{11} \left (78 b^2 c^2+65 a b c d+10 a^2 d^2\right ) x^{17}+\frac {10}{9} c^2 d^{12} \left (91 b^2 c^2+56 a b c d+6 a^2 d^2\right ) x^{18}+\frac {16}{19} c d^{13} \left (35 b^2 c^2+15 a b c d+a^2 d^2\right ) x^{19}+\frac {1}{20} d^{14} \left (120 b^2 c^2+32 a b c d+a^2 d^2\right ) x^{20}+\frac {2}{21} b d^{15} (8 b c+a d) x^{21}+\frac {1}{22} b^2 d^{16} x^{22} \]
(a^2*c^16*x^4)/4 + (2*a*c^15*(b*c + 8*a*d)*x^5)/5 + (c^14*(b^2*c^2 + 32*a* b*c*d + 120*a^2*d^2)*x^6)/6 + (16*c^13*d*(b^2*c^2 + 15*a*b*c*d + 35*a^2*d^ 2)*x^7)/7 + (5*c^12*d^2*(6*b^2*c^2 + 56*a*b*c*d + 91*a^2*d^2)*x^8)/2 + (56 *c^11*d^3*(10*b^2*c^2 + 65*a*b*c*d + 78*a^2*d^2)*x^9)/9 + (182*c^10*d^4*(5 *b^2*c^2 + 24*a*b*c*d + 22*a^2*d^2)*x^10)/5 + (208*c^9*d^5*(21*b^2*c^2 + 7 7*a*b*c*d + 55*a^2*d^2)*x^11)/11 + (143*c^8*d^6*(28*b^2*c^2 + 80*a*b*c*d + 45*a^2*d^2)*x^12)/6 + 220*c^7*d^7*(4*b^2*c^2 + 9*a*b*c*d + 4*a^2*d^2)*x^1 3 + (143*c^6*d^8*(45*b^2*c^2 + 80*a*b*c*d + 28*a^2*d^2)*x^14)/7 + (208*c^5 *d^9*(55*b^2*c^2 + 77*a*b*c*d + 21*a^2*d^2)*x^15)/15 + (91*c^4*d^10*(22*b^ 2*c^2 + 24*a*b*c*d + 5*a^2*d^2)*x^16)/4 + (56*c^3*d^11*(78*b^2*c^2 + 65*a* b*c*d + 10*a^2*d^2)*x^17)/17 + (10*c^2*d^12*(91*b^2*c^2 + 56*a*b*c*d + 6*a ^2*d^2)*x^18)/9 + (16*c*d^13*(35*b^2*c^2 + 15*a*b*c*d + a^2*d^2)*x^19)/19 + (d^14*(120*b^2*c^2 + 32*a*b*c*d + a^2*d^2)*x^20)/20 + (2*b*d^15*(8*b*c + a*d)*x^21)/21 + (b^2*d^16*x^22)/22
Time = 0.64 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {99, 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.
| \(\displaystyle \int x^3 (a+b x)^2 (c+d x)^{16} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {(c+d x)^{19} \left (a^2 d^2-8 a b c d+10 b^2 c^2\right )}{d^5}-\frac {c (c+d x)^{18} \left (3 a^2 d^2-12 a b c d+10 b^2 c^2\right )}{d^5}-\frac {c^3 (c+d x)^{16} (b c-a d)^2}{d^5}+\frac {c^2 (c+d x)^{17} (5 b c-3 a d) (b c-a d)}{d^5}-\frac {b (c+d x)^{20} (5 b c-2 a d)}{d^5}+\frac {b^2 (c+d x)^{21}}{d^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(c+d x)^{20} \left (a^2 d^2-8 a b c d+10 b^2 c^2\right )}{20 d^6}-\frac {c (c+d x)^{19} \left (3 a^2 d^2-12 a b c d+10 b^2 c^2\right )}{19 d^6}-\frac {c^3 (c+d x)^{17} (b c-a d)^2}{17 d^6}+\frac {c^2 (c+d x)^{18} (5 b c-3 a d) (b c-a d)}{18 d^6}-\frac {b (c+d x)^{21} (5 b c-2 a d)}{21 d^6}+\frac {b^2 (c+d x)^{22}}{22 d^6}\) |
-1/17*(c^3*(b*c - a*d)^2*(c + d*x)^17)/d^6 + (c^2*(5*b*c - 3*a*d)*(b*c - a *d)*(c + d*x)^18)/(18*d^6) - (c*(10*b^2*c^2 - 12*a*b*c*d + 3*a^2*d^2)*(c + d*x)^19)/(19*d^6) + ((10*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*(c + d*x)^20)/(20 *d^6) - (b*(5*b*c - 2*a*d)*(c + d*x)^21)/(21*d^6) + (b^2*(c + d*x)^22)/(22 *d^6)
3.3.2.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Leaf count of result is larger than twice the leaf count of optimal. \(606\) vs. \(2(165)=330\).
Time = 0.43 (sec) , antiderivative size = 607, normalized size of antiderivative = 3.43
| method | result | size |
| norman | \(\frac {a^{2} c^{16} x^{4}}{4}+\left (\frac {16}{5} a^{2} c^{15} d +\frac {2}{5} a b \,c^{16}\right ) x^{5}+\left (20 a^{2} c^{14} d^{2}+\frac {16}{3} a b \,c^{15} d +\frac {1}{6} b^{2} c^{16}\right ) x^{6}+\left (80 a^{2} c^{13} d^{3}+\frac {240}{7} a b \,c^{14} d^{2}+\frac {16}{7} b^{2} c^{15} d \right ) x^{7}+\left (\frac {455}{2} a^{2} c^{12} d^{4}+140 a b \,c^{13} d^{3}+15 b^{2} c^{14} d^{2}\right ) x^{8}+\left (\frac {1456}{3} a^{2} c^{11} d^{5}+\frac {3640}{9} a b \,c^{12} d^{4}+\frac {560}{9} b^{2} c^{13} d^{3}\right ) x^{9}+\left (\frac {4004}{5} a^{2} c^{10} d^{6}+\frac {4368}{5} a b \,c^{11} d^{5}+182 b^{2} c^{12} d^{4}\right ) x^{10}+\left (1040 a^{2} c^{9} d^{7}+1456 a b \,c^{10} d^{6}+\frac {4368}{11} b^{2} c^{11} d^{5}\right ) x^{11}+\left (\frac {2145}{2} a^{2} c^{8} d^{8}+\frac {5720}{3} a b \,c^{9} d^{7}+\frac {2002}{3} b^{2} c^{10} d^{6}\right ) x^{12}+\left (880 a^{2} c^{7} d^{9}+1980 a b \,c^{8} d^{8}+880 b^{2} c^{9} d^{7}\right ) x^{13}+\left (572 a^{2} c^{6} d^{10}+\frac {11440}{7} a b \,c^{7} d^{9}+\frac {6435}{7} b^{2} c^{8} d^{8}\right ) x^{14}+\left (\frac {1456}{5} a^{2} c^{5} d^{11}+\frac {16016}{15} a b \,c^{6} d^{10}+\frac {2288}{3} b^{2} c^{7} d^{9}\right ) x^{15}+\left (\frac {455}{4} a^{2} c^{4} d^{12}+546 a b \,c^{5} d^{11}+\frac {1001}{2} b^{2} c^{6} d^{10}\right ) x^{16}+\left (\frac {560}{17} a^{2} c^{3} d^{13}+\frac {3640}{17} a b \,c^{4} d^{12}+\frac {4368}{17} b^{2} c^{5} d^{11}\right ) x^{17}+\left (\frac {20}{3} a^{2} c^{2} d^{14}+\frac {560}{9} a b \,c^{3} d^{13}+\frac {910}{9} b^{2} c^{4} d^{12}\right ) x^{18}+\left (\frac {16}{19} a^{2} c \,d^{15}+\frac {240}{19} a b \,c^{2} d^{14}+\frac {560}{19} b^{2} c^{3} d^{13}\right ) x^{19}+\left (\frac {1}{20} a^{2} d^{16}+\frac {8}{5} a b c \,d^{15}+6 b^{2} c^{2} d^{14}\right ) x^{20}+\left (\frac {2}{21} a b \,d^{16}+\frac {16}{21} b^{2} c \,d^{15}\right ) x^{21}+\frac {b^{2} d^{16} x^{22}}{22}\) | \(607\) |
| default | \(\frac {b^{2} d^{16} x^{22}}{22}+\frac {\left (2 a b \,d^{16}+16 b^{2} c \,d^{15}\right ) x^{21}}{21}+\frac {\left (a^{2} d^{16}+32 a b c \,d^{15}+120 b^{2} c^{2} d^{14}\right ) x^{20}}{20}+\frac {\left (16 a^{2} c \,d^{15}+240 a b \,c^{2} d^{14}+560 b^{2} c^{3} d^{13}\right ) x^{19}}{19}+\frac {\left (120 a^{2} c^{2} d^{14}+1120 a b \,c^{3} d^{13}+1820 b^{2} c^{4} d^{12}\right ) x^{18}}{18}+\frac {\left (560 a^{2} c^{3} d^{13}+3640 a b \,c^{4} d^{12}+4368 b^{2} c^{5} d^{11}\right ) x^{17}}{17}+\frac {\left (1820 a^{2} c^{4} d^{12}+8736 a b \,c^{5} d^{11}+8008 b^{2} c^{6} d^{10}\right ) x^{16}}{16}+\frac {\left (4368 a^{2} c^{5} d^{11}+16016 a b \,c^{6} d^{10}+11440 b^{2} c^{7} d^{9}\right ) x^{15}}{15}+\frac {\left (8008 a^{2} c^{6} d^{10}+22880 a b \,c^{7} d^{9}+12870 b^{2} c^{8} d^{8}\right ) x^{14}}{14}+\frac {\left (11440 a^{2} c^{7} d^{9}+25740 a b \,c^{8} d^{8}+11440 b^{2} c^{9} d^{7}\right ) x^{13}}{13}+\frac {\left (12870 a^{2} c^{8} d^{8}+22880 a b \,c^{9} d^{7}+8008 b^{2} c^{10} d^{6}\right ) x^{12}}{12}+\frac {\left (11440 a^{2} c^{9} d^{7}+16016 a b \,c^{10} d^{6}+4368 b^{2} c^{11} d^{5}\right ) x^{11}}{11}+\frac {\left (8008 a^{2} c^{10} d^{6}+8736 a b \,c^{11} d^{5}+1820 b^{2} c^{12} d^{4}\right ) x^{10}}{10}+\frac {\left (4368 a^{2} c^{11} d^{5}+3640 a b \,c^{12} d^{4}+560 b^{2} c^{13} d^{3}\right ) x^{9}}{9}+\frac {\left (1820 a^{2} c^{12} d^{4}+1120 a b \,c^{13} d^{3}+120 b^{2} c^{14} d^{2}\right ) x^{8}}{8}+\frac {\left (560 a^{2} c^{13} d^{3}+240 a b \,c^{14} d^{2}+16 b^{2} c^{15} d \right ) x^{7}}{7}+\frac {\left (120 a^{2} c^{14} d^{2}+32 a b \,c^{15} d +b^{2} c^{16}\right ) x^{6}}{6}+\frac {\left (16 a^{2} c^{15} d +2 a b \,c^{16}\right ) x^{5}}{5}+\frac {a^{2} c^{16} x^{4}}{4}\) | \(622\) |
| gosper | \(\frac {1}{6} x^{6} b^{2} c^{16}+\frac {1}{20} x^{20} a^{2} d^{16}+\frac {1}{4} a^{2} c^{16} x^{4}+\frac {1}{22} b^{2} d^{16} x^{22}+572 x^{14} a^{2} c^{6} d^{10}+\frac {6435}{7} x^{14} b^{2} c^{8} d^{8}+\frac {1456}{5} x^{15} a^{2} c^{5} d^{11}+\frac {2288}{3} x^{15} b^{2} c^{7} d^{9}+\frac {455}{4} x^{16} a^{2} c^{4} d^{12}+\frac {1001}{2} x^{16} b^{2} c^{6} d^{10}+\frac {560}{17} x^{17} a^{2} c^{3} d^{13}+\frac {4368}{17} x^{17} b^{2} c^{5} d^{11}+\frac {20}{3} x^{18} a^{2} c^{2} d^{14}+\frac {910}{9} x^{18} b^{2} c^{4} d^{12}+\frac {16}{19} x^{19} a^{2} c \,d^{15}+\frac {560}{19} x^{19} b^{2} c^{3} d^{13}+6 x^{20} b^{2} c^{2} d^{14}+\frac {2}{21} x^{21} a b \,d^{16}+\frac {16}{21} x^{21} b^{2} c \,d^{15}+880 a^{2} c^{7} d^{9} x^{13}+880 b^{2} c^{9} d^{7} x^{13}+\frac {16}{5} x^{5} a^{2} c^{15} d +\frac {2}{5} x^{5} a b \,c^{16}+20 x^{6} a^{2} c^{14} d^{2}+80 x^{7} a^{2} c^{13} d^{3}+\frac {16}{7} x^{7} b^{2} c^{15} d +\frac {455}{2} x^{8} a^{2} c^{12} d^{4}+15 x^{8} b^{2} c^{14} d^{2}+\frac {1456}{3} x^{9} a^{2} c^{11} d^{5}+\frac {560}{9} x^{9} b^{2} c^{13} d^{3}+\frac {4004}{5} x^{10} a^{2} c^{10} d^{6}+182 x^{10} b^{2} c^{12} d^{4}+1040 x^{11} a^{2} c^{9} d^{7}+\frac {4368}{11} x^{11} b^{2} c^{11} d^{5}+\frac {2145}{2} x^{12} a^{2} c^{8} d^{8}+\frac {2002}{3} x^{12} b^{2} c^{10} d^{6}+\frac {240}{7} x^{7} a b \,c^{14} d^{2}+140 x^{8} a b \,c^{13} d^{3}+\frac {3640}{9} x^{9} a b \,c^{12} d^{4}+\frac {4368}{5} x^{10} a b \,c^{11} d^{5}+1456 x^{11} a b \,c^{10} d^{6}+\frac {5720}{3} x^{12} a b \,c^{9} d^{7}+\frac {11440}{7} x^{14} a b \,c^{7} d^{9}+\frac {16016}{15} x^{15} a b \,c^{6} d^{10}+\frac {16}{3} x^{6} a b \,c^{15} d +546 x^{16} a b \,c^{5} d^{11}+\frac {3640}{17} x^{17} a b \,c^{4} d^{12}+\frac {560}{9} x^{18} a b \,c^{3} d^{13}+\frac {240}{19} x^{19} a b \,c^{2} d^{14}+\frac {8}{5} x^{20} a b c \,d^{15}+1980 a b \,c^{8} d^{8} x^{13}\) | \(669\) |
| risch | \(\frac {1}{6} x^{6} b^{2} c^{16}+\frac {1}{20} x^{20} a^{2} d^{16}+\frac {1}{4} a^{2} c^{16} x^{4}+\frac {1}{22} b^{2} d^{16} x^{22}+572 x^{14} a^{2} c^{6} d^{10}+\frac {6435}{7} x^{14} b^{2} c^{8} d^{8}+\frac {1456}{5} x^{15} a^{2} c^{5} d^{11}+\frac {2288}{3} x^{15} b^{2} c^{7} d^{9}+\frac {455}{4} x^{16} a^{2} c^{4} d^{12}+\frac {1001}{2} x^{16} b^{2} c^{6} d^{10}+\frac {560}{17} x^{17} a^{2} c^{3} d^{13}+\frac {4368}{17} x^{17} b^{2} c^{5} d^{11}+\frac {20}{3} x^{18} a^{2} c^{2} d^{14}+\frac {910}{9} x^{18} b^{2} c^{4} d^{12}+\frac {16}{19} x^{19} a^{2} c \,d^{15}+\frac {560}{19} x^{19} b^{2} c^{3} d^{13}+6 x^{20} b^{2} c^{2} d^{14}+\frac {2}{21} x^{21} a b \,d^{16}+\frac {16}{21} x^{21} b^{2} c \,d^{15}+880 a^{2} c^{7} d^{9} x^{13}+880 b^{2} c^{9} d^{7} x^{13}+\frac {16}{5} x^{5} a^{2} c^{15} d +\frac {2}{5} x^{5} a b \,c^{16}+20 x^{6} a^{2} c^{14} d^{2}+80 x^{7} a^{2} c^{13} d^{3}+\frac {16}{7} x^{7} b^{2} c^{15} d +\frac {455}{2} x^{8} a^{2} c^{12} d^{4}+15 x^{8} b^{2} c^{14} d^{2}+\frac {1456}{3} x^{9} a^{2} c^{11} d^{5}+\frac {560}{9} x^{9} b^{2} c^{13} d^{3}+\frac {4004}{5} x^{10} a^{2} c^{10} d^{6}+182 x^{10} b^{2} c^{12} d^{4}+1040 x^{11} a^{2} c^{9} d^{7}+\frac {4368}{11} x^{11} b^{2} c^{11} d^{5}+\frac {2145}{2} x^{12} a^{2} c^{8} d^{8}+\frac {2002}{3} x^{12} b^{2} c^{10} d^{6}+\frac {240}{7} x^{7} a b \,c^{14} d^{2}+140 x^{8} a b \,c^{13} d^{3}+\frac {3640}{9} x^{9} a b \,c^{12} d^{4}+\frac {4368}{5} x^{10} a b \,c^{11} d^{5}+1456 x^{11} a b \,c^{10} d^{6}+\frac {5720}{3} x^{12} a b \,c^{9} d^{7}+\frac {11440}{7} x^{14} a b \,c^{7} d^{9}+\frac {16016}{15} x^{15} a b \,c^{6} d^{10}+\frac {16}{3} x^{6} a b \,c^{15} d +546 x^{16} a b \,c^{5} d^{11}+\frac {3640}{17} x^{17} a b \,c^{4} d^{12}+\frac {560}{9} x^{18} a b \,c^{3} d^{13}+\frac {240}{19} x^{19} a b \,c^{2} d^{14}+\frac {8}{5} x^{20} a b c \,d^{15}+1980 a b \,c^{8} d^{8} x^{13}\) | \(669\) |
| parallelrisch | \(\frac {1}{6} x^{6} b^{2} c^{16}+\frac {1}{20} x^{20} a^{2} d^{16}+\frac {1}{4} a^{2} c^{16} x^{4}+\frac {1}{22} b^{2} d^{16} x^{22}+572 x^{14} a^{2} c^{6} d^{10}+\frac {6435}{7} x^{14} b^{2} c^{8} d^{8}+\frac {1456}{5} x^{15} a^{2} c^{5} d^{11}+\frac {2288}{3} x^{15} b^{2} c^{7} d^{9}+\frac {455}{4} x^{16} a^{2} c^{4} d^{12}+\frac {1001}{2} x^{16} b^{2} c^{6} d^{10}+\frac {560}{17} x^{17} a^{2} c^{3} d^{13}+\frac {4368}{17} x^{17} b^{2} c^{5} d^{11}+\frac {20}{3} x^{18} a^{2} c^{2} d^{14}+\frac {910}{9} x^{18} b^{2} c^{4} d^{12}+\frac {16}{19} x^{19} a^{2} c \,d^{15}+\frac {560}{19} x^{19} b^{2} c^{3} d^{13}+6 x^{20} b^{2} c^{2} d^{14}+\frac {2}{21} x^{21} a b \,d^{16}+\frac {16}{21} x^{21} b^{2} c \,d^{15}+880 a^{2} c^{7} d^{9} x^{13}+880 b^{2} c^{9} d^{7} x^{13}+\frac {16}{5} x^{5} a^{2} c^{15} d +\frac {2}{5} x^{5} a b \,c^{16}+20 x^{6} a^{2} c^{14} d^{2}+80 x^{7} a^{2} c^{13} d^{3}+\frac {16}{7} x^{7} b^{2} c^{15} d +\frac {455}{2} x^{8} a^{2} c^{12} d^{4}+15 x^{8} b^{2} c^{14} d^{2}+\frac {1456}{3} x^{9} a^{2} c^{11} d^{5}+\frac {560}{9} x^{9} b^{2} c^{13} d^{3}+\frac {4004}{5} x^{10} a^{2} c^{10} d^{6}+182 x^{10} b^{2} c^{12} d^{4}+1040 x^{11} a^{2} c^{9} d^{7}+\frac {4368}{11} x^{11} b^{2} c^{11} d^{5}+\frac {2145}{2} x^{12} a^{2} c^{8} d^{8}+\frac {2002}{3} x^{12} b^{2} c^{10} d^{6}+\frac {240}{7} x^{7} a b \,c^{14} d^{2}+140 x^{8} a b \,c^{13} d^{3}+\frac {3640}{9} x^{9} a b \,c^{12} d^{4}+\frac {4368}{5} x^{10} a b \,c^{11} d^{5}+1456 x^{11} a b \,c^{10} d^{6}+\frac {5720}{3} x^{12} a b \,c^{9} d^{7}+\frac {11440}{7} x^{14} a b \,c^{7} d^{9}+\frac {16016}{15} x^{15} a b \,c^{6} d^{10}+\frac {16}{3} x^{6} a b \,c^{15} d +546 x^{16} a b \,c^{5} d^{11}+\frac {3640}{17} x^{17} a b \,c^{4} d^{12}+\frac {560}{9} x^{18} a b \,c^{3} d^{13}+\frac {240}{19} x^{19} a b \,c^{2} d^{14}+\frac {8}{5} x^{20} a b c \,d^{15}+1980 a b \,c^{8} d^{8} x^{13}\) | \(669\) |
1/4*a^2*c^16*x^4+(16/5*a^2*c^15*d+2/5*a*b*c^16)*x^5+(20*a^2*c^14*d^2+16/3* a*b*c^15*d+1/6*b^2*c^16)*x^6+(80*a^2*c^13*d^3+240/7*a*b*c^14*d^2+16/7*b^2* c^15*d)*x^7+(455/2*a^2*c^12*d^4+140*a*b*c^13*d^3+15*b^2*c^14*d^2)*x^8+(145 6/3*a^2*c^11*d^5+3640/9*a*b*c^12*d^4+560/9*b^2*c^13*d^3)*x^9+(4004/5*a^2*c ^10*d^6+4368/5*a*b*c^11*d^5+182*b^2*c^12*d^4)*x^10+(1040*a^2*c^9*d^7+1456* a*b*c^10*d^6+4368/11*b^2*c^11*d^5)*x^11+(2145/2*a^2*c^8*d^8+5720/3*a*b*c^9 *d^7+2002/3*b^2*c^10*d^6)*x^12+(880*a^2*c^7*d^9+1980*a*b*c^8*d^8+880*b^2*c ^9*d^7)*x^13+(572*a^2*c^6*d^10+11440/7*a*b*c^7*d^9+6435/7*b^2*c^8*d^8)*x^1 4+(1456/5*a^2*c^5*d^11+16016/15*a*b*c^6*d^10+2288/3*b^2*c^7*d^9)*x^15+(455 /4*a^2*c^4*d^12+546*a*b*c^5*d^11+1001/2*b^2*c^6*d^10)*x^16+(560/17*a^2*c^3 *d^13+3640/17*a*b*c^4*d^12+4368/17*b^2*c^5*d^11)*x^17+(20/3*a^2*c^2*d^14+5 60/9*a*b*c^3*d^13+910/9*b^2*c^4*d^12)*x^18+(16/19*a^2*c*d^15+240/19*a*b*c^ 2*d^14+560/19*b^2*c^3*d^13)*x^19+(1/20*a^2*d^16+8/5*a*b*c*d^15+6*b^2*c^2*d ^14)*x^20+(2/21*a*b*d^16+16/21*b^2*c*d^15)*x^21+1/22*b^2*d^16*x^22
Leaf count of result is larger than twice the leaf count of optimal. 617 vs. \(2 (165) = 330\).
Time = 0.23 (sec) , antiderivative size = 617, normalized size of antiderivative = 3.49 \[ \int x^3 (a+b x)^2 (c+d x)^{16} \, dx=\frac {1}{22} \, b^{2} d^{16} x^{22} + \frac {1}{4} \, a^{2} c^{16} x^{4} + \frac {2}{21} \, {\left (8 \, b^{2} c d^{15} + a b d^{16}\right )} x^{21} + \frac {1}{20} \, {\left (120 \, b^{2} c^{2} d^{14} + 32 \, a b c d^{15} + a^{2} d^{16}\right )} x^{20} + \frac {16}{19} \, {\left (35 \, b^{2} c^{3} d^{13} + 15 \, a b c^{2} d^{14} + a^{2} c d^{15}\right )} x^{19} + \frac {10}{9} \, {\left (91 \, b^{2} c^{4} d^{12} + 56 \, a b c^{3} d^{13} + 6 \, a^{2} c^{2} d^{14}\right )} x^{18} + \frac {56}{17} \, {\left (78 \, b^{2} c^{5} d^{11} + 65 \, a b c^{4} d^{12} + 10 \, a^{2} c^{3} d^{13}\right )} x^{17} + \frac {91}{4} \, {\left (22 \, b^{2} c^{6} d^{10} + 24 \, a b c^{5} d^{11} + 5 \, a^{2} c^{4} d^{12}\right )} x^{16} + \frac {208}{15} \, {\left (55 \, b^{2} c^{7} d^{9} + 77 \, a b c^{6} d^{10} + 21 \, a^{2} c^{5} d^{11}\right )} x^{15} + \frac {143}{7} \, {\left (45 \, b^{2} c^{8} d^{8} + 80 \, a b c^{7} d^{9} + 28 \, a^{2} c^{6} d^{10}\right )} x^{14} + 220 \, {\left (4 \, b^{2} c^{9} d^{7} + 9 \, a b c^{8} d^{8} + 4 \, a^{2} c^{7} d^{9}\right )} x^{13} + \frac {143}{6} \, {\left (28 \, b^{2} c^{10} d^{6} + 80 \, a b c^{9} d^{7} + 45 \, a^{2} c^{8} d^{8}\right )} x^{12} + \frac {208}{11} \, {\left (21 \, b^{2} c^{11} d^{5} + 77 \, a b c^{10} d^{6} + 55 \, a^{2} c^{9} d^{7}\right )} x^{11} + \frac {182}{5} \, {\left (5 \, b^{2} c^{12} d^{4} + 24 \, a b c^{11} d^{5} + 22 \, a^{2} c^{10} d^{6}\right )} x^{10} + \frac {56}{9} \, {\left (10 \, b^{2} c^{13} d^{3} + 65 \, a b c^{12} d^{4} + 78 \, a^{2} c^{11} d^{5}\right )} x^{9} + \frac {5}{2} \, {\left (6 \, b^{2} c^{14} d^{2} + 56 \, a b c^{13} d^{3} + 91 \, a^{2} c^{12} d^{4}\right )} x^{8} + \frac {16}{7} \, {\left (b^{2} c^{15} d + 15 \, a b c^{14} d^{2} + 35 \, a^{2} c^{13} d^{3}\right )} x^{7} + \frac {1}{6} \, {\left (b^{2} c^{16} + 32 \, a b c^{15} d + 120 \, a^{2} c^{14} d^{2}\right )} x^{6} + \frac {2}{5} \, {\left (a b c^{16} + 8 \, a^{2} c^{15} d\right )} x^{5} \]
1/22*b^2*d^16*x^22 + 1/4*a^2*c^16*x^4 + 2/21*(8*b^2*c*d^15 + a*b*d^16)*x^2 1 + 1/20*(120*b^2*c^2*d^14 + 32*a*b*c*d^15 + a^2*d^16)*x^20 + 16/19*(35*b^ 2*c^3*d^13 + 15*a*b*c^2*d^14 + a^2*c*d^15)*x^19 + 10/9*(91*b^2*c^4*d^12 + 56*a*b*c^3*d^13 + 6*a^2*c^2*d^14)*x^18 + 56/17*(78*b^2*c^5*d^11 + 65*a*b*c ^4*d^12 + 10*a^2*c^3*d^13)*x^17 + 91/4*(22*b^2*c^6*d^10 + 24*a*b*c^5*d^11 + 5*a^2*c^4*d^12)*x^16 + 208/15*(55*b^2*c^7*d^9 + 77*a*b*c^6*d^10 + 21*a^2 *c^5*d^11)*x^15 + 143/7*(45*b^2*c^8*d^8 + 80*a*b*c^7*d^9 + 28*a^2*c^6*d^10 )*x^14 + 220*(4*b^2*c^9*d^7 + 9*a*b*c^8*d^8 + 4*a^2*c^7*d^9)*x^13 + 143/6* (28*b^2*c^10*d^6 + 80*a*b*c^9*d^7 + 45*a^2*c^8*d^8)*x^12 + 208/11*(21*b^2* c^11*d^5 + 77*a*b*c^10*d^6 + 55*a^2*c^9*d^7)*x^11 + 182/5*(5*b^2*c^12*d^4 + 24*a*b*c^11*d^5 + 22*a^2*c^10*d^6)*x^10 + 56/9*(10*b^2*c^13*d^3 + 65*a*b *c^12*d^4 + 78*a^2*c^11*d^5)*x^9 + 5/2*(6*b^2*c^14*d^2 + 56*a*b*c^13*d^3 + 91*a^2*c^12*d^4)*x^8 + 16/7*(b^2*c^15*d + 15*a*b*c^14*d^2 + 35*a^2*c^13*d ^3)*x^7 + 1/6*(b^2*c^16 + 32*a*b*c^15*d + 120*a^2*c^14*d^2)*x^6 + 2/5*(a*b *c^16 + 8*a^2*c^15*d)*x^5
Leaf count of result is larger than twice the leaf count of optimal. 697 vs. \(2 (170) = 340\).
Time = 0.07 (sec) , antiderivative size = 697, normalized size of antiderivative = 3.94 \[ \int x^3 (a+b x)^2 (c+d x)^{16} \, dx=\frac {a^{2} c^{16} x^{4}}{4} + \frac {b^{2} d^{16} x^{22}}{22} + x^{21} \cdot \left (\frac {2 a b d^{16}}{21} + \frac {16 b^{2} c d^{15}}{21}\right ) + x^{20} \left (\frac {a^{2} d^{16}}{20} + \frac {8 a b c d^{15}}{5} + 6 b^{2} c^{2} d^{14}\right ) + x^{19} \cdot \left (\frac {16 a^{2} c d^{15}}{19} + \frac {240 a b c^{2} d^{14}}{19} + \frac {560 b^{2} c^{3} d^{13}}{19}\right ) + x^{18} \cdot \left (\frac {20 a^{2} c^{2} d^{14}}{3} + \frac {560 a b c^{3} d^{13}}{9} + \frac {910 b^{2} c^{4} d^{12}}{9}\right ) + x^{17} \cdot \left (\frac {560 a^{2} c^{3} d^{13}}{17} + \frac {3640 a b c^{4} d^{12}}{17} + \frac {4368 b^{2} c^{5} d^{11}}{17}\right ) + x^{16} \cdot \left (\frac {455 a^{2} c^{4} d^{12}}{4} + 546 a b c^{5} d^{11} + \frac {1001 b^{2} c^{6} d^{10}}{2}\right ) + x^{15} \cdot \left (\frac {1456 a^{2} c^{5} d^{11}}{5} + \frac {16016 a b c^{6} d^{10}}{15} + \frac {2288 b^{2} c^{7} d^{9}}{3}\right ) + x^{14} \cdot \left (572 a^{2} c^{6} d^{10} + \frac {11440 a b c^{7} d^{9}}{7} + \frac {6435 b^{2} c^{8} d^{8}}{7}\right ) + x^{13} \cdot \left (880 a^{2} c^{7} d^{9} + 1980 a b c^{8} d^{8} + 880 b^{2} c^{9} d^{7}\right ) + x^{12} \cdot \left (\frac {2145 a^{2} c^{8} d^{8}}{2} + \frac {5720 a b c^{9} d^{7}}{3} + \frac {2002 b^{2} c^{10} d^{6}}{3}\right ) + x^{11} \cdot \left (1040 a^{2} c^{9} d^{7} + 1456 a b c^{10} d^{6} + \frac {4368 b^{2} c^{11} d^{5}}{11}\right ) + x^{10} \cdot \left (\frac {4004 a^{2} c^{10} d^{6}}{5} + \frac {4368 a b c^{11} d^{5}}{5} + 182 b^{2} c^{12} d^{4}\right ) + x^{9} \cdot \left (\frac {1456 a^{2} c^{11} d^{5}}{3} + \frac {3640 a b c^{12} d^{4}}{9} + \frac {560 b^{2} c^{13} d^{3}}{9}\right ) + x^{8} \cdot \left (\frac {455 a^{2} c^{12} d^{4}}{2} + 140 a b c^{13} d^{3} + 15 b^{2} c^{14} d^{2}\right ) + x^{7} \cdot \left (80 a^{2} c^{13} d^{3} + \frac {240 a b c^{14} d^{2}}{7} + \frac {16 b^{2} c^{15} d}{7}\right ) + x^{6} \cdot \left (20 a^{2} c^{14} d^{2} + \frac {16 a b c^{15} d}{3} + \frac {b^{2} c^{16}}{6}\right ) + x^{5} \cdot \left (\frac {16 a^{2} c^{15} d}{5} + \frac {2 a b c^{16}}{5}\right ) \]
a**2*c**16*x**4/4 + b**2*d**16*x**22/22 + x**21*(2*a*b*d**16/21 + 16*b**2* c*d**15/21) + x**20*(a**2*d**16/20 + 8*a*b*c*d**15/5 + 6*b**2*c**2*d**14) + x**19*(16*a**2*c*d**15/19 + 240*a*b*c**2*d**14/19 + 560*b**2*c**3*d**13/ 19) + x**18*(20*a**2*c**2*d**14/3 + 560*a*b*c**3*d**13/9 + 910*b**2*c**4*d **12/9) + x**17*(560*a**2*c**3*d**13/17 + 3640*a*b*c**4*d**12/17 + 4368*b* *2*c**5*d**11/17) + x**16*(455*a**2*c**4*d**12/4 + 546*a*b*c**5*d**11 + 10 01*b**2*c**6*d**10/2) + x**15*(1456*a**2*c**5*d**11/5 + 16016*a*b*c**6*d** 10/15 + 2288*b**2*c**7*d**9/3) + x**14*(572*a**2*c**6*d**10 + 11440*a*b*c* *7*d**9/7 + 6435*b**2*c**8*d**8/7) + x**13*(880*a**2*c**7*d**9 + 1980*a*b* c**8*d**8 + 880*b**2*c**9*d**7) + x**12*(2145*a**2*c**8*d**8/2 + 5720*a*b* c**9*d**7/3 + 2002*b**2*c**10*d**6/3) + x**11*(1040*a**2*c**9*d**7 + 1456* a*b*c**10*d**6 + 4368*b**2*c**11*d**5/11) + x**10*(4004*a**2*c**10*d**6/5 + 4368*a*b*c**11*d**5/5 + 182*b**2*c**12*d**4) + x**9*(1456*a**2*c**11*d** 5/3 + 3640*a*b*c**12*d**4/9 + 560*b**2*c**13*d**3/9) + x**8*(455*a**2*c**1 2*d**4/2 + 140*a*b*c**13*d**3 + 15*b**2*c**14*d**2) + x**7*(80*a**2*c**13* d**3 + 240*a*b*c**14*d**2/7 + 16*b**2*c**15*d/7) + x**6*(20*a**2*c**14*d** 2 + 16*a*b*c**15*d/3 + b**2*c**16/6) + x**5*(16*a**2*c**15*d/5 + 2*a*b*c** 16/5)
Leaf count of result is larger than twice the leaf count of optimal. 617 vs. \(2 (165) = 330\).
Time = 0.19 (sec) , antiderivative size = 617, normalized size of antiderivative = 3.49 \[ \int x^3 (a+b x)^2 (c+d x)^{16} \, dx=\frac {1}{22} \, b^{2} d^{16} x^{22} + \frac {1}{4} \, a^{2} c^{16} x^{4} + \frac {2}{21} \, {\left (8 \, b^{2} c d^{15} + a b d^{16}\right )} x^{21} + \frac {1}{20} \, {\left (120 \, b^{2} c^{2} d^{14} + 32 \, a b c d^{15} + a^{2} d^{16}\right )} x^{20} + \frac {16}{19} \, {\left (35 \, b^{2} c^{3} d^{13} + 15 \, a b c^{2} d^{14} + a^{2} c d^{15}\right )} x^{19} + \frac {10}{9} \, {\left (91 \, b^{2} c^{4} d^{12} + 56 \, a b c^{3} d^{13} + 6 \, a^{2} c^{2} d^{14}\right )} x^{18} + \frac {56}{17} \, {\left (78 \, b^{2} c^{5} d^{11} + 65 \, a b c^{4} d^{12} + 10 \, a^{2} c^{3} d^{13}\right )} x^{17} + \frac {91}{4} \, {\left (22 \, b^{2} c^{6} d^{10} + 24 \, a b c^{5} d^{11} + 5 \, a^{2} c^{4} d^{12}\right )} x^{16} + \frac {208}{15} \, {\left (55 \, b^{2} c^{7} d^{9} + 77 \, a b c^{6} d^{10} + 21 \, a^{2} c^{5} d^{11}\right )} x^{15} + \frac {143}{7} \, {\left (45 \, b^{2} c^{8} d^{8} + 80 \, a b c^{7} d^{9} + 28 \, a^{2} c^{6} d^{10}\right )} x^{14} + 220 \, {\left (4 \, b^{2} c^{9} d^{7} + 9 \, a b c^{8} d^{8} + 4 \, a^{2} c^{7} d^{9}\right )} x^{13} + \frac {143}{6} \, {\left (28 \, b^{2} c^{10} d^{6} + 80 \, a b c^{9} d^{7} + 45 \, a^{2} c^{8} d^{8}\right )} x^{12} + \frac {208}{11} \, {\left (21 \, b^{2} c^{11} d^{5} + 77 \, a b c^{10} d^{6} + 55 \, a^{2} c^{9} d^{7}\right )} x^{11} + \frac {182}{5} \, {\left (5 \, b^{2} c^{12} d^{4} + 24 \, a b c^{11} d^{5} + 22 \, a^{2} c^{10} d^{6}\right )} x^{10} + \frac {56}{9} \, {\left (10 \, b^{2} c^{13} d^{3} + 65 \, a b c^{12} d^{4} + 78 \, a^{2} c^{11} d^{5}\right )} x^{9} + \frac {5}{2} \, {\left (6 \, b^{2} c^{14} d^{2} + 56 \, a b c^{13} d^{3} + 91 \, a^{2} c^{12} d^{4}\right )} x^{8} + \frac {16}{7} \, {\left (b^{2} c^{15} d + 15 \, a b c^{14} d^{2} + 35 \, a^{2} c^{13} d^{3}\right )} x^{7} + \frac {1}{6} \, {\left (b^{2} c^{16} + 32 \, a b c^{15} d + 120 \, a^{2} c^{14} d^{2}\right )} x^{6} + \frac {2}{5} \, {\left (a b c^{16} + 8 \, a^{2} c^{15} d\right )} x^{5} \]
1/22*b^2*d^16*x^22 + 1/4*a^2*c^16*x^4 + 2/21*(8*b^2*c*d^15 + a*b*d^16)*x^2 1 + 1/20*(120*b^2*c^2*d^14 + 32*a*b*c*d^15 + a^2*d^16)*x^20 + 16/19*(35*b^ 2*c^3*d^13 + 15*a*b*c^2*d^14 + a^2*c*d^15)*x^19 + 10/9*(91*b^2*c^4*d^12 + 56*a*b*c^3*d^13 + 6*a^2*c^2*d^14)*x^18 + 56/17*(78*b^2*c^5*d^11 + 65*a*b*c ^4*d^12 + 10*a^2*c^3*d^13)*x^17 + 91/4*(22*b^2*c^6*d^10 + 24*a*b*c^5*d^11 + 5*a^2*c^4*d^12)*x^16 + 208/15*(55*b^2*c^7*d^9 + 77*a*b*c^6*d^10 + 21*a^2 *c^5*d^11)*x^15 + 143/7*(45*b^2*c^8*d^8 + 80*a*b*c^7*d^9 + 28*a^2*c^6*d^10 )*x^14 + 220*(4*b^2*c^9*d^7 + 9*a*b*c^8*d^8 + 4*a^2*c^7*d^9)*x^13 + 143/6* (28*b^2*c^10*d^6 + 80*a*b*c^9*d^7 + 45*a^2*c^8*d^8)*x^12 + 208/11*(21*b^2* c^11*d^5 + 77*a*b*c^10*d^6 + 55*a^2*c^9*d^7)*x^11 + 182/5*(5*b^2*c^12*d^4 + 24*a*b*c^11*d^5 + 22*a^2*c^10*d^6)*x^10 + 56/9*(10*b^2*c^13*d^3 + 65*a*b *c^12*d^4 + 78*a^2*c^11*d^5)*x^9 + 5/2*(6*b^2*c^14*d^2 + 56*a*b*c^13*d^3 + 91*a^2*c^12*d^4)*x^8 + 16/7*(b^2*c^15*d + 15*a*b*c^14*d^2 + 35*a^2*c^13*d ^3)*x^7 + 1/6*(b^2*c^16 + 32*a*b*c^15*d + 120*a^2*c^14*d^2)*x^6 + 2/5*(a*b *c^16 + 8*a^2*c^15*d)*x^5
Leaf count of result is larger than twice the leaf count of optimal. 668 vs. \(2 (165) = 330\).
Time = 0.28 (sec) , antiderivative size = 668, normalized size of antiderivative = 3.77 \[ \int x^3 (a+b x)^2 (c+d x)^{16} \, dx=\frac {1}{22} \, b^{2} d^{16} x^{22} + \frac {16}{21} \, b^{2} c d^{15} x^{21} + \frac {2}{21} \, a b d^{16} x^{21} + 6 \, b^{2} c^{2} d^{14} x^{20} + \frac {8}{5} \, a b c d^{15} x^{20} + \frac {1}{20} \, a^{2} d^{16} x^{20} + \frac {560}{19} \, b^{2} c^{3} d^{13} x^{19} + \frac {240}{19} \, a b c^{2} d^{14} x^{19} + \frac {16}{19} \, a^{2} c d^{15} x^{19} + \frac {910}{9} \, b^{2} c^{4} d^{12} x^{18} + \frac {560}{9} \, a b c^{3} d^{13} x^{18} + \frac {20}{3} \, a^{2} c^{2} d^{14} x^{18} + \frac {4368}{17} \, b^{2} c^{5} d^{11} x^{17} + \frac {3640}{17} \, a b c^{4} d^{12} x^{17} + \frac {560}{17} \, a^{2} c^{3} d^{13} x^{17} + \frac {1001}{2} \, b^{2} c^{6} d^{10} x^{16} + 546 \, a b c^{5} d^{11} x^{16} + \frac {455}{4} \, a^{2} c^{4} d^{12} x^{16} + \frac {2288}{3} \, b^{2} c^{7} d^{9} x^{15} + \frac {16016}{15} \, a b c^{6} d^{10} x^{15} + \frac {1456}{5} \, a^{2} c^{5} d^{11} x^{15} + \frac {6435}{7} \, b^{2} c^{8} d^{8} x^{14} + \frac {11440}{7} \, a b c^{7} d^{9} x^{14} + 572 \, a^{2} c^{6} d^{10} x^{14} + 880 \, b^{2} c^{9} d^{7} x^{13} + 1980 \, a b c^{8} d^{8} x^{13} + 880 \, a^{2} c^{7} d^{9} x^{13} + \frac {2002}{3} \, b^{2} c^{10} d^{6} x^{12} + \frac {5720}{3} \, a b c^{9} d^{7} x^{12} + \frac {2145}{2} \, a^{2} c^{8} d^{8} x^{12} + \frac {4368}{11} \, b^{2} c^{11} d^{5} x^{11} + 1456 \, a b c^{10} d^{6} x^{11} + 1040 \, a^{2} c^{9} d^{7} x^{11} + 182 \, b^{2} c^{12} d^{4} x^{10} + \frac {4368}{5} \, a b c^{11} d^{5} x^{10} + \frac {4004}{5} \, a^{2} c^{10} d^{6} x^{10} + \frac {560}{9} \, b^{2} c^{13} d^{3} x^{9} + \frac {3640}{9} \, a b c^{12} d^{4} x^{9} + \frac {1456}{3} \, a^{2} c^{11} d^{5} x^{9} + 15 \, b^{2} c^{14} d^{2} x^{8} + 140 \, a b c^{13} d^{3} x^{8} + \frac {455}{2} \, a^{2} c^{12} d^{4} x^{8} + \frac {16}{7} \, b^{2} c^{15} d x^{7} + \frac {240}{7} \, a b c^{14} d^{2} x^{7} + 80 \, a^{2} c^{13} d^{3} x^{7} + \frac {1}{6} \, b^{2} c^{16} x^{6} + \frac {16}{3} \, a b c^{15} d x^{6} + 20 \, a^{2} c^{14} d^{2} x^{6} + \frac {2}{5} \, a b c^{16} x^{5} + \frac {16}{5} \, a^{2} c^{15} d x^{5} + \frac {1}{4} \, a^{2} c^{16} x^{4} \]
1/22*b^2*d^16*x^22 + 16/21*b^2*c*d^15*x^21 + 2/21*a*b*d^16*x^21 + 6*b^2*c^ 2*d^14*x^20 + 8/5*a*b*c*d^15*x^20 + 1/20*a^2*d^16*x^20 + 560/19*b^2*c^3*d^ 13*x^19 + 240/19*a*b*c^2*d^14*x^19 + 16/19*a^2*c*d^15*x^19 + 910/9*b^2*c^4 *d^12*x^18 + 560/9*a*b*c^3*d^13*x^18 + 20/3*a^2*c^2*d^14*x^18 + 4368/17*b^ 2*c^5*d^11*x^17 + 3640/17*a*b*c^4*d^12*x^17 + 560/17*a^2*c^3*d^13*x^17 + 1 001/2*b^2*c^6*d^10*x^16 + 546*a*b*c^5*d^11*x^16 + 455/4*a^2*c^4*d^12*x^16 + 2288/3*b^2*c^7*d^9*x^15 + 16016/15*a*b*c^6*d^10*x^15 + 1456/5*a^2*c^5*d^ 11*x^15 + 6435/7*b^2*c^8*d^8*x^14 + 11440/7*a*b*c^7*d^9*x^14 + 572*a^2*c^6 *d^10*x^14 + 880*b^2*c^9*d^7*x^13 + 1980*a*b*c^8*d^8*x^13 + 880*a^2*c^7*d^ 9*x^13 + 2002/3*b^2*c^10*d^6*x^12 + 5720/3*a*b*c^9*d^7*x^12 + 2145/2*a^2*c ^8*d^8*x^12 + 4368/11*b^2*c^11*d^5*x^11 + 1456*a*b*c^10*d^6*x^11 + 1040*a^ 2*c^9*d^7*x^11 + 182*b^2*c^12*d^4*x^10 + 4368/5*a*b*c^11*d^5*x^10 + 4004/5 *a^2*c^10*d^6*x^10 + 560/9*b^2*c^13*d^3*x^9 + 3640/9*a*b*c^12*d^4*x^9 + 14 56/3*a^2*c^11*d^5*x^9 + 15*b^2*c^14*d^2*x^8 + 140*a*b*c^13*d^3*x^8 + 455/2 *a^2*c^12*d^4*x^8 + 16/7*b^2*c^15*d*x^7 + 240/7*a*b*c^14*d^2*x^7 + 80*a^2* c^13*d^3*x^7 + 1/6*b^2*c^16*x^6 + 16/3*a*b*c^15*d*x^6 + 20*a^2*c^14*d^2*x^ 6 + 2/5*a*b*c^16*x^5 + 16/5*a^2*c^15*d*x^5 + 1/4*a^2*c^16*x^4
Time = 0.43 (sec) , antiderivative size = 557, normalized size of antiderivative = 3.15 \[ \int x^3 (a+b x)^2 (c+d x)^{16} \, dx=x^6\,\left (20\,a^2\,c^{14}\,d^2+\frac {16\,a\,b\,c^{15}\,d}{3}+\frac {b^2\,c^{16}}{6}\right )+x^{20}\,\left (\frac {a^2\,d^{16}}{20}+\frac {8\,a\,b\,c\,d^{15}}{5}+6\,b^2\,c^2\,d^{14}\right )+\frac {a^2\,c^{16}\,x^4}{4}+\frac {b^2\,d^{16}\,x^{22}}{22}+\frac {2\,a\,c^{15}\,x^5\,\left (8\,a\,d+b\,c\right )}{5}+\frac {2\,b\,d^{15}\,x^{21}\,\left (a\,d+8\,b\,c\right )}{21}+\frac {16\,c^{13}\,d\,x^7\,\left (35\,a^2\,d^2+15\,a\,b\,c\,d+b^2\,c^2\right )}{7}+\frac {16\,c\,d^{13}\,x^{19}\,\left (a^2\,d^2+15\,a\,b\,c\,d+35\,b^2\,c^2\right )}{19}+220\,c^7\,d^7\,x^{13}\,\left (4\,a^2\,d^2+9\,a\,b\,c\,d+4\,b^2\,c^2\right )+\frac {182\,c^{10}\,d^4\,x^{10}\,\left (22\,a^2\,d^2+24\,a\,b\,c\,d+5\,b^2\,c^2\right )}{5}+\frac {91\,c^4\,d^{10}\,x^{16}\,\left (5\,a^2\,d^2+24\,a\,b\,c\,d+22\,b^2\,c^2\right )}{4}+\frac {5\,c^{12}\,d^2\,x^8\,\left (91\,a^2\,d^2+56\,a\,b\,c\,d+6\,b^2\,c^2\right )}{2}+\frac {56\,c^{11}\,d^3\,x^9\,\left (78\,a^2\,d^2+65\,a\,b\,c\,d+10\,b^2\,c^2\right )}{9}+\frac {208\,c^9\,d^5\,x^{11}\,\left (55\,a^2\,d^2+77\,a\,b\,c\,d+21\,b^2\,c^2\right )}{11}+\frac {143\,c^8\,d^6\,x^{12}\,\left (45\,a^2\,d^2+80\,a\,b\,c\,d+28\,b^2\,c^2\right )}{6}+\frac {143\,c^6\,d^8\,x^{14}\,\left (28\,a^2\,d^2+80\,a\,b\,c\,d+45\,b^2\,c^2\right )}{7}+\frac {208\,c^5\,d^9\,x^{15}\,\left (21\,a^2\,d^2+77\,a\,b\,c\,d+55\,b^2\,c^2\right )}{15}+\frac {56\,c^3\,d^{11}\,x^{17}\,\left (10\,a^2\,d^2+65\,a\,b\,c\,d+78\,b^2\,c^2\right )}{17}+\frac {10\,c^2\,d^{12}\,x^{18}\,\left (6\,a^2\,d^2+56\,a\,b\,c\,d+91\,b^2\,c^2\right )}{9} \]
x^6*((b^2*c^16)/6 + 20*a^2*c^14*d^2 + (16*a*b*c^15*d)/3) + x^20*((a^2*d^16 )/20 + 6*b^2*c^2*d^14 + (8*a*b*c*d^15)/5) + (a^2*c^16*x^4)/4 + (b^2*d^16*x ^22)/22 + (2*a*c^15*x^5*(8*a*d + b*c))/5 + (2*b*d^15*x^21*(a*d + 8*b*c))/2 1 + (16*c^13*d*x^7*(35*a^2*d^2 + b^2*c^2 + 15*a*b*c*d))/7 + (16*c*d^13*x^1 9*(a^2*d^2 + 35*b^2*c^2 + 15*a*b*c*d))/19 + 220*c^7*d^7*x^13*(4*a^2*d^2 + 4*b^2*c^2 + 9*a*b*c*d) + (182*c^10*d^4*x^10*(22*a^2*d^2 + 5*b^2*c^2 + 24*a *b*c*d))/5 + (91*c^4*d^10*x^16*(5*a^2*d^2 + 22*b^2*c^2 + 24*a*b*c*d))/4 + (5*c^12*d^2*x^8*(91*a^2*d^2 + 6*b^2*c^2 + 56*a*b*c*d))/2 + (56*c^11*d^3*x^ 9*(78*a^2*d^2 + 10*b^2*c^2 + 65*a*b*c*d))/9 + (208*c^9*d^5*x^11*(55*a^2*d^ 2 + 21*b^2*c^2 + 77*a*b*c*d))/11 + (143*c^8*d^6*x^12*(45*a^2*d^2 + 28*b^2* c^2 + 80*a*b*c*d))/6 + (143*c^6*d^8*x^14*(28*a^2*d^2 + 45*b^2*c^2 + 80*a*b *c*d))/7 + (208*c^5*d^9*x^15*(21*a^2*d^2 + 55*b^2*c^2 + 77*a*b*c*d))/15 + (56*c^3*d^11*x^17*(10*a^2*d^2 + 78*b^2*c^2 + 65*a*b*c*d))/17 + (10*c^2*d^1 2*x^18*(6*a^2*d^2 + 91*b^2*c^2 + 56*a*b*c*d))/9