Integrand size = 18, antiderivative size = 137 \[ \int x^2 (a+b x)^2 (c+d x)^{16} \, dx=\frac {c^2 (b c-a d)^2 (c+d x)^{17}}{17 d^5}-\frac {c (b c-a d) (2 b c-a d) (c+d x)^{18}}{9 d^5}+\frac {\left (6 b^2 c^2-6 a b c d+a^2 d^2\right ) (c+d x)^{19}}{19 d^5}-\frac {b (2 b c-a d) (c+d x)^{20}}{10 d^5}+\frac {b^2 (c+d x)^{21}}{21 d^5} \] Output:
1/17*c^2*(-a*d+b*c)^2*(d*x+c)^17/d^5-1/9*c*(-a*d+b*c)*(-a*d+2*b*c)*(d*x+c) ^18/d^5+1/19*(a^2*d^2-6*a*b*c*d+6*b^2*c^2)*(d*x+c)^19/d^5-1/10*b*(-a*d+2*b *c)*(d*x+c)^20/d^5+1/21*b^2*(d*x+c)^21/d^5
Leaf count is larger than twice the leaf count of optimal. \(585\) vs. \(2(137)=274\).
Time = 0.06 (sec) , antiderivative size = 585, normalized size of antiderivative = 4.27 \[ \int x^2 (a+b x)^2 (c+d x)^{16} \, dx=\frac {1}{3} a^2 c^{16} x^3+\frac {1}{2} a c^{15} (b c+8 a d) x^4+\frac {1}{5} c^{14} \left (b^2 c^2+32 a b c d+120 a^2 d^2\right ) x^5+\frac {8}{3} c^{13} d \left (b^2 c^2+15 a b c d+35 a^2 d^2\right ) x^6+\frac {20}{7} c^{12} d^2 \left (6 b^2 c^2+56 a b c d+91 a^2 d^2\right ) x^7+7 c^{11} d^3 \left (10 b^2 c^2+65 a b c d+78 a^2 d^2\right ) x^8+\frac {364}{9} c^{10} d^4 \left (5 b^2 c^2+24 a b c d+22 a^2 d^2\right ) x^9+\frac {104}{5} c^9 d^5 \left (21 b^2 c^2+77 a b c d+55 a^2 d^2\right ) x^{10}+26 c^8 d^6 \left (28 b^2 c^2+80 a b c d+45 a^2 d^2\right ) x^{11}+\frac {715}{3} c^7 d^7 \left (4 b^2 c^2+9 a b c d+4 a^2 d^2\right ) x^{12}+22 c^6 d^8 \left (45 b^2 c^2+80 a b c d+28 a^2 d^2\right ) x^{13}+\frac {104}{7} c^5 d^9 \left (55 b^2 c^2+77 a b c d+21 a^2 d^2\right ) x^{14}+\frac {364}{15} c^4 d^{10} \left (22 b^2 c^2+24 a b c d+5 a^2 d^2\right ) x^{15}+\frac {7}{2} c^3 d^{11} \left (78 b^2 c^2+65 a b c d+10 a^2 d^2\right ) x^{16}+\frac {20}{17} c^2 d^{12} \left (91 b^2 c^2+56 a b c d+6 a^2 d^2\right ) x^{17}+\frac {8}{9} c d^{13} \left (35 b^2 c^2+15 a b c d+a^2 d^2\right ) x^{18}+\frac {1}{19} d^{14} \left (120 b^2 c^2+32 a b c d+a^2 d^2\right ) x^{19}+\frac {1}{10} b d^{15} (8 b c+a d) x^{20}+\frac {1}{21} b^2 d^{16} x^{21} \] Input:
Integrate[x^2*(a + b*x)^2*(c + d*x)^16,x]
Output:
(a^2*c^16*x^3)/3 + (a*c^15*(b*c + 8*a*d)*x^4)/2 + (c^14*(b^2*c^2 + 32*a*b* c*d + 120*a^2*d^2)*x^5)/5 + (8*c^13*d*(b^2*c^2 + 15*a*b*c*d + 35*a^2*d^2)* x^6)/3 + (20*c^12*d^2*(6*b^2*c^2 + 56*a*b*c*d + 91*a^2*d^2)*x^7)/7 + 7*c^1 1*d^3*(10*b^2*c^2 + 65*a*b*c*d + 78*a^2*d^2)*x^8 + (364*c^10*d^4*(5*b^2*c^ 2 + 24*a*b*c*d + 22*a^2*d^2)*x^9)/9 + (104*c^9*d^5*(21*b^2*c^2 + 77*a*b*c* d + 55*a^2*d^2)*x^10)/5 + 26*c^8*d^6*(28*b^2*c^2 + 80*a*b*c*d + 45*a^2*d^2 )*x^11 + (715*c^7*d^7*(4*b^2*c^2 + 9*a*b*c*d + 4*a^2*d^2)*x^12)/3 + 22*c^6 *d^8*(45*b^2*c^2 + 80*a*b*c*d + 28*a^2*d^2)*x^13 + (104*c^5*d^9*(55*b^2*c^ 2 + 77*a*b*c*d + 21*a^2*d^2)*x^14)/7 + (364*c^4*d^10*(22*b^2*c^2 + 24*a*b* c*d + 5*a^2*d^2)*x^15)/15 + (7*c^3*d^11*(78*b^2*c^2 + 65*a*b*c*d + 10*a^2* d^2)*x^16)/2 + (20*c^2*d^12*(91*b^2*c^2 + 56*a*b*c*d + 6*a^2*d^2)*x^17)/17 + (8*c*d^13*(35*b^2*c^2 + 15*a*b*c*d + a^2*d^2)*x^18)/9 + (d^14*(120*b^2* c^2 + 32*a*b*c*d + a^2*d^2)*x^19)/19 + (b*d^15*(8*b*c + a*d)*x^20)/10 + (b ^2*d^16*x^21)/21
Time = 0.53 (sec) , antiderivative size = 137, 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.
| \(\displaystyle \int x^2 (a+b x)^2 (c+d x)^{16} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {(c+d x)^{18} \left (a^2 d^2-6 a b c d+6 b^2 c^2\right )}{d^4}+\frac {c^2 (c+d x)^{16} (b c-a d)^2}{d^4}-\frac {2 b (c+d x)^{19} (2 b c-a d)}{d^4}+\frac {2 c (c+d x)^{17} (b c-a d) (a d-2 b c)}{d^4}+\frac {b^2 (c+d x)^{20}}{d^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(c+d x)^{19} \left (a^2 d^2-6 a b c d+6 b^2 c^2\right )}{19 d^5}+\frac {c^2 (c+d x)^{17} (b c-a d)^2}{17 d^5}-\frac {b (c+d x)^{20} (2 b c-a d)}{10 d^5}-\frac {c (c+d x)^{18} (b c-a d) (2 b c-a d)}{9 d^5}+\frac {b^2 (c+d x)^{21}}{21 d^5}\) |
Input:
Int[x^2*(a + b*x)^2*(c + d*x)^16,x]
Output:
(c^2*(b*c - a*d)^2*(c + d*x)^17)/(17*d^5) - (c*(b*c - a*d)*(2*b*c - a*d)*( c + d*x)^18)/(9*d^5) + ((6*b^2*c^2 - 6*a*b*c*d + a^2*d^2)*(c + d*x)^19)/(1 9*d^5) - (b*(2*b*c - a*d)*(c + d*x)^20)/(10*d^5) + (b^2*(c + d*x)^21)/(21* d^5)
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. \(2(127)=254\).
Time = 0.15 (sec) , antiderivative size = 607, normalized size of antiderivative = 4.43
| method | result | size |
| norman | \(\frac {a^{2} c^{16} x^{3}}{3}+\left (4 a^{2} c^{15} d +\frac {1}{2} a b \,c^{16}\right ) x^{4}+\left (24 a^{2} c^{14} d^{2}+\frac {32}{5} a b \,c^{15} d +\frac {1}{5} b^{2} c^{16}\right ) x^{5}+\left (\frac {280}{3} a^{2} c^{13} d^{3}+40 a b \,c^{14} d^{2}+\frac {8}{3} b^{2} c^{15} d \right ) x^{6}+\left (260 a^{2} c^{12} d^{4}+160 a b \,c^{13} d^{3}+\frac {120}{7} b^{2} c^{14} d^{2}\right ) x^{7}+\left (546 a^{2} c^{11} d^{5}+455 a b \,c^{12} d^{4}+70 b^{2} c^{13} d^{3}\right ) x^{8}+\left (\frac {8008}{9} a^{2} c^{10} d^{6}+\frac {2912}{3} a b \,c^{11} d^{5}+\frac {1820}{9} b^{2} c^{12} d^{4}\right ) x^{9}+\left (1144 a^{2} c^{9} d^{7}+\frac {8008}{5} a b \,c^{10} d^{6}+\frac {2184}{5} b^{2} c^{11} d^{5}\right ) x^{10}+\left (1170 a^{2} c^{8} d^{8}+2080 a b \,c^{9} d^{7}+728 b^{2} c^{10} d^{6}\right ) x^{11}+\left (\frac {2860}{3} a^{2} c^{7} d^{9}+2145 a b \,c^{8} d^{8}+\frac {2860}{3} b^{2} c^{9} d^{7}\right ) x^{12}+\left (616 a^{2} c^{6} d^{10}+1760 a b \,c^{7} d^{9}+990 b^{2} c^{8} d^{8}\right ) x^{13}+\left (312 a^{2} c^{5} d^{11}+1144 a b \,c^{6} d^{10}+\frac {5720}{7} b^{2} c^{7} d^{9}\right ) x^{14}+\left (\frac {364}{3} a^{2} c^{4} d^{12}+\frac {2912}{5} a b \,c^{5} d^{11}+\frac {8008}{15} b^{2} c^{6} d^{10}\right ) x^{15}+\left (35 a^{2} c^{3} d^{13}+\frac {455}{2} a b \,c^{4} d^{12}+273 b^{2} c^{5} d^{11}\right ) x^{16}+\left (\frac {120}{17} a^{2} c^{2} d^{14}+\frac {1120}{17} a b \,c^{3} d^{13}+\frac {1820}{17} b^{2} c^{4} d^{12}\right ) x^{17}+\left (\frac {8}{9} a^{2} c \,d^{15}+\frac {40}{3} a b \,c^{2} d^{14}+\frac {280}{9} b^{2} c^{3} d^{13}\right ) x^{18}+\left (\frac {1}{19} a^{2} d^{16}+\frac {32}{19} a b c \,d^{15}+\frac {120}{19} b^{2} c^{2} d^{14}\right ) x^{19}+\left (\frac {1}{10} a b \,d^{16}+\frac {4}{5} b^{2} c \,d^{15}\right ) x^{20}+\frac {b^{2} d^{16} x^{21}}{21}\) | \(607\) |
| default | \(\frac {b^{2} d^{16} x^{21}}{21}+\frac {\left (2 a b \,d^{16}+16 b^{2} c \,d^{15}\right ) x^{20}}{20}+\frac {\left (a^{2} d^{16}+32 a b c \,d^{15}+120 b^{2} c^{2} d^{14}\right ) x^{19}}{19}+\frac {\left (16 a^{2} c \,d^{15}+240 a b \,c^{2} d^{14}+560 b^{2} c^{3} d^{13}\right ) x^{18}}{18}+\frac {\left (120 a^{2} c^{2} d^{14}+1120 a b \,c^{3} d^{13}+1820 b^{2} c^{4} d^{12}\right ) x^{17}}{17}+\frac {\left (560 a^{2} c^{3} d^{13}+3640 a b \,c^{4} d^{12}+4368 b^{2} c^{5} d^{11}\right ) x^{16}}{16}+\frac {\left (1820 a^{2} c^{4} d^{12}+8736 a b \,c^{5} d^{11}+8008 b^{2} c^{6} d^{10}\right ) x^{15}}{15}+\frac {\left (4368 a^{2} c^{5} d^{11}+16016 a b \,c^{6} d^{10}+11440 b^{2} c^{7} d^{9}\right ) x^{14}}{14}+\frac {\left (8008 a^{2} c^{6} d^{10}+22880 a b \,c^{7} d^{9}+12870 b^{2} c^{8} d^{8}\right ) x^{13}}{13}+\frac {\left (11440 a^{2} c^{7} d^{9}+25740 a b \,c^{8} d^{8}+11440 b^{2} c^{9} d^{7}\right ) x^{12}}{12}+\frac {\left (12870 a^{2} c^{8} d^{8}+22880 a b \,c^{9} d^{7}+8008 b^{2} c^{10} d^{6}\right ) x^{11}}{11}+\frac {\left (11440 a^{2} c^{9} d^{7}+16016 a b \,c^{10} d^{6}+4368 b^{2} c^{11} d^{5}\right ) x^{10}}{10}+\frac {\left (8008 a^{2} c^{10} d^{6}+8736 a b \,c^{11} d^{5}+1820 b^{2} c^{12} d^{4}\right ) x^{9}}{9}+\frac {\left (4368 a^{2} c^{11} d^{5}+3640 a b \,c^{12} d^{4}+560 b^{2} c^{13} d^{3}\right ) x^{8}}{8}+\frac {\left (1820 a^{2} c^{12} d^{4}+1120 a b \,c^{13} d^{3}+120 b^{2} c^{14} d^{2}\right ) x^{7}}{7}+\frac {\left (560 a^{2} c^{13} d^{3}+240 a b \,c^{14} d^{2}+16 b^{2} c^{15} d \right ) x^{6}}{6}+\frac {\left (120 a^{2} c^{14} d^{2}+32 a b \,c^{15} d +b^{2} c^{16}\right ) x^{5}}{5}+\frac {\left (16 a^{2} c^{15} d +2 a b \,c^{16}\right ) x^{4}}{4}+\frac {a^{2} c^{16} x^{3}}{3}\) | \(622\) |
| orering | \(\frac {x^{3} \left (9690 b^{2} d^{16} x^{18}+20349 a b \,d^{16} x^{17}+162792 b^{2} c \,d^{15} x^{17}+10710 a^{2} d^{16} x^{16}+342720 a b c \,d^{15} x^{16}+1285200 b^{2} c^{2} d^{14} x^{16}+180880 a^{2} c \,d^{15} x^{15}+2713200 a b \,c^{2} d^{14} x^{15}+6330800 b^{2} c^{3} d^{13} x^{15}+1436400 a^{2} c^{2} d^{14} x^{14}+13406400 a b \,c^{3} d^{13} x^{14}+21785400 b^{2} c^{4} d^{12} x^{14}+7122150 a^{2} c^{3} d^{13} x^{13}+46293975 a b \,c^{4} d^{12} x^{13}+55552770 b^{2} c^{5} d^{11} x^{13}+24690120 a^{2} c^{4} d^{12} x^{12}+118512576 a b \,c^{5} d^{11} x^{12}+108636528 b^{2} c^{6} d^{10} x^{12}+63488880 a^{2} c^{5} d^{11} x^{11}+232792560 a b \,c^{6} d^{10} x^{11}+166280400 b^{2} c^{7} d^{9} x^{11}+125349840 a^{2} c^{6} d^{10} x^{10}+358142400 a b \,c^{7} d^{9} x^{10}+201455100 b^{2} c^{8} d^{8} x^{10}+193993800 a^{2} c^{7} d^{9} x^{9}+436486050 a b \,c^{8} d^{8} x^{9}+193993800 b^{2} c^{9} d^{7} x^{9}+238083300 a^{2} c^{8} d^{8} x^{8}+423259200 a b \,c^{9} d^{7} x^{8}+148140720 b^{2} c^{10} d^{6} x^{8}+232792560 a^{2} c^{9} d^{7} x^{7}+325909584 a b \,c^{10} d^{6} x^{7}+88884432 b^{2} c^{11} d^{5} x^{7}+181060880 a^{2} c^{10} d^{6} x^{6}+197520960 a b \,c^{11} d^{5} x^{6}+41150200 b^{2} c^{12} d^{4} x^{6}+111105540 a^{2} c^{11} d^{5} x^{5}+92587950 a b \,c^{12} d^{4} x^{5}+14244300 b^{2} c^{13} d^{3} x^{5}+52907400 a^{2} c^{12} d^{4} x^{4}+32558400 a b \,c^{13} d^{3} x^{4}+3488400 b^{2} c^{14} d^{2} x^{4}+18992400 a^{2} c^{13} d^{3} x^{3}+8139600 a b \,c^{14} d^{2} x^{3}+542640 b^{2} c^{15} d \,x^{3}+4883760 a^{2} c^{14} d^{2} x^{2}+1302336 a b \,c^{15} d \,x^{2}+40698 b^{2} c^{16} x^{2}+813960 a^{2} c^{15} d x +101745 a b \,c^{16} x +67830 a^{2} c^{16}\right )}{203490}\) | \(667\) |
| gosper | \(4 x^{4} a^{2} c^{15} d +\frac {1}{2} x^{4} a b \,c^{16}+\frac {8}{9} x^{18} a^{2} c \,d^{15}+\frac {280}{9} x^{18} b^{2} c^{3} d^{13}+\frac {120}{19} x^{19} b^{2} c^{2} d^{14}+\frac {1}{10} x^{20} a b \,d^{16}+\frac {4}{5} x^{20} b^{2} c \,d^{15}+546 a^{2} c^{11} d^{5} x^{8}+70 b^{2} c^{13} d^{3} x^{8}+1170 a^{2} c^{8} d^{8} x^{11}+728 b^{2} c^{10} d^{6} x^{11}+616 a^{2} c^{6} d^{10} x^{13}+990 b^{2} c^{8} d^{8} x^{13}+\frac {1}{5} x^{5} b^{2} c^{16}+\frac {32}{5} x^{5} a b \,c^{15} d +40 x^{6} a b \,c^{14} d^{2}+160 x^{7} a b \,c^{13} d^{3}+\frac {2912}{3} x^{9} a b \,c^{11} d^{5}+\frac {8008}{5} x^{10} a b \,c^{10} d^{6}+2145 x^{12} a b \,c^{8} d^{8}+1144 x^{14} a b \,c^{6} d^{10}+\frac {455}{2} x^{16} a b \,c^{4} d^{12}+\frac {1120}{17} x^{17} a b \,c^{3} d^{13}+\frac {40}{3} x^{18} a b \,c^{2} d^{14}+\frac {32}{19} x^{19} a b c \,d^{15}+455 a b \,c^{12} d^{4} x^{8}+2080 a b \,c^{9} d^{7} x^{11}+1760 a b \,c^{7} d^{9} x^{13}+\frac {1}{19} x^{19} a^{2} d^{16}+\frac {1820}{9} x^{9} b^{2} c^{12} d^{4}+1144 x^{10} a^{2} c^{9} d^{7}+\frac {2184}{5} x^{10} b^{2} c^{11} d^{5}+\frac {2860}{3} x^{12} a^{2} c^{7} d^{9}+\frac {2860}{3} x^{12} b^{2} c^{9} d^{7}+312 x^{14} a^{2} c^{5} d^{11}+\frac {5720}{7} x^{14} b^{2} c^{7} d^{9}+\frac {364}{3} x^{15} a^{2} c^{4} d^{12}+\frac {8008}{15} x^{15} b^{2} c^{6} d^{10}+35 x^{16} a^{2} c^{3} d^{13}+273 x^{16} b^{2} c^{5} d^{11}+\frac {120}{17} x^{17} a^{2} c^{2} d^{14}+\frac {1820}{17} x^{17} b^{2} c^{4} d^{12}+24 x^{5} a^{2} c^{14} d^{2}+\frac {280}{3} x^{6} a^{2} c^{13} d^{3}+\frac {8}{3} x^{6} b^{2} c^{15} d +260 x^{7} a^{2} c^{12} d^{4}+\frac {120}{7} x^{7} b^{2} c^{14} d^{2}+\frac {8008}{9} x^{9} a^{2} c^{10} d^{6}+\frac {1}{21} b^{2} d^{16} x^{21}+\frac {1}{3} a^{2} c^{16} x^{3}+\frac {2912}{5} x^{15} a b \,c^{5} d^{11}\) | \(669\) |
| risch | \(4 x^{4} a^{2} c^{15} d +\frac {1}{2} x^{4} a b \,c^{16}+\frac {8}{9} x^{18} a^{2} c \,d^{15}+\frac {280}{9} x^{18} b^{2} c^{3} d^{13}+\frac {120}{19} x^{19} b^{2} c^{2} d^{14}+\frac {1}{10} x^{20} a b \,d^{16}+\frac {4}{5} x^{20} b^{2} c \,d^{15}+546 a^{2} c^{11} d^{5} x^{8}+70 b^{2} c^{13} d^{3} x^{8}+1170 a^{2} c^{8} d^{8} x^{11}+728 b^{2} c^{10} d^{6} x^{11}+616 a^{2} c^{6} d^{10} x^{13}+990 b^{2} c^{8} d^{8} x^{13}+\frac {1}{5} x^{5} b^{2} c^{16}+\frac {32}{5} x^{5} a b \,c^{15} d +40 x^{6} a b \,c^{14} d^{2}+160 x^{7} a b \,c^{13} d^{3}+\frac {2912}{3} x^{9} a b \,c^{11} d^{5}+\frac {8008}{5} x^{10} a b \,c^{10} d^{6}+2145 x^{12} a b \,c^{8} d^{8}+1144 x^{14} a b \,c^{6} d^{10}+\frac {455}{2} x^{16} a b \,c^{4} d^{12}+\frac {1120}{17} x^{17} a b \,c^{3} d^{13}+\frac {40}{3} x^{18} a b \,c^{2} d^{14}+\frac {32}{19} x^{19} a b c \,d^{15}+455 a b \,c^{12} d^{4} x^{8}+2080 a b \,c^{9} d^{7} x^{11}+1760 a b \,c^{7} d^{9} x^{13}+\frac {1}{19} x^{19} a^{2} d^{16}+\frac {1820}{9} x^{9} b^{2} c^{12} d^{4}+1144 x^{10} a^{2} c^{9} d^{7}+\frac {2184}{5} x^{10} b^{2} c^{11} d^{5}+\frac {2860}{3} x^{12} a^{2} c^{7} d^{9}+\frac {2860}{3} x^{12} b^{2} c^{9} d^{7}+312 x^{14} a^{2} c^{5} d^{11}+\frac {5720}{7} x^{14} b^{2} c^{7} d^{9}+\frac {364}{3} x^{15} a^{2} c^{4} d^{12}+\frac {8008}{15} x^{15} b^{2} c^{6} d^{10}+35 x^{16} a^{2} c^{3} d^{13}+273 x^{16} b^{2} c^{5} d^{11}+\frac {120}{17} x^{17} a^{2} c^{2} d^{14}+\frac {1820}{17} x^{17} b^{2} c^{4} d^{12}+24 x^{5} a^{2} c^{14} d^{2}+\frac {280}{3} x^{6} a^{2} c^{13} d^{3}+\frac {8}{3} x^{6} b^{2} c^{15} d +260 x^{7} a^{2} c^{12} d^{4}+\frac {120}{7} x^{7} b^{2} c^{14} d^{2}+\frac {8008}{9} x^{9} a^{2} c^{10} d^{6}+\frac {1}{21} b^{2} d^{16} x^{21}+\frac {1}{3} a^{2} c^{16} x^{3}+\frac {2912}{5} x^{15} a b \,c^{5} d^{11}\) | \(669\) |
| parallelrisch | \(4 x^{4} a^{2} c^{15} d +\frac {1}{2} x^{4} a b \,c^{16}+\frac {8}{9} x^{18} a^{2} c \,d^{15}+\frac {280}{9} x^{18} b^{2} c^{3} d^{13}+\frac {120}{19} x^{19} b^{2} c^{2} d^{14}+\frac {1}{10} x^{20} a b \,d^{16}+\frac {4}{5} x^{20} b^{2} c \,d^{15}+546 a^{2} c^{11} d^{5} x^{8}+70 b^{2} c^{13} d^{3} x^{8}+1170 a^{2} c^{8} d^{8} x^{11}+728 b^{2} c^{10} d^{6} x^{11}+616 a^{2} c^{6} d^{10} x^{13}+990 b^{2} c^{8} d^{8} x^{13}+\frac {1}{5} x^{5} b^{2} c^{16}+\frac {32}{5} x^{5} a b \,c^{15} d +40 x^{6} a b \,c^{14} d^{2}+160 x^{7} a b \,c^{13} d^{3}+\frac {2912}{3} x^{9} a b \,c^{11} d^{5}+\frac {8008}{5} x^{10} a b \,c^{10} d^{6}+2145 x^{12} a b \,c^{8} d^{8}+1144 x^{14} a b \,c^{6} d^{10}+\frac {455}{2} x^{16} a b \,c^{4} d^{12}+\frac {1120}{17} x^{17} a b \,c^{3} d^{13}+\frac {40}{3} x^{18} a b \,c^{2} d^{14}+\frac {32}{19} x^{19} a b c \,d^{15}+455 a b \,c^{12} d^{4} x^{8}+2080 a b \,c^{9} d^{7} x^{11}+1760 a b \,c^{7} d^{9} x^{13}+\frac {1}{19} x^{19} a^{2} d^{16}+\frac {1820}{9} x^{9} b^{2} c^{12} d^{4}+1144 x^{10} a^{2} c^{9} d^{7}+\frac {2184}{5} x^{10} b^{2} c^{11} d^{5}+\frac {2860}{3} x^{12} a^{2} c^{7} d^{9}+\frac {2860}{3} x^{12} b^{2} c^{9} d^{7}+312 x^{14} a^{2} c^{5} d^{11}+\frac {5720}{7} x^{14} b^{2} c^{7} d^{9}+\frac {364}{3} x^{15} a^{2} c^{4} d^{12}+\frac {8008}{15} x^{15} b^{2} c^{6} d^{10}+35 x^{16} a^{2} c^{3} d^{13}+273 x^{16} b^{2} c^{5} d^{11}+\frac {120}{17} x^{17} a^{2} c^{2} d^{14}+\frac {1820}{17} x^{17} b^{2} c^{4} d^{12}+24 x^{5} a^{2} c^{14} d^{2}+\frac {280}{3} x^{6} a^{2} c^{13} d^{3}+\frac {8}{3} x^{6} b^{2} c^{15} d +260 x^{7} a^{2} c^{12} d^{4}+\frac {120}{7} x^{7} b^{2} c^{14} d^{2}+\frac {8008}{9} x^{9} a^{2} c^{10} d^{6}+\frac {1}{21} b^{2} d^{16} x^{21}+\frac {1}{3} a^{2} c^{16} x^{3}+\frac {2912}{5} x^{15} a b \,c^{5} d^{11}\) | \(669\) |
Input:
int(x^2*(b*x+a)^2*(d*x+c)^16,x,method=_RETURNVERBOSE)
Output:
1/3*a^2*c^16*x^3+(4*a^2*c^15*d+1/2*a*b*c^16)*x^4+(24*a^2*c^14*d^2+32/5*a*b *c^15*d+1/5*b^2*c^16)*x^5+(280/3*a^2*c^13*d^3+40*a*b*c^14*d^2+8/3*b^2*c^15 *d)*x^6+(260*a^2*c^12*d^4+160*a*b*c^13*d^3+120/7*b^2*c^14*d^2)*x^7+(546*a^ 2*c^11*d^5+455*a*b*c^12*d^4+70*b^2*c^13*d^3)*x^8+(8008/9*a^2*c^10*d^6+2912 /3*a*b*c^11*d^5+1820/9*b^2*c^12*d^4)*x^9+(1144*a^2*c^9*d^7+8008/5*a*b*c^10 *d^6+2184/5*b^2*c^11*d^5)*x^10+(1170*a^2*c^8*d^8+2080*a*b*c^9*d^7+728*b^2* c^10*d^6)*x^11+(2860/3*a^2*c^7*d^9+2145*a*b*c^8*d^8+2860/3*b^2*c^9*d^7)*x^ 12+(616*a^2*c^6*d^10+1760*a*b*c^7*d^9+990*b^2*c^8*d^8)*x^13+(312*a^2*c^5*d ^11+1144*a*b*c^6*d^10+5720/7*b^2*c^7*d^9)*x^14+(364/3*a^2*c^4*d^12+2912/5* a*b*c^5*d^11+8008/15*b^2*c^6*d^10)*x^15+(35*a^2*c^3*d^13+455/2*a*b*c^4*d^1 2+273*b^2*c^5*d^11)*x^16+(120/17*a^2*c^2*d^14+1120/17*a*b*c^3*d^13+1820/17 *b^2*c^4*d^12)*x^17+(8/9*a^2*c*d^15+40/3*a*b*c^2*d^14+280/9*b^2*c^3*d^13)* x^18+(1/19*a^2*d^16+32/19*a*b*c*d^15+120/19*b^2*c^2*d^14)*x^19+(1/10*a*b*d ^16+4/5*b^2*c*d^15)*x^20+1/21*b^2*d^16*x^21
Leaf count of result is larger than twice the leaf count of optimal. 617 vs. \(2 (127) = 254\).
Time = 0.11 (sec) , antiderivative size = 617, normalized size of antiderivative = 4.50 \[ \int x^2 (a+b x)^2 (c+d x)^{16} \, dx=\frac {1}{21} \, b^{2} d^{16} x^{21} + \frac {1}{3} \, a^{2} c^{16} x^{3} + \frac {1}{10} \, {\left (8 \, b^{2} c d^{15} + a b d^{16}\right )} x^{20} + \frac {1}{19} \, {\left (120 \, b^{2} c^{2} d^{14} + 32 \, a b c d^{15} + a^{2} d^{16}\right )} x^{19} + \frac {8}{9} \, {\left (35 \, b^{2} c^{3} d^{13} + 15 \, a b c^{2} d^{14} + a^{2} c d^{15}\right )} x^{18} + \frac {20}{17} \, {\left (91 \, b^{2} c^{4} d^{12} + 56 \, a b c^{3} d^{13} + 6 \, a^{2} c^{2} d^{14}\right )} x^{17} + \frac {7}{2} \, {\left (78 \, b^{2} c^{5} d^{11} + 65 \, a b c^{4} d^{12} + 10 \, a^{2} c^{3} d^{13}\right )} x^{16} + \frac {364}{15} \, {\left (22 \, b^{2} c^{6} d^{10} + 24 \, a b c^{5} d^{11} + 5 \, a^{2} c^{4} d^{12}\right )} x^{15} + \frac {104}{7} \, {\left (55 \, b^{2} c^{7} d^{9} + 77 \, a b c^{6} d^{10} + 21 \, a^{2} c^{5} d^{11}\right )} x^{14} + 22 \, {\left (45 \, b^{2} c^{8} d^{8} + 80 \, a b c^{7} d^{9} + 28 \, a^{2} c^{6} d^{10}\right )} x^{13} + \frac {715}{3} \, {\left (4 \, b^{2} c^{9} d^{7} + 9 \, a b c^{8} d^{8} + 4 \, a^{2} c^{7} d^{9}\right )} x^{12} + 26 \, {\left (28 \, b^{2} c^{10} d^{6} + 80 \, a b c^{9} d^{7} + 45 \, a^{2} c^{8} d^{8}\right )} x^{11} + \frac {104}{5} \, {\left (21 \, b^{2} c^{11} d^{5} + 77 \, a b c^{10} d^{6} + 55 \, a^{2} c^{9} d^{7}\right )} x^{10} + \frac {364}{9} \, {\left (5 \, b^{2} c^{12} d^{4} + 24 \, a b c^{11} d^{5} + 22 \, a^{2} c^{10} d^{6}\right )} x^{9} + 7 \, {\left (10 \, b^{2} c^{13} d^{3} + 65 \, a b c^{12} d^{4} + 78 \, a^{2} c^{11} d^{5}\right )} x^{8} + \frac {20}{7} \, {\left (6 \, b^{2} c^{14} d^{2} + 56 \, a b c^{13} d^{3} + 91 \, a^{2} c^{12} d^{4}\right )} x^{7} + \frac {8}{3} \, {\left (b^{2} c^{15} d + 15 \, a b c^{14} d^{2} + 35 \, a^{2} c^{13} d^{3}\right )} x^{6} + \frac {1}{5} \, {\left (b^{2} c^{16} + 32 \, a b c^{15} d + 120 \, a^{2} c^{14} d^{2}\right )} x^{5} + \frac {1}{2} \, {\left (a b c^{16} + 8 \, a^{2} c^{15} d\right )} x^{4} \] Input:
integrate(x^2*(b*x+a)^2*(d*x+c)^16,x, algorithm="fricas")
Output:
1/21*b^2*d^16*x^21 + 1/3*a^2*c^16*x^3 + 1/10*(8*b^2*c*d^15 + a*b*d^16)*x^2 0 + 1/19*(120*b^2*c^2*d^14 + 32*a*b*c*d^15 + a^2*d^16)*x^19 + 8/9*(35*b^2* c^3*d^13 + 15*a*b*c^2*d^14 + a^2*c*d^15)*x^18 + 20/17*(91*b^2*c^4*d^12 + 5 6*a*b*c^3*d^13 + 6*a^2*c^2*d^14)*x^17 + 7/2*(78*b^2*c^5*d^11 + 65*a*b*c^4* d^12 + 10*a^2*c^3*d^13)*x^16 + 364/15*(22*b^2*c^6*d^10 + 24*a*b*c^5*d^11 + 5*a^2*c^4*d^12)*x^15 + 104/7*(55*b^2*c^7*d^9 + 77*a*b*c^6*d^10 + 21*a^2*c ^5*d^11)*x^14 + 22*(45*b^2*c^8*d^8 + 80*a*b*c^7*d^9 + 28*a^2*c^6*d^10)*x^1 3 + 715/3*(4*b^2*c^9*d^7 + 9*a*b*c^8*d^8 + 4*a^2*c^7*d^9)*x^12 + 26*(28*b^ 2*c^10*d^6 + 80*a*b*c^9*d^7 + 45*a^2*c^8*d^8)*x^11 + 104/5*(21*b^2*c^11*d^ 5 + 77*a*b*c^10*d^6 + 55*a^2*c^9*d^7)*x^10 + 364/9*(5*b^2*c^12*d^4 + 24*a* b*c^11*d^5 + 22*a^2*c^10*d^6)*x^9 + 7*(10*b^2*c^13*d^3 + 65*a*b*c^12*d^4 + 78*a^2*c^11*d^5)*x^8 + 20/7*(6*b^2*c^14*d^2 + 56*a*b*c^13*d^3 + 91*a^2*c^ 12*d^4)*x^7 + 8/3*(b^2*c^15*d + 15*a*b*c^14*d^2 + 35*a^2*c^13*d^3)*x^6 + 1 /5*(b^2*c^16 + 32*a*b*c^15*d + 120*a^2*c^14*d^2)*x^5 + 1/2*(a*b*c^16 + 8*a ^2*c^15*d)*x^4
Leaf count of result is larger than twice the leaf count of optimal. 682 vs. \(2 (128) = 256\).
Time = 0.07 (sec) , antiderivative size = 682, normalized size of antiderivative = 4.98 \[ \int x^2 (a+b x)^2 (c+d x)^{16} \, dx=\frac {a^{2} c^{16} x^{3}}{3} + \frac {b^{2} d^{16} x^{21}}{21} + x^{20} \left (\frac {a b d^{16}}{10} + \frac {4 b^{2} c d^{15}}{5}\right ) + x^{19} \left (\frac {a^{2} d^{16}}{19} + \frac {32 a b c d^{15}}{19} + \frac {120 b^{2} c^{2} d^{14}}{19}\right ) + x^{18} \cdot \left (\frac {8 a^{2} c d^{15}}{9} + \frac {40 a b c^{2} d^{14}}{3} + \frac {280 b^{2} c^{3} d^{13}}{9}\right ) + x^{17} \cdot \left (\frac {120 a^{2} c^{2} d^{14}}{17} + \frac {1120 a b c^{3} d^{13}}{17} + \frac {1820 b^{2} c^{4} d^{12}}{17}\right ) + x^{16} \cdot \left (35 a^{2} c^{3} d^{13} + \frac {455 a b c^{4} d^{12}}{2} + 273 b^{2} c^{5} d^{11}\right ) + x^{15} \cdot \left (\frac {364 a^{2} c^{4} d^{12}}{3} + \frac {2912 a b c^{5} d^{11}}{5} + \frac {8008 b^{2} c^{6} d^{10}}{15}\right ) + x^{14} \cdot \left (312 a^{2} c^{5} d^{11} + 1144 a b c^{6} d^{10} + \frac {5720 b^{2} c^{7} d^{9}}{7}\right ) + x^{13} \cdot \left (616 a^{2} c^{6} d^{10} + 1760 a b c^{7} d^{9} + 990 b^{2} c^{8} d^{8}\right ) + x^{12} \cdot \left (\frac {2860 a^{2} c^{7} d^{9}}{3} + 2145 a b c^{8} d^{8} + \frac {2860 b^{2} c^{9} d^{7}}{3}\right ) + x^{11} \cdot \left (1170 a^{2} c^{8} d^{8} + 2080 a b c^{9} d^{7} + 728 b^{2} c^{10} d^{6}\right ) + x^{10} \cdot \left (1144 a^{2} c^{9} d^{7} + \frac {8008 a b c^{10} d^{6}}{5} + \frac {2184 b^{2} c^{11} d^{5}}{5}\right ) + x^{9} \cdot \left (\frac {8008 a^{2} c^{10} d^{6}}{9} + \frac {2912 a b c^{11} d^{5}}{3} + \frac {1820 b^{2} c^{12} d^{4}}{9}\right ) + x^{8} \cdot \left (546 a^{2} c^{11} d^{5} + 455 a b c^{12} d^{4} + 70 b^{2} c^{13} d^{3}\right ) + x^{7} \cdot \left (260 a^{2} c^{12} d^{4} + 160 a b c^{13} d^{3} + \frac {120 b^{2} c^{14} d^{2}}{7}\right ) + x^{6} \cdot \left (\frac {280 a^{2} c^{13} d^{3}}{3} + 40 a b c^{14} d^{2} + \frac {8 b^{2} c^{15} d}{3}\right ) + x^{5} \cdot \left (24 a^{2} c^{14} d^{2} + \frac {32 a b c^{15} d}{5} + \frac {b^{2} c^{16}}{5}\right ) + x^{4} \cdot \left (4 a^{2} c^{15} d + \frac {a b c^{16}}{2}\right ) \] Input:
integrate(x**2*(b*x+a)**2*(d*x+c)**16,x)
Output:
a**2*c**16*x**3/3 + b**2*d**16*x**21/21 + x**20*(a*b*d**16/10 + 4*b**2*c*d **15/5) + x**19*(a**2*d**16/19 + 32*a*b*c*d**15/19 + 120*b**2*c**2*d**14/1 9) + x**18*(8*a**2*c*d**15/9 + 40*a*b*c**2*d**14/3 + 280*b**2*c**3*d**13/9 ) + x**17*(120*a**2*c**2*d**14/17 + 1120*a*b*c**3*d**13/17 + 1820*b**2*c** 4*d**12/17) + x**16*(35*a**2*c**3*d**13 + 455*a*b*c**4*d**12/2 + 273*b**2* c**5*d**11) + x**15*(364*a**2*c**4*d**12/3 + 2912*a*b*c**5*d**11/5 + 8008* b**2*c**6*d**10/15) + x**14*(312*a**2*c**5*d**11 + 1144*a*b*c**6*d**10 + 5 720*b**2*c**7*d**9/7) + x**13*(616*a**2*c**6*d**10 + 1760*a*b*c**7*d**9 + 990*b**2*c**8*d**8) + x**12*(2860*a**2*c**7*d**9/3 + 2145*a*b*c**8*d**8 + 2860*b**2*c**9*d**7/3) + x**11*(1170*a**2*c**8*d**8 + 2080*a*b*c**9*d**7 + 728*b**2*c**10*d**6) + x**10*(1144*a**2*c**9*d**7 + 8008*a*b*c**10*d**6/5 + 2184*b**2*c**11*d**5/5) + x**9*(8008*a**2*c**10*d**6/9 + 2912*a*b*c**11 *d**5/3 + 1820*b**2*c**12*d**4/9) + x**8*(546*a**2*c**11*d**5 + 455*a*b*c* *12*d**4 + 70*b**2*c**13*d**3) + x**7*(260*a**2*c**12*d**4 + 160*a*b*c**13 *d**3 + 120*b**2*c**14*d**2/7) + x**6*(280*a**2*c**13*d**3/3 + 40*a*b*c**1 4*d**2 + 8*b**2*c**15*d/3) + x**5*(24*a**2*c**14*d**2 + 32*a*b*c**15*d/5 + b**2*c**16/5) + x**4*(4*a**2*c**15*d + a*b*c**16/2)
Leaf count of result is larger than twice the leaf count of optimal. 617 vs. \(2 (127) = 254\).
Time = 0.03 (sec) , antiderivative size = 617, normalized size of antiderivative = 4.50 \[ \int x^2 (a+b x)^2 (c+d x)^{16} \, dx=\frac {1}{21} \, b^{2} d^{16} x^{21} + \frac {1}{3} \, a^{2} c^{16} x^{3} + \frac {1}{10} \, {\left (8 \, b^{2} c d^{15} + a b d^{16}\right )} x^{20} + \frac {1}{19} \, {\left (120 \, b^{2} c^{2} d^{14} + 32 \, a b c d^{15} + a^{2} d^{16}\right )} x^{19} + \frac {8}{9} \, {\left (35 \, b^{2} c^{3} d^{13} + 15 \, a b c^{2} d^{14} + a^{2} c d^{15}\right )} x^{18} + \frac {20}{17} \, {\left (91 \, b^{2} c^{4} d^{12} + 56 \, a b c^{3} d^{13} + 6 \, a^{2} c^{2} d^{14}\right )} x^{17} + \frac {7}{2} \, {\left (78 \, b^{2} c^{5} d^{11} + 65 \, a b c^{4} d^{12} + 10 \, a^{2} c^{3} d^{13}\right )} x^{16} + \frac {364}{15} \, {\left (22 \, b^{2} c^{6} d^{10} + 24 \, a b c^{5} d^{11} + 5 \, a^{2} c^{4} d^{12}\right )} x^{15} + \frac {104}{7} \, {\left (55 \, b^{2} c^{7} d^{9} + 77 \, a b c^{6} d^{10} + 21 \, a^{2} c^{5} d^{11}\right )} x^{14} + 22 \, {\left (45 \, b^{2} c^{8} d^{8} + 80 \, a b c^{7} d^{9} + 28 \, a^{2} c^{6} d^{10}\right )} x^{13} + \frac {715}{3} \, {\left (4 \, b^{2} c^{9} d^{7} + 9 \, a b c^{8} d^{8} + 4 \, a^{2} c^{7} d^{9}\right )} x^{12} + 26 \, {\left (28 \, b^{2} c^{10} d^{6} + 80 \, a b c^{9} d^{7} + 45 \, a^{2} c^{8} d^{8}\right )} x^{11} + \frac {104}{5} \, {\left (21 \, b^{2} c^{11} d^{5} + 77 \, a b c^{10} d^{6} + 55 \, a^{2} c^{9} d^{7}\right )} x^{10} + \frac {364}{9} \, {\left (5 \, b^{2} c^{12} d^{4} + 24 \, a b c^{11} d^{5} + 22 \, a^{2} c^{10} d^{6}\right )} x^{9} + 7 \, {\left (10 \, b^{2} c^{13} d^{3} + 65 \, a b c^{12} d^{4} + 78 \, a^{2} c^{11} d^{5}\right )} x^{8} + \frac {20}{7} \, {\left (6 \, b^{2} c^{14} d^{2} + 56 \, a b c^{13} d^{3} + 91 \, a^{2} c^{12} d^{4}\right )} x^{7} + \frac {8}{3} \, {\left (b^{2} c^{15} d + 15 \, a b c^{14} d^{2} + 35 \, a^{2} c^{13} d^{3}\right )} x^{6} + \frac {1}{5} \, {\left (b^{2} c^{16} + 32 \, a b c^{15} d + 120 \, a^{2} c^{14} d^{2}\right )} x^{5} + \frac {1}{2} \, {\left (a b c^{16} + 8 \, a^{2} c^{15} d\right )} x^{4} \] Input:
integrate(x^2*(b*x+a)^2*(d*x+c)^16,x, algorithm="maxima")
Output:
1/21*b^2*d^16*x^21 + 1/3*a^2*c^16*x^3 + 1/10*(8*b^2*c*d^15 + a*b*d^16)*x^2 0 + 1/19*(120*b^2*c^2*d^14 + 32*a*b*c*d^15 + a^2*d^16)*x^19 + 8/9*(35*b^2* c^3*d^13 + 15*a*b*c^2*d^14 + a^2*c*d^15)*x^18 + 20/17*(91*b^2*c^4*d^12 + 5 6*a*b*c^3*d^13 + 6*a^2*c^2*d^14)*x^17 + 7/2*(78*b^2*c^5*d^11 + 65*a*b*c^4* d^12 + 10*a^2*c^3*d^13)*x^16 + 364/15*(22*b^2*c^6*d^10 + 24*a*b*c^5*d^11 + 5*a^2*c^4*d^12)*x^15 + 104/7*(55*b^2*c^7*d^9 + 77*a*b*c^6*d^10 + 21*a^2*c ^5*d^11)*x^14 + 22*(45*b^2*c^8*d^8 + 80*a*b*c^7*d^9 + 28*a^2*c^6*d^10)*x^1 3 + 715/3*(4*b^2*c^9*d^7 + 9*a*b*c^8*d^8 + 4*a^2*c^7*d^9)*x^12 + 26*(28*b^ 2*c^10*d^6 + 80*a*b*c^9*d^7 + 45*a^2*c^8*d^8)*x^11 + 104/5*(21*b^2*c^11*d^ 5 + 77*a*b*c^10*d^6 + 55*a^2*c^9*d^7)*x^10 + 364/9*(5*b^2*c^12*d^4 + 24*a* b*c^11*d^5 + 22*a^2*c^10*d^6)*x^9 + 7*(10*b^2*c^13*d^3 + 65*a*b*c^12*d^4 + 78*a^2*c^11*d^5)*x^8 + 20/7*(6*b^2*c^14*d^2 + 56*a*b*c^13*d^3 + 91*a^2*c^ 12*d^4)*x^7 + 8/3*(b^2*c^15*d + 15*a*b*c^14*d^2 + 35*a^2*c^13*d^3)*x^6 + 1 /5*(b^2*c^16 + 32*a*b*c^15*d + 120*a^2*c^14*d^2)*x^5 + 1/2*(a*b*c^16 + 8*a ^2*c^15*d)*x^4
Leaf count of result is larger than twice the leaf count of optimal. 668 vs. \(2 (127) = 254\).
Time = 0.12 (sec) , antiderivative size = 668, normalized size of antiderivative = 4.88 \[ \int x^2 (a+b x)^2 (c+d x)^{16} \, dx=\frac {1}{21} \, b^{2} d^{16} x^{21} + \frac {4}{5} \, b^{2} c d^{15} x^{20} + \frac {1}{10} \, a b d^{16} x^{20} + \frac {120}{19} \, b^{2} c^{2} d^{14} x^{19} + \frac {32}{19} \, a b c d^{15} x^{19} + \frac {1}{19} \, a^{2} d^{16} x^{19} + \frac {280}{9} \, b^{2} c^{3} d^{13} x^{18} + \frac {40}{3} \, a b c^{2} d^{14} x^{18} + \frac {8}{9} \, a^{2} c d^{15} x^{18} + \frac {1820}{17} \, b^{2} c^{4} d^{12} x^{17} + \frac {1120}{17} \, a b c^{3} d^{13} x^{17} + \frac {120}{17} \, a^{2} c^{2} d^{14} x^{17} + 273 \, b^{2} c^{5} d^{11} x^{16} + \frac {455}{2} \, a b c^{4} d^{12} x^{16} + 35 \, a^{2} c^{3} d^{13} x^{16} + \frac {8008}{15} \, b^{2} c^{6} d^{10} x^{15} + \frac {2912}{5} \, a b c^{5} d^{11} x^{15} + \frac {364}{3} \, a^{2} c^{4} d^{12} x^{15} + \frac {5720}{7} \, b^{2} c^{7} d^{9} x^{14} + 1144 \, a b c^{6} d^{10} x^{14} + 312 \, a^{2} c^{5} d^{11} x^{14} + 990 \, b^{2} c^{8} d^{8} x^{13} + 1760 \, a b c^{7} d^{9} x^{13} + 616 \, a^{2} c^{6} d^{10} x^{13} + \frac {2860}{3} \, b^{2} c^{9} d^{7} x^{12} + 2145 \, a b c^{8} d^{8} x^{12} + \frac {2860}{3} \, a^{2} c^{7} d^{9} x^{12} + 728 \, b^{2} c^{10} d^{6} x^{11} + 2080 \, a b c^{9} d^{7} x^{11} + 1170 \, a^{2} c^{8} d^{8} x^{11} + \frac {2184}{5} \, b^{2} c^{11} d^{5} x^{10} + \frac {8008}{5} \, a b c^{10} d^{6} x^{10} + 1144 \, a^{2} c^{9} d^{7} x^{10} + \frac {1820}{9} \, b^{2} c^{12} d^{4} x^{9} + \frac {2912}{3} \, a b c^{11} d^{5} x^{9} + \frac {8008}{9} \, a^{2} c^{10} d^{6} x^{9} + 70 \, b^{2} c^{13} d^{3} x^{8} + 455 \, a b c^{12} d^{4} x^{8} + 546 \, a^{2} c^{11} d^{5} x^{8} + \frac {120}{7} \, b^{2} c^{14} d^{2} x^{7} + 160 \, a b c^{13} d^{3} x^{7} + 260 \, a^{2} c^{12} d^{4} x^{7} + \frac {8}{3} \, b^{2} c^{15} d x^{6} + 40 \, a b c^{14} d^{2} x^{6} + \frac {280}{3} \, a^{2} c^{13} d^{3} x^{6} + \frac {1}{5} \, b^{2} c^{16} x^{5} + \frac {32}{5} \, a b c^{15} d x^{5} + 24 \, a^{2} c^{14} d^{2} x^{5} + \frac {1}{2} \, a b c^{16} x^{4} + 4 \, a^{2} c^{15} d x^{4} + \frac {1}{3} \, a^{2} c^{16} x^{3} \] Input:
integrate(x^2*(b*x+a)^2*(d*x+c)^16,x, algorithm="giac")
Output:
1/21*b^2*d^16*x^21 + 4/5*b^2*c*d^15*x^20 + 1/10*a*b*d^16*x^20 + 120/19*b^2 *c^2*d^14*x^19 + 32/19*a*b*c*d^15*x^19 + 1/19*a^2*d^16*x^19 + 280/9*b^2*c^ 3*d^13*x^18 + 40/3*a*b*c^2*d^14*x^18 + 8/9*a^2*c*d^15*x^18 + 1820/17*b^2*c ^4*d^12*x^17 + 1120/17*a*b*c^3*d^13*x^17 + 120/17*a^2*c^2*d^14*x^17 + 273* b^2*c^5*d^11*x^16 + 455/2*a*b*c^4*d^12*x^16 + 35*a^2*c^3*d^13*x^16 + 8008/ 15*b^2*c^6*d^10*x^15 + 2912/5*a*b*c^5*d^11*x^15 + 364/3*a^2*c^4*d^12*x^15 + 5720/7*b^2*c^7*d^9*x^14 + 1144*a*b*c^6*d^10*x^14 + 312*a^2*c^5*d^11*x^14 + 990*b^2*c^8*d^8*x^13 + 1760*a*b*c^7*d^9*x^13 + 616*a^2*c^6*d^10*x^13 + 2860/3*b^2*c^9*d^7*x^12 + 2145*a*b*c^8*d^8*x^12 + 2860/3*a^2*c^7*d^9*x^12 + 728*b^2*c^10*d^6*x^11 + 2080*a*b*c^9*d^7*x^11 + 1170*a^2*c^8*d^8*x^11 + 2184/5*b^2*c^11*d^5*x^10 + 8008/5*a*b*c^10*d^6*x^10 + 1144*a^2*c^9*d^7*x^1 0 + 1820/9*b^2*c^12*d^4*x^9 + 2912/3*a*b*c^11*d^5*x^9 + 8008/9*a^2*c^10*d^ 6*x^9 + 70*b^2*c^13*d^3*x^8 + 455*a*b*c^12*d^4*x^8 + 546*a^2*c^11*d^5*x^8 + 120/7*b^2*c^14*d^2*x^7 + 160*a*b*c^13*d^3*x^7 + 260*a^2*c^12*d^4*x^7 + 8 /3*b^2*c^15*d*x^6 + 40*a*b*c^14*d^2*x^6 + 280/3*a^2*c^13*d^3*x^6 + 1/5*b^2 *c^16*x^5 + 32/5*a*b*c^15*d*x^5 + 24*a^2*c^14*d^2*x^5 + 1/2*a*b*c^16*x^4 + 4*a^2*c^15*d*x^4 + 1/3*a^2*c^16*x^3
Time = 0.70 (sec) , antiderivative size = 557, normalized size of antiderivative = 4.07 \[ \int x^2 (a+b x)^2 (c+d x)^{16} \, dx=x^5\,\left (24\,a^2\,c^{14}\,d^2+\frac {32\,a\,b\,c^{15}\,d}{5}+\frac {b^2\,c^{16}}{5}\right )+x^{19}\,\left (\frac {a^2\,d^{16}}{19}+\frac {32\,a\,b\,c\,d^{15}}{19}+\frac {120\,b^2\,c^2\,d^{14}}{19}\right )+\frac {a^2\,c^{16}\,x^3}{3}+\frac {b^2\,d^{16}\,x^{21}}{21}+\frac {a\,c^{15}\,x^4\,\left (8\,a\,d+b\,c\right )}{2}+\frac {b\,d^{15}\,x^{20}\,\left (a\,d+8\,b\,c\right )}{10}+\frac {8\,c^{13}\,d\,x^6\,\left (35\,a^2\,d^2+15\,a\,b\,c\,d+b^2\,c^2\right )}{3}+\frac {8\,c\,d^{13}\,x^{18}\,\left (a^2\,d^2+15\,a\,b\,c\,d+35\,b^2\,c^2\right )}{9}+\frac {715\,c^7\,d^7\,x^{12}\,\left (4\,a^2\,d^2+9\,a\,b\,c\,d+4\,b^2\,c^2\right )}{3}+\frac {364\,c^{10}\,d^4\,x^9\,\left (22\,a^2\,d^2+24\,a\,b\,c\,d+5\,b^2\,c^2\right )}{9}+\frac {364\,c^4\,d^{10}\,x^{15}\,\left (5\,a^2\,d^2+24\,a\,b\,c\,d+22\,b^2\,c^2\right )}{15}+\frac {20\,c^{12}\,d^2\,x^7\,\left (91\,a^2\,d^2+56\,a\,b\,c\,d+6\,b^2\,c^2\right )}{7}+7\,c^{11}\,d^3\,x^8\,\left (78\,a^2\,d^2+65\,a\,b\,c\,d+10\,b^2\,c^2\right )+\frac {104\,c^9\,d^5\,x^{10}\,\left (55\,a^2\,d^2+77\,a\,b\,c\,d+21\,b^2\,c^2\right )}{5}+26\,c^8\,d^6\,x^{11}\,\left (45\,a^2\,d^2+80\,a\,b\,c\,d+28\,b^2\,c^2\right )+22\,c^6\,d^8\,x^{13}\,\left (28\,a^2\,d^2+80\,a\,b\,c\,d+45\,b^2\,c^2\right )+\frac {104\,c^5\,d^9\,x^{14}\,\left (21\,a^2\,d^2+77\,a\,b\,c\,d+55\,b^2\,c^2\right )}{7}+\frac {7\,c^3\,d^{11}\,x^{16}\,\left (10\,a^2\,d^2+65\,a\,b\,c\,d+78\,b^2\,c^2\right )}{2}+\frac {20\,c^2\,d^{12}\,x^{17}\,\left (6\,a^2\,d^2+56\,a\,b\,c\,d+91\,b^2\,c^2\right )}{17} \] Input:
int(x^2*(a + b*x)^2*(c + d*x)^16,x)
Output:
x^5*((b^2*c^16)/5 + 24*a^2*c^14*d^2 + (32*a*b*c^15*d)/5) + x^19*((a^2*d^16 )/19 + (120*b^2*c^2*d^14)/19 + (32*a*b*c*d^15)/19) + (a^2*c^16*x^3)/3 + (b ^2*d^16*x^21)/21 + (a*c^15*x^4*(8*a*d + b*c))/2 + (b*d^15*x^20*(a*d + 8*b* c))/10 + (8*c^13*d*x^6*(35*a^2*d^2 + b^2*c^2 + 15*a*b*c*d))/3 + (8*c*d^13* x^18*(a^2*d^2 + 35*b^2*c^2 + 15*a*b*c*d))/9 + (715*c^7*d^7*x^12*(4*a^2*d^2 + 4*b^2*c^2 + 9*a*b*c*d))/3 + (364*c^10*d^4*x^9*(22*a^2*d^2 + 5*b^2*c^2 + 24*a*b*c*d))/9 + (364*c^4*d^10*x^15*(5*a^2*d^2 + 22*b^2*c^2 + 24*a*b*c*d) )/15 + (20*c^12*d^2*x^7*(91*a^2*d^2 + 6*b^2*c^2 + 56*a*b*c*d))/7 + 7*c^11* d^3*x^8*(78*a^2*d^2 + 10*b^2*c^2 + 65*a*b*c*d) + (104*c^9*d^5*x^10*(55*a^2 *d^2 + 21*b^2*c^2 + 77*a*b*c*d))/5 + 26*c^8*d^6*x^11*(45*a^2*d^2 + 28*b^2* c^2 + 80*a*b*c*d) + 22*c^6*d^8*x^13*(28*a^2*d^2 + 45*b^2*c^2 + 80*a*b*c*d) + (104*c^5*d^9*x^14*(21*a^2*d^2 + 55*b^2*c^2 + 77*a*b*c*d))/7 + (7*c^3*d^ 11*x^16*(10*a^2*d^2 + 78*b^2*c^2 + 65*a*b*c*d))/2 + (20*c^2*d^12*x^17*(6*a ^2*d^2 + 91*b^2*c^2 + 56*a*b*c*d))/17
Time = 0.17 (sec) , antiderivative size = 666, normalized size of antiderivative = 4.86 \[ \int x^2 (a+b x)^2 (c+d x)^{16} \, dx=\frac {x^{3} \left (9690 b^{2} d^{16} x^{18}+20349 a b \,d^{16} x^{17}+162792 b^{2} c \,d^{15} x^{17}+10710 a^{2} d^{16} x^{16}+342720 a b c \,d^{15} x^{16}+1285200 b^{2} c^{2} d^{14} x^{16}+180880 a^{2} c \,d^{15} x^{15}+2713200 a b \,c^{2} d^{14} x^{15}+6330800 b^{2} c^{3} d^{13} x^{15}+1436400 a^{2} c^{2} d^{14} x^{14}+13406400 a b \,c^{3} d^{13} x^{14}+21785400 b^{2} c^{4} d^{12} x^{14}+7122150 a^{2} c^{3} d^{13} x^{13}+46293975 a b \,c^{4} d^{12} x^{13}+55552770 b^{2} c^{5} d^{11} x^{13}+24690120 a^{2} c^{4} d^{12} x^{12}+118512576 a b \,c^{5} d^{11} x^{12}+108636528 b^{2} c^{6} d^{10} x^{12}+63488880 a^{2} c^{5} d^{11} x^{11}+232792560 a b \,c^{6} d^{10} x^{11}+166280400 b^{2} c^{7} d^{9} x^{11}+125349840 a^{2} c^{6} d^{10} x^{10}+358142400 a b \,c^{7} d^{9} x^{10}+201455100 b^{2} c^{8} d^{8} x^{10}+193993800 a^{2} c^{7} d^{9} x^{9}+436486050 a b \,c^{8} d^{8} x^{9}+193993800 b^{2} c^{9} d^{7} x^{9}+238083300 a^{2} c^{8} d^{8} x^{8}+423259200 a b \,c^{9} d^{7} x^{8}+148140720 b^{2} c^{10} d^{6} x^{8}+232792560 a^{2} c^{9} d^{7} x^{7}+325909584 a b \,c^{10} d^{6} x^{7}+88884432 b^{2} c^{11} d^{5} x^{7}+181060880 a^{2} c^{10} d^{6} x^{6}+197520960 a b \,c^{11} d^{5} x^{6}+41150200 b^{2} c^{12} d^{4} x^{6}+111105540 a^{2} c^{11} d^{5} x^{5}+92587950 a b \,c^{12} d^{4} x^{5}+14244300 b^{2} c^{13} d^{3} x^{5}+52907400 a^{2} c^{12} d^{4} x^{4}+32558400 a b \,c^{13} d^{3} x^{4}+3488400 b^{2} c^{14} d^{2} x^{4}+18992400 a^{2} c^{13} d^{3} x^{3}+8139600 a b \,c^{14} d^{2} x^{3}+542640 b^{2} c^{15} d \,x^{3}+4883760 a^{2} c^{14} d^{2} x^{2}+1302336 a b \,c^{15} d \,x^{2}+40698 b^{2} c^{16} x^{2}+813960 a^{2} c^{15} d x +101745 a b \,c^{16} x +67830 a^{2} c^{16}\right )}{203490} \] Input:
int(x^2*(b*x+a)^2*(d*x+c)^16,x)
Output:
(x**3*(67830*a**2*c**16 + 813960*a**2*c**15*d*x + 4883760*a**2*c**14*d**2* x**2 + 18992400*a**2*c**13*d**3*x**3 + 52907400*a**2*c**12*d**4*x**4 + 111 105540*a**2*c**11*d**5*x**5 + 181060880*a**2*c**10*d**6*x**6 + 232792560*a **2*c**9*d**7*x**7 + 238083300*a**2*c**8*d**8*x**8 + 193993800*a**2*c**7*d **9*x**9 + 125349840*a**2*c**6*d**10*x**10 + 63488880*a**2*c**5*d**11*x**1 1 + 24690120*a**2*c**4*d**12*x**12 + 7122150*a**2*c**3*d**13*x**13 + 14364 00*a**2*c**2*d**14*x**14 + 180880*a**2*c*d**15*x**15 + 10710*a**2*d**16*x* *16 + 101745*a*b*c**16*x + 1302336*a*b*c**15*d*x**2 + 8139600*a*b*c**14*d* *2*x**3 + 32558400*a*b*c**13*d**3*x**4 + 92587950*a*b*c**12*d**4*x**5 + 19 7520960*a*b*c**11*d**5*x**6 + 325909584*a*b*c**10*d**6*x**7 + 423259200*a* b*c**9*d**7*x**8 + 436486050*a*b*c**8*d**8*x**9 + 358142400*a*b*c**7*d**9* x**10 + 232792560*a*b*c**6*d**10*x**11 + 118512576*a*b*c**5*d**11*x**12 + 46293975*a*b*c**4*d**12*x**13 + 13406400*a*b*c**3*d**13*x**14 + 2713200*a* b*c**2*d**14*x**15 + 342720*a*b*c*d**15*x**16 + 20349*a*b*d**16*x**17 + 40 698*b**2*c**16*x**2 + 542640*b**2*c**15*d*x**3 + 3488400*b**2*c**14*d**2*x **4 + 14244300*b**2*c**13*d**3*x**5 + 41150200*b**2*c**12*d**4*x**6 + 8888 4432*b**2*c**11*d**5*x**7 + 148140720*b**2*c**10*d**6*x**8 + 193993800*b** 2*c**9*d**7*x**9 + 201455100*b**2*c**8*d**8*x**10 + 166280400*b**2*c**7*d* *9*x**11 + 108636528*b**2*c**6*d**10*x**12 + 55552770*b**2*c**5*d**11*x**1 3 + 21785400*b**2*c**4*d**12*x**14 + 6330800*b**2*c**3*d**13*x**15 + 12...