Integrand size = 16, antiderivative size = 114 \[ \int x^3 (a+b x) (c+d x)^{16} \, dx=\frac {c^3 (b c-a d) (c+d x)^{17}}{17 d^5}-\frac {c^2 (4 b c-3 a d) (c+d x)^{18}}{18 d^5}+\frac {3 c (2 b c-a d) (c+d x)^{19}}{19 d^5}-\frac {(4 b c-a d) (c+d x)^{20}}{20 d^5}+\frac {b (c+d x)^{21}}{21 d^5} \] Output:
1/17*c^3*(-a*d+b*c)*(d*x+c)^17/d^5-1/18*c^2*(-3*a*d+4*b*c)*(d*x+c)^18/d^5+ 3/19*c*(-a*d+2*b*c)*(d*x+c)^19/d^5-1/20*(-a*d+4*b*c)*(d*x+c)^20/d^5+1/21*b *(d*x+c)^21/d^5
Leaf count is larger than twice the leaf count of optimal. \(359\) vs. \(2(114)=228\).
Time = 0.05 (sec) , antiderivative size = 359, normalized size of antiderivative = 3.15 \[ \int x^3 (a+b x) (c+d x)^{16} \, dx=\frac {1}{4} a c^{16} x^4+\frac {1}{5} c^{15} (b c+16 a d) x^5+\frac {4}{3} c^{14} d (2 b c+15 a d) x^6+\frac {40}{7} c^{13} d^2 (3 b c+14 a d) x^7+\frac {35}{2} c^{12} d^3 (4 b c+13 a d) x^8+\frac {364}{9} c^{11} d^4 (5 b c+12 a d) x^9+\frac {364}{5} c^{10} d^5 (6 b c+11 a d) x^{10}+104 c^9 d^6 (7 b c+10 a d) x^{11}+\frac {715}{6} c^8 d^7 (8 b c+9 a d) x^{12}+110 c^7 d^8 (9 b c+8 a d) x^{13}+\frac {572}{7} c^6 d^9 (10 b c+7 a d) x^{14}+\frac {728}{15} c^5 d^{10} (11 b c+6 a d) x^{15}+\frac {91}{4} c^4 d^{11} (12 b c+5 a d) x^{16}+\frac {140}{17} c^3 d^{12} (13 b c+4 a d) x^{17}+\frac {20}{9} c^2 d^{13} (14 b c+3 a d) x^{18}+\frac {8}{19} c d^{14} (15 b c+2 a d) x^{19}+\frac {1}{20} d^{15} (16 b c+a d) x^{20}+\frac {1}{21} b d^{16} x^{21} \] Input:
Integrate[x^3*(a + b*x)*(c + d*x)^16,x]
Output:
(a*c^16*x^4)/4 + (c^15*(b*c + 16*a*d)*x^5)/5 + (4*c^14*d*(2*b*c + 15*a*d)* x^6)/3 + (40*c^13*d^2*(3*b*c + 14*a*d)*x^7)/7 + (35*c^12*d^3*(4*b*c + 13*a *d)*x^8)/2 + (364*c^11*d^4*(5*b*c + 12*a*d)*x^9)/9 + (364*c^10*d^5*(6*b*c + 11*a*d)*x^10)/5 + 104*c^9*d^6*(7*b*c + 10*a*d)*x^11 + (715*c^8*d^7*(8*b* c + 9*a*d)*x^12)/6 + 110*c^7*d^8*(9*b*c + 8*a*d)*x^13 + (572*c^6*d^9*(10*b *c + 7*a*d)*x^14)/7 + (728*c^5*d^10*(11*b*c + 6*a*d)*x^15)/15 + (91*c^4*d^ 11*(12*b*c + 5*a*d)*x^16)/4 + (140*c^3*d^12*(13*b*c + 4*a*d)*x^17)/17 + (2 0*c^2*d^13*(14*b*c + 3*a*d)*x^18)/9 + (8*c*d^14*(15*b*c + 2*a*d)*x^19)/19 + (d^15*(16*b*c + a*d)*x^20)/20 + (b*d^16*x^21)/21
Time = 0.46 (sec) , antiderivative size = 114, 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.
\(\displaystyle \int x^3 (a+b x) (c+d x)^{16} \, dx\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \int \left (\frac {c^3 (c+d x)^{16} (b c-a d)}{d^4}-\frac {c^2 (c+d x)^{17} (4 b c-3 a d)}{d^4}+\frac {(c+d x)^{19} (a d-4 b c)}{d^4}+\frac {3 c (c+d x)^{18} (2 b c-a d)}{d^4}+\frac {b (c+d x)^{20}}{d^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {c^3 (c+d x)^{17} (b c-a d)}{17 d^5}-\frac {c^2 (c+d x)^{18} (4 b c-3 a d)}{18 d^5}-\frac {(c+d x)^{20} (4 b c-a d)}{20 d^5}+\frac {3 c (c+d x)^{19} (2 b c-a d)}{19 d^5}+\frac {b (c+d x)^{21}}{21 d^5}\) |
Input:
Int[x^3*(a + b*x)*(c + d*x)^16,x]
Output:
(c^3*(b*c - a*d)*(c + d*x)^17)/(17*d^5) - (c^2*(4*b*c - 3*a*d)*(c + d*x)^1 8)/(18*d^5) + (3*c*(2*b*c - a*d)*(c + d*x)^19)/(19*d^5) - ((4*b*c - a*d)*( c + d*x)^20)/(20*d^5) + (b*(c + d*x)^21)/(21*d^5)
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. \(2(104)=208\).
Time = 0.13 (sec) , antiderivative size = 374, normalized size of antiderivative = 3.28
method | result | size |
norman | \(\frac {a \,c^{16} x^{4}}{4}+\left (\frac {16}{5} a \,c^{15} d +\frac {1}{5} b \,c^{16}\right ) x^{5}+\left (20 a \,c^{14} d^{2}+\frac {8}{3} b \,c^{15} d \right ) x^{6}+\left (80 a \,c^{13} d^{3}+\frac {120}{7} b \,c^{14} d^{2}\right ) x^{7}+\left (\frac {455}{2} a \,c^{12} d^{4}+70 b \,c^{13} d^{3}\right ) x^{8}+\left (\frac {1456}{3} a \,c^{11} d^{5}+\frac {1820}{9} b \,c^{12} d^{4}\right ) x^{9}+\left (\frac {4004}{5} a \,c^{10} d^{6}+\frac {2184}{5} b \,c^{11} d^{5}\right ) x^{10}+\left (1040 a \,c^{9} d^{7}+728 b \,c^{10} d^{6}\right ) x^{11}+\left (\frac {2145}{2} a \,c^{8} d^{8}+\frac {2860}{3} b \,c^{9} d^{7}\right ) x^{12}+\left (880 a \,c^{7} d^{9}+990 b \,c^{8} d^{8}\right ) x^{13}+\left (572 a \,c^{6} d^{10}+\frac {5720}{7} b \,c^{7} d^{9}\right ) x^{14}+\left (\frac {1456}{5} a \,c^{5} d^{11}+\frac {8008}{15} b \,c^{6} d^{10}\right ) x^{15}+\left (\frac {455}{4} a \,c^{4} d^{12}+273 b \,c^{5} d^{11}\right ) x^{16}+\left (\frac {560}{17} a \,c^{3} d^{13}+\frac {1820}{17} b \,c^{4} d^{12}\right ) x^{17}+\left (\frac {20}{3} a \,c^{2} d^{14}+\frac {280}{9} b \,c^{3} d^{13}\right ) x^{18}+\left (\frac {16}{19} a c \,d^{15}+\frac {120}{19} b \,c^{2} d^{14}\right ) x^{19}+\left (\frac {1}{20} a \,d^{16}+\frac {4}{5} b c \,d^{15}\right ) x^{20}+\frac {b \,d^{16} x^{21}}{21}\) | \(374\) |
default | \(\frac {b \,d^{16} x^{21}}{21}+\frac {\left (a \,d^{16}+16 b c \,d^{15}\right ) x^{20}}{20}+\frac {\left (16 a c \,d^{15}+120 b \,c^{2} d^{14}\right ) x^{19}}{19}+\frac {\left (120 a \,c^{2} d^{14}+560 b \,c^{3} d^{13}\right ) x^{18}}{18}+\frac {\left (560 a \,c^{3} d^{13}+1820 b \,c^{4} d^{12}\right ) x^{17}}{17}+\frac {\left (1820 a \,c^{4} d^{12}+4368 b \,c^{5} d^{11}\right ) x^{16}}{16}+\frac {\left (4368 a \,c^{5} d^{11}+8008 b \,c^{6} d^{10}\right ) x^{15}}{15}+\frac {\left (8008 a \,c^{6} d^{10}+11440 b \,c^{7} d^{9}\right ) x^{14}}{14}+\frac {\left (11440 a \,c^{7} d^{9}+12870 b \,c^{8} d^{8}\right ) x^{13}}{13}+\frac {\left (12870 a \,c^{8} d^{8}+11440 b \,c^{9} d^{7}\right ) x^{12}}{12}+\frac {\left (11440 a \,c^{9} d^{7}+8008 b \,c^{10} d^{6}\right ) x^{11}}{11}+\frac {\left (8008 a \,c^{10} d^{6}+4368 b \,c^{11} d^{5}\right ) x^{10}}{10}+\frac {\left (4368 a \,c^{11} d^{5}+1820 b \,c^{12} d^{4}\right ) x^{9}}{9}+\frac {\left (1820 a \,c^{12} d^{4}+560 b \,c^{13} d^{3}\right ) x^{8}}{8}+\frac {\left (560 a \,c^{13} d^{3}+120 b \,c^{14} d^{2}\right ) x^{7}}{7}+\frac {\left (120 a \,c^{14} d^{2}+16 b \,c^{15} d \right ) x^{6}}{6}+\frac {\left (16 a \,c^{15} d +b \,c^{16}\right ) x^{5}}{5}+\frac {a \,c^{16} x^{4}}{4}\) | \(388\) |
orering | \(\frac {x^{4} \left (19380 b \,d^{16} x^{17}+20349 a \,d^{16} x^{16}+325584 b c \,d^{15} x^{16}+342720 a c \,d^{15} x^{15}+2570400 b \,c^{2} d^{14} x^{15}+2713200 a \,c^{2} d^{14} x^{14}+12661600 b \,c^{3} d^{13} x^{14}+13406400 a \,c^{3} d^{13} x^{13}+43570800 b \,c^{4} d^{12} x^{13}+46293975 a \,c^{4} d^{12} x^{12}+111105540 b \,c^{5} d^{11} x^{12}+118512576 a \,c^{5} d^{11} x^{11}+217273056 b \,c^{6} d^{10} x^{11}+232792560 a \,c^{6} d^{10} x^{10}+332560800 b \,c^{7} d^{9} x^{10}+358142400 a \,c^{7} d^{9} x^{9}+402910200 b \,c^{8} d^{8} x^{9}+436486050 a \,c^{8} d^{8} x^{8}+387987600 b \,c^{9} d^{7} x^{8}+423259200 a \,c^{9} d^{7} x^{7}+296281440 b \,c^{10} d^{6} x^{7}+325909584 a \,c^{10} d^{6} x^{6}+177768864 b \,c^{11} d^{5} x^{6}+197520960 a \,c^{11} d^{5} x^{5}+82300400 b \,c^{12} d^{4} x^{5}+92587950 a \,c^{12} d^{4} x^{4}+28488600 b \,c^{13} d^{3} x^{4}+32558400 a \,c^{13} d^{3} x^{3}+6976800 b \,c^{14} d^{2} x^{3}+8139600 a \,c^{14} d^{2} x^{2}+1085280 b \,c^{15} d \,x^{2}+1302336 a \,c^{15} d x +81396 b \,c^{16} x +101745 a \,c^{16}\right )}{406980}\) | \(388\) |
gosper | \(\frac {1}{4} a \,c^{16} x^{4}+80 x^{7} a \,c^{13} d^{3}+\frac {120}{7} x^{7} b \,c^{14} d^{2}+\frac {455}{2} x^{8} a \,c^{12} d^{4}+70 x^{8} b \,c^{13} d^{3}+\frac {1456}{3} x^{9} a \,c^{11} d^{5}+\frac {1820}{9} x^{9} b \,c^{12} d^{4}+\frac {4004}{5} x^{10} a \,c^{10} d^{6}+\frac {2184}{5} x^{10} b \,c^{11} d^{5}+\frac {2145}{2} x^{12} a \,c^{8} d^{8}+\frac {2860}{3} x^{12} b \,c^{9} d^{7}+572 x^{14} a \,c^{6} d^{10}+\frac {5720}{7} x^{14} b \,c^{7} d^{9}+\frac {1456}{5} x^{15} a \,c^{5} d^{11}+273 x^{16} b \,c^{5} d^{11}+\frac {560}{17} x^{17} a \,c^{3} d^{13}+\frac {4}{5} x^{20} b c \,d^{15}+1040 a \,c^{9} d^{7} x^{11}+728 b \,c^{10} d^{6} x^{11}+880 a \,c^{7} d^{9} x^{13}+990 b \,c^{8} d^{8} x^{13}+\frac {1820}{17} x^{17} b \,c^{4} d^{12}+\frac {20}{3} x^{18} a \,c^{2} d^{14}+\frac {280}{9} x^{18} b \,c^{3} d^{13}+\frac {16}{19} x^{19} a c \,d^{15}+\frac {120}{19} x^{19} b \,c^{2} d^{14}+\frac {8008}{15} x^{15} b \,c^{6} d^{10}+\frac {455}{4} x^{16} a \,c^{4} d^{12}+\frac {16}{5} x^{5} a \,c^{15} d +20 x^{6} a \,c^{14} d^{2}+\frac {8}{3} x^{6} b \,c^{15} d +\frac {1}{5} x^{5} b \,c^{16}+\frac {1}{20} x^{20} a \,d^{16}+\frac {1}{21} b \,d^{16} x^{21}\) | \(390\) |
risch | \(\frac {1}{4} a \,c^{16} x^{4}+80 x^{7} a \,c^{13} d^{3}+\frac {120}{7} x^{7} b \,c^{14} d^{2}+\frac {455}{2} x^{8} a \,c^{12} d^{4}+70 x^{8} b \,c^{13} d^{3}+\frac {1456}{3} x^{9} a \,c^{11} d^{5}+\frac {1820}{9} x^{9} b \,c^{12} d^{4}+\frac {4004}{5} x^{10} a \,c^{10} d^{6}+\frac {2184}{5} x^{10} b \,c^{11} d^{5}+\frac {2145}{2} x^{12} a \,c^{8} d^{8}+\frac {2860}{3} x^{12} b \,c^{9} d^{7}+572 x^{14} a \,c^{6} d^{10}+\frac {5720}{7} x^{14} b \,c^{7} d^{9}+\frac {1456}{5} x^{15} a \,c^{5} d^{11}+273 x^{16} b \,c^{5} d^{11}+\frac {560}{17} x^{17} a \,c^{3} d^{13}+\frac {4}{5} x^{20} b c \,d^{15}+1040 a \,c^{9} d^{7} x^{11}+728 b \,c^{10} d^{6} x^{11}+880 a \,c^{7} d^{9} x^{13}+990 b \,c^{8} d^{8} x^{13}+\frac {1820}{17} x^{17} b \,c^{4} d^{12}+\frac {20}{3} x^{18} a \,c^{2} d^{14}+\frac {280}{9} x^{18} b \,c^{3} d^{13}+\frac {16}{19} x^{19} a c \,d^{15}+\frac {120}{19} x^{19} b \,c^{2} d^{14}+\frac {8008}{15} x^{15} b \,c^{6} d^{10}+\frac {455}{4} x^{16} a \,c^{4} d^{12}+\frac {16}{5} x^{5} a \,c^{15} d +20 x^{6} a \,c^{14} d^{2}+\frac {8}{3} x^{6} b \,c^{15} d +\frac {1}{5} x^{5} b \,c^{16}+\frac {1}{20} x^{20} a \,d^{16}+\frac {1}{21} b \,d^{16} x^{21}\) | \(390\) |
parallelrisch | \(\frac {1}{4} a \,c^{16} x^{4}+80 x^{7} a \,c^{13} d^{3}+\frac {120}{7} x^{7} b \,c^{14} d^{2}+\frac {455}{2} x^{8} a \,c^{12} d^{4}+70 x^{8} b \,c^{13} d^{3}+\frac {1456}{3} x^{9} a \,c^{11} d^{5}+\frac {1820}{9} x^{9} b \,c^{12} d^{4}+\frac {4004}{5} x^{10} a \,c^{10} d^{6}+\frac {2184}{5} x^{10} b \,c^{11} d^{5}+\frac {2145}{2} x^{12} a \,c^{8} d^{8}+\frac {2860}{3} x^{12} b \,c^{9} d^{7}+572 x^{14} a \,c^{6} d^{10}+\frac {5720}{7} x^{14} b \,c^{7} d^{9}+\frac {1456}{5} x^{15} a \,c^{5} d^{11}+273 x^{16} b \,c^{5} d^{11}+\frac {560}{17} x^{17} a \,c^{3} d^{13}+\frac {4}{5} x^{20} b c \,d^{15}+1040 a \,c^{9} d^{7} x^{11}+728 b \,c^{10} d^{6} x^{11}+880 a \,c^{7} d^{9} x^{13}+990 b \,c^{8} d^{8} x^{13}+\frac {1820}{17} x^{17} b \,c^{4} d^{12}+\frac {20}{3} x^{18} a \,c^{2} d^{14}+\frac {280}{9} x^{18} b \,c^{3} d^{13}+\frac {16}{19} x^{19} a c \,d^{15}+\frac {120}{19} x^{19} b \,c^{2} d^{14}+\frac {8008}{15} x^{15} b \,c^{6} d^{10}+\frac {455}{4} x^{16} a \,c^{4} d^{12}+\frac {16}{5} x^{5} a \,c^{15} d +20 x^{6} a \,c^{14} d^{2}+\frac {8}{3} x^{6} b \,c^{15} d +\frac {1}{5} x^{5} b \,c^{16}+\frac {1}{20} x^{20} a \,d^{16}+\frac {1}{21} b \,d^{16} x^{21}\) | \(390\) |
Input:
int(x^3*(b*x+a)*(d*x+c)^16,x,method=_RETURNVERBOSE)
Output:
1/4*a*c^16*x^4+(16/5*a*c^15*d+1/5*b*c^16)*x^5+(20*a*c^14*d^2+8/3*b*c^15*d) *x^6+(80*a*c^13*d^3+120/7*b*c^14*d^2)*x^7+(455/2*a*c^12*d^4+70*b*c^13*d^3) *x^8+(1456/3*a*c^11*d^5+1820/9*b*c^12*d^4)*x^9+(4004/5*a*c^10*d^6+2184/5*b *c^11*d^5)*x^10+(1040*a*c^9*d^7+728*b*c^10*d^6)*x^11+(2145/2*a*c^8*d^8+286 0/3*b*c^9*d^7)*x^12+(880*a*c^7*d^9+990*b*c^8*d^8)*x^13+(572*a*c^6*d^10+572 0/7*b*c^7*d^9)*x^14+(1456/5*a*c^5*d^11+8008/15*b*c^6*d^10)*x^15+(455/4*a*c ^4*d^12+273*b*c^5*d^11)*x^16+(560/17*a*c^3*d^13+1820/17*b*c^4*d^12)*x^17+( 20/3*a*c^2*d^14+280/9*b*c^3*d^13)*x^18+(16/19*a*c*d^15+120/19*b*c^2*d^14)* x^19+(1/20*a*d^16+4/5*b*c*d^15)*x^20+1/21*b*d^16*x^21
Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (104) = 208\).
Time = 0.08 (sec) , antiderivative size = 387, normalized size of antiderivative = 3.39 \[ \int x^3 (a+b x) (c+d x)^{16} \, dx=\frac {1}{21} \, b d^{16} x^{21} + \frac {1}{4} \, a c^{16} x^{4} + \frac {1}{20} \, {\left (16 \, b c d^{15} + a d^{16}\right )} x^{20} + \frac {8}{19} \, {\left (15 \, b c^{2} d^{14} + 2 \, a c d^{15}\right )} x^{19} + \frac {20}{9} \, {\left (14 \, b c^{3} d^{13} + 3 \, a c^{2} d^{14}\right )} x^{18} + \frac {140}{17} \, {\left (13 \, b c^{4} d^{12} + 4 \, a c^{3} d^{13}\right )} x^{17} + \frac {91}{4} \, {\left (12 \, b c^{5} d^{11} + 5 \, a c^{4} d^{12}\right )} x^{16} + \frac {728}{15} \, {\left (11 \, b c^{6} d^{10} + 6 \, a c^{5} d^{11}\right )} x^{15} + \frac {572}{7} \, {\left (10 \, b c^{7} d^{9} + 7 \, a c^{6} d^{10}\right )} x^{14} + 110 \, {\left (9 \, b c^{8} d^{8} + 8 \, a c^{7} d^{9}\right )} x^{13} + \frac {715}{6} \, {\left (8 \, b c^{9} d^{7} + 9 \, a c^{8} d^{8}\right )} x^{12} + 104 \, {\left (7 \, b c^{10} d^{6} + 10 \, a c^{9} d^{7}\right )} x^{11} + \frac {364}{5} \, {\left (6 \, b c^{11} d^{5} + 11 \, a c^{10} d^{6}\right )} x^{10} + \frac {364}{9} \, {\left (5 \, b c^{12} d^{4} + 12 \, a c^{11} d^{5}\right )} x^{9} + \frac {35}{2} \, {\left (4 \, b c^{13} d^{3} + 13 \, a c^{12} d^{4}\right )} x^{8} + \frac {40}{7} \, {\left (3 \, b c^{14} d^{2} + 14 \, a c^{13} d^{3}\right )} x^{7} + \frac {4}{3} \, {\left (2 \, b c^{15} d + 15 \, a c^{14} d^{2}\right )} x^{6} + \frac {1}{5} \, {\left (b c^{16} + 16 \, a c^{15} d\right )} x^{5} \] Input:
integrate(x^3*(b*x+a)*(d*x+c)^16,x, algorithm="fricas")
Output:
1/21*b*d^16*x^21 + 1/4*a*c^16*x^4 + 1/20*(16*b*c*d^15 + a*d^16)*x^20 + 8/1 9*(15*b*c^2*d^14 + 2*a*c*d^15)*x^19 + 20/9*(14*b*c^3*d^13 + 3*a*c^2*d^14)* x^18 + 140/17*(13*b*c^4*d^12 + 4*a*c^3*d^13)*x^17 + 91/4*(12*b*c^5*d^11 + 5*a*c^4*d^12)*x^16 + 728/15*(11*b*c^6*d^10 + 6*a*c^5*d^11)*x^15 + 572/7*(1 0*b*c^7*d^9 + 7*a*c^6*d^10)*x^14 + 110*(9*b*c^8*d^8 + 8*a*c^7*d^9)*x^13 + 715/6*(8*b*c^9*d^7 + 9*a*c^8*d^8)*x^12 + 104*(7*b*c^10*d^6 + 10*a*c^9*d^7) *x^11 + 364/5*(6*b*c^11*d^5 + 11*a*c^10*d^6)*x^10 + 364/9*(5*b*c^12*d^4 + 12*a*c^11*d^5)*x^9 + 35/2*(4*b*c^13*d^3 + 13*a*c^12*d^4)*x^8 + 40/7*(3*b*c ^14*d^2 + 14*a*c^13*d^3)*x^7 + 4/3*(2*b*c^15*d + 15*a*c^14*d^2)*x^6 + 1/5* (b*c^16 + 16*a*c^15*d)*x^5
Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (104) = 208\).
Time = 0.06 (sec) , antiderivative size = 422, normalized size of antiderivative = 3.70 \[ \int x^3 (a+b x) (c+d x)^{16} \, dx=\frac {a c^{16} x^{4}}{4} + \frac {b d^{16} x^{21}}{21} + x^{20} \left (\frac {a d^{16}}{20} + \frac {4 b c d^{15}}{5}\right ) + x^{19} \cdot \left (\frac {16 a c d^{15}}{19} + \frac {120 b c^{2} d^{14}}{19}\right ) + x^{18} \cdot \left (\frac {20 a c^{2} d^{14}}{3} + \frac {280 b c^{3} d^{13}}{9}\right ) + x^{17} \cdot \left (\frac {560 a c^{3} d^{13}}{17} + \frac {1820 b c^{4} d^{12}}{17}\right ) + x^{16} \cdot \left (\frac {455 a c^{4} d^{12}}{4} + 273 b c^{5} d^{11}\right ) + x^{15} \cdot \left (\frac {1456 a c^{5} d^{11}}{5} + \frac {8008 b c^{6} d^{10}}{15}\right ) + x^{14} \cdot \left (572 a c^{6} d^{10} + \frac {5720 b c^{7} d^{9}}{7}\right ) + x^{13} \cdot \left (880 a c^{7} d^{9} + 990 b c^{8} d^{8}\right ) + x^{12} \cdot \left (\frac {2145 a c^{8} d^{8}}{2} + \frac {2860 b c^{9} d^{7}}{3}\right ) + x^{11} \cdot \left (1040 a c^{9} d^{7} + 728 b c^{10} d^{6}\right ) + x^{10} \cdot \left (\frac {4004 a c^{10} d^{6}}{5} + \frac {2184 b c^{11} d^{5}}{5}\right ) + x^{9} \cdot \left (\frac {1456 a c^{11} d^{5}}{3} + \frac {1820 b c^{12} d^{4}}{9}\right ) + x^{8} \cdot \left (\frac {455 a c^{12} d^{4}}{2} + 70 b c^{13} d^{3}\right ) + x^{7} \cdot \left (80 a c^{13} d^{3} + \frac {120 b c^{14} d^{2}}{7}\right ) + x^{6} \cdot \left (20 a c^{14} d^{2} + \frac {8 b c^{15} d}{3}\right ) + x^{5} \cdot \left (\frac {16 a c^{15} d}{5} + \frac {b c^{16}}{5}\right ) \] Input:
integrate(x**3*(b*x+a)*(d*x+c)**16,x)
Output:
a*c**16*x**4/4 + b*d**16*x**21/21 + x**20*(a*d**16/20 + 4*b*c*d**15/5) + x **19*(16*a*c*d**15/19 + 120*b*c**2*d**14/19) + x**18*(20*a*c**2*d**14/3 + 280*b*c**3*d**13/9) + x**17*(560*a*c**3*d**13/17 + 1820*b*c**4*d**12/17) + x**16*(455*a*c**4*d**12/4 + 273*b*c**5*d**11) + x**15*(1456*a*c**5*d**11/ 5 + 8008*b*c**6*d**10/15) + x**14*(572*a*c**6*d**10 + 5720*b*c**7*d**9/7) + x**13*(880*a*c**7*d**9 + 990*b*c**8*d**8) + x**12*(2145*a*c**8*d**8/2 + 2860*b*c**9*d**7/3) + x**11*(1040*a*c**9*d**7 + 728*b*c**10*d**6) + x**10* (4004*a*c**10*d**6/5 + 2184*b*c**11*d**5/5) + x**9*(1456*a*c**11*d**5/3 + 1820*b*c**12*d**4/9) + x**8*(455*a*c**12*d**4/2 + 70*b*c**13*d**3) + x**7* (80*a*c**13*d**3 + 120*b*c**14*d**2/7) + x**6*(20*a*c**14*d**2 + 8*b*c**15 *d/3) + x**5*(16*a*c**15*d/5 + b*c**16/5)
Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (104) = 208\).
Time = 0.04 (sec) , antiderivative size = 387, normalized size of antiderivative = 3.39 \[ \int x^3 (a+b x) (c+d x)^{16} \, dx=\frac {1}{21} \, b d^{16} x^{21} + \frac {1}{4} \, a c^{16} x^{4} + \frac {1}{20} \, {\left (16 \, b c d^{15} + a d^{16}\right )} x^{20} + \frac {8}{19} \, {\left (15 \, b c^{2} d^{14} + 2 \, a c d^{15}\right )} x^{19} + \frac {20}{9} \, {\left (14 \, b c^{3} d^{13} + 3 \, a c^{2} d^{14}\right )} x^{18} + \frac {140}{17} \, {\left (13 \, b c^{4} d^{12} + 4 \, a c^{3} d^{13}\right )} x^{17} + \frac {91}{4} \, {\left (12 \, b c^{5} d^{11} + 5 \, a c^{4} d^{12}\right )} x^{16} + \frac {728}{15} \, {\left (11 \, b c^{6} d^{10} + 6 \, a c^{5} d^{11}\right )} x^{15} + \frac {572}{7} \, {\left (10 \, b c^{7} d^{9} + 7 \, a c^{6} d^{10}\right )} x^{14} + 110 \, {\left (9 \, b c^{8} d^{8} + 8 \, a c^{7} d^{9}\right )} x^{13} + \frac {715}{6} \, {\left (8 \, b c^{9} d^{7} + 9 \, a c^{8} d^{8}\right )} x^{12} + 104 \, {\left (7 \, b c^{10} d^{6} + 10 \, a c^{9} d^{7}\right )} x^{11} + \frac {364}{5} \, {\left (6 \, b c^{11} d^{5} + 11 \, a c^{10} d^{6}\right )} x^{10} + \frac {364}{9} \, {\left (5 \, b c^{12} d^{4} + 12 \, a c^{11} d^{5}\right )} x^{9} + \frac {35}{2} \, {\left (4 \, b c^{13} d^{3} + 13 \, a c^{12} d^{4}\right )} x^{8} + \frac {40}{7} \, {\left (3 \, b c^{14} d^{2} + 14 \, a c^{13} d^{3}\right )} x^{7} + \frac {4}{3} \, {\left (2 \, b c^{15} d + 15 \, a c^{14} d^{2}\right )} x^{6} + \frac {1}{5} \, {\left (b c^{16} + 16 \, a c^{15} d\right )} x^{5} \] Input:
integrate(x^3*(b*x+a)*(d*x+c)^16,x, algorithm="maxima")
Output:
1/21*b*d^16*x^21 + 1/4*a*c^16*x^4 + 1/20*(16*b*c*d^15 + a*d^16)*x^20 + 8/1 9*(15*b*c^2*d^14 + 2*a*c*d^15)*x^19 + 20/9*(14*b*c^3*d^13 + 3*a*c^2*d^14)* x^18 + 140/17*(13*b*c^4*d^12 + 4*a*c^3*d^13)*x^17 + 91/4*(12*b*c^5*d^11 + 5*a*c^4*d^12)*x^16 + 728/15*(11*b*c^6*d^10 + 6*a*c^5*d^11)*x^15 + 572/7*(1 0*b*c^7*d^9 + 7*a*c^6*d^10)*x^14 + 110*(9*b*c^8*d^8 + 8*a*c^7*d^9)*x^13 + 715/6*(8*b*c^9*d^7 + 9*a*c^8*d^8)*x^12 + 104*(7*b*c^10*d^6 + 10*a*c^9*d^7) *x^11 + 364/5*(6*b*c^11*d^5 + 11*a*c^10*d^6)*x^10 + 364/9*(5*b*c^12*d^4 + 12*a*c^11*d^5)*x^9 + 35/2*(4*b*c^13*d^3 + 13*a*c^12*d^4)*x^8 + 40/7*(3*b*c ^14*d^2 + 14*a*c^13*d^3)*x^7 + 4/3*(2*b*c^15*d + 15*a*c^14*d^2)*x^6 + 1/5* (b*c^16 + 16*a*c^15*d)*x^5
Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (104) = 208\).
Time = 0.12 (sec) , antiderivative size = 389, normalized size of antiderivative = 3.41 \[ \int x^3 (a+b x) (c+d x)^{16} \, dx=\frac {1}{21} \, b d^{16} x^{21} + \frac {4}{5} \, b c d^{15} x^{20} + \frac {1}{20} \, a d^{16} x^{20} + \frac {120}{19} \, b c^{2} d^{14} x^{19} + \frac {16}{19} \, a c d^{15} x^{19} + \frac {280}{9} \, b c^{3} d^{13} x^{18} + \frac {20}{3} \, a c^{2} d^{14} x^{18} + \frac {1820}{17} \, b c^{4} d^{12} x^{17} + \frac {560}{17} \, a c^{3} d^{13} x^{17} + 273 \, b c^{5} d^{11} x^{16} + \frac {455}{4} \, a c^{4} d^{12} x^{16} + \frac {8008}{15} \, b c^{6} d^{10} x^{15} + \frac {1456}{5} \, a c^{5} d^{11} x^{15} + \frac {5720}{7} \, b c^{7} d^{9} x^{14} + 572 \, a c^{6} d^{10} x^{14} + 990 \, b c^{8} d^{8} x^{13} + 880 \, a c^{7} d^{9} x^{13} + \frac {2860}{3} \, b c^{9} d^{7} x^{12} + \frac {2145}{2} \, a c^{8} d^{8} x^{12} + 728 \, b c^{10} d^{6} x^{11} + 1040 \, a c^{9} d^{7} x^{11} + \frac {2184}{5} \, b c^{11} d^{5} x^{10} + \frac {4004}{5} \, a c^{10} d^{6} x^{10} + \frac {1820}{9} \, b c^{12} d^{4} x^{9} + \frac {1456}{3} \, a c^{11} d^{5} x^{9} + 70 \, b c^{13} d^{3} x^{8} + \frac {455}{2} \, a c^{12} d^{4} x^{8} + \frac {120}{7} \, b c^{14} d^{2} x^{7} + 80 \, a c^{13} d^{3} x^{7} + \frac {8}{3} \, b c^{15} d x^{6} + 20 \, a c^{14} d^{2} x^{6} + \frac {1}{5} \, b c^{16} x^{5} + \frac {16}{5} \, a c^{15} d x^{5} + \frac {1}{4} \, a c^{16} x^{4} \] Input:
integrate(x^3*(b*x+a)*(d*x+c)^16,x, algorithm="giac")
Output:
1/21*b*d^16*x^21 + 4/5*b*c*d^15*x^20 + 1/20*a*d^16*x^20 + 120/19*b*c^2*d^1 4*x^19 + 16/19*a*c*d^15*x^19 + 280/9*b*c^3*d^13*x^18 + 20/3*a*c^2*d^14*x^1 8 + 1820/17*b*c^4*d^12*x^17 + 560/17*a*c^3*d^13*x^17 + 273*b*c^5*d^11*x^16 + 455/4*a*c^4*d^12*x^16 + 8008/15*b*c^6*d^10*x^15 + 1456/5*a*c^5*d^11*x^1 5 + 5720/7*b*c^7*d^9*x^14 + 572*a*c^6*d^10*x^14 + 990*b*c^8*d^8*x^13 + 880 *a*c^7*d^9*x^13 + 2860/3*b*c^9*d^7*x^12 + 2145/2*a*c^8*d^8*x^12 + 728*b*c^ 10*d^6*x^11 + 1040*a*c^9*d^7*x^11 + 2184/5*b*c^11*d^5*x^10 + 4004/5*a*c^10 *d^6*x^10 + 1820/9*b*c^12*d^4*x^9 + 1456/3*a*c^11*d^5*x^9 + 70*b*c^13*d^3* x^8 + 455/2*a*c^12*d^4*x^8 + 120/7*b*c^14*d^2*x^7 + 80*a*c^13*d^3*x^7 + 8/ 3*b*c^15*d*x^6 + 20*a*c^14*d^2*x^6 + 1/5*b*c^16*x^5 + 16/5*a*c^15*d*x^5 + 1/4*a*c^16*x^4
Time = 0.20 (sec) , antiderivative size = 331, normalized size of antiderivative = 2.90 \[ \int x^3 (a+b x) (c+d x)^{16} \, dx=x^5\,\left (\frac {b\,c^{16}}{5}+\frac {16\,a\,d\,c^{15}}{5}\right )+x^{20}\,\left (\frac {a\,d^{16}}{20}+\frac {4\,b\,c\,d^{15}}{5}\right )+\frac {a\,c^{16}\,x^4}{4}+\frac {b\,d^{16}\,x^{21}}{21}+\frac {4\,c^{14}\,d\,x^6\,\left (15\,a\,d+2\,b\,c\right )}{3}+\frac {8\,c\,d^{14}\,x^{19}\,\left (2\,a\,d+15\,b\,c\right )}{19}+\frac {40\,c^{13}\,d^2\,x^7\,\left (14\,a\,d+3\,b\,c\right )}{7}+\frac {35\,c^{12}\,d^3\,x^8\,\left (13\,a\,d+4\,b\,c\right )}{2}+\frac {364\,c^{11}\,d^4\,x^9\,\left (12\,a\,d+5\,b\,c\right )}{9}+\frac {364\,c^{10}\,d^5\,x^{10}\,\left (11\,a\,d+6\,b\,c\right )}{5}+104\,c^9\,d^6\,x^{11}\,\left (10\,a\,d+7\,b\,c\right )+\frac {715\,c^8\,d^7\,x^{12}\,\left (9\,a\,d+8\,b\,c\right )}{6}+110\,c^7\,d^8\,x^{13}\,\left (8\,a\,d+9\,b\,c\right )+\frac {572\,c^6\,d^9\,x^{14}\,\left (7\,a\,d+10\,b\,c\right )}{7}+\frac {728\,c^5\,d^{10}\,x^{15}\,\left (6\,a\,d+11\,b\,c\right )}{15}+\frac {91\,c^4\,d^{11}\,x^{16}\,\left (5\,a\,d+12\,b\,c\right )}{4}+\frac {140\,c^3\,d^{12}\,x^{17}\,\left (4\,a\,d+13\,b\,c\right )}{17}+\frac {20\,c^2\,d^{13}\,x^{18}\,\left (3\,a\,d+14\,b\,c\right )}{9} \] Input:
int(x^3*(a + b*x)*(c + d*x)^16,x)
Output:
x^5*((b*c^16)/5 + (16*a*c^15*d)/5) + x^20*((a*d^16)/20 + (4*b*c*d^15)/5) + (a*c^16*x^4)/4 + (b*d^16*x^21)/21 + (4*c^14*d*x^6*(15*a*d + 2*b*c))/3 + ( 8*c*d^14*x^19*(2*a*d + 15*b*c))/19 + (40*c^13*d^2*x^7*(14*a*d + 3*b*c))/7 + (35*c^12*d^3*x^8*(13*a*d + 4*b*c))/2 + (364*c^11*d^4*x^9*(12*a*d + 5*b*c ))/9 + (364*c^10*d^5*x^10*(11*a*d + 6*b*c))/5 + 104*c^9*d^6*x^11*(10*a*d + 7*b*c) + (715*c^8*d^7*x^12*(9*a*d + 8*b*c))/6 + 110*c^7*d^8*x^13*(8*a*d + 9*b*c) + (572*c^6*d^9*x^14*(7*a*d + 10*b*c))/7 + (728*c^5*d^10*x^15*(6*a* d + 11*b*c))/15 + (91*c^4*d^11*x^16*(5*a*d + 12*b*c))/4 + (140*c^3*d^12*x^ 17*(4*a*d + 13*b*c))/17 + (20*c^2*d^13*x^18*(3*a*d + 14*b*c))/9
Time = 0.16 (sec) , antiderivative size = 387, normalized size of antiderivative = 3.39 \[ \int x^3 (a+b x) (c+d x)^{16} \, dx=\frac {x^{4} \left (19380 b \,d^{16} x^{17}+20349 a \,d^{16} x^{16}+325584 b c \,d^{15} x^{16}+342720 a c \,d^{15} x^{15}+2570400 b \,c^{2} d^{14} x^{15}+2713200 a \,c^{2} d^{14} x^{14}+12661600 b \,c^{3} d^{13} x^{14}+13406400 a \,c^{3} d^{13} x^{13}+43570800 b \,c^{4} d^{12} x^{13}+46293975 a \,c^{4} d^{12} x^{12}+111105540 b \,c^{5} d^{11} x^{12}+118512576 a \,c^{5} d^{11} x^{11}+217273056 b \,c^{6} d^{10} x^{11}+232792560 a \,c^{6} d^{10} x^{10}+332560800 b \,c^{7} d^{9} x^{10}+358142400 a \,c^{7} d^{9} x^{9}+402910200 b \,c^{8} d^{8} x^{9}+436486050 a \,c^{8} d^{8} x^{8}+387987600 b \,c^{9} d^{7} x^{8}+423259200 a \,c^{9} d^{7} x^{7}+296281440 b \,c^{10} d^{6} x^{7}+325909584 a \,c^{10} d^{6} x^{6}+177768864 b \,c^{11} d^{5} x^{6}+197520960 a \,c^{11} d^{5} x^{5}+82300400 b \,c^{12} d^{4} x^{5}+92587950 a \,c^{12} d^{4} x^{4}+28488600 b \,c^{13} d^{3} x^{4}+32558400 a \,c^{13} d^{3} x^{3}+6976800 b \,c^{14} d^{2} x^{3}+8139600 a \,c^{14} d^{2} x^{2}+1085280 b \,c^{15} d \,x^{2}+1302336 a \,c^{15} d x +81396 b \,c^{16} x +101745 a \,c^{16}\right )}{406980} \] Input:
int(x^3*(b*x+a)*(d*x+c)^16,x)
Output:
(x**4*(101745*a*c**16 + 1302336*a*c**15*d*x + 8139600*a*c**14*d**2*x**2 + 32558400*a*c**13*d**3*x**3 + 92587950*a*c**12*d**4*x**4 + 197520960*a*c**1 1*d**5*x**5 + 325909584*a*c**10*d**6*x**6 + 423259200*a*c**9*d**7*x**7 + 4 36486050*a*c**8*d**8*x**8 + 358142400*a*c**7*d**9*x**9 + 232792560*a*c**6* d**10*x**10 + 118512576*a*c**5*d**11*x**11 + 46293975*a*c**4*d**12*x**12 + 13406400*a*c**3*d**13*x**13 + 2713200*a*c**2*d**14*x**14 + 342720*a*c*d** 15*x**15 + 20349*a*d**16*x**16 + 81396*b*c**16*x + 1085280*b*c**15*d*x**2 + 6976800*b*c**14*d**2*x**3 + 28488600*b*c**13*d**3*x**4 + 82300400*b*c**1 2*d**4*x**5 + 177768864*b*c**11*d**5*x**6 + 296281440*b*c**10*d**6*x**7 + 387987600*b*c**9*d**7*x**8 + 402910200*b*c**8*d**8*x**9 + 332560800*b*c**7 *d**9*x**10 + 217273056*b*c**6*d**10*x**11 + 111105540*b*c**5*d**11*x**12 + 43570800*b*c**4*d**12*x**13 + 12661600*b*c**3*d**13*x**14 + 2570400*b*c* *2*d**14*x**15 + 325584*b*c*d**15*x**16 + 19380*b*d**16*x**17))/406980