Integrand size = 17, antiderivative size = 80 \[ \int x^8 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{12} (1+x)^{12}-\frac {8}{13} (1+x)^{13}+2 (1+x)^{14}-\frac {56}{15} (1+x)^{15}+\frac {35}{8} (1+x)^{16}-\frac {56}{17} (1+x)^{17}+\frac {14}{9} (1+x)^{18}-\frac {8}{19} (1+x)^{19}+\frac {1}{20} (1+x)^{20} \] Output:
1/12*(1+x)^12-8/13*(1+x)^13+2*(1+x)^14-56/15*(1+x)^15+35/8*(1+x)^16-56/17* (1+x)^17+14/9*(1+x)^18-8/19*(1+x)^19+1/20*(1+x)^20
Time = 0.00 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.01 \[ \int x^8 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^9}{9}+\frac {11 x^{10}}{10}+5 x^{11}+\frac {55 x^{12}}{4}+\frac {330 x^{13}}{13}+33 x^{14}+\frac {154 x^{15}}{5}+\frac {165 x^{16}}{8}+\frac {165 x^{17}}{17}+\frac {55 x^{18}}{18}+\frac {11 x^{19}}{19}+\frac {x^{20}}{20} \] Input:
Integrate[x^8*(1 + x)*(1 + 2*x + x^2)^5,x]
Output:
x^9/9 + (11*x^10)/10 + 5*x^11 + (55*x^12)/4 + (330*x^13)/13 + 33*x^14 + (1 54*x^15)/5 + (165*x^16)/8 + (165*x^17)/17 + (55*x^18)/18 + (11*x^19)/19 + x^20/20
Time = 0.38 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1184, 49, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^8 (x+1) \left (x^2+2 x+1\right )^5 \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int x^8 (x+1)^{11}dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left ((x+1)^{19}-8 (x+1)^{18}+28 (x+1)^{17}-56 (x+1)^{16}+70 (x+1)^{15}-56 (x+1)^{14}+28 (x+1)^{13}-8 (x+1)^{12}+(x+1)^{11}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{20} (x+1)^{20}-\frac {8}{19} (x+1)^{19}+\frac {14}{9} (x+1)^{18}-\frac {56}{17} (x+1)^{17}+\frac {35}{8} (x+1)^{16}-\frac {56}{15} (x+1)^{15}+2 (x+1)^{14}-\frac {8}{13} (x+1)^{13}+\frac {1}{12} (x+1)^{12}\) |
Input:
Int[x^8*(1 + x)*(1 + 2*x + x^2)^5,x]
Output:
(1 + x)^12/12 - (8*(1 + x)^13)/13 + 2*(1 + x)^14 - (56*(1 + x)^15)/15 + (3 5*(1 + x)^16)/8 - (56*(1 + x)^17)/17 + (14*(1 + x)^18)/9 - (8*(1 + x)^19)/ 19 + (1 + x)^20/20
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.82 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.76
method | result | size |
gosper | \(\frac {x^{9} \left (75582 x^{11}+875160 x^{10}+4618900 x^{9}+14671800 x^{8}+31177575 x^{7}+46558512 x^{6}+49884120 x^{5}+38372400 x^{4}+20785050 x^{3}+7558200 x^{2}+1662804 x +167960\right )}{1511640}\) | \(61\) |
default | \(\frac {1}{9} x^{9}+\frac {11}{10} x^{10}+5 x^{11}+\frac {55}{4} x^{12}+\frac {330}{13} x^{13}+33 x^{14}+\frac {154}{5} x^{15}+\frac {165}{8} x^{16}+\frac {165}{17} x^{17}+\frac {55}{18} x^{18}+\frac {11}{19} x^{19}+\frac {1}{20} x^{20}\) | \(62\) |
norman | \(\frac {1}{9} x^{9}+\frac {11}{10} x^{10}+5 x^{11}+\frac {55}{4} x^{12}+\frac {330}{13} x^{13}+33 x^{14}+\frac {154}{5} x^{15}+\frac {165}{8} x^{16}+\frac {165}{17} x^{17}+\frac {55}{18} x^{18}+\frac {11}{19} x^{19}+\frac {1}{20} x^{20}\) | \(62\) |
risch | \(\frac {1}{9} x^{9}+\frac {11}{10} x^{10}+5 x^{11}+\frac {55}{4} x^{12}+\frac {330}{13} x^{13}+33 x^{14}+\frac {154}{5} x^{15}+\frac {165}{8} x^{16}+\frac {165}{17} x^{17}+\frac {55}{18} x^{18}+\frac {11}{19} x^{19}+\frac {1}{20} x^{20}\) | \(62\) |
parallelrisch | \(\frac {1}{9} x^{9}+\frac {11}{10} x^{10}+5 x^{11}+\frac {55}{4} x^{12}+\frac {330}{13} x^{13}+33 x^{14}+\frac {154}{5} x^{15}+\frac {165}{8} x^{16}+\frac {165}{17} x^{17}+\frac {55}{18} x^{18}+\frac {11}{19} x^{19}+\frac {1}{20} x^{20}\) | \(62\) |
orering | \(\frac {x^{9} \left (75582 x^{11}+875160 x^{10}+4618900 x^{9}+14671800 x^{8}+31177575 x^{7}+46558512 x^{6}+49884120 x^{5}+38372400 x^{4}+20785050 x^{3}+7558200 x^{2}+1662804 x +167960\right ) \left (x^{2}+2 x +1\right )^{5}}{1511640 \left (x +1\right )^{10}}\) | \(76\) |
Input:
int(x^8*(x+1)*(x^2+2*x+1)^5,x,method=_RETURNVERBOSE)
Output:
1/1511640*x^9*(75582*x^11+875160*x^10+4618900*x^9+14671800*x^8+31177575*x^ 7+46558512*x^6+49884120*x^5+38372400*x^4+20785050*x^3+7558200*x^2+1662804* x+167960)
Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.76 \[ \int x^8 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{20} \, x^{20} + \frac {11}{19} \, x^{19} + \frac {55}{18} \, x^{18} + \frac {165}{17} \, x^{17} + \frac {165}{8} \, x^{16} + \frac {154}{5} \, x^{15} + 33 \, x^{14} + \frac {330}{13} \, x^{13} + \frac {55}{4} \, x^{12} + 5 \, x^{11} + \frac {11}{10} \, x^{10} + \frac {1}{9} \, x^{9} \] Input:
integrate(x^8*(1+x)*(x^2+2*x+1)^5,x, algorithm="fricas")
Output:
1/20*x^20 + 11/19*x^19 + 55/18*x^18 + 165/17*x^17 + 165/8*x^16 + 154/5*x^1 5 + 33*x^14 + 330/13*x^13 + 55/4*x^12 + 5*x^11 + 11/10*x^10 + 1/9*x^9
Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.91 \[ \int x^8 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{20}}{20} + \frac {11 x^{19}}{19} + \frac {55 x^{18}}{18} + \frac {165 x^{17}}{17} + \frac {165 x^{16}}{8} + \frac {154 x^{15}}{5} + 33 x^{14} + \frac {330 x^{13}}{13} + \frac {55 x^{12}}{4} + 5 x^{11} + \frac {11 x^{10}}{10} + \frac {x^{9}}{9} \] Input:
integrate(x**8*(1+x)*(x**2+2*x+1)**5,x)
Output:
x**20/20 + 11*x**19/19 + 55*x**18/18 + 165*x**17/17 + 165*x**16/8 + 154*x* *15/5 + 33*x**14 + 330*x**13/13 + 55*x**12/4 + 5*x**11 + 11*x**10/10 + x** 9/9
Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.76 \[ \int x^8 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{20} \, x^{20} + \frac {11}{19} \, x^{19} + \frac {55}{18} \, x^{18} + \frac {165}{17} \, x^{17} + \frac {165}{8} \, x^{16} + \frac {154}{5} \, x^{15} + 33 \, x^{14} + \frac {330}{13} \, x^{13} + \frac {55}{4} \, x^{12} + 5 \, x^{11} + \frac {11}{10} \, x^{10} + \frac {1}{9} \, x^{9} \] Input:
integrate(x^8*(1+x)*(x^2+2*x+1)^5,x, algorithm="maxima")
Output:
1/20*x^20 + 11/19*x^19 + 55/18*x^18 + 165/17*x^17 + 165/8*x^16 + 154/5*x^1 5 + 33*x^14 + 330/13*x^13 + 55/4*x^12 + 5*x^11 + 11/10*x^10 + 1/9*x^9
Time = 0.18 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.76 \[ \int x^8 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{20} \, x^{20} + \frac {11}{19} \, x^{19} + \frac {55}{18} \, x^{18} + \frac {165}{17} \, x^{17} + \frac {165}{8} \, x^{16} + \frac {154}{5} \, x^{15} + 33 \, x^{14} + \frac {330}{13} \, x^{13} + \frac {55}{4} \, x^{12} + 5 \, x^{11} + \frac {11}{10} \, x^{10} + \frac {1}{9} \, x^{9} \] Input:
integrate(x^8*(1+x)*(x^2+2*x+1)^5,x, algorithm="giac")
Output:
1/20*x^20 + 11/19*x^19 + 55/18*x^18 + 165/17*x^17 + 165/8*x^16 + 154/5*x^1 5 + 33*x^14 + 330/13*x^13 + 55/4*x^12 + 5*x^11 + 11/10*x^10 + 1/9*x^9
Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.76 \[ \int x^8 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{20}}{20}+\frac {11\,x^{19}}{19}+\frac {55\,x^{18}}{18}+\frac {165\,x^{17}}{17}+\frac {165\,x^{16}}{8}+\frac {154\,x^{15}}{5}+33\,x^{14}+\frac {330\,x^{13}}{13}+\frac {55\,x^{12}}{4}+5\,x^{11}+\frac {11\,x^{10}}{10}+\frac {x^9}{9} \] Input:
int(x^8*(x + 1)*(2*x + x^2 + 1)^5,x)
Output:
x^9/9 + (11*x^10)/10 + 5*x^11 + (55*x^12)/4 + (330*x^13)/13 + 33*x^14 + (1 54*x^15)/5 + (165*x^16)/8 + (165*x^17)/17 + (55*x^18)/18 + (11*x^19)/19 + x^20/20
Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.75 \[ \int x^8 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{9} \left (75582 x^{11}+875160 x^{10}+4618900 x^{9}+14671800 x^{8}+31177575 x^{7}+46558512 x^{6}+49884120 x^{5}+38372400 x^{4}+20785050 x^{3}+7558200 x^{2}+1662804 x +167960\right )}{1511640} \] Input:
int(x^8*(1+x)*(x^2+2*x+1)^5,x)
Output:
(x**9*(75582*x**11 + 875160*x**10 + 4618900*x**9 + 14671800*x**8 + 3117757 5*x**7 + 46558512*x**6 + 49884120*x**5 + 38372400*x**4 + 20785050*x**3 + 7 558200*x**2 + 1662804*x + 167960))/1511640