Integrand size = 17, antiderivative size = 73 \[ \int x^7 (1+x) \left (1+2 x+x^2\right )^5 \, dx=-\frac {1}{12} (1+x)^{12}+\frac {7}{13} (1+x)^{13}-\frac {3}{2} (1+x)^{14}+\frac {7}{3} (1+x)^{15}-\frac {35}{16} (1+x)^{16}+\frac {21}{17} (1+x)^{17}-\frac {7}{18} (1+x)^{18}+\frac {1}{19} (1+x)^{19} \] Output:
-1/12*(1+x)^12+7/13*(1+x)^13-3/2*(1+x)^14+7/3*(1+x)^15-35/16*(1+x)^16+21/1 7*(1+x)^17-7/18*(1+x)^18+1/19*(1+x)^19
Time = 0.00 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.08 \[ \int x^7 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^8}{8}+\frac {11 x^9}{9}+\frac {11 x^{10}}{2}+15 x^{11}+\frac {55 x^{12}}{2}+\frac {462 x^{13}}{13}+33 x^{14}+22 x^{15}+\frac {165 x^{16}}{16}+\frac {55 x^{17}}{17}+\frac {11 x^{18}}{18}+\frac {x^{19}}{19} \] Input:
Integrate[x^7*(1 + x)*(1 + 2*x + x^2)^5,x]
Output:
x^8/8 + (11*x^9)/9 + (11*x^10)/2 + 15*x^11 + (55*x^12)/2 + (462*x^13)/13 + 33*x^14 + 22*x^15 + (165*x^16)/16 + (55*x^17)/17 + (11*x^18)/18 + x^19/19
Time = 0.37 (sec) , antiderivative size = 73, 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.
\(\displaystyle \int x^7 (x+1) \left (x^2+2 x+1\right )^5 \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int x^7 (x+1)^{11}dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left ((x+1)^{18}-7 (x+1)^{17}+21 (x+1)^{16}-35 (x+1)^{15}+35 (x+1)^{14}-21 (x+1)^{13}+7 (x+1)^{12}-(x+1)^{11}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{19} (x+1)^{19}-\frac {7}{18} (x+1)^{18}+\frac {21}{17} (x+1)^{17}-\frac {35}{16} (x+1)^{16}+\frac {7}{3} (x+1)^{15}-\frac {3}{2} (x+1)^{14}+\frac {7}{13} (x+1)^{13}-\frac {1}{12} (x+1)^{12}\) |
Input:
Int[x^7*(1 + x)*(1 + 2*x + x^2)^5,x]
Output:
-1/12*(1 + x)^12 + (7*(1 + x)^13)/13 - (3*(1 + x)^14)/2 + (7*(1 + x)^15)/3 - (35*(1 + x)^16)/16 + (21*(1 + x)^17)/17 - (7*(1 + x)^18)/18 + (1 + x)^1 9/19
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 = 0.94 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84
method | result | size |
gosper | \(\frac {x^{8} \left (31824 x^{11}+369512 x^{10}+1956240 x^{9}+6235515 x^{8}+13302432 x^{7}+19953648 x^{6}+21488544 x^{5}+16628040 x^{4}+9069840 x^{3}+3325608 x^{2}+739024 x +75582\right )}{604656}\) | \(61\) |
default | \(\frac {1}{8} x^{8}+\frac {11}{9} x^{9}+\frac {11}{2} x^{10}+15 x^{11}+\frac {55}{2} x^{12}+\frac {462}{13} x^{13}+33 x^{14}+22 x^{15}+\frac {165}{16} x^{16}+\frac {55}{17} x^{17}+\frac {11}{18} x^{18}+\frac {1}{19} x^{19}\) | \(62\) |
norman | \(\frac {1}{8} x^{8}+\frac {11}{9} x^{9}+\frac {11}{2} x^{10}+15 x^{11}+\frac {55}{2} x^{12}+\frac {462}{13} x^{13}+33 x^{14}+22 x^{15}+\frac {165}{16} x^{16}+\frac {55}{17} x^{17}+\frac {11}{18} x^{18}+\frac {1}{19} x^{19}\) | \(62\) |
risch | \(\frac {1}{8} x^{8}+\frac {11}{9} x^{9}+\frac {11}{2} x^{10}+15 x^{11}+\frac {55}{2} x^{12}+\frac {462}{13} x^{13}+33 x^{14}+22 x^{15}+\frac {165}{16} x^{16}+\frac {55}{17} x^{17}+\frac {11}{18} x^{18}+\frac {1}{19} x^{19}\) | \(62\) |
parallelrisch | \(\frac {1}{8} x^{8}+\frac {11}{9} x^{9}+\frac {11}{2} x^{10}+15 x^{11}+\frac {55}{2} x^{12}+\frac {462}{13} x^{13}+33 x^{14}+22 x^{15}+\frac {165}{16} x^{16}+\frac {55}{17} x^{17}+\frac {11}{18} x^{18}+\frac {1}{19} x^{19}\) | \(62\) |
orering | \(\frac {x^{8} \left (31824 x^{11}+369512 x^{10}+1956240 x^{9}+6235515 x^{8}+13302432 x^{7}+19953648 x^{6}+21488544 x^{5}+16628040 x^{4}+9069840 x^{3}+3325608 x^{2}+739024 x +75582\right ) \left (x^{2}+2 x +1\right )^{5}}{604656 \left (x +1\right )^{10}}\) | \(76\) |
Input:
int(x^7*(x+1)*(x^2+2*x+1)^5,x,method=_RETURNVERBOSE)
Output:
1/604656*x^8*(31824*x^11+369512*x^10+1956240*x^9+6235515*x^8+13302432*x^7+ 19953648*x^6+21488544*x^5+16628040*x^4+9069840*x^3+3325608*x^2+739024*x+75 582)
Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84 \[ \int x^7 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{19} \, x^{19} + \frac {11}{18} \, x^{18} + \frac {55}{17} \, x^{17} + \frac {165}{16} \, x^{16} + 22 \, x^{15} + 33 \, x^{14} + \frac {462}{13} \, x^{13} + \frac {55}{2} \, x^{12} + 15 \, x^{11} + \frac {11}{2} \, x^{10} + \frac {11}{9} \, x^{9} + \frac {1}{8} \, x^{8} \] Input:
integrate(x^7*(1+x)*(x^2+2*x+1)^5,x, algorithm="fricas")
Output:
1/19*x^19 + 11/18*x^18 + 55/17*x^17 + 165/16*x^16 + 22*x^15 + 33*x^14 + 46 2/13*x^13 + 55/2*x^12 + 15*x^11 + 11/2*x^10 + 11/9*x^9 + 1/8*x^8
Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.97 \[ \int x^7 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{19}}{19} + \frac {11 x^{18}}{18} + \frac {55 x^{17}}{17} + \frac {165 x^{16}}{16} + 22 x^{15} + 33 x^{14} + \frac {462 x^{13}}{13} + \frac {55 x^{12}}{2} + 15 x^{11} + \frac {11 x^{10}}{2} + \frac {11 x^{9}}{9} + \frac {x^{8}}{8} \] Input:
integrate(x**7*(1+x)*(x**2+2*x+1)**5,x)
Output:
x**19/19 + 11*x**18/18 + 55*x**17/17 + 165*x**16/16 + 22*x**15 + 33*x**14 + 462*x**13/13 + 55*x**12/2 + 15*x**11 + 11*x**10/2 + 11*x**9/9 + x**8/8
Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84 \[ \int x^7 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{19} \, x^{19} + \frac {11}{18} \, x^{18} + \frac {55}{17} \, x^{17} + \frac {165}{16} \, x^{16} + 22 \, x^{15} + 33 \, x^{14} + \frac {462}{13} \, x^{13} + \frac {55}{2} \, x^{12} + 15 \, x^{11} + \frac {11}{2} \, x^{10} + \frac {11}{9} \, x^{9} + \frac {1}{8} \, x^{8} \] Input:
integrate(x^7*(1+x)*(x^2+2*x+1)^5,x, algorithm="maxima")
Output:
1/19*x^19 + 11/18*x^18 + 55/17*x^17 + 165/16*x^16 + 22*x^15 + 33*x^14 + 46 2/13*x^13 + 55/2*x^12 + 15*x^11 + 11/2*x^10 + 11/9*x^9 + 1/8*x^8
Time = 0.16 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84 \[ \int x^7 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{19} \, x^{19} + \frac {11}{18} \, x^{18} + \frac {55}{17} \, x^{17} + \frac {165}{16} \, x^{16} + 22 \, x^{15} + 33 \, x^{14} + \frac {462}{13} \, x^{13} + \frac {55}{2} \, x^{12} + 15 \, x^{11} + \frac {11}{2} \, x^{10} + \frac {11}{9} \, x^{9} + \frac {1}{8} \, x^{8} \] Input:
integrate(x^7*(1+x)*(x^2+2*x+1)^5,x, algorithm="giac")
Output:
1/19*x^19 + 11/18*x^18 + 55/17*x^17 + 165/16*x^16 + 22*x^15 + 33*x^14 + 46 2/13*x^13 + 55/2*x^12 + 15*x^11 + 11/2*x^10 + 11/9*x^9 + 1/8*x^8
Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84 \[ \int x^7 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{19}}{19}+\frac {11\,x^{18}}{18}+\frac {55\,x^{17}}{17}+\frac {165\,x^{16}}{16}+22\,x^{15}+33\,x^{14}+\frac {462\,x^{13}}{13}+\frac {55\,x^{12}}{2}+15\,x^{11}+\frac {11\,x^{10}}{2}+\frac {11\,x^9}{9}+\frac {x^8}{8} \] Input:
int(x^7*(x + 1)*(2*x + x^2 + 1)^5,x)
Output:
x^8/8 + (11*x^9)/9 + (11*x^10)/2 + 15*x^11 + (55*x^12)/2 + (462*x^13)/13 + 33*x^14 + 22*x^15 + (165*x^16)/16 + (55*x^17)/17 + (11*x^18)/18 + x^19/19
Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.82 \[ \int x^7 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{8} \left (31824 x^{11}+369512 x^{10}+1956240 x^{9}+6235515 x^{8}+13302432 x^{7}+19953648 x^{6}+21488544 x^{5}+16628040 x^{4}+9069840 x^{3}+3325608 x^{2}+739024 x +75582\right )}{604656} \] Input:
int(x^7*(1+x)*(x^2+2*x+1)^5,x)
Output:
(x**8*(31824*x**11 + 369512*x**10 + 1956240*x**9 + 6235515*x**8 + 13302432 *x**7 + 19953648*x**6 + 21488544*x**5 + 16628040*x**4 + 9069840*x**3 + 332 5608*x**2 + 739024*x + 75582))/604656