Integrand size = 17, antiderivative size = 83 \[ \int x^{11} (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{12}}{12}+\frac {11 x^{13}}{13}+\frac {55 x^{14}}{14}+11 x^{15}+\frac {165 x^{16}}{8}+\frac {462 x^{17}}{17}+\frac {77 x^{18}}{3}+\frac {330 x^{19}}{19}+\frac {33 x^{20}}{4}+\frac {55 x^{21}}{21}+\frac {x^{22}}{2}+\frac {x^{23}}{23} \] Output:
1/12*x^12+11/13*x^13+55/14*x^14+11*x^15+165/8*x^16+462/17*x^17+77/3*x^18+3 30/19*x^19+33/4*x^20+55/21*x^21+1/2*x^22+1/23*x^23
Time = 0.00 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00 \[ \int x^{11} (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{12}}{12}+\frac {11 x^{13}}{13}+\frac {55 x^{14}}{14}+11 x^{15}+\frac {165 x^{16}}{8}+\frac {462 x^{17}}{17}+\frac {77 x^{18}}{3}+\frac {330 x^{19}}{19}+\frac {33 x^{20}}{4}+\frac {55 x^{21}}{21}+\frac {x^{22}}{2}+\frac {x^{23}}{23} \] Input:
Integrate[x^11*(1 + x)*(1 + 2*x + x^2)^5,x]
Output:
x^12/12 + (11*x^13)/13 + (55*x^14)/14 + 11*x^15 + (165*x^16)/8 + (462*x^17 )/17 + (77*x^18)/3 + (330*x^19)/19 + (33*x^20)/4 + (55*x^21)/21 + x^22/2 + x^23/23
Time = 0.38 (sec) , antiderivative size = 83, 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.
\(\displaystyle \int x^{11} (x+1) \left (x^2+2 x+1\right )^5 \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int x^{11} (x+1)^{11}dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (x^{22}+11 x^{21}+55 x^{20}+165 x^{19}+330 x^{18}+462 x^{17}+462 x^{16}+330 x^{15}+165 x^{14}+55 x^{13}+11 x^{12}+x^{11}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x^{23}}{23}+\frac {x^{22}}{2}+\frac {55 x^{21}}{21}+\frac {33 x^{20}}{4}+\frac {330 x^{19}}{19}+\frac {77 x^{18}}{3}+\frac {462 x^{17}}{17}+\frac {165 x^{16}}{8}+11 x^{15}+\frac {55 x^{14}}{14}+\frac {11 x^{13}}{13}+\frac {x^{12}}{12}\) |
Input:
Int[x^11*(1 + x)*(1 + 2*x + x^2)^5,x]
Output:
x^12/12 + (11*x^13)/13 + (55*x^14)/14 + 11*x^15 + (165*x^16)/8 + (462*x^17 )/17 + (77*x^18)/3 + (330*x^19)/19 + (33*x^20)/4 + (55*x^21)/21 + x^22/2 + x^23/23
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 = 0.93 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.73
method | result | size |
gosper | \(\frac {x^{12} \left (705432 x^{11}+8112468 x^{10}+42493880 x^{9}+133855722 x^{8}+281801520 x^{7}+416440024 x^{6}+440936496 x^{5}+334639305 x^{4}+178474296 x^{3}+63740820 x^{2}+13728792 x +1352078\right )}{16224936}\) | \(61\) |
default | \(\frac {1}{12} x^{12}+\frac {11}{13} x^{13}+\frac {55}{14} x^{14}+11 x^{15}+\frac {165}{8} x^{16}+\frac {462}{17} x^{17}+\frac {77}{3} x^{18}+\frac {330}{19} x^{19}+\frac {33}{4} x^{20}+\frac {55}{21} x^{21}+\frac {1}{2} x^{22}+\frac {1}{23} x^{23}\) | \(62\) |
norman | \(\frac {1}{12} x^{12}+\frac {11}{13} x^{13}+\frac {55}{14} x^{14}+11 x^{15}+\frac {165}{8} x^{16}+\frac {462}{17} x^{17}+\frac {77}{3} x^{18}+\frac {330}{19} x^{19}+\frac {33}{4} x^{20}+\frac {55}{21} x^{21}+\frac {1}{2} x^{22}+\frac {1}{23} x^{23}\) | \(62\) |
risch | \(\frac {1}{12} x^{12}+\frac {11}{13} x^{13}+\frac {55}{14} x^{14}+11 x^{15}+\frac {165}{8} x^{16}+\frac {462}{17} x^{17}+\frac {77}{3} x^{18}+\frac {330}{19} x^{19}+\frac {33}{4} x^{20}+\frac {55}{21} x^{21}+\frac {1}{2} x^{22}+\frac {1}{23} x^{23}\) | \(62\) |
parallelrisch | \(\frac {1}{12} x^{12}+\frac {11}{13} x^{13}+\frac {55}{14} x^{14}+11 x^{15}+\frac {165}{8} x^{16}+\frac {462}{17} x^{17}+\frac {77}{3} x^{18}+\frac {330}{19} x^{19}+\frac {33}{4} x^{20}+\frac {55}{21} x^{21}+\frac {1}{2} x^{22}+\frac {1}{23} x^{23}\) | \(62\) |
orering | \(\frac {x^{12} \left (705432 x^{11}+8112468 x^{10}+42493880 x^{9}+133855722 x^{8}+281801520 x^{7}+416440024 x^{6}+440936496 x^{5}+334639305 x^{4}+178474296 x^{3}+63740820 x^{2}+13728792 x +1352078\right ) \left (x^{2}+2 x +1\right )^{5}}{16224936 \left (x +1\right )^{10}}\) | \(76\) |
Input:
int(x^11*(x+1)*(x^2+2*x+1)^5,x,method=_RETURNVERBOSE)
Output:
1/16224936*x^12*(705432*x^11+8112468*x^10+42493880*x^9+133855722*x^8+28180 1520*x^7+416440024*x^6+440936496*x^5+334639305*x^4+178474296*x^3+63740820* x^2+13728792*x+1352078)
Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.73 \[ \int x^{11} (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{23} \, x^{23} + \frac {1}{2} \, x^{22} + \frac {55}{21} \, x^{21} + \frac {33}{4} \, x^{20} + \frac {330}{19} \, x^{19} + \frac {77}{3} \, x^{18} + \frac {462}{17} \, x^{17} + \frac {165}{8} \, x^{16} + 11 \, x^{15} + \frac {55}{14} \, x^{14} + \frac {11}{13} \, x^{13} + \frac {1}{12} \, x^{12} \] Input:
integrate(x^11*(1+x)*(x^2+2*x+1)^5,x, algorithm="fricas")
Output:
1/23*x^23 + 1/2*x^22 + 55/21*x^21 + 33/4*x^20 + 330/19*x^19 + 77/3*x^18 + 462/17*x^17 + 165/8*x^16 + 11*x^15 + 55/14*x^14 + 11/13*x^13 + 1/12*x^12
Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.88 \[ \int x^{11} (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{23}}{23} + \frac {x^{22}}{2} + \frac {55 x^{21}}{21} + \frac {33 x^{20}}{4} + \frac {330 x^{19}}{19} + \frac {77 x^{18}}{3} + \frac {462 x^{17}}{17} + \frac {165 x^{16}}{8} + 11 x^{15} + \frac {55 x^{14}}{14} + \frac {11 x^{13}}{13} + \frac {x^{12}}{12} \] Input:
integrate(x**11*(1+x)*(x**2+2*x+1)**5,x)
Output:
x**23/23 + x**22/2 + 55*x**21/21 + 33*x**20/4 + 330*x**19/19 + 77*x**18/3 + 462*x**17/17 + 165*x**16/8 + 11*x**15 + 55*x**14/14 + 11*x**13/13 + x**1 2/12
Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.73 \[ \int x^{11} (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{23} \, x^{23} + \frac {1}{2} \, x^{22} + \frac {55}{21} \, x^{21} + \frac {33}{4} \, x^{20} + \frac {330}{19} \, x^{19} + \frac {77}{3} \, x^{18} + \frac {462}{17} \, x^{17} + \frac {165}{8} \, x^{16} + 11 \, x^{15} + \frac {55}{14} \, x^{14} + \frac {11}{13} \, x^{13} + \frac {1}{12} \, x^{12} \] Input:
integrate(x^11*(1+x)*(x^2+2*x+1)^5,x, algorithm="maxima")
Output:
1/23*x^23 + 1/2*x^22 + 55/21*x^21 + 33/4*x^20 + 330/19*x^19 + 77/3*x^18 + 462/17*x^17 + 165/8*x^16 + 11*x^15 + 55/14*x^14 + 11/13*x^13 + 1/12*x^12
Time = 0.16 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.73 \[ \int x^{11} (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{23} \, x^{23} + \frac {1}{2} \, x^{22} + \frac {55}{21} \, x^{21} + \frac {33}{4} \, x^{20} + \frac {330}{19} \, x^{19} + \frac {77}{3} \, x^{18} + \frac {462}{17} \, x^{17} + \frac {165}{8} \, x^{16} + 11 \, x^{15} + \frac {55}{14} \, x^{14} + \frac {11}{13} \, x^{13} + \frac {1}{12} \, x^{12} \] Input:
integrate(x^11*(1+x)*(x^2+2*x+1)^5,x, algorithm="giac")
Output:
1/23*x^23 + 1/2*x^22 + 55/21*x^21 + 33/4*x^20 + 330/19*x^19 + 77/3*x^18 + 462/17*x^17 + 165/8*x^16 + 11*x^15 + 55/14*x^14 + 11/13*x^13 + 1/12*x^12
Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.73 \[ \int x^{11} (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{23}}{23}+\frac {x^{22}}{2}+\frac {55\,x^{21}}{21}+\frac {33\,x^{20}}{4}+\frac {330\,x^{19}}{19}+\frac {77\,x^{18}}{3}+\frac {462\,x^{17}}{17}+\frac {165\,x^{16}}{8}+11\,x^{15}+\frac {55\,x^{14}}{14}+\frac {11\,x^{13}}{13}+\frac {x^{12}}{12} \] Input:
int(x^11*(x + 1)*(2*x + x^2 + 1)^5,x)
Output:
x^12/12 + (11*x^13)/13 + (55*x^14)/14 + 11*x^15 + (165*x^16)/8 + (462*x^17 )/17 + (77*x^18)/3 + (330*x^19)/19 + (33*x^20)/4 + (55*x^21)/21 + x^22/2 + x^23/23
Time = 0.30 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.72 \[ \int x^{11} (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{12} \left (705432 x^{11}+8112468 x^{10}+42493880 x^{9}+133855722 x^{8}+281801520 x^{7}+416440024 x^{6}+440936496 x^{5}+334639305 x^{4}+178474296 x^{3}+63740820 x^{2}+13728792 x +1352078\right )}{16224936} \] Input:
int(x^11*(1+x)*(x^2+2*x+1)^5,x)
Output:
(x**12*(705432*x**11 + 8112468*x**10 + 42493880*x**9 + 133855722*x**8 + 28 1801520*x**7 + 416440024*x**6 + 440936496*x**5 + 334639305*x**4 + 17847429 6*x**3 + 63740820*x**2 + 13728792*x + 1352078))/16224936