Integrand size = 17, antiderivative size = 55 \[ \int x^5 (1+x) \left (1+2 x+x^2\right )^5 \, dx=-\frac {1}{12} (1+x)^{12}+\frac {5}{13} (1+x)^{13}-\frac {5}{7} (1+x)^{14}+\frac {2}{3} (1+x)^{15}-\frac {5}{16} (1+x)^{16}+\frac {1}{17} (1+x)^{17} \] Output:
-1/12*(1+x)^12+5/13*(1+x)^13-5/7*(1+x)^14+2/3*(1+x)^15-5/16*(1+x)^16+1/17* (1+x)^17
Time = 0.00 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.47 \[ \int x^5 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^6}{6}+\frac {11 x^7}{7}+\frac {55 x^8}{8}+\frac {55 x^9}{3}+33 x^{10}+42 x^{11}+\frac {77 x^{12}}{2}+\frac {330 x^{13}}{13}+\frac {165 x^{14}}{14}+\frac {11 x^{15}}{3}+\frac {11 x^{16}}{16}+\frac {x^{17}}{17} \] Input:
Integrate[x^5*(1 + x)*(1 + 2*x + x^2)^5,x]
Output:
x^6/6 + (11*x^7)/7 + (55*x^8)/8 + (55*x^9)/3 + 33*x^10 + 42*x^11 + (77*x^1 2)/2 + (330*x^13)/13 + (165*x^14)/14 + (11*x^15)/3 + (11*x^16)/16 + x^17/1 7
Time = 0.35 (sec) , antiderivative size = 55, 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.
\(\displaystyle \int x^5 (x+1) \left (x^2+2 x+1\right )^5 \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int x^5 (x+1)^{11}dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left ((x+1)^{16}-5 (x+1)^{15}+10 (x+1)^{14}-10 (x+1)^{13}+5 (x+1)^{12}-(x+1)^{11}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{17} (x+1)^{17}-\frac {5}{16} (x+1)^{16}+\frac {2}{3} (x+1)^{15}-\frac {5}{7} (x+1)^{14}+\frac {5}{13} (x+1)^{13}-\frac {1}{12} (x+1)^{12}\) |
Input:
Int[x^5*(1 + x)*(1 + 2*x + x^2)^5,x]
Output:
-1/12*(1 + x)^12 + (5*(1 + x)^13)/13 - (5*(1 + x)^14)/7 + (2*(1 + x)^15)/3 - (5*(1 + x)^16)/16 + (1 + x)^17/17
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 = 0.88 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.11
method | result | size |
gosper | \(\frac {x^{6} \left (4368 x^{11}+51051 x^{10}+272272 x^{9}+875160 x^{8}+1884960 x^{7}+2858856 x^{6}+3118752 x^{5}+2450448 x^{4}+1361360 x^{3}+510510 x^{2}+116688 x +12376\right )}{74256}\) | \(61\) |
default | \(\frac {1}{6} x^{6}+\frac {11}{7} x^{7}+\frac {55}{8} x^{8}+\frac {55}{3} x^{9}+33 x^{10}+42 x^{11}+\frac {77}{2} x^{12}+\frac {330}{13} x^{13}+\frac {165}{14} x^{14}+\frac {11}{3} x^{15}+\frac {11}{16} x^{16}+\frac {1}{17} x^{17}\) | \(62\) |
norman | \(\frac {1}{6} x^{6}+\frac {11}{7} x^{7}+\frac {55}{8} x^{8}+\frac {55}{3} x^{9}+33 x^{10}+42 x^{11}+\frac {77}{2} x^{12}+\frac {330}{13} x^{13}+\frac {165}{14} x^{14}+\frac {11}{3} x^{15}+\frac {11}{16} x^{16}+\frac {1}{17} x^{17}\) | \(62\) |
risch | \(\frac {1}{6} x^{6}+\frac {11}{7} x^{7}+\frac {55}{8} x^{8}+\frac {55}{3} x^{9}+33 x^{10}+42 x^{11}+\frac {77}{2} x^{12}+\frac {330}{13} x^{13}+\frac {165}{14} x^{14}+\frac {11}{3} x^{15}+\frac {11}{16} x^{16}+\frac {1}{17} x^{17}\) | \(62\) |
parallelrisch | \(\frac {1}{6} x^{6}+\frac {11}{7} x^{7}+\frac {55}{8} x^{8}+\frac {55}{3} x^{9}+33 x^{10}+42 x^{11}+\frac {77}{2} x^{12}+\frac {330}{13} x^{13}+\frac {165}{14} x^{14}+\frac {11}{3} x^{15}+\frac {11}{16} x^{16}+\frac {1}{17} x^{17}\) | \(62\) |
orering | \(\frac {x^{6} \left (4368 x^{11}+51051 x^{10}+272272 x^{9}+875160 x^{8}+1884960 x^{7}+2858856 x^{6}+3118752 x^{5}+2450448 x^{4}+1361360 x^{3}+510510 x^{2}+116688 x +12376\right ) \left (x^{2}+2 x +1\right )^{5}}{74256 \left (x +1\right )^{10}}\) | \(76\) |
Input:
int(x^5*(x+1)*(x^2+2*x+1)^5,x,method=_RETURNVERBOSE)
Output:
1/74256*x^6*(4368*x^11+51051*x^10+272272*x^9+875160*x^8+1884960*x^7+285885 6*x^6+3118752*x^5+2450448*x^4+1361360*x^3+510510*x^2+116688*x+12376)
Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.11 \[ \int x^5 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{17} \, x^{17} + \frac {11}{16} \, x^{16} + \frac {11}{3} \, x^{15} + \frac {165}{14} \, x^{14} + \frac {330}{13} \, x^{13} + \frac {77}{2} \, x^{12} + 42 \, x^{11} + 33 \, x^{10} + \frac {55}{3} \, x^{9} + \frac {55}{8} \, x^{8} + \frac {11}{7} \, x^{7} + \frac {1}{6} \, x^{6} \] Input:
integrate(x^5*(1+x)*(x^2+2*x+1)^5,x, algorithm="fricas")
Output:
1/17*x^17 + 11/16*x^16 + 11/3*x^15 + 165/14*x^14 + 330/13*x^13 + 77/2*x^12 + 42*x^11 + 33*x^10 + 55/3*x^9 + 55/8*x^8 + 11/7*x^7 + 1/6*x^6
Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.33 \[ \int x^5 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{17}}{17} + \frac {11 x^{16}}{16} + \frac {11 x^{15}}{3} + \frac {165 x^{14}}{14} + \frac {330 x^{13}}{13} + \frac {77 x^{12}}{2} + 42 x^{11} + 33 x^{10} + \frac {55 x^{9}}{3} + \frac {55 x^{8}}{8} + \frac {11 x^{7}}{7} + \frac {x^{6}}{6} \] Input:
integrate(x**5*(1+x)*(x**2+2*x+1)**5,x)
Output:
x**17/17 + 11*x**16/16 + 11*x**15/3 + 165*x**14/14 + 330*x**13/13 + 77*x** 12/2 + 42*x**11 + 33*x**10 + 55*x**9/3 + 55*x**8/8 + 11*x**7/7 + x**6/6
Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.11 \[ \int x^5 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{17} \, x^{17} + \frac {11}{16} \, x^{16} + \frac {11}{3} \, x^{15} + \frac {165}{14} \, x^{14} + \frac {330}{13} \, x^{13} + \frac {77}{2} \, x^{12} + 42 \, x^{11} + 33 \, x^{10} + \frac {55}{3} \, x^{9} + \frac {55}{8} \, x^{8} + \frac {11}{7} \, x^{7} + \frac {1}{6} \, x^{6} \] Input:
integrate(x^5*(1+x)*(x^2+2*x+1)^5,x, algorithm="maxima")
Output:
1/17*x^17 + 11/16*x^16 + 11/3*x^15 + 165/14*x^14 + 330/13*x^13 + 77/2*x^12 + 42*x^11 + 33*x^10 + 55/3*x^9 + 55/8*x^8 + 11/7*x^7 + 1/6*x^6
Time = 0.15 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.11 \[ \int x^5 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{17} \, x^{17} + \frac {11}{16} \, x^{16} + \frac {11}{3} \, x^{15} + \frac {165}{14} \, x^{14} + \frac {330}{13} \, x^{13} + \frac {77}{2} \, x^{12} + 42 \, x^{11} + 33 \, x^{10} + \frac {55}{3} \, x^{9} + \frac {55}{8} \, x^{8} + \frac {11}{7} \, x^{7} + \frac {1}{6} \, x^{6} \] Input:
integrate(x^5*(1+x)*(x^2+2*x+1)^5,x, algorithm="giac")
Output:
1/17*x^17 + 11/16*x^16 + 11/3*x^15 + 165/14*x^14 + 330/13*x^13 + 77/2*x^12 + 42*x^11 + 33*x^10 + 55/3*x^9 + 55/8*x^8 + 11/7*x^7 + 1/6*x^6
Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.11 \[ \int x^5 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{17}}{17}+\frac {11\,x^{16}}{16}+\frac {11\,x^{15}}{3}+\frac {165\,x^{14}}{14}+\frac {330\,x^{13}}{13}+\frac {77\,x^{12}}{2}+42\,x^{11}+33\,x^{10}+\frac {55\,x^9}{3}+\frac {55\,x^8}{8}+\frac {11\,x^7}{7}+\frac {x^6}{6} \] Input:
int(x^5*(x + 1)*(2*x + x^2 + 1)^5,x)
Output:
x^6/6 + (11*x^7)/7 + (55*x^8)/8 + (55*x^9)/3 + 33*x^10 + 42*x^11 + (77*x^1 2)/2 + (330*x^13)/13 + (165*x^14)/14 + (11*x^15)/3 + (11*x^16)/16 + x^17/1 7
Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.09 \[ \int x^5 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{6} \left (4368 x^{11}+51051 x^{10}+272272 x^{9}+875160 x^{8}+1884960 x^{7}+2858856 x^{6}+3118752 x^{5}+2450448 x^{4}+1361360 x^{3}+510510 x^{2}+116688 x +12376\right )}{74256} \] Input:
int(x^5*(1+x)*(x^2+2*x+1)^5,x)
Output:
(x**6*(4368*x**11 + 51051*x**10 + 272272*x**9 + 875160*x**8 + 1884960*x**7 + 2858856*x**6 + 3118752*x**5 + 2450448*x**4 + 1361360*x**3 + 510510*x**2 + 116688*x + 12376))/74256