Integrand size = 17, antiderivative size = 64 \[ \int x^6 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{12} (1+x)^{12}-\frac {6}{13} (1+x)^{13}+\frac {15}{14} (1+x)^{14}-\frac {4}{3} (1+x)^{15}+\frac {15}{16} (1+x)^{16}-\frac {6}{17} (1+x)^{17}+\frac {1}{18} (1+x)^{18} \] Output:
1/12*(1+x)^12-6/13*(1+x)^13+15/14*(1+x)^14-4/3*(1+x)^15+15/16*(1+x)^16-6/1 7*(1+x)^17+1/18*(1+x)^18
Time = 0.00 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.27 \[ \int x^6 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^7}{7}+\frac {11 x^8}{8}+\frac {55 x^9}{9}+\frac {33 x^{10}}{2}+30 x^{11}+\frac {77 x^{12}}{2}+\frac {462 x^{13}}{13}+\frac {165 x^{14}}{7}+11 x^{15}+\frac {55 x^{16}}{16}+\frac {11 x^{17}}{17}+\frac {x^{18}}{18} \] Input:
Integrate[x^6*(1 + x)*(1 + 2*x + x^2)^5,x]
Output:
x^7/7 + (11*x^8)/8 + (55*x^9)/9 + (33*x^10)/2 + 30*x^11 + (77*x^12)/2 + (4 62*x^13)/13 + (165*x^14)/7 + 11*x^15 + (55*x^16)/16 + (11*x^17)/17 + x^18/ 18
Time = 0.35 (sec) , antiderivative size = 64, 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.
\(\displaystyle \int x^6 (x+1) \left (x^2+2 x+1\right )^5 \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int x^6 (x+1)^{11}dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left ((x+1)^{17}-6 (x+1)^{16}+15 (x+1)^{15}-20 (x+1)^{14}+15 (x+1)^{13}-6 (x+1)^{12}+(x+1)^{11}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{18} (x+1)^{18}-\frac {6}{17} (x+1)^{17}+\frac {15}{16} (x+1)^{16}-\frac {4}{3} (x+1)^{15}+\frac {15}{14} (x+1)^{14}-\frac {6}{13} (x+1)^{13}+\frac {1}{12} (x+1)^{12}\) |
Input:
Int[x^6*(1 + x)*(1 + 2*x + x^2)^5,x]
Output:
(1 + x)^12/12 - (6*(1 + x)^13)/13 + (15*(1 + x)^14)/14 - (4*(1 + x)^15)/3 + (15*(1 + x)^16)/16 - (6*(1 + x)^17)/17 + (1 + x)^18/18
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 = 0.95
method | result | size |
gosper | \(\frac {x^{7} \left (12376 x^{11}+144144 x^{10}+765765 x^{9}+2450448 x^{8}+5250960 x^{7}+7916832 x^{6}+8576568 x^{5}+6683040 x^{4}+3675672 x^{3}+1361360 x^{2}+306306 x +31824\right )}{222768}\) | \(61\) |
default | \(\frac {1}{7} x^{7}+\frac {11}{8} x^{8}+\frac {55}{9} x^{9}+\frac {33}{2} x^{10}+30 x^{11}+\frac {77}{2} x^{12}+\frac {462}{13} x^{13}+\frac {165}{7} x^{14}+11 x^{15}+\frac {55}{16} x^{16}+\frac {11}{17} x^{17}+\frac {1}{18} x^{18}\) | \(62\) |
norman | \(\frac {1}{7} x^{7}+\frac {11}{8} x^{8}+\frac {55}{9} x^{9}+\frac {33}{2} x^{10}+30 x^{11}+\frac {77}{2} x^{12}+\frac {462}{13} x^{13}+\frac {165}{7} x^{14}+11 x^{15}+\frac {55}{16} x^{16}+\frac {11}{17} x^{17}+\frac {1}{18} x^{18}\) | \(62\) |
risch | \(\frac {1}{7} x^{7}+\frac {11}{8} x^{8}+\frac {55}{9} x^{9}+\frac {33}{2} x^{10}+30 x^{11}+\frac {77}{2} x^{12}+\frac {462}{13} x^{13}+\frac {165}{7} x^{14}+11 x^{15}+\frac {55}{16} x^{16}+\frac {11}{17} x^{17}+\frac {1}{18} x^{18}\) | \(62\) |
parallelrisch | \(\frac {1}{7} x^{7}+\frac {11}{8} x^{8}+\frac {55}{9} x^{9}+\frac {33}{2} x^{10}+30 x^{11}+\frac {77}{2} x^{12}+\frac {462}{13} x^{13}+\frac {165}{7} x^{14}+11 x^{15}+\frac {55}{16} x^{16}+\frac {11}{17} x^{17}+\frac {1}{18} x^{18}\) | \(62\) |
orering | \(\frac {x^{7} \left (12376 x^{11}+144144 x^{10}+765765 x^{9}+2450448 x^{8}+5250960 x^{7}+7916832 x^{6}+8576568 x^{5}+6683040 x^{4}+3675672 x^{3}+1361360 x^{2}+306306 x +31824\right ) \left (x^{2}+2 x +1\right )^{5}}{222768 \left (x +1\right )^{10}}\) | \(76\) |
Input:
int(x^6*(x+1)*(x^2+2*x+1)^5,x,method=_RETURNVERBOSE)
Output:
1/222768*x^7*(12376*x^11+144144*x^10+765765*x^9+2450448*x^8+5250960*x^7+79 16832*x^6+8576568*x^5+6683040*x^4+3675672*x^3+1361360*x^2+306306*x+31824)
Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95 \[ \int x^6 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{18} \, x^{18} + \frac {11}{17} \, x^{17} + \frac {55}{16} \, x^{16} + 11 \, x^{15} + \frac {165}{7} \, x^{14} + \frac {462}{13} \, x^{13} + \frac {77}{2} \, x^{12} + 30 \, x^{11} + \frac {33}{2} \, x^{10} + \frac {55}{9} \, x^{9} + \frac {11}{8} \, x^{8} + \frac {1}{7} \, x^{7} \] Input:
integrate(x^6*(1+x)*(x^2+2*x+1)^5,x, algorithm="fricas")
Output:
1/18*x^18 + 11/17*x^17 + 55/16*x^16 + 11*x^15 + 165/7*x^14 + 462/13*x^13 + 77/2*x^12 + 30*x^11 + 33/2*x^10 + 55/9*x^9 + 11/8*x^8 + 1/7*x^7
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.14 \[ \int x^6 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{18}}{18} + \frac {11 x^{17}}{17} + \frac {55 x^{16}}{16} + 11 x^{15} + \frac {165 x^{14}}{7} + \frac {462 x^{13}}{13} + \frac {77 x^{12}}{2} + 30 x^{11} + \frac {33 x^{10}}{2} + \frac {55 x^{9}}{9} + \frac {11 x^{8}}{8} + \frac {x^{7}}{7} \] Input:
integrate(x**6*(1+x)*(x**2+2*x+1)**5,x)
Output:
x**18/18 + 11*x**17/17 + 55*x**16/16 + 11*x**15 + 165*x**14/7 + 462*x**13/ 13 + 77*x**12/2 + 30*x**11 + 33*x**10/2 + 55*x**9/9 + 11*x**8/8 + x**7/7
Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95 \[ \int x^6 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{18} \, x^{18} + \frac {11}{17} \, x^{17} + \frac {55}{16} \, x^{16} + 11 \, x^{15} + \frac {165}{7} \, x^{14} + \frac {462}{13} \, x^{13} + \frac {77}{2} \, x^{12} + 30 \, x^{11} + \frac {33}{2} \, x^{10} + \frac {55}{9} \, x^{9} + \frac {11}{8} \, x^{8} + \frac {1}{7} \, x^{7} \] Input:
integrate(x^6*(1+x)*(x^2+2*x+1)^5,x, algorithm="maxima")
Output:
1/18*x^18 + 11/17*x^17 + 55/16*x^16 + 11*x^15 + 165/7*x^14 + 462/13*x^13 + 77/2*x^12 + 30*x^11 + 33/2*x^10 + 55/9*x^9 + 11/8*x^8 + 1/7*x^7
Time = 0.18 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95 \[ \int x^6 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{18} \, x^{18} + \frac {11}{17} \, x^{17} + \frac {55}{16} \, x^{16} + 11 \, x^{15} + \frac {165}{7} \, x^{14} + \frac {462}{13} \, x^{13} + \frac {77}{2} \, x^{12} + 30 \, x^{11} + \frac {33}{2} \, x^{10} + \frac {55}{9} \, x^{9} + \frac {11}{8} \, x^{8} + \frac {1}{7} \, x^{7} \] Input:
integrate(x^6*(1+x)*(x^2+2*x+1)^5,x, algorithm="giac")
Output:
1/18*x^18 + 11/17*x^17 + 55/16*x^16 + 11*x^15 + 165/7*x^14 + 462/13*x^13 + 77/2*x^12 + 30*x^11 + 33/2*x^10 + 55/9*x^9 + 11/8*x^8 + 1/7*x^7
Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95 \[ \int x^6 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{18}}{18}+\frac {11\,x^{17}}{17}+\frac {55\,x^{16}}{16}+11\,x^{15}+\frac {165\,x^{14}}{7}+\frac {462\,x^{13}}{13}+\frac {77\,x^{12}}{2}+30\,x^{11}+\frac {33\,x^{10}}{2}+\frac {55\,x^9}{9}+\frac {11\,x^8}{8}+\frac {x^7}{7} \] Input:
int(x^6*(x + 1)*(2*x + x^2 + 1)^5,x)
Output:
x^7/7 + (11*x^8)/8 + (55*x^9)/9 + (33*x^10)/2 + 30*x^11 + (77*x^12)/2 + (4 62*x^13)/13 + (165*x^14)/7 + 11*x^15 + (55*x^16)/16 + (11*x^17)/17 + x^18/ 18
Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.94 \[ \int x^6 (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{7} \left (12376 x^{11}+144144 x^{10}+765765 x^{9}+2450448 x^{8}+5250960 x^{7}+7916832 x^{6}+8576568 x^{5}+6683040 x^{4}+3675672 x^{3}+1361360 x^{2}+306306 x +31824\right )}{222768} \] Input:
int(x^6*(1+x)*(x^2+2*x+1)^5,x)
Output:
(x**7*(12376*x**11 + 144144*x**10 + 765765*x**9 + 2450448*x**8 + 5250960*x **7 + 7916832*x**6 + 8576568*x**5 + 6683040*x**4 + 3675672*x**3 + 1361360* x**2 + 306306*x + 31824))/222768