Integrand size = 17, antiderivative size = 83 \[ \int x^{10} (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{11}}{11}+\frac {11 x^{12}}{12}+\frac {55 x^{13}}{13}+\frac {165 x^{14}}{14}+22 x^{15}+\frac {231 x^{16}}{8}+\frac {462 x^{17}}{17}+\frac {55 x^{18}}{3}+\frac {165 x^{19}}{19}+\frac {11 x^{20}}{4}+\frac {11 x^{21}}{21}+\frac {x^{22}}{22} \] Output:
1/11*x^11+11/12*x^12+55/13*x^13+165/14*x^14+22*x^15+231/8*x^16+462/17*x^17 +55/3*x^18+165/19*x^19+11/4*x^20+11/21*x^21+1/22*x^22
Time = 0.00 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00 \[ \int x^{10} (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{11}}{11}+\frac {11 x^{12}}{12}+\frac {55 x^{13}}{13}+\frac {165 x^{14}}{14}+22 x^{15}+\frac {231 x^{16}}{8}+\frac {462 x^{17}}{17}+\frac {55 x^{18}}{3}+\frac {165 x^{19}}{19}+\frac {11 x^{20}}{4}+\frac {11 x^{21}}{21}+\frac {x^{22}}{22} \] Input:
Integrate[x^10*(1 + x)*(1 + 2*x + x^2)^5,x]
Output:
x^11/11 + (11*x^12)/12 + (55*x^13)/13 + (165*x^14)/14 + 22*x^15 + (231*x^1 6)/8 + (462*x^17)/17 + (55*x^18)/3 + (165*x^19)/19 + (11*x^20)/4 + (11*x^2 1)/21 + x^22/22
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.
\(\displaystyle \int x^{10} (x+1) \left (x^2+2 x+1\right )^5 \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int x^{10} (x+1)^{11}dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (x^{21}+11 x^{20}+55 x^{19}+165 x^{18}+330 x^{17}+462 x^{16}+462 x^{15}+330 x^{14}+165 x^{13}+55 x^{12}+11 x^{11}+x^{10}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x^{22}}{22}+\frac {11 x^{21}}{21}+\frac {11 x^{20}}{4}+\frac {165 x^{19}}{19}+\frac {55 x^{18}}{3}+\frac {462 x^{17}}{17}+\frac {231 x^{16}}{8}+22 x^{15}+\frac {165 x^{14}}{14}+\frac {55 x^{13}}{13}+\frac {11 x^{12}}{12}+\frac {x^{11}}{11}\) |
Input:
Int[x^10*(1 + x)*(1 + 2*x + x^2)^5,x]
Output:
x^11/11 + (11*x^12)/12 + (55*x^13)/13 + (165*x^14)/14 + 22*x^15 + (231*x^1 6)/8 + (462*x^17)/17 + (55*x^18)/3 + (165*x^19)/19 + (11*x^20)/4 + (11*x^2 1)/21 + x^22/22
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 = 0.73
method | result | size |
gosper | \(\frac {x^{11} \left (352716 x^{11}+4064632 x^{10}+21339318 x^{9}+67387320 x^{8}+142262120 x^{7}+210882672 x^{6}+224062839 x^{5}+170714544 x^{4}+91454220 x^{3}+32829720 x^{2}+7113106 x +705432\right )}{7759752}\) | \(61\) |
default | \(\frac {1}{11} x^{11}+\frac {11}{12} x^{12}+\frac {55}{13} x^{13}+\frac {165}{14} x^{14}+22 x^{15}+\frac {231}{8} x^{16}+\frac {462}{17} x^{17}+\frac {55}{3} x^{18}+\frac {165}{19} x^{19}+\frac {11}{4} x^{20}+\frac {11}{21} x^{21}+\frac {1}{22} x^{22}\) | \(62\) |
norman | \(\frac {1}{11} x^{11}+\frac {11}{12} x^{12}+\frac {55}{13} x^{13}+\frac {165}{14} x^{14}+22 x^{15}+\frac {231}{8} x^{16}+\frac {462}{17} x^{17}+\frac {55}{3} x^{18}+\frac {165}{19} x^{19}+\frac {11}{4} x^{20}+\frac {11}{21} x^{21}+\frac {1}{22} x^{22}\) | \(62\) |
risch | \(\frac {1}{11} x^{11}+\frac {11}{12} x^{12}+\frac {55}{13} x^{13}+\frac {165}{14} x^{14}+22 x^{15}+\frac {231}{8} x^{16}+\frac {462}{17} x^{17}+\frac {55}{3} x^{18}+\frac {165}{19} x^{19}+\frac {11}{4} x^{20}+\frac {11}{21} x^{21}+\frac {1}{22} x^{22}\) | \(62\) |
parallelrisch | \(\frac {1}{11} x^{11}+\frac {11}{12} x^{12}+\frac {55}{13} x^{13}+\frac {165}{14} x^{14}+22 x^{15}+\frac {231}{8} x^{16}+\frac {462}{17} x^{17}+\frac {55}{3} x^{18}+\frac {165}{19} x^{19}+\frac {11}{4} x^{20}+\frac {11}{21} x^{21}+\frac {1}{22} x^{22}\) | \(62\) |
orering | \(\frac {x^{11} \left (352716 x^{11}+4064632 x^{10}+21339318 x^{9}+67387320 x^{8}+142262120 x^{7}+210882672 x^{6}+224062839 x^{5}+170714544 x^{4}+91454220 x^{3}+32829720 x^{2}+7113106 x +705432\right ) \left (x^{2}+2 x +1\right )^{5}}{7759752 \left (x +1\right )^{10}}\) | \(76\) |
Input:
int(x^10*(x+1)*(x^2+2*x+1)^5,x,method=_RETURNVERBOSE)
Output:
1/7759752*x^11*(352716*x^11+4064632*x^10+21339318*x^9+67387320*x^8+1422621 20*x^7+210882672*x^6+224062839*x^5+170714544*x^4+91454220*x^3+32829720*x^2 +7113106*x+705432)
Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.73 \[ \int x^{10} (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{22} \, x^{22} + \frac {11}{21} \, x^{21} + \frac {11}{4} \, x^{20} + \frac {165}{19} \, x^{19} + \frac {55}{3} \, x^{18} + \frac {462}{17} \, x^{17} + \frac {231}{8} \, x^{16} + 22 \, x^{15} + \frac {165}{14} \, x^{14} + \frac {55}{13} \, x^{13} + \frac {11}{12} \, x^{12} + \frac {1}{11} \, x^{11} \] Input:
integrate(x^10*(1+x)*(x^2+2*x+1)^5,x, algorithm="fricas")
Output:
1/22*x^22 + 11/21*x^21 + 11/4*x^20 + 165/19*x^19 + 55/3*x^18 + 462/17*x^17 + 231/8*x^16 + 22*x^15 + 165/14*x^14 + 55/13*x^13 + 11/12*x^12 + 1/11*x^1 1
Time = 0.02 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.90 \[ \int x^{10} (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{22}}{22} + \frac {11 x^{21}}{21} + \frac {11 x^{20}}{4} + \frac {165 x^{19}}{19} + \frac {55 x^{18}}{3} + \frac {462 x^{17}}{17} + \frac {231 x^{16}}{8} + 22 x^{15} + \frac {165 x^{14}}{14} + \frac {55 x^{13}}{13} + \frac {11 x^{12}}{12} + \frac {x^{11}}{11} \] Input:
integrate(x**10*(1+x)*(x**2+2*x+1)**5,x)
Output:
x**22/22 + 11*x**21/21 + 11*x**20/4 + 165*x**19/19 + 55*x**18/3 + 462*x**1 7/17 + 231*x**16/8 + 22*x**15 + 165*x**14/14 + 55*x**13/13 + 11*x**12/12 + x**11/11
Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.73 \[ \int x^{10} (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{22} \, x^{22} + \frac {11}{21} \, x^{21} + \frac {11}{4} \, x^{20} + \frac {165}{19} \, x^{19} + \frac {55}{3} \, x^{18} + \frac {462}{17} \, x^{17} + \frac {231}{8} \, x^{16} + 22 \, x^{15} + \frac {165}{14} \, x^{14} + \frac {55}{13} \, x^{13} + \frac {11}{12} \, x^{12} + \frac {1}{11} \, x^{11} \] Input:
integrate(x^10*(1+x)*(x^2+2*x+1)^5,x, algorithm="maxima")
Output:
1/22*x^22 + 11/21*x^21 + 11/4*x^20 + 165/19*x^19 + 55/3*x^18 + 462/17*x^17 + 231/8*x^16 + 22*x^15 + 165/14*x^14 + 55/13*x^13 + 11/12*x^12 + 1/11*x^1 1
Time = 0.19 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.73 \[ \int x^{10} (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{22} \, x^{22} + \frac {11}{21} \, x^{21} + \frac {11}{4} \, x^{20} + \frac {165}{19} \, x^{19} + \frac {55}{3} \, x^{18} + \frac {462}{17} \, x^{17} + \frac {231}{8} \, x^{16} + 22 \, x^{15} + \frac {165}{14} \, x^{14} + \frac {55}{13} \, x^{13} + \frac {11}{12} \, x^{12} + \frac {1}{11} \, x^{11} \] Input:
integrate(x^10*(1+x)*(x^2+2*x+1)^5,x, algorithm="giac")
Output:
1/22*x^22 + 11/21*x^21 + 11/4*x^20 + 165/19*x^19 + 55/3*x^18 + 462/17*x^17 + 231/8*x^16 + 22*x^15 + 165/14*x^14 + 55/13*x^13 + 11/12*x^12 + 1/11*x^1 1
Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.73 \[ \int x^{10} (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{22}}{22}+\frac {11\,x^{21}}{21}+\frac {11\,x^{20}}{4}+\frac {165\,x^{19}}{19}+\frac {55\,x^{18}}{3}+\frac {462\,x^{17}}{17}+\frac {231\,x^{16}}{8}+22\,x^{15}+\frac {165\,x^{14}}{14}+\frac {55\,x^{13}}{13}+\frac {11\,x^{12}}{12}+\frac {x^{11}}{11} \] Input:
int(x^10*(x + 1)*(2*x + x^2 + 1)^5,x)
Output:
x^11/11 + (11*x^12)/12 + (55*x^13)/13 + (165*x^14)/14 + 22*x^15 + (231*x^1 6)/8 + (462*x^17)/17 + (55*x^18)/3 + (165*x^19)/19 + (11*x^20)/4 + (11*x^2 1)/21 + x^22/22
Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.72 \[ \int x^{10} (1+x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{11} \left (352716 x^{11}+4064632 x^{10}+21339318 x^{9}+67387320 x^{8}+142262120 x^{7}+210882672 x^{6}+224062839 x^{5}+170714544 x^{4}+91454220 x^{3}+32829720 x^{2}+7113106 x +705432\right )}{7759752} \] Input:
int(x^10*(1+x)*(x^2+2*x+1)^5,x)
Output:
(x**11*(352716*x**11 + 4064632*x**10 + 21339318*x**9 + 67387320*x**8 + 142 262120*x**7 + 210882672*x**6 + 224062839*x**5 + 170714544*x**4 + 91454220* x**3 + 32829720*x**2 + 7113106*x + 705432))/7759752