Integrand size = 17, antiderivative size = 83 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{22}} \, dx=-\frac {1}{21 x^{21}}-\frac {11}{20 x^{20}}-\frac {55}{19 x^{19}}-\frac {55}{6 x^{18}}-\frac {330}{17 x^{17}}-\frac {231}{8 x^{16}}-\frac {154}{5 x^{15}}-\frac {165}{7 x^{14}}-\frac {165}{13 x^{13}}-\frac {55}{12 x^{12}}-\frac {1}{x^{11}}-\frac {1}{10 x^{10}} \] Output:
-1/21/x^21-11/20/x^20-55/19/x^19-55/6/x^18-330/17/x^17-231/8/x^16-154/5/x^ 15-165/7/x^14-165/13/x^13-55/12/x^12-1/x^11-1/10/x^10
Time = 0.01 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{22}} \, dx=-\frac {1}{21 x^{21}}-\frac {11}{20 x^{20}}-\frac {55}{19 x^{19}}-\frac {55}{6 x^{18}}-\frac {330}{17 x^{17}}-\frac {231}{8 x^{16}}-\frac {154}{5 x^{15}}-\frac {165}{7 x^{14}}-\frac {165}{13 x^{13}}-\frac {55}{12 x^{12}}-\frac {1}{x^{11}}-\frac {1}{10 x^{10}} \] Input:
Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^22,x]
Output:
-1/21*1/x^21 - 11/(20*x^20) - 55/(19*x^19) - 55/(6*x^18) - 330/(17*x^17) - 231/(8*x^16) - 154/(5*x^15) - 165/(7*x^14) - 165/(13*x^13) - 55/(12*x^12) - x^(-11) - 1/(10*x^10)
Time = 0.35 (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, 53, 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.
\(\displaystyle \int \frac {(x+1) \left (x^2+2 x+1\right )^5}{x^{22}} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int \frac {(x+1)^{11}}{x^{22}}dx\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \int \left (\frac {1}{x^{22}}+\frac {11}{x^{21}}+\frac {55}{x^{20}}+\frac {165}{x^{19}}+\frac {330}{x^{18}}+\frac {462}{x^{17}}+\frac {462}{x^{16}}+\frac {330}{x^{15}}+\frac {165}{x^{14}}+\frac {55}{x^{13}}+\frac {11}{x^{12}}+\frac {1}{x^{11}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{21 x^{21}}-\frac {11}{20 x^{20}}-\frac {55}{19 x^{19}}-\frac {55}{6 x^{18}}-\frac {330}{17 x^{17}}-\frac {231}{8 x^{16}}-\frac {154}{5 x^{15}}-\frac {165}{7 x^{14}}-\frac {165}{13 x^{13}}-\frac {55}{12 x^{12}}-\frac {1}{x^{11}}-\frac {1}{10 x^{10}}\) |
Input:
Int[((1 + x)*(1 + 2*x + x^2)^5)/x^22,x]
Output:
-1/21*1/x^21 - 11/(20*x^20) - 55/(19*x^19) - 55/(6*x^18) - 330/(17*x^17) - 231/(8*x^16) - 154/(5*x^15) - 165/(7*x^14) - 165/(13*x^13) - 55/(12*x^12) - x^(-11) - 1/(10*x^10)
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, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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 = 0.76 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.72
method | result | size |
norman | \(\frac {-\frac {1}{21}-\frac {11}{20} x -\frac {55}{19} x^{2}-\frac {55}{6} x^{3}-\frac {330}{17} x^{4}-\frac {231}{8} x^{5}-\frac {154}{5} x^{6}-\frac {165}{7} x^{7}-\frac {165}{13} x^{8}-\frac {55}{12} x^{9}-x^{10}-\frac {1}{10} x^{11}}{x^{21}}\) | \(60\) |
risch | \(\frac {-\frac {1}{21}-\frac {11}{20} x -\frac {55}{19} x^{2}-\frac {55}{6} x^{3}-\frac {330}{17} x^{4}-\frac {231}{8} x^{5}-\frac {154}{5} x^{6}-\frac {165}{7} x^{7}-\frac {165}{13} x^{8}-\frac {55}{12} x^{9}-x^{10}-\frac {1}{10} x^{11}}{x^{21}}\) | \(60\) |
gosper | \(-\frac {352716 x^{11}+3527160 x^{10}+16166150 x^{9}+44767800 x^{8}+83140200 x^{7}+108636528 x^{6}+101846745 x^{5}+68468400 x^{4}+32332300 x^{3}+10210200 x^{2}+1939938 x +167960}{3527160 x^{21}}\) | \(61\) |
parallelrisch | \(\frac {-352716 x^{11}-3527160 x^{10}-16166150 x^{9}-44767800 x^{8}-83140200 x^{7}-108636528 x^{6}-101846745 x^{5}-68468400 x^{4}-32332300 x^{3}-10210200 x^{2}-1939938 x -167960}{3527160 x^{21}}\) | \(61\) |
default | \(-\frac {1}{21 x^{21}}-\frac {11}{20 x^{20}}-\frac {55}{19 x^{19}}-\frac {55}{6 x^{18}}-\frac {330}{17 x^{17}}-\frac {231}{8 x^{16}}-\frac {154}{5 x^{15}}-\frac {165}{7 x^{14}}-\frac {165}{13 x^{13}}-\frac {55}{12 x^{12}}-\frac {1}{x^{11}}-\frac {1}{10 x^{10}}\) | \(62\) |
orering | \(-\frac {\left (352716 x^{11}+3527160 x^{10}+16166150 x^{9}+44767800 x^{8}+83140200 x^{7}+108636528 x^{6}+101846745 x^{5}+68468400 x^{4}+32332300 x^{3}+10210200 x^{2}+1939938 x +167960\right ) \left (x^{2}+2 x +1\right )^{5}}{3527160 x^{21} \left (x +1\right )^{10}}\) | \(76\) |
Input:
int((x+1)*(x^2+2*x+1)^5/x^22,x,method=_RETURNVERBOSE)
Output:
(-1/21-11/20*x-55/19*x^2-55/6*x^3-330/17*x^4-231/8*x^5-154/5*x^6-165/7*x^7 -165/13*x^8-55/12*x^9-x^10-1/10*x^11)/x^21
Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.72 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{22}} \, dx=-\frac {352716 \, x^{11} + 3527160 \, x^{10} + 16166150 \, x^{9} + 44767800 \, x^{8} + 83140200 \, x^{7} + 108636528 \, x^{6} + 101846745 \, x^{5} + 68468400 \, x^{4} + 32332300 \, x^{3} + 10210200 \, x^{2} + 1939938 \, x + 167960}{3527160 \, x^{21}} \] Input:
integrate((1+x)*(x^2+2*x+1)^5/x^22,x, algorithm="fricas")
Output:
-1/3527160*(352716*x^11 + 3527160*x^10 + 16166150*x^9 + 44767800*x^8 + 831 40200*x^7 + 108636528*x^6 + 101846745*x^5 + 68468400*x^4 + 32332300*x^3 + 10210200*x^2 + 1939938*x + 167960)/x^21
Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.73 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{22}} \, dx=\frac {- 352716 x^{11} - 3527160 x^{10} - 16166150 x^{9} - 44767800 x^{8} - 83140200 x^{7} - 108636528 x^{6} - 101846745 x^{5} - 68468400 x^{4} - 32332300 x^{3} - 10210200 x^{2} - 1939938 x - 167960}{3527160 x^{21}} \] Input:
integrate((1+x)*(x**2+2*x+1)**5/x**22,x)
Output:
(-352716*x**11 - 3527160*x**10 - 16166150*x**9 - 44767800*x**8 - 83140200* x**7 - 108636528*x**6 - 101846745*x**5 - 68468400*x**4 - 32332300*x**3 - 1 0210200*x**2 - 1939938*x - 167960)/(3527160*x**21)
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.72 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{22}} \, dx=-\frac {352716 \, x^{11} + 3527160 \, x^{10} + 16166150 \, x^{9} + 44767800 \, x^{8} + 83140200 \, x^{7} + 108636528 \, x^{6} + 101846745 \, x^{5} + 68468400 \, x^{4} + 32332300 \, x^{3} + 10210200 \, x^{2} + 1939938 \, x + 167960}{3527160 \, x^{21}} \] Input:
integrate((1+x)*(x^2+2*x+1)^5/x^22,x, algorithm="maxima")
Output:
-1/3527160*(352716*x^11 + 3527160*x^10 + 16166150*x^9 + 44767800*x^8 + 831 40200*x^7 + 108636528*x^6 + 101846745*x^5 + 68468400*x^4 + 32332300*x^3 + 10210200*x^2 + 1939938*x + 167960)/x^21
Time = 0.19 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.72 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{22}} \, dx=-\frac {352716 \, x^{11} + 3527160 \, x^{10} + 16166150 \, x^{9} + 44767800 \, x^{8} + 83140200 \, x^{7} + 108636528 \, x^{6} + 101846745 \, x^{5} + 68468400 \, x^{4} + 32332300 \, x^{3} + 10210200 \, x^{2} + 1939938 \, x + 167960}{3527160 \, x^{21}} \] Input:
integrate((1+x)*(x^2+2*x+1)^5/x^22,x, algorithm="giac")
Output:
-1/3527160*(352716*x^11 + 3527160*x^10 + 16166150*x^9 + 44767800*x^8 + 831 40200*x^7 + 108636528*x^6 + 101846745*x^5 + 68468400*x^4 + 32332300*x^3 + 10210200*x^2 + 1939938*x + 167960)/x^21
Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.70 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{22}} \, dx=-\frac {\frac {x^{11}}{10}+x^{10}+\frac {55\,x^9}{12}+\frac {165\,x^8}{13}+\frac {165\,x^7}{7}+\frac {154\,x^6}{5}+\frac {231\,x^5}{8}+\frac {330\,x^4}{17}+\frac {55\,x^3}{6}+\frac {55\,x^2}{19}+\frac {11\,x}{20}+\frac {1}{21}}{x^{21}} \] Input:
int(((x + 1)*(2*x + x^2 + 1)^5)/x^22,x)
Output:
-((11*x)/20 + (55*x^2)/19 + (55*x^3)/6 + (330*x^4)/17 + (231*x^5)/8 + (154 *x^6)/5 + (165*x^7)/7 + (165*x^8)/13 + (55*x^9)/12 + x^10 + x^11/10 + 1/21 )/x^21
Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.72 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{22}} \, dx=\frac {-352716 x^{11}-3527160 x^{10}-16166150 x^{9}-44767800 x^{8}-83140200 x^{7}-108636528 x^{6}-101846745 x^{5}-68468400 x^{4}-32332300 x^{3}-10210200 x^{2}-1939938 x -167960}{3527160 x^{21}} \] Input:
int((1+x)*(x^2+2*x+1)^5/x^22,x)
Output:
( - 352716*x**11 - 3527160*x**10 - 16166150*x**9 - 44767800*x**8 - 8314020 0*x**7 - 108636528*x**6 - 101846745*x**5 - 68468400*x**4 - 32332300*x**3 - 10210200*x**2 - 1939938*x - 167960)/(3527160*x**21)