Integrand size = 17, antiderivative size = 81 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=-\frac {1}{20 x^{20}}-\frac {11}{19 x^{19}}-\frac {55}{18 x^{18}}-\frac {165}{17 x^{17}}-\frac {165}{8 x^{16}}-\frac {154}{5 x^{15}}-\frac {33}{x^{14}}-\frac {330}{13 x^{13}}-\frac {55}{4 x^{12}}-\frac {5}{x^{11}}-\frac {11}{10 x^{10}}-\frac {1}{9 x^9} \] Output:
-1/20/x^20-11/19/x^19-55/18/x^18-165/17/x^17-165/8/x^16-154/5/x^15-33/x^14 -330/13/x^13-55/4/x^12-5/x^11-11/10/x^10-1/9/x^9
Time = 0.00 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=-\frac {1}{20 x^{20}}-\frac {11}{19 x^{19}}-\frac {55}{18 x^{18}}-\frac {165}{17 x^{17}}-\frac {165}{8 x^{16}}-\frac {154}{5 x^{15}}-\frac {33}{x^{14}}-\frac {330}{13 x^{13}}-\frac {55}{4 x^{12}}-\frac {5}{x^{11}}-\frac {11}{10 x^{10}}-\frac {1}{9 x^9} \] Input:
Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^21,x]
Output:
-1/20*1/x^20 - 11/(19*x^19) - 55/(18*x^18) - 165/(17*x^17) - 165/(8*x^16) - 154/(5*x^15) - 33/x^14 - 330/(13*x^13) - 55/(4*x^12) - 5/x^11 - 11/(10*x ^10) - 1/(9*x^9)
Time = 0.34 (sec) , antiderivative size = 81, 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.
\(\displaystyle \int \frac {(x+1) \left (x^2+2 x+1\right )^5}{x^{21}} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int \frac {(x+1)^{11}}{x^{21}}dx\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \int \left (\frac {1}{x^{21}}+\frac {11}{x^{20}}+\frac {55}{x^{19}}+\frac {165}{x^{18}}+\frac {330}{x^{17}}+\frac {462}{x^{16}}+\frac {462}{x^{15}}+\frac {330}{x^{14}}+\frac {165}{x^{13}}+\frac {55}{x^{12}}+\frac {11}{x^{11}}+\frac {1}{x^{10}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{20 x^{20}}-\frac {11}{19 x^{19}}-\frac {55}{18 x^{18}}-\frac {165}{17 x^{17}}-\frac {165}{8 x^{16}}-\frac {154}{5 x^{15}}-\frac {33}{x^{14}}-\frac {330}{13 x^{13}}-\frac {55}{4 x^{12}}-\frac {5}{x^{11}}-\frac {11}{10 x^{10}}-\frac {1}{9 x^9}\) |
Input:
Int[((1 + x)*(1 + 2*x + x^2)^5)/x^21,x]
Output:
-1/20*1/x^20 - 11/(19*x^19) - 55/(18*x^18) - 165/(17*x^17) - 165/(8*x^16) - 154/(5*x^15) - 33/x^14 - 330/(13*x^13) - 55/(4*x^12) - 5/x^11 - 11/(10*x ^10) - 1/(9*x^9)
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 = 0.86 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.74
method | result | size |
norman | \(\frac {-\frac {1}{20}-\frac {11}{19} x -\frac {55}{18} x^{2}-\frac {165}{17} x^{3}-\frac {165}{8} x^{4}-\frac {154}{5} x^{5}-33 x^{6}-\frac {330}{13} x^{7}-\frac {55}{4} x^{8}-5 x^{9}-\frac {11}{10} x^{10}-\frac {1}{9} x^{11}}{x^{20}}\) | \(60\) |
risch | \(\frac {-\frac {1}{20}-\frac {11}{19} x -\frac {55}{18} x^{2}-\frac {165}{17} x^{3}-\frac {165}{8} x^{4}-\frac {154}{5} x^{5}-33 x^{6}-\frac {330}{13} x^{7}-\frac {55}{4} x^{8}-5 x^{9}-\frac {11}{10} x^{10}-\frac {1}{9} x^{11}}{x^{20}}\) | \(60\) |
gosper | \(-\frac {167960 x^{11}+1662804 x^{10}+7558200 x^{9}+20785050 x^{8}+38372400 x^{7}+49884120 x^{6}+46558512 x^{5}+31177575 x^{4}+14671800 x^{3}+4618900 x^{2}+875160 x +75582}{1511640 x^{20}}\) | \(61\) |
parallelrisch | \(\frac {-167960 x^{11}-1662804 x^{10}-7558200 x^{9}-20785050 x^{8}-38372400 x^{7}-49884120 x^{6}-46558512 x^{5}-31177575 x^{4}-14671800 x^{3}-4618900 x^{2}-875160 x -75582}{1511640 x^{20}}\) | \(61\) |
default | \(-\frac {1}{20 x^{20}}-\frac {11}{19 x^{19}}-\frac {55}{18 x^{18}}-\frac {165}{17 x^{17}}-\frac {165}{8 x^{16}}-\frac {154}{5 x^{15}}-\frac {33}{x^{14}}-\frac {330}{13 x^{13}}-\frac {55}{4 x^{12}}-\frac {5}{x^{11}}-\frac {11}{10 x^{10}}-\frac {1}{9 x^{9}}\) | \(62\) |
orering | \(-\frac {\left (167960 x^{11}+1662804 x^{10}+7558200 x^{9}+20785050 x^{8}+38372400 x^{7}+49884120 x^{6}+46558512 x^{5}+31177575 x^{4}+14671800 x^{3}+4618900 x^{2}+875160 x +75582\right ) \left (x^{2}+2 x +1\right )^{5}}{1511640 x^{20} \left (x +1\right )^{10}}\) | \(76\) |
Input:
int((x+1)*(x^2+2*x+1)^5/x^21,x,method=_RETURNVERBOSE)
Output:
(-1/20-11/19*x-55/18*x^2-165/17*x^3-165/8*x^4-154/5*x^5-33*x^6-330/13*x^7- 55/4*x^8-5*x^9-11/10*x^10-1/9*x^11)/x^20
Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.74 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=-\frac {167960 \, x^{11} + 1662804 \, x^{10} + 7558200 \, x^{9} + 20785050 \, x^{8} + 38372400 \, x^{7} + 49884120 \, x^{6} + 46558512 \, x^{5} + 31177575 \, x^{4} + 14671800 \, x^{3} + 4618900 \, x^{2} + 875160 \, x + 75582}{1511640 \, x^{20}} \] Input:
integrate((1+x)*(x^2+2*x+1)^5/x^21,x, algorithm="fricas")
Output:
-1/1511640*(167960*x^11 + 1662804*x^10 + 7558200*x^9 + 20785050*x^8 + 3837 2400*x^7 + 49884120*x^6 + 46558512*x^5 + 31177575*x^4 + 14671800*x^3 + 461 8900*x^2 + 875160*x + 75582)/x^20
Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.75 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=\frac {- 167960 x^{11} - 1662804 x^{10} - 7558200 x^{9} - 20785050 x^{8} - 38372400 x^{7} - 49884120 x^{6} - 46558512 x^{5} - 31177575 x^{4} - 14671800 x^{3} - 4618900 x^{2} - 875160 x - 75582}{1511640 x^{20}} \] Input:
integrate((1+x)*(x**2+2*x+1)**5/x**21,x)
Output:
(-167960*x**11 - 1662804*x**10 - 7558200*x**9 - 20785050*x**8 - 38372400*x **7 - 49884120*x**6 - 46558512*x**5 - 31177575*x**4 - 14671800*x**3 - 4618 900*x**2 - 875160*x - 75582)/(1511640*x**20)
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.74 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=-\frac {167960 \, x^{11} + 1662804 \, x^{10} + 7558200 \, x^{9} + 20785050 \, x^{8} + 38372400 \, x^{7} + 49884120 \, x^{6} + 46558512 \, x^{5} + 31177575 \, x^{4} + 14671800 \, x^{3} + 4618900 \, x^{2} + 875160 \, x + 75582}{1511640 \, x^{20}} \] Input:
integrate((1+x)*(x^2+2*x+1)^5/x^21,x, algorithm="maxima")
Output:
-1/1511640*(167960*x^11 + 1662804*x^10 + 7558200*x^9 + 20785050*x^8 + 3837 2400*x^7 + 49884120*x^6 + 46558512*x^5 + 31177575*x^4 + 14671800*x^3 + 461 8900*x^2 + 875160*x + 75582)/x^20
Time = 0.20 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.74 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=-\frac {167960 \, x^{11} + 1662804 \, x^{10} + 7558200 \, x^{9} + 20785050 \, x^{8} + 38372400 \, x^{7} + 49884120 \, x^{6} + 46558512 \, x^{5} + 31177575 \, x^{4} + 14671800 \, x^{3} + 4618900 \, x^{2} + 875160 \, x + 75582}{1511640 \, x^{20}} \] Input:
integrate((1+x)*(x^2+2*x+1)^5/x^21,x, algorithm="giac")
Output:
-1/1511640*(167960*x^11 + 1662804*x^10 + 7558200*x^9 + 20785050*x^8 + 3837 2400*x^7 + 49884120*x^6 + 46558512*x^5 + 31177575*x^4 + 14671800*x^3 + 461 8900*x^2 + 875160*x + 75582)/x^20
Time = 10.70 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.74 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=-\frac {\frac {x^{11}}{9}+\frac {11\,x^{10}}{10}+5\,x^9+\frac {55\,x^8}{4}+\frac {330\,x^7}{13}+33\,x^6+\frac {154\,x^5}{5}+\frac {165\,x^4}{8}+\frac {165\,x^3}{17}+\frac {55\,x^2}{18}+\frac {11\,x}{19}+\frac {1}{20}}{x^{20}} \] Input:
int(((x + 1)*(2*x + x^2 + 1)^5)/x^21,x)
Output:
-((11*x)/19 + (55*x^2)/18 + (165*x^3)/17 + (165*x^4)/8 + (154*x^5)/5 + 33* x^6 + (330*x^7)/13 + (55*x^8)/4 + 5*x^9 + (11*x^10)/10 + x^11/9 + 1/20)/x^ 20
Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.74 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=\frac {-167960 x^{11}-1662804 x^{10}-7558200 x^{9}-20785050 x^{8}-38372400 x^{7}-49884120 x^{6}-46558512 x^{5}-31177575 x^{4}-14671800 x^{3}-4618900 x^{2}-875160 x -75582}{1511640 x^{20}} \] Input:
int((1+x)*(x^2+2*x+1)^5/x^21,x)
Output:
( - 167960*x**11 - 1662804*x**10 - 7558200*x**9 - 20785050*x**8 - 38372400 *x**7 - 49884120*x**6 - 46558512*x**5 - 31177575*x**4 - 14671800*x**3 - 46 18900*x**2 - 875160*x - 75582)/(1511640*x**20)