Integrand size = 17, antiderivative size = 97 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{20}} \, dx=-\frac {(1+x)^{12}}{19 x^{19}}+\frac {7 (1+x)^{12}}{342 x^{18}}-\frac {7 (1+x)^{12}}{969 x^{17}}+\frac {35 (1+x)^{12}}{15504 x^{16}}-\frac {7 (1+x)^{12}}{11628 x^{15}}+\frac {(1+x)^{12}}{7752 x^{14}}-\frac {(1+x)^{12}}{50388 x^{13}}+\frac {(1+x)^{12}}{604656 x^{12}} \]
-1/19*(1+x)^12/x^19+7/342*(1+x)^12/x^18-7/969*(1+x)^12/x^17+35/15504*(1+x) ^12/x^16-7/11628*(1+x)^12/x^15+1/7752*(1+x)^12/x^14-1/50388*(1+x)^12/x^13+ 1/604656*(1+x)^12/x^12
Time = 0.00 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.81 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{20}} \, dx=-\frac {1}{19 x^{19}}-\frac {11}{18 x^{18}}-\frac {55}{17 x^{17}}-\frac {165}{16 x^{16}}-\frac {22}{x^{15}}-\frac {33}{x^{14}}-\frac {462}{13 x^{13}}-\frac {55}{2 x^{12}}-\frac {15}{x^{11}}-\frac {11}{2 x^{10}}-\frac {11}{9 x^9}-\frac {1}{8 x^8} \]
-1/19*1/x^19 - 11/(18*x^18) - 55/(17*x^17) - 165/(16*x^16) - 22/x^15 - 33/ x^14 - 462/(13*x^13) - 55/(2*x^12) - 15/x^11 - 11/(2*x^10) - 11/(9*x^9) - 1/(8*x^8)
Time = 0.21 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.31, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {1184, 55, 55, 55, 55, 55, 55, 55, 48}
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.
\(\displaystyle \int \frac {(x+1) \left (x^2+2 x+1\right )^5}{x^{20}} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int \frac {(x+1)^{11}}{x^{20}}dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {7}{19} \int \frac {(x+1)^{11}}{x^{19}}dx-\frac {(x+1)^{12}}{19 x^{19}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {7}{19} \left (-\frac {1}{3} \int \frac {(x+1)^{11}}{x^{18}}dx-\frac {(x+1)^{12}}{18 x^{18}}\right )-\frac {(x+1)^{12}}{19 x^{19}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {7}{19} \left (\frac {1}{3} \left (\frac {5}{17} \int \frac {(x+1)^{11}}{x^{17}}dx+\frac {(x+1)^{12}}{17 x^{17}}\right )-\frac {(x+1)^{12}}{18 x^{18}}\right )-\frac {(x+1)^{12}}{19 x^{19}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {7}{19} \left (\frac {1}{3} \left (\frac {5}{17} \left (-\frac {1}{4} \int \frac {(x+1)^{11}}{x^{16}}dx-\frac {(x+1)^{12}}{16 x^{16}}\right )+\frac {(x+1)^{12}}{17 x^{17}}\right )-\frac {(x+1)^{12}}{18 x^{18}}\right )-\frac {(x+1)^{12}}{19 x^{19}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {7}{19} \left (\frac {1}{3} \left (\frac {5}{17} \left (\frac {1}{4} \left (\frac {1}{5} \int \frac {(x+1)^{11}}{x^{15}}dx+\frac {(x+1)^{12}}{15 x^{15}}\right )-\frac {(x+1)^{12}}{16 x^{16}}\right )+\frac {(x+1)^{12}}{17 x^{17}}\right )-\frac {(x+1)^{12}}{18 x^{18}}\right )-\frac {(x+1)^{12}}{19 x^{19}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {7}{19} \left (\frac {1}{3} \left (\frac {5}{17} \left (\frac {1}{4} \left (\frac {1}{5} \left (-\frac {1}{7} \int \frac {(x+1)^{11}}{x^{14}}dx-\frac {(x+1)^{12}}{14 x^{14}}\right )+\frac {(x+1)^{12}}{15 x^{15}}\right )-\frac {(x+1)^{12}}{16 x^{16}}\right )+\frac {(x+1)^{12}}{17 x^{17}}\right )-\frac {(x+1)^{12}}{18 x^{18}}\right )-\frac {(x+1)^{12}}{19 x^{19}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {7}{19} \left (\frac {1}{3} \left (\frac {5}{17} \left (\frac {1}{4} \left (\frac {1}{5} \left (\frac {1}{7} \left (\frac {1}{13} \int \frac {(x+1)^{11}}{x^{13}}dx+\frac {(x+1)^{12}}{13 x^{13}}\right )-\frac {(x+1)^{12}}{14 x^{14}}\right )+\frac {(x+1)^{12}}{15 x^{15}}\right )-\frac {(x+1)^{12}}{16 x^{16}}\right )+\frac {(x+1)^{12}}{17 x^{17}}\right )-\frac {(x+1)^{12}}{18 x^{18}}\right )-\frac {(x+1)^{12}}{19 x^{19}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {(x+1)^{12}}{19 x^{19}}-\frac {7}{19} \left (\frac {1}{3} \left (\frac {(x+1)^{12}}{17 x^{17}}+\frac {5}{17} \left (\frac {1}{4} \left (\frac {(x+1)^{12}}{15 x^{15}}+\frac {1}{5} \left (\frac {1}{7} \left (\frac {(x+1)^{12}}{13 x^{13}}-\frac {(x+1)^{12}}{156 x^{12}}\right )-\frac {(x+1)^{12}}{14 x^{14}}\right )\right )-\frac {(x+1)^{12}}{16 x^{16}}\right )\right )-\frac {(x+1)^{12}}{18 x^{18}}\right )\) |
-1/19*(1 + x)^12/x^19 - (7*(-1/18*(1 + x)^12/x^18 + ((1 + x)^12/(17*x^17) + (5*(-1/16*(1 + x)^12/x^16 + ((1 + x)^12/(15*x^15) + (-1/14*(1 + x)^12/x^ 14 + ((1 + x)^12/(13*x^13) - (1 + x)^12/(156*x^12))/7)/5)/4))/17)/3))/19
3.7.19.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
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 = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.62
method | result | size |
norman | \(\frac {-\frac {1}{19}-\frac {11}{18} x -\frac {55}{17} x^{2}-\frac {165}{16} x^{3}-22 x^{4}-33 x^{5}-\frac {462}{13} x^{6}-\frac {55}{2} x^{7}-15 x^{8}-\frac {11}{2} x^{9}-\frac {11}{9} x^{10}-\frac {1}{8} x^{11}}{x^{19}}\) | \(60\) |
risch | \(\frac {-\frac {1}{19}-\frac {11}{18} x -\frac {55}{17} x^{2}-\frac {165}{16} x^{3}-22 x^{4}-33 x^{5}-\frac {462}{13} x^{6}-\frac {55}{2} x^{7}-15 x^{8}-\frac {11}{2} x^{9}-\frac {11}{9} x^{10}-\frac {1}{8} x^{11}}{x^{19}}\) | \(60\) |
gosper | \(-\frac {75582 x^{11}+739024 x^{10}+3325608 x^{9}+9069840 x^{8}+16628040 x^{7}+21488544 x^{6}+19953648 x^{5}+13302432 x^{4}+6235515 x^{3}+1956240 x^{2}+369512 x +31824}{604656 x^{19}}\) | \(61\) |
parallelrisch | \(\frac {-75582 x^{11}-739024 x^{10}-3325608 x^{9}-9069840 x^{8}-16628040 x^{7}-21488544 x^{6}-19953648 x^{5}-13302432 x^{4}-6235515 x^{3}-1956240 x^{2}-369512 x -31824}{604656 x^{19}}\) | \(61\) |
default | \(-\frac {11}{18 x^{18}}-\frac {33}{x^{14}}-\frac {11}{2 x^{10}}-\frac {462}{13 x^{13}}-\frac {11}{9 x^{9}}-\frac {55}{17 x^{17}}-\frac {1}{19 x^{19}}-\frac {165}{16 x^{16}}-\frac {22}{x^{15}}-\frac {15}{x^{11}}-\frac {55}{2 x^{12}}-\frac {1}{8 x^{8}}\) | \(62\) |
(-1/19-11/18*x-55/17*x^2-165/16*x^3-22*x^4-33*x^5-462/13*x^6-55/2*x^7-15*x ^8-11/2*x^9-11/9*x^10-1/8*x^11)/x^19
Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.62 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{20}} \, dx=-\frac {75582 \, x^{11} + 739024 \, x^{10} + 3325608 \, x^{9} + 9069840 \, x^{8} + 16628040 \, x^{7} + 21488544 \, x^{6} + 19953648 \, x^{5} + 13302432 \, x^{4} + 6235515 \, x^{3} + 1956240 \, x^{2} + 369512 \, x + 31824}{604656 \, x^{19}} \]
-1/604656*(75582*x^11 + 739024*x^10 + 3325608*x^9 + 9069840*x^8 + 16628040 *x^7 + 21488544*x^6 + 19953648*x^5 + 13302432*x^4 + 6235515*x^3 + 1956240* x^2 + 369512*x + 31824)/x^19
Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.63 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{20}} \, dx=\frac {- 75582 x^{11} - 739024 x^{10} - 3325608 x^{9} - 9069840 x^{8} - 16628040 x^{7} - 21488544 x^{6} - 19953648 x^{5} - 13302432 x^{4} - 6235515 x^{3} - 1956240 x^{2} - 369512 x - 31824}{604656 x^{19}} \]
(-75582*x**11 - 739024*x**10 - 3325608*x**9 - 9069840*x**8 - 16628040*x**7 - 21488544*x**6 - 19953648*x**5 - 13302432*x**4 - 6235515*x**3 - 1956240* x**2 - 369512*x - 31824)/(604656*x**19)
Time = 0.18 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.62 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{20}} \, dx=-\frac {75582 \, x^{11} + 739024 \, x^{10} + 3325608 \, x^{9} + 9069840 \, x^{8} + 16628040 \, x^{7} + 21488544 \, x^{6} + 19953648 \, x^{5} + 13302432 \, x^{4} + 6235515 \, x^{3} + 1956240 \, x^{2} + 369512 \, x + 31824}{604656 \, x^{19}} \]
-1/604656*(75582*x^11 + 739024*x^10 + 3325608*x^9 + 9069840*x^8 + 16628040 *x^7 + 21488544*x^6 + 19953648*x^5 + 13302432*x^4 + 6235515*x^3 + 1956240* x^2 + 369512*x + 31824)/x^19
Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.62 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{20}} \, dx=-\frac {75582 \, x^{11} + 739024 \, x^{10} + 3325608 \, x^{9} + 9069840 \, x^{8} + 16628040 \, x^{7} + 21488544 \, x^{6} + 19953648 \, x^{5} + 13302432 \, x^{4} + 6235515 \, x^{3} + 1956240 \, x^{2} + 369512 \, x + 31824}{604656 \, x^{19}} \]
-1/604656*(75582*x^11 + 739024*x^10 + 3325608*x^9 + 9069840*x^8 + 16628040 *x^7 + 21488544*x^6 + 19953648*x^5 + 13302432*x^4 + 6235515*x^3 + 1956240* x^2 + 369512*x + 31824)/x^19
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.62 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{20}} \, dx=-\frac {\frac {x^{11}}{8}+\frac {11\,x^{10}}{9}+\frac {11\,x^9}{2}+15\,x^8+\frac {55\,x^7}{2}+\frac {462\,x^6}{13}+33\,x^5+22\,x^4+\frac {165\,x^3}{16}+\frac {55\,x^2}{17}+\frac {11\,x}{18}+\frac {1}{19}}{x^{19}} \]