Integrand size = 17, antiderivative size = 73 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{18}} \, dx=-\frac {(1+x)^{12}}{17 x^{17}}+\frac {5 (1+x)^{12}}{272 x^{16}}-\frac {(1+x)^{12}}{204 x^{15}}+\frac {(1+x)^{12}}{952 x^{14}}-\frac {(1+x)^{12}}{6188 x^{13}}+\frac {(1+x)^{12}}{74256 x^{12}} \] Output:
-1/17*(1+x)^12/x^17+5/272*(1+x)^12/x^16-1/204*(1+x)^12/x^15+1/952*(1+x)^12 /x^14-1/6188*(1+x)^12/x^13+1/74256*(1+x)^12/x^12
Time = 0.00 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.11 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{18}} \, dx=-\frac {1}{17 x^{17}}-\frac {11}{16 x^{16}}-\frac {11}{3 x^{15}}-\frac {165}{14 x^{14}}-\frac {330}{13 x^{13}}-\frac {77}{2 x^{12}}-\frac {42}{x^{11}}-\frac {33}{x^{10}}-\frac {55}{3 x^9}-\frac {55}{8 x^8}-\frac {11}{7 x^7}-\frac {1}{6 x^6} \] Input:
Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^18,x]
Output:
-1/17*1/x^17 - 11/(16*x^16) - 11/(3*x^15) - 165/(14*x^14) - 330/(13*x^13) - 77/(2*x^12) - 42/x^11 - 33/x^10 - 55/(3*x^9) - 55/(8*x^8) - 11/(7*x^7) - 1/(6*x^6)
Time = 0.31 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.27, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {1184, 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.
\(\displaystyle \int \frac {(x+1) \left (x^2+2 x+1\right )^5}{x^{18}} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int \frac {(x+1)^{11}}{x^{18}}dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {5}{17} \int \frac {(x+1)^{11}}{x^{17}}dx-\frac {(x+1)^{12}}{17 x^{17}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {5}{17} \left (-\frac {1}{4} \int \frac {(x+1)^{11}}{x^{16}}dx-\frac {(x+1)^{12}}{16 x^{16}}\right )-\frac {(x+1)^{12}}{17 x^{17}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {5}{17} \left (\frac {1}{4} \left (\frac {1}{5} \int \frac {(x+1)^{11}}{x^{15}}dx+\frac {(x+1)^{12}}{15 x^{15}}\right )-\frac {(x+1)^{12}}{16 x^{16}}\right )-\frac {(x+1)^{12}}{17 x^{17}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {5}{17} \left (\frac {1}{4} \left (\frac {1}{5} \left (-\frac {1}{7} \int \frac {(x+1)^{11}}{x^{14}}dx-\frac {(x+1)^{12}}{14 x^{14}}\right )+\frac {(x+1)^{12}}{15 x^{15}}\right )-\frac {(x+1)^{12}}{16 x^{16}}\right )-\frac {(x+1)^{12}}{17 x^{17}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {5}{17} \left (\frac {1}{4} \left (\frac {1}{5} \left (\frac {1}{7} \left (\frac {1}{13} \int \frac {(x+1)^{11}}{x^{13}}dx+\frac {(x+1)^{12}}{13 x^{13}}\right )-\frac {(x+1)^{12}}{14 x^{14}}\right )+\frac {(x+1)^{12}}{15 x^{15}}\right )-\frac {(x+1)^{12}}{16 x^{16}}\right )-\frac {(x+1)^{12}}{17 x^{17}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {(x+1)^{12}}{17 x^{17}}-\frac {5}{17} \left (\frac {1}{4} \left (\frac {(x+1)^{12}}{15 x^{15}}+\frac {1}{5} \left (\frac {1}{7} \left (\frac {(x+1)^{12}}{13 x^{13}}-\frac {(x+1)^{12}}{156 x^{12}}\right )-\frac {(x+1)^{12}}{14 x^{14}}\right )\right )-\frac {(x+1)^{12}}{16 x^{16}}\right )\) |
Input:
Int[((1 + x)*(1 + 2*x + x^2)^5)/x^18,x]
Output:
-1/17*(1 + x)^12/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
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 = 0.83 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.82
method | result | size |
norman | \(\frac {-\frac {1}{17}-\frac {11}{16} x -\frac {11}{3} x^{2}-\frac {165}{14} x^{3}-\frac {330}{13} x^{4}-\frac {77}{2} x^{5}-42 x^{6}-33 x^{7}-\frac {55}{3} x^{8}-\frac {55}{8} x^{9}-\frac {11}{7} x^{10}-\frac {1}{6} x^{11}}{x^{17}}\) | \(60\) |
risch | \(\frac {-\frac {1}{17}-\frac {11}{16} x -\frac {11}{3} x^{2}-\frac {165}{14} x^{3}-\frac {330}{13} x^{4}-\frac {77}{2} x^{5}-42 x^{6}-33 x^{7}-\frac {55}{3} x^{8}-\frac {55}{8} x^{9}-\frac {11}{7} x^{10}-\frac {1}{6} x^{11}}{x^{17}}\) | \(60\) |
gosper | \(-\frac {12376 x^{11}+116688 x^{10}+510510 x^{9}+1361360 x^{8}+2450448 x^{7}+3118752 x^{6}+2858856 x^{5}+1884960 x^{4}+875160 x^{3}+272272 x^{2}+51051 x +4368}{74256 x^{17}}\) | \(61\) |
parallelrisch | \(\frac {-12376 x^{11}-116688 x^{10}-510510 x^{9}-1361360 x^{8}-2450448 x^{7}-3118752 x^{6}-2858856 x^{5}-1884960 x^{4}-875160 x^{3}-272272 x^{2}-51051 x -4368}{74256 x^{17}}\) | \(61\) |
default | \(-\frac {1}{6 x^{6}}-\frac {330}{13 x^{13}}-\frac {55}{8 x^{8}}-\frac {165}{14 x^{14}}-\frac {55}{3 x^{9}}-\frac {11}{7 x^{7}}-\frac {33}{x^{10}}-\frac {1}{17 x^{17}}-\frac {11}{3 x^{15}}-\frac {42}{x^{11}}-\frac {11}{16 x^{16}}-\frac {77}{2 x^{12}}\) | \(62\) |
orering | \(-\frac {\left (12376 x^{11}+116688 x^{10}+510510 x^{9}+1361360 x^{8}+2450448 x^{7}+3118752 x^{6}+2858856 x^{5}+1884960 x^{4}+875160 x^{3}+272272 x^{2}+51051 x +4368\right ) \left (x^{2}+2 x +1\right )^{5}}{74256 x^{17} \left (x +1\right )^{10}}\) | \(76\) |
Input:
int((x+1)*(x^2+2*x+1)^5/x^18,x,method=_RETURNVERBOSE)
Output:
(-1/17-11/16*x-11/3*x^2-165/14*x^3-330/13*x^4-77/2*x^5-42*x^6-33*x^7-55/3* x^8-55/8*x^9-11/7*x^10-1/6*x^11)/x^17
Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.82 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{18}} \, dx=-\frac {12376 \, x^{11} + 116688 \, x^{10} + 510510 \, x^{9} + 1361360 \, x^{8} + 2450448 \, x^{7} + 3118752 \, x^{6} + 2858856 \, x^{5} + 1884960 \, x^{4} + 875160 \, x^{3} + 272272 \, x^{2} + 51051 \, x + 4368}{74256 \, x^{17}} \] Input:
integrate((1+x)*(x^2+2*x+1)^5/x^18,x, algorithm="fricas")
Output:
-1/74256*(12376*x^11 + 116688*x^10 + 510510*x^9 + 1361360*x^8 + 2450448*x^ 7 + 3118752*x^6 + 2858856*x^5 + 1884960*x^4 + 875160*x^3 + 272272*x^2 + 51 051*x + 4368)/x^17
Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{18}} \, dx=\frac {- 12376 x^{11} - 116688 x^{10} - 510510 x^{9} - 1361360 x^{8} - 2450448 x^{7} - 3118752 x^{6} - 2858856 x^{5} - 1884960 x^{4} - 875160 x^{3} - 272272 x^{2} - 51051 x - 4368}{74256 x^{17}} \] Input:
integrate((1+x)*(x**2+2*x+1)**5/x**18,x)
Output:
(-12376*x**11 - 116688*x**10 - 510510*x**9 - 1361360*x**8 - 2450448*x**7 - 3118752*x**6 - 2858856*x**5 - 1884960*x**4 - 875160*x**3 - 272272*x**2 - 51051*x - 4368)/(74256*x**17)
Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.82 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{18}} \, dx=-\frac {12376 \, x^{11} + 116688 \, x^{10} + 510510 \, x^{9} + 1361360 \, x^{8} + 2450448 \, x^{7} + 3118752 \, x^{6} + 2858856 \, x^{5} + 1884960 \, x^{4} + 875160 \, x^{3} + 272272 \, x^{2} + 51051 \, x + 4368}{74256 \, x^{17}} \] Input:
integrate((1+x)*(x^2+2*x+1)^5/x^18,x, algorithm="maxima")
Output:
-1/74256*(12376*x^11 + 116688*x^10 + 510510*x^9 + 1361360*x^8 + 2450448*x^ 7 + 3118752*x^6 + 2858856*x^5 + 1884960*x^4 + 875160*x^3 + 272272*x^2 + 51 051*x + 4368)/x^17
Time = 0.17 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.82 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{18}} \, dx=-\frac {12376 \, x^{11} + 116688 \, x^{10} + 510510 \, x^{9} + 1361360 \, x^{8} + 2450448 \, x^{7} + 3118752 \, x^{6} + 2858856 \, x^{5} + 1884960 \, x^{4} + 875160 \, x^{3} + 272272 \, x^{2} + 51051 \, x + 4368}{74256 \, x^{17}} \] Input:
integrate((1+x)*(x^2+2*x+1)^5/x^18,x, algorithm="giac")
Output:
-1/74256*(12376*x^11 + 116688*x^10 + 510510*x^9 + 1361360*x^8 + 2450448*x^ 7 + 3118752*x^6 + 2858856*x^5 + 1884960*x^4 + 875160*x^3 + 272272*x^2 + 51 051*x + 4368)/x^17
Time = 10.67 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.82 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{18}} \, dx=-\frac {\frac {x^{11}}{6}+\frac {11\,x^{10}}{7}+\frac {55\,x^9}{8}+\frac {55\,x^8}{3}+33\,x^7+42\,x^6+\frac {77\,x^5}{2}+\frac {330\,x^4}{13}+\frac {165\,x^3}{14}+\frac {11\,x^2}{3}+\frac {11\,x}{16}+\frac {1}{17}}{x^{17}} \] Input:
int(((x + 1)*(2*x + x^2 + 1)^5)/x^18,x)
Output:
-((11*x)/16 + (11*x^2)/3 + (165*x^3)/14 + (330*x^4)/13 + (77*x^5)/2 + 42*x ^6 + 33*x^7 + (55*x^8)/3 + (55*x^9)/8 + (11*x^10)/7 + x^11/6 + 1/17)/x^17
Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.82 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{18}} \, dx=\frac {-12376 x^{11}-116688 x^{10}-510510 x^{9}-1361360 x^{8}-2450448 x^{7}-3118752 x^{6}-2858856 x^{5}-1884960 x^{4}-875160 x^{3}-272272 x^{2}-51051 x -4368}{74256 x^{17}} \] Input:
int((1+x)*(x^2+2*x+1)^5/x^18,x)
Output:
( - 12376*x**11 - 116688*x**10 - 510510*x**9 - 1361360*x**8 - 2450448*x**7 - 3118752*x**6 - 2858856*x**5 - 1884960*x**4 - 875160*x**3 - 272272*x**2 - 51051*x - 4368)/(74256*x**17)