Integrand size = 17, antiderivative size = 61 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{17}} \, dx=-\frac {(1+x)^{12}}{16 x^{16}}+\frac {(1+x)^{12}}{60 x^{15}}-\frac {(1+x)^{12}}{280 x^{14}}+\frac {(1+x)^{12}}{1820 x^{13}}-\frac {(1+x)^{12}}{21840 x^{12}} \] Output:
-1/16*(1+x)^12/x^16+1/60*(1+x)^12/x^15-1/280*(1+x)^12/x^14+1/1820*(1+x)^12 /x^13-1/21840*(1+x)^12/x^12
Time = 0.00 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.36 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{17}} \, dx=-\frac {1}{16 x^{16}}-\frac {11}{15 x^{15}}-\frac {55}{14 x^{14}}-\frac {165}{13 x^{13}}-\frac {55}{2 x^{12}}-\frac {42}{x^{11}}-\frac {231}{5 x^{10}}-\frac {110}{3 x^9}-\frac {165}{8 x^8}-\frac {55}{7 x^7}-\frac {11}{6 x^6}-\frac {1}{5 x^5} \] Input:
Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^17,x]
Output:
-1/16*1/x^16 - 11/(15*x^15) - 55/(14*x^14) - 165/(13*x^13) - 55/(2*x^12) - 42/x^11 - 231/(5*x^10) - 110/(3*x^9) - 165/(8*x^8) - 55/(7*x^7) - 11/(6*x ^6) - 1/(5*x^5)
Time = 0.30 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.25, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {1184, 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.
\(\displaystyle \int \frac {(x+1) \left (x^2+2 x+1\right )^5}{x^{17}} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int \frac {(x+1)^{11}}{x^{17}}dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {1}{4} \int \frac {(x+1)^{11}}{x^{16}}dx-\frac {(x+1)^{12}}{16 x^{16}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{5} \int \frac {(x+1)^{11}}{x^{15}}dx+\frac {(x+1)^{12}}{15 x^{15}}\right )-\frac {(x+1)^{12}}{16 x^{16}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{5} \left (-\frac {1}{7} \int \frac {(x+1)^{11}}{x^{14}}dx-\frac {(x+1)^{12}}{14 x^{14}}\right )+\frac {(x+1)^{12}}{15 x^{15}}\right )-\frac {(x+1)^{12}}{16 x^{16}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{5} \left (\frac {1}{7} \left (\frac {1}{13} \int \frac {(x+1)^{11}}{x^{13}}dx+\frac {(x+1)^{12}}{13 x^{13}}\right )-\frac {(x+1)^{12}}{14 x^{14}}\right )+\frac {(x+1)^{12}}{15 x^{15}}\right )-\frac {(x+1)^{12}}{16 x^{16}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {1}{4} \left (\frac {(x+1)^{12}}{15 x^{15}}+\frac {1}{5} \left (\frac {1}{7} \left (\frac {(x+1)^{12}}{13 x^{13}}-\frac {(x+1)^{12}}{156 x^{12}}\right )-\frac {(x+1)^{12}}{14 x^{14}}\right )\right )-\frac {(x+1)^{12}}{16 x^{16}}\) |
Input:
Int[((1 + x)*(1 + 2*x + x^2)^5)/x^17,x]
Output:
-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
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 = 0.81 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.98
method | result | size |
norman | \(\frac {-\frac {1}{16}-\frac {11}{15} x -\frac {55}{14} x^{2}-\frac {165}{13} x^{3}-\frac {55}{2} x^{4}-42 x^{5}-\frac {231}{5} x^{6}-\frac {110}{3} x^{7}-\frac {165}{8} x^{8}-\frac {55}{7} x^{9}-\frac {11}{6} x^{10}-\frac {1}{5} x^{11}}{x^{16}}\) | \(60\) |
risch | \(\frac {-\frac {1}{16}-\frac {11}{15} x -\frac {55}{14} x^{2}-\frac {165}{13} x^{3}-\frac {55}{2} x^{4}-42 x^{5}-\frac {231}{5} x^{6}-\frac {110}{3} x^{7}-\frac {165}{8} x^{8}-\frac {55}{7} x^{9}-\frac {11}{6} x^{10}-\frac {1}{5} x^{11}}{x^{16}}\) | \(60\) |
gosper | \(-\frac {4368 x^{11}+40040 x^{10}+171600 x^{9}+450450 x^{8}+800800 x^{7}+1009008 x^{6}+917280 x^{5}+600600 x^{4}+277200 x^{3}+85800 x^{2}+16016 x +1365}{21840 x^{16}}\) | \(61\) |
parallelrisch | \(\frac {-4368 x^{11}-40040 x^{10}-171600 x^{9}-450450 x^{8}-800800 x^{7}-1009008 x^{6}-917280 x^{5}-600600 x^{4}-277200 x^{3}-85800 x^{2}-16016 x -1365}{21840 x^{16}}\) | \(61\) |
default | \(-\frac {1}{5 x^{5}}-\frac {11}{6 x^{6}}-\frac {165}{13 x^{13}}-\frac {165}{8 x^{8}}-\frac {55}{14 x^{14}}-\frac {110}{3 x^{9}}-\frac {55}{7 x^{7}}-\frac {231}{5 x^{10}}-\frac {11}{15 x^{15}}-\frac {42}{x^{11}}-\frac {1}{16 x^{16}}-\frac {55}{2 x^{12}}\) | \(62\) |
orering | \(-\frac {\left (4368 x^{11}+40040 x^{10}+171600 x^{9}+450450 x^{8}+800800 x^{7}+1009008 x^{6}+917280 x^{5}+600600 x^{4}+277200 x^{3}+85800 x^{2}+16016 x +1365\right ) \left (x^{2}+2 x +1\right )^{5}}{21840 x^{16} \left (x +1\right )^{10}}\) | \(76\) |
Input:
int((x+1)*(x^2+2*x+1)^5/x^17,x,method=_RETURNVERBOSE)
Output:
(-1/16-11/15*x-55/14*x^2-165/13*x^3-55/2*x^4-42*x^5-231/5*x^6-110/3*x^7-16 5/8*x^8-55/7*x^9-11/6*x^10-1/5*x^11)/x^16
Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.98 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{17}} \, dx=-\frac {4368 \, x^{11} + 40040 \, x^{10} + 171600 \, x^{9} + 450450 \, x^{8} + 800800 \, x^{7} + 1009008 \, x^{6} + 917280 \, x^{5} + 600600 \, x^{4} + 277200 \, x^{3} + 85800 \, x^{2} + 16016 \, x + 1365}{21840 \, x^{16}} \] Input:
integrate((1+x)*(x^2+2*x+1)^5/x^17,x, algorithm="fricas")
Output:
-1/21840*(4368*x^11 + 40040*x^10 + 171600*x^9 + 450450*x^8 + 800800*x^7 + 1009008*x^6 + 917280*x^5 + 600600*x^4 + 277200*x^3 + 85800*x^2 + 16016*x + 1365)/x^16
Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{17}} \, dx=\frac {- 4368 x^{11} - 40040 x^{10} - 171600 x^{9} - 450450 x^{8} - 800800 x^{7} - 1009008 x^{6} - 917280 x^{5} - 600600 x^{4} - 277200 x^{3} - 85800 x^{2} - 16016 x - 1365}{21840 x^{16}} \] Input:
integrate((1+x)*(x**2+2*x+1)**5/x**17,x)
Output:
(-4368*x**11 - 40040*x**10 - 171600*x**9 - 450450*x**8 - 800800*x**7 - 100 9008*x**6 - 917280*x**5 - 600600*x**4 - 277200*x**3 - 85800*x**2 - 16016*x - 1365)/(21840*x**16)
Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.98 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{17}} \, dx=-\frac {4368 \, x^{11} + 40040 \, x^{10} + 171600 \, x^{9} + 450450 \, x^{8} + 800800 \, x^{7} + 1009008 \, x^{6} + 917280 \, x^{5} + 600600 \, x^{4} + 277200 \, x^{3} + 85800 \, x^{2} + 16016 \, x + 1365}{21840 \, x^{16}} \] Input:
integrate((1+x)*(x^2+2*x+1)^5/x^17,x, algorithm="maxima")
Output:
-1/21840*(4368*x^11 + 40040*x^10 + 171600*x^9 + 450450*x^8 + 800800*x^7 + 1009008*x^6 + 917280*x^5 + 600600*x^4 + 277200*x^3 + 85800*x^2 + 16016*x + 1365)/x^16
Time = 0.22 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.98 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{17}} \, dx=-\frac {4368 \, x^{11} + 40040 \, x^{10} + 171600 \, x^{9} + 450450 \, x^{8} + 800800 \, x^{7} + 1009008 \, x^{6} + 917280 \, x^{5} + 600600 \, x^{4} + 277200 \, x^{3} + 85800 \, x^{2} + 16016 \, x + 1365}{21840 \, x^{16}} \] Input:
integrate((1+x)*(x^2+2*x+1)^5/x^17,x, algorithm="giac")
Output:
-1/21840*(4368*x^11 + 40040*x^10 + 171600*x^9 + 450450*x^8 + 800800*x^7 + 1009008*x^6 + 917280*x^5 + 600600*x^4 + 277200*x^3 + 85800*x^2 + 16016*x + 1365)/x^16
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.98 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{17}} \, dx=-\frac {\frac {x^{11}}{5}+\frac {11\,x^{10}}{6}+\frac {55\,x^9}{7}+\frac {165\,x^8}{8}+\frac {110\,x^7}{3}+\frac {231\,x^6}{5}+42\,x^5+\frac {55\,x^4}{2}+\frac {165\,x^3}{13}+\frac {55\,x^2}{14}+\frac {11\,x}{15}+\frac {1}{16}}{x^{16}} \] Input:
int(((x + 1)*(2*x + x^2 + 1)^5)/x^17,x)
Output:
-((11*x)/15 + (55*x^2)/14 + (165*x^3)/13 + (55*x^4)/2 + 42*x^5 + (231*x^6) /5 + (110*x^7)/3 + (165*x^8)/8 + (55*x^9)/7 + (11*x^10)/6 + x^11/5 + 1/16) /x^16
Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.98 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{17}} \, dx=\frac {-4368 x^{11}-40040 x^{10}-171600 x^{9}-450450 x^{8}-800800 x^{7}-1009008 x^{6}-917280 x^{5}-600600 x^{4}-277200 x^{3}-85800 x^{2}-16016 x -1365}{21840 x^{16}} \] Input:
int((1+x)*(x^2+2*x+1)^5/x^17,x)
Output:
( - 4368*x**11 - 40040*x**10 - 171600*x**9 - 450450*x**8 - 800800*x**7 - 1 009008*x**6 - 917280*x**5 - 600600*x**4 - 277200*x**3 - 85800*x**2 - 16016 *x - 1365)/(21840*x**16)