Integrand size = 17, antiderivative size = 49 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{16}} \, dx=-\frac {(1+x)^{12}}{15 x^{15}}+\frac {(1+x)^{12}}{70 x^{14}}-\frac {(1+x)^{12}}{455 x^{13}}+\frac {(1+x)^{12}}{5460 x^{12}} \] Output:
-1/15*(1+x)^12/x^15+1/70*(1+x)^12/x^14-1/455*(1+x)^12/x^13+1/5460*(1+x)^12 /x^12
Time = 0.00 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.69 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{16}} \, dx=-\frac {1}{15 x^{15}}-\frac {11}{14 x^{14}}-\frac {55}{13 x^{13}}-\frac {55}{4 x^{12}}-\frac {30}{x^{11}}-\frac {231}{5 x^{10}}-\frac {154}{3 x^9}-\frac {165}{4 x^8}-\frac {165}{7 x^7}-\frac {55}{6 x^6}-\frac {11}{5 x^5}-\frac {1}{4 x^4} \] Input:
Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^16,x]
Output:
-1/15*1/x^15 - 11/(14*x^14) - 55/(13*x^13) - 55/(4*x^12) - 30/x^11 - 231/( 5*x^10) - 154/(3*x^9) - 165/(4*x^8) - 165/(7*x^7) - 55/(6*x^6) - 11/(5*x^5 ) - 1/(4*x^4)
Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.20, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {1184, 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.
\(\displaystyle \int \frac {(x+1) \left (x^2+2 x+1\right )^5}{x^{16}} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int \frac {(x+1)^{11}}{x^{16}}dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {1}{5} \int \frac {(x+1)^{11}}{x^{15}}dx-\frac {(x+1)^{12}}{15 x^{15}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{5} \left (\frac {1}{7} \int \frac {(x+1)^{11}}{x^{14}}dx+\frac {(x+1)^{12}}{14 x^{14}}\right )-\frac {(x+1)^{12}}{15 x^{15}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{5} \left (\frac {1}{7} \left (-\frac {1}{13} \int \frac {(x+1)^{11}}{x^{13}}dx-\frac {(x+1)^{12}}{13 x^{13}}\right )+\frac {(x+1)^{12}}{14 x^{14}}\right )-\frac {(x+1)^{12}}{15 x^{15}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {1}{5} \left (\frac {(x+1)^{12}}{14 x^{14}}+\frac {1}{7} \left (\frac {(x+1)^{12}}{156 x^{12}}-\frac {(x+1)^{12}}{13 x^{13}}\right )\right )-\frac {(x+1)^{12}}{15 x^{15}}\) |
Input:
Int[((1 + x)*(1 + 2*x + x^2)^5)/x^16,x]
Output:
-1/15*(1 + x)^12/x^15 + ((1 + x)^12/(14*x^14) + (-1/13*(1 + x)^12/x^13 + ( 1 + x)^12/(156*x^12))/7)/5
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 = 0.75 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.22
method | result | size |
norman | \(\frac {-\frac {1}{15}-\frac {11}{14} x -\frac {55}{13} x^{2}-\frac {55}{4} x^{3}-30 x^{4}-\frac {231}{5} x^{5}-\frac {154}{3} x^{6}-\frac {165}{4} x^{7}-\frac {165}{7} x^{8}-\frac {55}{6} x^{9}-\frac {11}{5} x^{10}-\frac {1}{4} x^{11}}{x^{15}}\) | \(60\) |
risch | \(\frac {-\frac {1}{15}-\frac {11}{14} x -\frac {55}{13} x^{2}-\frac {55}{4} x^{3}-30 x^{4}-\frac {231}{5} x^{5}-\frac {154}{3} x^{6}-\frac {165}{4} x^{7}-\frac {165}{7} x^{8}-\frac {55}{6} x^{9}-\frac {11}{5} x^{10}-\frac {1}{4} x^{11}}{x^{15}}\) | \(60\) |
gosper | \(-\frac {1365 x^{11}+12012 x^{10}+50050 x^{9}+128700 x^{8}+225225 x^{7}+280280 x^{6}+252252 x^{5}+163800 x^{4}+75075 x^{3}+23100 x^{2}+4290 x +364}{5460 x^{15}}\) | \(61\) |
parallelrisch | \(\frac {-1365 x^{11}-12012 x^{10}-50050 x^{9}-128700 x^{8}-225225 x^{7}-280280 x^{6}-252252 x^{5}-163800 x^{4}-75075 x^{3}-23100 x^{2}-4290 x -364}{5460 x^{15}}\) | \(61\) |
default | \(-\frac {11}{5 x^{5}}-\frac {55}{6 x^{6}}-\frac {55}{13 x^{13}}-\frac {165}{4 x^{8}}-\frac {1}{4 x^{4}}-\frac {11}{14 x^{14}}-\frac {154}{3 x^{9}}-\frac {165}{7 x^{7}}-\frac {231}{5 x^{10}}-\frac {1}{15 x^{15}}-\frac {30}{x^{11}}-\frac {55}{4 x^{12}}\) | \(62\) |
orering | \(-\frac {\left (1365 x^{11}+12012 x^{10}+50050 x^{9}+128700 x^{8}+225225 x^{7}+280280 x^{6}+252252 x^{5}+163800 x^{4}+75075 x^{3}+23100 x^{2}+4290 x +364\right ) \left (x^{2}+2 x +1\right )^{5}}{5460 x^{15} \left (x +1\right )^{10}}\) | \(76\) |
Input:
int((x+1)*(x^2+2*x+1)^5/x^16,x,method=_RETURNVERBOSE)
Output:
(-1/15-11/14*x-55/13*x^2-55/4*x^3-30*x^4-231/5*x^5-154/3*x^6-165/4*x^7-165 /7*x^8-55/6*x^9-11/5*x^10-1/4*x^11)/x^15
Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.22 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{16}} \, dx=-\frac {1365 \, x^{11} + 12012 \, x^{10} + 50050 \, x^{9} + 128700 \, x^{8} + 225225 \, x^{7} + 280280 \, x^{6} + 252252 \, x^{5} + 163800 \, x^{4} + 75075 \, x^{3} + 23100 \, x^{2} + 4290 \, x + 364}{5460 \, x^{15}} \] Input:
integrate((1+x)*(x^2+2*x+1)^5/x^16,x, algorithm="fricas")
Output:
-1/5460*(1365*x^11 + 12012*x^10 + 50050*x^9 + 128700*x^8 + 225225*x^7 + 28 0280*x^6 + 252252*x^5 + 163800*x^4 + 75075*x^3 + 23100*x^2 + 4290*x + 364) /x^15
Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.24 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{16}} \, dx=\frac {- 1365 x^{11} - 12012 x^{10} - 50050 x^{9} - 128700 x^{8} - 225225 x^{7} - 280280 x^{6} - 252252 x^{5} - 163800 x^{4} - 75075 x^{3} - 23100 x^{2} - 4290 x - 364}{5460 x^{15}} \] Input:
integrate((1+x)*(x**2+2*x+1)**5/x**16,x)
Output:
(-1365*x**11 - 12012*x**10 - 50050*x**9 - 128700*x**8 - 225225*x**7 - 2802 80*x**6 - 252252*x**5 - 163800*x**4 - 75075*x**3 - 23100*x**2 - 4290*x - 3 64)/(5460*x**15)
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.22 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{16}} \, dx=-\frac {1365 \, x^{11} + 12012 \, x^{10} + 50050 \, x^{9} + 128700 \, x^{8} + 225225 \, x^{7} + 280280 \, x^{6} + 252252 \, x^{5} + 163800 \, x^{4} + 75075 \, x^{3} + 23100 \, x^{2} + 4290 \, x + 364}{5460 \, x^{15}} \] Input:
integrate((1+x)*(x^2+2*x+1)^5/x^16,x, algorithm="maxima")
Output:
-1/5460*(1365*x^11 + 12012*x^10 + 50050*x^9 + 128700*x^8 + 225225*x^7 + 28 0280*x^6 + 252252*x^5 + 163800*x^4 + 75075*x^3 + 23100*x^2 + 4290*x + 364) /x^15
Time = 0.18 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.22 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{16}} \, dx=-\frac {1365 \, x^{11} + 12012 \, x^{10} + 50050 \, x^{9} + 128700 \, x^{8} + 225225 \, x^{7} + 280280 \, x^{6} + 252252 \, x^{5} + 163800 \, x^{4} + 75075 \, x^{3} + 23100 \, x^{2} + 4290 \, x + 364}{5460 \, x^{15}} \] Input:
integrate((1+x)*(x^2+2*x+1)^5/x^16,x, algorithm="giac")
Output:
-1/5460*(1365*x^11 + 12012*x^10 + 50050*x^9 + 128700*x^8 + 225225*x^7 + 28 0280*x^6 + 252252*x^5 + 163800*x^4 + 75075*x^3 + 23100*x^2 + 4290*x + 364) /x^15
Time = 10.52 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.22 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{16}} \, dx=-\frac {\frac {x^{11}}{4}+\frac {11\,x^{10}}{5}+\frac {55\,x^9}{6}+\frac {165\,x^8}{7}+\frac {165\,x^7}{4}+\frac {154\,x^6}{3}+\frac {231\,x^5}{5}+30\,x^4+\frac {55\,x^3}{4}+\frac {55\,x^2}{13}+\frac {11\,x}{14}+\frac {1}{15}}{x^{15}} \] Input:
int(((x + 1)*(2*x + x^2 + 1)^5)/x^16,x)
Output:
-((11*x)/14 + (55*x^2)/13 + (55*x^3)/4 + 30*x^4 + (231*x^5)/5 + (154*x^6)/ 3 + (165*x^7)/4 + (165*x^8)/7 + (55*x^9)/6 + (11*x^10)/5 + x^11/4 + 1/15)/ x^15
Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.22 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{16}} \, dx=\frac {-1365 x^{11}-12012 x^{10}-50050 x^{9}-128700 x^{8}-225225 x^{7}-280280 x^{6}-252252 x^{5}-163800 x^{4}-75075 x^{3}-23100 x^{2}-4290 x -364}{5460 x^{15}} \] Input:
int((1+x)*(x^2+2*x+1)^5/x^16,x)
Output:
( - 1365*x**11 - 12012*x**10 - 50050*x**9 - 128700*x**8 - 225225*x**7 - 28 0280*x**6 - 252252*x**5 - 163800*x**4 - 75075*x**3 - 23100*x**2 - 4290*x - 364)/(5460*x**15)