Integrand size = 17, antiderivative size = 85 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{19}} \, dx=-\frac {(1+x)^{12}}{18 x^{18}}+\frac {(1+x)^{12}}{51 x^{17}}-\frac {5 (1+x)^{12}}{816 x^{16}}+\frac {(1+x)^{12}}{612 x^{15}}-\frac {(1+x)^{12}}{2856 x^{14}}+\frac {(1+x)^{12}}{18564 x^{13}}-\frac {(1+x)^{12}}{222768 x^{12}} \] Output:
-1/18*(1+x)^12/x^18+1/51*(1+x)^12/x^17-5/816*(1+x)^12/x^16+1/612*(1+x)^12/ x^15-1/2856*(1+x)^12/x^14+1/18564*(1+x)^12/x^13-1/222768*(1+x)^12/x^12
Time = 0.00 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.95 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{19}} \, dx=-\frac {1}{18 x^{18}}-\frac {11}{17 x^{17}}-\frac {55}{16 x^{16}}-\frac {11}{x^{15}}-\frac {165}{7 x^{14}}-\frac {462}{13 x^{13}}-\frac {77}{2 x^{12}}-\frac {30}{x^{11}}-\frac {33}{2 x^{10}}-\frac {55}{9 x^9}-\frac {11}{8 x^8}-\frac {1}{7 x^7} \] Input:
Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^19,x]
Output:
-1/18*1/x^18 - 11/(17*x^17) - 55/(16*x^16) - 11/x^15 - 165/(7*x^14) - 462/ (13*x^13) - 77/(2*x^12) - 30/x^11 - 33/(2*x^10) - 55/(9*x^9) - 11/(8*x^8) - 1/(7*x^7)
Time = 0.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.29, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {1184, 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.
\(\displaystyle \int \frac {(x+1) \left (x^2+2 x+1\right )^5}{x^{19}} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int \frac {(x+1)^{11}}{x^{19}}dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {1}{3} \int \frac {(x+1)^{11}}{x^{18}}dx-\frac {(x+1)^{12}}{18 x^{18}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{3} \left (\frac {5}{17} \int \frac {(x+1)^{11}}{x^{17}}dx+\frac {(x+1)^{12}}{17 x^{17}}\right )-\frac {(x+1)^{12}}{18 x^{18}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{3} \left (\frac {5}{17} \left (-\frac {1}{4} \int \frac {(x+1)^{11}}{x^{16}}dx-\frac {(x+1)^{12}}{16 x^{16}}\right )+\frac {(x+1)^{12}}{17 x^{17}}\right )-\frac {(x+1)^{12}}{18 x^{18}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{3} \left (\frac {5}{17} \left (\frac {1}{4} \left (\frac {1}{5} \int \frac {(x+1)^{11}}{x^{15}}dx+\frac {(x+1)^{12}}{15 x^{15}}\right )-\frac {(x+1)^{12}}{16 x^{16}}\right )+\frac {(x+1)^{12}}{17 x^{17}}\right )-\frac {(x+1)^{12}}{18 x^{18}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{3} \left (\frac {5}{17} \left (\frac {1}{4} \left (\frac {1}{5} \left (-\frac {1}{7} \int \frac {(x+1)^{11}}{x^{14}}dx-\frac {(x+1)^{12}}{14 x^{14}}\right )+\frac {(x+1)^{12}}{15 x^{15}}\right )-\frac {(x+1)^{12}}{16 x^{16}}\right )+\frac {(x+1)^{12}}{17 x^{17}}\right )-\frac {(x+1)^{12}}{18 x^{18}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {1}{3} \left (\frac {5}{17} \left (\frac {1}{4} \left (\frac {1}{5} \left (\frac {1}{7} \left (\frac {1}{13} \int \frac {(x+1)^{11}}{x^{13}}dx+\frac {(x+1)^{12}}{13 x^{13}}\right )-\frac {(x+1)^{12}}{14 x^{14}}\right )+\frac {(x+1)^{12}}{15 x^{15}}\right )-\frac {(x+1)^{12}}{16 x^{16}}\right )+\frac {(x+1)^{12}}{17 x^{17}}\right )-\frac {(x+1)^{12}}{18 x^{18}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {1}{3} \left (\frac {(x+1)^{12}}{17 x^{17}}+\frac {5}{17} \left (\frac {1}{4} \left (\frac {(x+1)^{12}}{15 x^{15}}+\frac {1}{5} \left (\frac {1}{7} \left (\frac {(x+1)^{12}}{13 x^{13}}-\frac {(x+1)^{12}}{156 x^{12}}\right )-\frac {(x+1)^{12}}{14 x^{14}}\right )\right )-\frac {(x+1)^{12}}{16 x^{16}}\right )\right )-\frac {(x+1)^{12}}{18 x^{18}}\) |
Input:
Int[((1 + x)*(1 + 2*x + x^2)^5)/x^19,x]
Output:
-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
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 = 0.76 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.71
method | result | size |
norman | \(\frac {-\frac {1}{18}-\frac {11}{17} x -\frac {55}{16} x^{2}-11 x^{3}-\frac {165}{7} x^{4}-\frac {462}{13} x^{5}-\frac {77}{2} x^{6}-30 x^{7}-\frac {33}{2} x^{8}-\frac {55}{9} x^{9}-\frac {11}{8} x^{10}-\frac {1}{7} x^{11}}{x^{18}}\) | \(60\) |
risch | \(\frac {-\frac {1}{18}-\frac {11}{17} x -\frac {55}{16} x^{2}-11 x^{3}-\frac {165}{7} x^{4}-\frac {462}{13} x^{5}-\frac {77}{2} x^{6}-30 x^{7}-\frac {33}{2} x^{8}-\frac {55}{9} x^{9}-\frac {11}{8} x^{10}-\frac {1}{7} x^{11}}{x^{18}}\) | \(60\) |
gosper | \(-\frac {31824 x^{11}+306306 x^{10}+1361360 x^{9}+3675672 x^{8}+6683040 x^{7}+8576568 x^{6}+7916832 x^{5}+5250960 x^{4}+2450448 x^{3}+765765 x^{2}+144144 x +12376}{222768 x^{18}}\) | \(61\) |
parallelrisch | \(\frac {-31824 x^{11}-306306 x^{10}-1361360 x^{9}-3675672 x^{8}-6683040 x^{7}-8576568 x^{6}-7916832 x^{5}-5250960 x^{4}-2450448 x^{3}-765765 x^{2}-144144 x -12376}{222768 x^{18}}\) | \(61\) |
default | \(-\frac {1}{18 x^{18}}-\frac {462}{13 x^{13}}-\frac {11}{8 x^{8}}-\frac {165}{7 x^{14}}-\frac {55}{9 x^{9}}-\frac {1}{7 x^{7}}-\frac {33}{2 x^{10}}-\frac {11}{17 x^{17}}-\frac {11}{x^{15}}-\frac {30}{x^{11}}-\frac {55}{16 x^{16}}-\frac {77}{2 x^{12}}\) | \(62\) |
orering | \(-\frac {\left (31824 x^{11}+306306 x^{10}+1361360 x^{9}+3675672 x^{8}+6683040 x^{7}+8576568 x^{6}+7916832 x^{5}+5250960 x^{4}+2450448 x^{3}+765765 x^{2}+144144 x +12376\right ) \left (x^{2}+2 x +1\right )^{5}}{222768 x^{18} \left (x +1\right )^{10}}\) | \(76\) |
Input:
int((x+1)*(x^2+2*x+1)^5/x^19,x,method=_RETURNVERBOSE)
Output:
(-1/18-11/17*x-55/16*x^2-11*x^3-165/7*x^4-462/13*x^5-77/2*x^6-30*x^7-33/2* x^8-55/9*x^9-11/8*x^10-1/7*x^11)/x^18
Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.71 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{19}} \, dx=-\frac {31824 \, x^{11} + 306306 \, x^{10} + 1361360 \, x^{9} + 3675672 \, x^{8} + 6683040 \, x^{7} + 8576568 \, x^{6} + 7916832 \, x^{5} + 5250960 \, x^{4} + 2450448 \, x^{3} + 765765 \, x^{2} + 144144 \, x + 12376}{222768 \, x^{18}} \] Input:
integrate((1+x)*(x^2+2*x+1)^5/x^19,x, algorithm="fricas")
Output:
-1/222768*(31824*x^11 + 306306*x^10 + 1361360*x^9 + 3675672*x^8 + 6683040* x^7 + 8576568*x^6 + 7916832*x^5 + 5250960*x^4 + 2450448*x^3 + 765765*x^2 + 144144*x + 12376)/x^18
Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.72 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{19}} \, dx=\frac {- 31824 x^{11} - 306306 x^{10} - 1361360 x^{9} - 3675672 x^{8} - 6683040 x^{7} - 8576568 x^{6} - 7916832 x^{5} - 5250960 x^{4} - 2450448 x^{3} - 765765 x^{2} - 144144 x - 12376}{222768 x^{18}} \] Input:
integrate((1+x)*(x**2+2*x+1)**5/x**19,x)
Output:
(-31824*x**11 - 306306*x**10 - 1361360*x**9 - 3675672*x**8 - 6683040*x**7 - 8576568*x**6 - 7916832*x**5 - 5250960*x**4 - 2450448*x**3 - 765765*x**2 - 144144*x - 12376)/(222768*x**18)
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.71 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{19}} \, dx=-\frac {31824 \, x^{11} + 306306 \, x^{10} + 1361360 \, x^{9} + 3675672 \, x^{8} + 6683040 \, x^{7} + 8576568 \, x^{6} + 7916832 \, x^{5} + 5250960 \, x^{4} + 2450448 \, x^{3} + 765765 \, x^{2} + 144144 \, x + 12376}{222768 \, x^{18}} \] Input:
integrate((1+x)*(x^2+2*x+1)^5/x^19,x, algorithm="maxima")
Output:
-1/222768*(31824*x^11 + 306306*x^10 + 1361360*x^9 + 3675672*x^8 + 6683040* x^7 + 8576568*x^6 + 7916832*x^5 + 5250960*x^4 + 2450448*x^3 + 765765*x^2 + 144144*x + 12376)/x^18
Time = 0.19 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.71 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{19}} \, dx=-\frac {31824 \, x^{11} + 306306 \, x^{10} + 1361360 \, x^{9} + 3675672 \, x^{8} + 6683040 \, x^{7} + 8576568 \, x^{6} + 7916832 \, x^{5} + 5250960 \, x^{4} + 2450448 \, x^{3} + 765765 \, x^{2} + 144144 \, x + 12376}{222768 \, x^{18}} \] Input:
integrate((1+x)*(x^2+2*x+1)^5/x^19,x, algorithm="giac")
Output:
-1/222768*(31824*x^11 + 306306*x^10 + 1361360*x^9 + 3675672*x^8 + 6683040* x^7 + 8576568*x^6 + 7916832*x^5 + 5250960*x^4 + 2450448*x^3 + 765765*x^2 + 144144*x + 12376)/x^18
Time = 10.64 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.71 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{19}} \, dx=-\frac {\frac {x^{11}}{7}+\frac {11\,x^{10}}{8}+\frac {55\,x^9}{9}+\frac {33\,x^8}{2}+30\,x^7+\frac {77\,x^6}{2}+\frac {462\,x^5}{13}+\frac {165\,x^4}{7}+11\,x^3+\frac {55\,x^2}{16}+\frac {11\,x}{17}+\frac {1}{18}}{x^{18}} \] Input:
int(((x + 1)*(2*x + x^2 + 1)^5)/x^19,x)
Output:
-((11*x)/17 + (55*x^2)/16 + 11*x^3 + (165*x^4)/7 + (462*x^5)/13 + (77*x^6) /2 + 30*x^7 + (33*x^8)/2 + (55*x^9)/9 + (11*x^10)/8 + x^11/7 + 1/18)/x^18
Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.71 \[ \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{19}} \, dx=\frac {-31824 x^{11}-306306 x^{10}-1361360 x^{9}-3675672 x^{8}-6683040 x^{7}-8576568 x^{6}-7916832 x^{5}-5250960 x^{4}-2450448 x^{3}-765765 x^{2}-144144 x -12376}{222768 x^{18}} \] Input:
int((1+x)*(x^2+2*x+1)^5/x^19,x)
Output:
( - 31824*x**11 - 306306*x**10 - 1361360*x**9 - 3675672*x**8 - 6683040*x** 7 - 8576568*x**6 - 7916832*x**5 - 5250960*x**4 - 2450448*x**3 - 765765*x** 2 - 144144*x - 12376)/(222768*x**18)