Integrand size = 19, antiderivative size = 128 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{18}} \, dx=-\frac {d (1+x)^{11}}{17 x^{17}}+\frac {(6 d-17 e) (1+x)^{11}}{272 x^{16}}-\frac {(6 d-17 e) (1+x)^{11}}{816 x^{15}}+\frac {(6 d-17 e) (1+x)^{11}}{2856 x^{14}}-\frac {(6 d-17 e) (1+x)^{11}}{12376 x^{13}}+\frac {(6 d-17 e) (1+x)^{11}}{74256 x^{12}}-\frac {(6 d-17 e) (1+x)^{11}}{816816 x^{11}} \] Output:
-1/17*d*(1+x)^11/x^17+1/272*(6*d-17*e)*(1+x)^11/x^16-1/816*(6*d-17*e)*(1+x )^11/x^15+1/2856*(6*d-17*e)*(1+x)^11/x^14-1/12376*(6*d-17*e)*(1+x)^11/x^13 +1/74256*(6*d-17*e)*(1+x)^11/x^12-1/816816*(6*d-17*e)*(1+x)^11/x^11
Time = 0.05 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.18 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{18}} \, dx=-\frac {d}{17 x^{17}}-\frac {10 d+e}{16 x^{16}}-\frac {9 d+2 e}{3 x^{15}}-\frac {15 (8 d+3 e)}{14 x^{14}}-\frac {30 (7 d+4 e)}{13 x^{13}}-\frac {7 (6 d+5 e)}{2 x^{12}}-\frac {42 (5 d+6 e)}{11 x^{11}}-\frac {3 (4 d+7 e)}{x^{10}}-\frac {5 (3 d+8 e)}{3 x^9}-\frac {5 (2 d+9 e)}{8 x^8}-\frac {d+10 e}{7 x^7}-\frac {e}{6 x^6} \] Input:
Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^18,x]
Output:
-1/17*d/x^17 - (10*d + e)/(16*x^16) - (9*d + 2*e)/(3*x^15) - (15*(8*d + 3* e))/(14*x^14) - (30*(7*d + 4*e))/(13*x^13) - (7*(6*d + 5*e))/(2*x^12) - (4 2*(5*d + 6*e))/(11*x^11) - (3*(4*d + 7*e))/x^10 - (5*(3*d + 8*e))/(3*x^9) - (5*(2*d + 9*e))/(8*x^8) - (d + 10*e)/(7*x^7) - e/(6*x^6)
Time = 0.36 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.92, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {1184, 87, 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.
| \(\displaystyle \int \frac {\left (x^2+2 x+1\right )^5 (d+e x)}{x^{18}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \int \frac {(x+1)^{10} (d+e x)}{x^{18}}dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {1}{17} (6 d-17 e) \int \frac {(x+1)^{10}}{x^{17}}dx-\frac {d (x+1)^{11}}{17 x^{17}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {1}{17} (6 d-17 e) \left (-\frac {5}{16} \int \frac {(x+1)^{10}}{x^{16}}dx-\frac {(x+1)^{11}}{16 x^{16}}\right )-\frac {d (x+1)^{11}}{17 x^{17}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {1}{17} (6 d-17 e) \left (-\frac {5}{16} \left (-\frac {4}{15} \int \frac {(x+1)^{10}}{x^{15}}dx-\frac {(x+1)^{11}}{15 x^{15}}\right )-\frac {(x+1)^{11}}{16 x^{16}}\right )-\frac {d (x+1)^{11}}{17 x^{17}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {1}{17} (6 d-17 e) \left (-\frac {5}{16} \left (-\frac {4}{15} \left (-\frac {3}{14} \int \frac {(x+1)^{10}}{x^{14}}dx-\frac {(x+1)^{11}}{14 x^{14}}\right )-\frac {(x+1)^{11}}{15 x^{15}}\right )-\frac {(x+1)^{11}}{16 x^{16}}\right )-\frac {d (x+1)^{11}}{17 x^{17}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {1}{17} (6 d-17 e) \left (-\frac {5}{16} \left (-\frac {4}{15} \left (-\frac {3}{14} \left (-\frac {2}{13} \int \frac {(x+1)^{10}}{x^{13}}dx-\frac {(x+1)^{11}}{13 x^{13}}\right )-\frac {(x+1)^{11}}{14 x^{14}}\right )-\frac {(x+1)^{11}}{15 x^{15}}\right )-\frac {(x+1)^{11}}{16 x^{16}}\right )-\frac {d (x+1)^{11}}{17 x^{17}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {1}{17} (6 d-17 e) \left (-\frac {5}{16} \left (-\frac {4}{15} \left (-\frac {3}{14} \left (-\frac {2}{13} \left (-\frac {1}{12} \int \frac {(x+1)^{10}}{x^{12}}dx-\frac {(x+1)^{11}}{12 x^{12}}\right )-\frac {(x+1)^{11}}{13 x^{13}}\right )-\frac {(x+1)^{11}}{14 x^{14}}\right )-\frac {(x+1)^{11}}{15 x^{15}}\right )-\frac {(x+1)^{11}}{16 x^{16}}\right )-\frac {d (x+1)^{11}}{17 x^{17}}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {1}{17} \left (-\frac {(x+1)^{11}}{16 x^{16}}-\frac {5}{16} \left (-\frac {(x+1)^{11}}{15 x^{15}}-\frac {4}{15} \left (-\frac {(x+1)^{11}}{14 x^{14}}-\frac {3}{14} \left (-\frac {(x+1)^{11}}{13 x^{13}}-\frac {2}{13} \left (\frac {(x+1)^{11}}{132 x^{11}}-\frac {(x+1)^{11}}{12 x^{12}}\right )\right )\right )\right )\right ) (6 d-17 e)-\frac {d (x+1)^{11}}{17 x^{17}}\) |
Input:
Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^18,x]
Output:
-1/17*(d*(1 + x)^11)/x^17 - ((6*d - 17*e)*(-1/16*(1 + x)^11/x^16 - (5*(-1/ 15*(1 + x)^11/x^15 - (4*(-1/14*(1 + x)^11/x^14 - (3*(-1/13*(1 + x)^11/x^13 - (2*(-1/12*(1 + x)^11/x^12 + (1 + x)^11/(132*x^11)))/13))/14))/15))/16)) /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[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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 = 123, normalized size of antiderivative = 0.96
| method | result | size |
| norman | \(\frac {-\frac {d}{17}+\left (-\frac {5 d}{8}-\frac {e}{16}\right ) x +\left (-3 d -\frac {2 e}{3}\right ) x^{2}+\left (-\frac {60 d}{7}-\frac {45 e}{14}\right ) x^{3}+\left (-\frac {210 d}{13}-\frac {120 e}{13}\right ) x^{4}+\left (-21 d -\frac {35 e}{2}\right ) x^{5}+\left (-\frac {210 d}{11}-\frac {252 e}{11}\right ) x^{6}+\left (-12 d -21 e \right ) x^{7}+\left (-5 d -\frac {40 e}{3}\right ) x^{8}+\left (-\frac {5 d}{4}-\frac {45 e}{8}\right ) x^{9}+\left (-\frac {d}{7}-\frac {10 e}{7}\right ) x^{10}-\frac {e \,x^{11}}{6}}{x^{17}}\) | \(123\) |
| risch | \(\frac {-\frac {d}{17}+\left (-\frac {5 d}{8}-\frac {e}{16}\right ) x +\left (-3 d -\frac {2 e}{3}\right ) x^{2}+\left (-\frac {60 d}{7}-\frac {45 e}{14}\right ) x^{3}+\left (-\frac {210 d}{13}-\frac {120 e}{13}\right ) x^{4}+\left (-21 d -\frac {35 e}{2}\right ) x^{5}+\left (-\frac {210 d}{11}-\frac {252 e}{11}\right ) x^{6}+\left (-12 d -21 e \right ) x^{7}+\left (-5 d -\frac {40 e}{3}\right ) x^{8}+\left (-\frac {5 d}{4}-\frac {45 e}{8}\right ) x^{9}+\left (-\frac {d}{7}-\frac {10 e}{7}\right ) x^{10}-\frac {e \,x^{11}}{6}}{x^{17}}\) | \(123\) |
| default | \(-\frac {e}{6 x^{6}}-\frac {210 d +120 e}{13 x^{13}}-\frac {10 d +45 e}{8 x^{8}}-\frac {120 d +45 e}{14 x^{14}}-\frac {45 d +120 e}{9 x^{9}}-\frac {d +10 e}{7 x^{7}}-\frac {120 d +210 e}{10 x^{10}}-\frac {d}{17 x^{17}}-\frac {45 d +10 e}{15 x^{15}}-\frac {210 d +252 e}{11 x^{11}}-\frac {10 d +e}{16 x^{16}}-\frac {252 d +210 e}{12 x^{12}}\) | \(130\) |
| gosper | \(-\frac {136136 e \,x^{11}+116688 d \,x^{10}+1166880 e \,x^{10}+1021020 d \,x^{9}+4594590 e \,x^{9}+4084080 d \,x^{8}+10890880 e \,x^{8}+9801792 d \,x^{7}+17153136 e \,x^{7}+15593760 d \,x^{6}+18712512 e \,x^{6}+17153136 d \,x^{5}+14294280 x^{5} e +13194720 d \,x^{4}+7539840 x^{4} e +7001280 d \,x^{3}+2625480 x^{3} e +2450448 d \,x^{2}+544544 e \,x^{2}+510510 d x +51051 e x +48048 d}{816816 x^{17}}\) | \(132\) |
| parallelrisch | \(\frac {-136136 e \,x^{11}-116688 d \,x^{10}-1166880 e \,x^{10}-1021020 d \,x^{9}-4594590 e \,x^{9}-4084080 d \,x^{8}-10890880 e \,x^{8}-9801792 d \,x^{7}-17153136 e \,x^{7}-15593760 d \,x^{6}-18712512 e \,x^{6}-17153136 d \,x^{5}-14294280 x^{5} e -13194720 d \,x^{4}-7539840 x^{4} e -7001280 d \,x^{3}-2625480 x^{3} e -2450448 d \,x^{2}-544544 e \,x^{2}-510510 d x -51051 e x -48048 d}{816816 x^{17}}\) | \(132\) |
| orering | \(-\frac {\left (136136 e \,x^{11}+116688 d \,x^{10}+1166880 e \,x^{10}+1021020 d \,x^{9}+4594590 e \,x^{9}+4084080 d \,x^{8}+10890880 e \,x^{8}+9801792 d \,x^{7}+17153136 e \,x^{7}+15593760 d \,x^{6}+18712512 e \,x^{6}+17153136 d \,x^{5}+14294280 x^{5} e +13194720 d \,x^{4}+7539840 x^{4} e +7001280 d \,x^{3}+2625480 x^{3} e +2450448 d \,x^{2}+544544 e \,x^{2}+510510 d x +51051 e x +48048 d \right ) \left (x^{2}+2 x +1\right )^{5}}{816816 x^{17} \left (x +1\right )^{10}}\) | \(147\) |
Input:
int((e*x+d)*(x^2+2*x+1)^5/x^18,x,method=_RETURNVERBOSE)
Output:
(-1/17*d+(-5/8*d-1/16*e)*x+(-3*d-2/3*e)*x^2+(-60/7*d-45/14*e)*x^3+(-210/13 *d-120/13*e)*x^4+(-21*d-35/2*e)*x^5+(-210/11*d-252/11*e)*x^6+(-12*d-21*e)* x^7+(-5*d-40/3*e)*x^8+(-5/4*d-45/8*e)*x^9+(-1/7*d-10/7*e)*x^10-1/6*e*x^11) /x^17
Time = 0.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{18}} \, dx=-\frac {136136 \, e x^{11} + 116688 \, {\left (d + 10 \, e\right )} x^{10} + 510510 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 1361360 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 2450448 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 3118752 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 2858856 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 1884960 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 875160 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 272272 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 51051 \, {\left (10 \, d + e\right )} x + 48048 \, d}{816816 \, x^{17}} \] Input:
integrate((e*x+d)*(x^2+2*x+1)^5/x^18,x, algorithm="fricas")
Output:
-1/816816*(136136*e*x^11 + 116688*(d + 10*e)*x^10 + 510510*(2*d + 9*e)*x^9 + 1361360*(3*d + 8*e)*x^8 + 2450448*(4*d + 7*e)*x^7 + 3118752*(5*d + 6*e) *x^6 + 2858856*(6*d + 5*e)*x^5 + 1884960*(7*d + 4*e)*x^4 + 875160*(8*d + 3 *e)*x^3 + 272272*(9*d + 2*e)*x^2 + 51051*(10*d + e)*x + 48048*d)/x^17
Time = 14.07 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.02 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{18}} \, dx=\frac {- 48048 d - 136136 e x^{11} + x^{10} \left (- 116688 d - 1166880 e\right ) + x^{9} \left (- 1021020 d - 4594590 e\right ) + x^{8} \left (- 4084080 d - 10890880 e\right ) + x^{7} \left (- 9801792 d - 17153136 e\right ) + x^{6} \left (- 15593760 d - 18712512 e\right ) + x^{5} \left (- 17153136 d - 14294280 e\right ) + x^{4} \left (- 13194720 d - 7539840 e\right ) + x^{3} \left (- 7001280 d - 2625480 e\right ) + x^{2} \left (- 2450448 d - 544544 e\right ) + x \left (- 510510 d - 51051 e\right )}{816816 x^{17}} \] Input:
integrate((e*x+d)*(x**2+2*x+1)**5/x**18,x)
Output:
(-48048*d - 136136*e*x**11 + x**10*(-116688*d - 1166880*e) + x**9*(-102102 0*d - 4594590*e) + x**8*(-4084080*d - 10890880*e) + x**7*(-9801792*d - 171 53136*e) + x**6*(-15593760*d - 18712512*e) + x**5*(-17153136*d - 14294280* e) + x**4*(-13194720*d - 7539840*e) + x**3*(-7001280*d - 2625480*e) + x**2 *(-2450448*d - 544544*e) + x*(-510510*d - 51051*e))/(816816*x**17)
Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{18}} \, dx=-\frac {136136 \, e x^{11} + 116688 \, {\left (d + 10 \, e\right )} x^{10} + 510510 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 1361360 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 2450448 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 3118752 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 2858856 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 1884960 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 875160 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 272272 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 51051 \, {\left (10 \, d + e\right )} x + 48048 \, d}{816816 \, x^{17}} \] Input:
integrate((e*x+d)*(x^2+2*x+1)^5/x^18,x, algorithm="maxima")
Output:
-1/816816*(136136*e*x^11 + 116688*(d + 10*e)*x^10 + 510510*(2*d + 9*e)*x^9 + 1361360*(3*d + 8*e)*x^8 + 2450448*(4*d + 7*e)*x^7 + 3118752*(5*d + 6*e) *x^6 + 2858856*(6*d + 5*e)*x^5 + 1884960*(7*d + 4*e)*x^4 + 875160*(8*d + 3 *e)*x^3 + 272272*(9*d + 2*e)*x^2 + 51051*(10*d + e)*x + 48048*d)/x^17
Time = 0.16 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.02 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{18}} \, dx=-\frac {136136 \, e x^{11} + 116688 \, d x^{10} + 1166880 \, e x^{10} + 1021020 \, d x^{9} + 4594590 \, e x^{9} + 4084080 \, d x^{8} + 10890880 \, e x^{8} + 9801792 \, d x^{7} + 17153136 \, e x^{7} + 15593760 \, d x^{6} + 18712512 \, e x^{6} + 17153136 \, d x^{5} + 14294280 \, e x^{5} + 13194720 \, d x^{4} + 7539840 \, e x^{4} + 7001280 \, d x^{3} + 2625480 \, e x^{3} + 2450448 \, d x^{2} + 544544 \, e x^{2} + 510510 \, d x + 51051 \, e x + 48048 \, d}{816816 \, x^{17}} \] Input:
integrate((e*x+d)*(x^2+2*x+1)^5/x^18,x, algorithm="giac")
Output:
-1/816816*(136136*e*x^11 + 116688*d*x^10 + 1166880*e*x^10 + 1021020*d*x^9 + 4594590*e*x^9 + 4084080*d*x^8 + 10890880*e*x^8 + 9801792*d*x^7 + 1715313 6*e*x^7 + 15593760*d*x^6 + 18712512*e*x^6 + 17153136*d*x^5 + 14294280*e*x^ 5 + 13194720*d*x^4 + 7539840*e*x^4 + 7001280*d*x^3 + 2625480*e*x^3 + 24504 48*d*x^2 + 544544*e*x^2 + 510510*d*x + 51051*e*x + 48048*d)/x^17
Time = 0.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.96 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{18}} \, dx=-\frac {\frac {e\,x^{11}}{6}+\left (\frac {d}{7}+\frac {10\,e}{7}\right )\,x^{10}+\left (\frac {5\,d}{4}+\frac {45\,e}{8}\right )\,x^9+\left (5\,d+\frac {40\,e}{3}\right )\,x^8+\left (12\,d+21\,e\right )\,x^7+\left (\frac {210\,d}{11}+\frac {252\,e}{11}\right )\,x^6+\left (21\,d+\frac {35\,e}{2}\right )\,x^5+\left (\frac {210\,d}{13}+\frac {120\,e}{13}\right )\,x^4+\left (\frac {60\,d}{7}+\frac {45\,e}{14}\right )\,x^3+\left (3\,d+\frac {2\,e}{3}\right )\,x^2+\left (\frac {5\,d}{8}+\frac {e}{16}\right )\,x+\frac {d}{17}}{x^{17}} \] Input:
int(((d + e*x)*(2*x + x^2 + 1)^5)/x^18,x)
Output:
-(d/17 + x^2*(3*d + (2*e)/3) + x^10*(d/7 + (10*e)/7) + x^7*(12*d + 21*e) + x^8*(5*d + (40*e)/3) + x^5*(21*d + (35*e)/2) + x^9*((5*d)/4 + (45*e)/8) + x^3*((60*d)/7 + (45*e)/14) + x^4*((210*d)/13 + (120*e)/13) + x^6*((210*d) /11 + (252*e)/11) + (e*x^11)/6 + x*((5*d)/8 + e/16))/x^17
Time = 0.26 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.02 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{18}} \, dx=\frac {-136136 e \,x^{11}-116688 d \,x^{10}-1166880 e \,x^{10}-1021020 d \,x^{9}-4594590 e \,x^{9}-4084080 d \,x^{8}-10890880 e \,x^{8}-9801792 d \,x^{7}-17153136 e \,x^{7}-15593760 d \,x^{6}-18712512 e \,x^{6}-17153136 d \,x^{5}-14294280 e \,x^{5}-13194720 d \,x^{4}-7539840 e \,x^{4}-7001280 d \,x^{3}-2625480 e \,x^{3}-2450448 d \,x^{2}-544544 e \,x^{2}-510510 d x -51051 e x -48048 d}{816816 x^{17}} \] Input:
int((e*x+d)*(x^2+2*x+1)^5/x^18,x)
Output:
( - 116688*d*x**10 - 1021020*d*x**9 - 4084080*d*x**8 - 9801792*d*x**7 - 15 593760*d*x**6 - 17153136*d*x**5 - 13194720*d*x**4 - 7001280*d*x**3 - 24504 48*d*x**2 - 510510*d*x - 48048*d - 136136*e*x**11 - 1166880*e*x**10 - 4594 590*e*x**9 - 10890880*e*x**8 - 17153136*e*x**7 - 18712512*e*x**6 - 1429428 0*e*x**5 - 7539840*e*x**4 - 2625480*e*x**3 - 544544*e*x**2 - 51051*e*x)/(8 16816*x**17)