Integrand size = 19, antiderivative size = 149 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{20}} \, dx=-\frac {d}{19 x^{19}}-\frac {10 d+e}{18 x^{18}}-\frac {5 (9 d+2 e)}{17 x^{17}}-\frac {15 (8 d+3 e)}{16 x^{16}}-\frac {2 (7 d+4 e)}{x^{15}}-\frac {3 (6 d+5 e)}{x^{14}}-\frac {42 (5 d+6 e)}{13 x^{13}}-\frac {5 (4 d+7 e)}{2 x^{12}}-\frac {15 (3 d+8 e)}{11 x^{11}}-\frac {2 d+9 e}{2 x^{10}}-\frac {d+10 e}{9 x^9}-\frac {e}{8 x^8} \] Output:
-1/19*d/x^19-1/18*(10*d+e)/x^18-5/17*(9*d+2*e)/x^17-15/16*(8*d+3*e)/x^16-2 *(7*d+4*e)/x^15-3*(6*d+5*e)/x^14-42/13*(5*d+6*e)/x^13-5/2*(4*d+7*e)/x^12-1 5/11*(3*d+8*e)/x^11-1/2*(2*d+9*e)/x^10-1/9*(d+10*e)/x^9-1/8*e/x^8
Time = 0.05 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{20}} \, dx=-\frac {d}{19 x^{19}}-\frac {10 d+e}{18 x^{18}}-\frac {5 (9 d+2 e)}{17 x^{17}}-\frac {15 (8 d+3 e)}{16 x^{16}}-\frac {2 (7 d+4 e)}{x^{15}}-\frac {3 (6 d+5 e)}{x^{14}}-\frac {42 (5 d+6 e)}{13 x^{13}}-\frac {5 (4 d+7 e)}{2 x^{12}}-\frac {15 (3 d+8 e)}{11 x^{11}}-\frac {2 d+9 e}{2 x^{10}}-\frac {d+10 e}{9 x^9}-\frac {e}{8 x^8} \] Input:
Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^20,x]
Output:
-1/19*d/x^19 - (10*d + e)/(18*x^18) - (5*(9*d + 2*e))/(17*x^17) - (15*(8*d + 3*e))/(16*x^16) - (2*(7*d + 4*e))/x^15 - (3*(6*d + 5*e))/x^14 - (42*(5* d + 6*e))/(13*x^13) - (5*(4*d + 7*e))/(2*x^12) - (15*(3*d + 8*e))/(11*x^11 ) - (2*d + 9*e)/(2*x^10) - (d + 10*e)/(9*x^9) - e/(8*x^8)
Time = 0.51 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1184, 85, 2009}
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.
| \(\displaystyle \int \frac {\left (x^2+2 x+1\right )^5 (d+e x)}{x^{20}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \int \frac {(x+1)^{10} (d+e x)}{x^{20}}dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left (\frac {10 d+e}{x^{19}}+\frac {5 (9 d+2 e)}{x^{18}}+\frac {15 (8 d+3 e)}{x^{17}}+\frac {30 (7 d+4 e)}{x^{16}}+\frac {42 (6 d+5 e)}{x^{15}}+\frac {42 (5 d+6 e)}{x^{14}}+\frac {30 (4 d+7 e)}{x^{13}}+\frac {15 (3 d+8 e)}{x^{12}}+\frac {5 (2 d+9 e)}{x^{11}}+\frac {d+10 e}{x^{10}}+\frac {d}{x^{20}}+\frac {e}{x^9}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {10 d+e}{18 x^{18}}-\frac {5 (9 d+2 e)}{17 x^{17}}-\frac {15 (8 d+3 e)}{16 x^{16}}-\frac {2 (7 d+4 e)}{x^{15}}-\frac {3 (6 d+5 e)}{x^{14}}-\frac {42 (5 d+6 e)}{13 x^{13}}-\frac {5 (4 d+7 e)}{2 x^{12}}-\frac {15 (3 d+8 e)}{11 x^{11}}-\frac {2 d+9 e}{2 x^{10}}-\frac {d+10 e}{9 x^9}-\frac {d}{19 x^{19}}-\frac {e}{8 x^8}\) |
Input:
Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^20,x]
Output:
-1/19*d/x^19 - (10*d + e)/(18*x^18) - (5*(9*d + 2*e))/(17*x^17) - (15*(8*d + 3*e))/(16*x^16) - (2*(7*d + 4*e))/x^15 - (3*(6*d + 5*e))/x^14 - (42*(5* d + 6*e))/(13*x^13) - (5*(4*d + 7*e))/(2*x^12) - (15*(3*d + 8*e))/(11*x^11 ) - (2*d + 9*e)/(2*x^10) - (d + 10*e)/(9*x^9) - e/(8*x^8)
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a* f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 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 = 123, normalized size of antiderivative = 0.83
| method | result | size |
| norman | \(\frac {-\frac {d}{19}+\left (-\frac {5 d}{9}-\frac {e}{18}\right ) x +\left (-\frac {45 d}{17}-\frac {10 e}{17}\right ) x^{2}+\left (-\frac {15 d}{2}-\frac {45 e}{16}\right ) x^{3}+\left (-14 d -8 e \right ) x^{4}+\left (-18 d -15 e \right ) x^{5}+\left (-\frac {210 d}{13}-\frac {252 e}{13}\right ) x^{6}+\left (-10 d -\frac {35 e}{2}\right ) x^{7}+\left (-\frac {45 d}{11}-\frac {120 e}{11}\right ) x^{8}+\left (-d -\frac {9 e}{2}\right ) x^{9}+\left (-\frac {d}{9}-\frac {10 e}{9}\right ) x^{10}-\frac {e \,x^{11}}{8}}{x^{19}}\) | \(123\) |
| risch | \(\frac {-\frac {d}{19}+\left (-\frac {5 d}{9}-\frac {e}{18}\right ) x +\left (-\frac {45 d}{17}-\frac {10 e}{17}\right ) x^{2}+\left (-\frac {15 d}{2}-\frac {45 e}{16}\right ) x^{3}+\left (-14 d -8 e \right ) x^{4}+\left (-18 d -15 e \right ) x^{5}+\left (-\frac {210 d}{13}-\frac {252 e}{13}\right ) x^{6}+\left (-10 d -\frac {35 e}{2}\right ) x^{7}+\left (-\frac {45 d}{11}-\frac {120 e}{11}\right ) x^{8}+\left (-d -\frac {9 e}{2}\right ) x^{9}+\left (-\frac {d}{9}-\frac {10 e}{9}\right ) x^{10}-\frac {e \,x^{11}}{8}}{x^{19}}\) | \(123\) |
| default | \(-\frac {10 d +e}{18 x^{18}}-\frac {210 d +252 e}{13 x^{13}}-\frac {e}{8 x^{8}}-\frac {252 d +210 e}{14 x^{14}}-\frac {d +10 e}{9 x^{9}}-\frac {10 d +45 e}{10 x^{10}}-\frac {45 d +10 e}{17 x^{17}}-\frac {210 d +120 e}{15 x^{15}}-\frac {45 d +120 e}{11 x^{11}}-\frac {d}{19 x^{19}}-\frac {120 d +45 e}{16 x^{16}}-\frac {120 d +210 e}{12 x^{12}}\) | \(130\) |
| gosper | \(-\frac {831402 e \,x^{11}+739024 d \,x^{10}+7390240 e \,x^{10}+6651216 d \,x^{9}+29930472 e \,x^{9}+27209520 d \,x^{8}+72558720 e \,x^{8}+66512160 d \,x^{7}+116396280 e \,x^{7}+107442720 d \,x^{6}+128931264 e \,x^{6}+119721888 d \,x^{5}+99768240 x^{5} e +93117024 d \,x^{4}+53209728 x^{4} e +49884120 d \,x^{3}+18706545 x^{3} e +17606160 d \,x^{2}+3912480 e \,x^{2}+3695120 d x +369512 e x +350064 d}{6651216 x^{19}}\) | \(132\) |
| parallelrisch | \(\frac {-831402 e \,x^{11}-739024 d \,x^{10}-7390240 e \,x^{10}-6651216 d \,x^{9}-29930472 e \,x^{9}-27209520 d \,x^{8}-72558720 e \,x^{8}-66512160 d \,x^{7}-116396280 e \,x^{7}-107442720 d \,x^{6}-128931264 e \,x^{6}-119721888 d \,x^{5}-99768240 x^{5} e -93117024 d \,x^{4}-53209728 x^{4} e -49884120 d \,x^{3}-18706545 x^{3} e -17606160 d \,x^{2}-3912480 e \,x^{2}-3695120 d x -369512 e x -350064 d}{6651216 x^{19}}\) | \(132\) |
| orering | \(-\frac {\left (831402 e \,x^{11}+739024 d \,x^{10}+7390240 e \,x^{10}+6651216 d \,x^{9}+29930472 e \,x^{9}+27209520 d \,x^{8}+72558720 e \,x^{8}+66512160 d \,x^{7}+116396280 e \,x^{7}+107442720 d \,x^{6}+128931264 e \,x^{6}+119721888 d \,x^{5}+99768240 x^{5} e +93117024 d \,x^{4}+53209728 x^{4} e +49884120 d \,x^{3}+18706545 x^{3} e +17606160 d \,x^{2}+3912480 e \,x^{2}+3695120 d x +369512 e x +350064 d \right ) \left (x^{2}+2 x +1\right )^{5}}{6651216 x^{19} \left (x +1\right )^{10}}\) | \(147\) |
Input:
int((e*x+d)*(x^2+2*x+1)^5/x^20,x,method=_RETURNVERBOSE)
Output:
(-1/19*d+(-5/9*d-1/18*e)*x+(-45/17*d-10/17*e)*x^2+(-15/2*d-45/16*e)*x^3+(- 14*d-8*e)*x^4+(-18*d-15*e)*x^5+(-210/13*d-252/13*e)*x^6+(-10*d-35/2*e)*x^7 +(-45/11*d-120/11*e)*x^8+(-d-9/2*e)*x^9+(-1/9*d-10/9*e)*x^10-1/8*e*x^11)/x ^19
Time = 0.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{20}} \, dx=-\frac {831402 \, e x^{11} + 739024 \, {\left (d + 10 \, e\right )} x^{10} + 3325608 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 9069840 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 16628040 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 21488544 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 19953648 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 13302432 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 6235515 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 1956240 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 369512 \, {\left (10 \, d + e\right )} x + 350064 \, d}{6651216 \, x^{19}} \] Input:
integrate((e*x+d)*(x^2+2*x+1)^5/x^20,x, algorithm="fricas")
Output:
-1/6651216*(831402*e*x^11 + 739024*(d + 10*e)*x^10 + 3325608*(2*d + 9*e)*x ^9 + 9069840*(3*d + 8*e)*x^8 + 16628040*(4*d + 7*e)*x^7 + 21488544*(5*d + 6*e)*x^6 + 19953648*(6*d + 5*e)*x^5 + 13302432*(7*d + 4*e)*x^4 + 6235515*( 8*d + 3*e)*x^3 + 1956240*(9*d + 2*e)*x^2 + 369512*(10*d + e)*x + 350064*d) /x^19
Time = 17.48 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{20}} \, dx=\frac {- 350064 d - 831402 e x^{11} + x^{10} \left (- 739024 d - 7390240 e\right ) + x^{9} \left (- 6651216 d - 29930472 e\right ) + x^{8} \left (- 27209520 d - 72558720 e\right ) + x^{7} \left (- 66512160 d - 116396280 e\right ) + x^{6} \left (- 107442720 d - 128931264 e\right ) + x^{5} \left (- 119721888 d - 99768240 e\right ) + x^{4} \left (- 93117024 d - 53209728 e\right ) + x^{3} \left (- 49884120 d - 18706545 e\right ) + x^{2} \left (- 17606160 d - 3912480 e\right ) + x \left (- 3695120 d - 369512 e\right )}{6651216 x^{19}} \] Input:
integrate((e*x+d)*(x**2+2*x+1)**5/x**20,x)
Output:
(-350064*d - 831402*e*x**11 + x**10*(-739024*d - 7390240*e) + x**9*(-66512 16*d - 29930472*e) + x**8*(-27209520*d - 72558720*e) + x**7*(-66512160*d - 116396280*e) + x**6*(-107442720*d - 128931264*e) + x**5*(-119721888*d - 9 9768240*e) + x**4*(-93117024*d - 53209728*e) + x**3*(-49884120*d - 1870654 5*e) + x**2*(-17606160*d - 3912480*e) + x*(-3695120*d - 369512*e))/(665121 6*x**19)
Time = 0.04 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{20}} \, dx=-\frac {831402 \, e x^{11} + 739024 \, {\left (d + 10 \, e\right )} x^{10} + 3325608 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 9069840 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 16628040 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 21488544 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 19953648 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 13302432 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 6235515 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 1956240 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 369512 \, {\left (10 \, d + e\right )} x + 350064 \, d}{6651216 \, x^{19}} \] Input:
integrate((e*x+d)*(x^2+2*x+1)^5/x^20,x, algorithm="maxima")
Output:
-1/6651216*(831402*e*x^11 + 739024*(d + 10*e)*x^10 + 3325608*(2*d + 9*e)*x ^9 + 9069840*(3*d + 8*e)*x^8 + 16628040*(4*d + 7*e)*x^7 + 21488544*(5*d + 6*e)*x^6 + 19953648*(6*d + 5*e)*x^5 + 13302432*(7*d + 4*e)*x^4 + 6235515*( 8*d + 3*e)*x^3 + 1956240*(9*d + 2*e)*x^2 + 369512*(10*d + e)*x + 350064*d) /x^19
Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{20}} \, dx=-\frac {831402 \, e x^{11} + 739024 \, d x^{10} + 7390240 \, e x^{10} + 6651216 \, d x^{9} + 29930472 \, e x^{9} + 27209520 \, d x^{8} + 72558720 \, e x^{8} + 66512160 \, d x^{7} + 116396280 \, e x^{7} + 107442720 \, d x^{6} + 128931264 \, e x^{6} + 119721888 \, d x^{5} + 99768240 \, e x^{5} + 93117024 \, d x^{4} + 53209728 \, e x^{4} + 49884120 \, d x^{3} + 18706545 \, e x^{3} + 17606160 \, d x^{2} + 3912480 \, e x^{2} + 3695120 \, d x + 369512 \, e x + 350064 \, d}{6651216 \, x^{19}} \] Input:
integrate((e*x+d)*(x^2+2*x+1)^5/x^20,x, algorithm="giac")
Output:
-1/6651216*(831402*e*x^11 + 739024*d*x^10 + 7390240*e*x^10 + 6651216*d*x^9 + 29930472*e*x^9 + 27209520*d*x^8 + 72558720*e*x^8 + 66512160*d*x^7 + 116 396280*e*x^7 + 107442720*d*x^6 + 128931264*e*x^6 + 119721888*d*x^5 + 99768 240*e*x^5 + 93117024*d*x^4 + 53209728*e*x^4 + 49884120*d*x^3 + 18706545*e* x^3 + 17606160*d*x^2 + 3912480*e*x^2 + 3695120*d*x + 369512*e*x + 350064*d )/x^19
Time = 0.12 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.81 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{20}} \, dx=-\frac {\frac {e\,x^{11}}{8}+\left (\frac {d}{9}+\frac {10\,e}{9}\right )\,x^{10}+\left (d+\frac {9\,e}{2}\right )\,x^9+\left (\frac {45\,d}{11}+\frac {120\,e}{11}\right )\,x^8+\left (10\,d+\frac {35\,e}{2}\right )\,x^7+\left (\frac {210\,d}{13}+\frac {252\,e}{13}\right )\,x^6+\left (18\,d+15\,e\right )\,x^5+\left (14\,d+8\,e\right )\,x^4+\left (\frac {15\,d}{2}+\frac {45\,e}{16}\right )\,x^3+\left (\frac {45\,d}{17}+\frac {10\,e}{17}\right )\,x^2+\left (\frac {5\,d}{9}+\frac {e}{18}\right )\,x+\frac {d}{19}}{x^{19}} \] Input:
int(((d + e*x)*(2*x + x^2 + 1)^5)/x^20,x)
Output:
-(d/19 + x^4*(14*d + 8*e) + x^5*(18*d + 15*e) + x^10*(d/9 + (10*e)/9) + x^ 7*(10*d + (35*e)/2) + x^3*((15*d)/2 + (45*e)/16) + x^2*((45*d)/17 + (10*e) /17) + x^8*((45*d)/11 + (120*e)/11) + x^6*((210*d)/13 + (252*e)/13) + (e*x ^11)/8 + x*((5*d)/9 + e/18) + x^9*(d + (9*e)/2))/x^19
Time = 0.27 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{20}} \, dx=\frac {-831402 e \,x^{11}-739024 d \,x^{10}-7390240 e \,x^{10}-6651216 d \,x^{9}-29930472 e \,x^{9}-27209520 d \,x^{8}-72558720 e \,x^{8}-66512160 d \,x^{7}-116396280 e \,x^{7}-107442720 d \,x^{6}-128931264 e \,x^{6}-119721888 d \,x^{5}-99768240 e \,x^{5}-93117024 d \,x^{4}-53209728 e \,x^{4}-49884120 d \,x^{3}-18706545 e \,x^{3}-17606160 d \,x^{2}-3912480 e \,x^{2}-3695120 d x -369512 e x -350064 d}{6651216 x^{19}} \] Input:
int((e*x+d)*(x^2+2*x+1)^5/x^20,x)
Output:
( - 739024*d*x**10 - 6651216*d*x**9 - 27209520*d*x**8 - 66512160*d*x**7 - 107442720*d*x**6 - 119721888*d*x**5 - 93117024*d*x**4 - 49884120*d*x**3 - 17606160*d*x**2 - 3695120*d*x - 350064*d - 831402*e*x**11 - 7390240*e*x**1 0 - 29930472*e*x**9 - 72558720*e*x**8 - 116396280*e*x**7 - 128931264*e*x** 6 - 99768240*e*x**5 - 53209728*e*x**4 - 18706545*e*x**3 - 3912480*e*x**2 - 369512*e*x)/(6651216*x**19)