Integrand size = 19, antiderivative size = 151 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=-\frac {d}{20 x^{20}}-\frac {10 d+e}{19 x^{19}}-\frac {5 (9 d+2 e)}{18 x^{18}}-\frac {15 (8 d+3 e)}{17 x^{17}}-\frac {15 (7 d+4 e)}{8 x^{16}}-\frac {14 (6 d+5 e)}{5 x^{15}}-\frac {3 (5 d+6 e)}{x^{14}}-\frac {30 (4 d+7 e)}{13 x^{13}}-\frac {5 (3 d+8 e)}{4 x^{12}}-\frac {5 (2 d+9 e)}{11 x^{11}}-\frac {d+10 e}{10 x^{10}}-\frac {e}{9 x^9} \] Output:
-1/20*d/x^20-1/19*(10*d+e)/x^19-5/18*(9*d+2*e)/x^18-15/17*(8*d+3*e)/x^17-1 5/8*(7*d+4*e)/x^16-14/5*(6*d+5*e)/x^15-3*(5*d+6*e)/x^14-30/13*(4*d+7*e)/x^ 13-5/4*(3*d+8*e)/x^12-5/11*(2*d+9*e)/x^11-1/10*(d+10*e)/x^10-1/9*e/x^9
Time = 0.05 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=-\frac {d}{20 x^{20}}-\frac {10 d+e}{19 x^{19}}-\frac {5 (9 d+2 e)}{18 x^{18}}-\frac {15 (8 d+3 e)}{17 x^{17}}-\frac {15 (7 d+4 e)}{8 x^{16}}-\frac {14 (6 d+5 e)}{5 x^{15}}-\frac {3 (5 d+6 e)}{x^{14}}-\frac {30 (4 d+7 e)}{13 x^{13}}-\frac {5 (3 d+8 e)}{4 x^{12}}-\frac {5 (2 d+9 e)}{11 x^{11}}-\frac {d+10 e}{10 x^{10}}-\frac {e}{9 x^9} \] Input:
Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^21,x]
Output:
-1/20*d/x^20 - (10*d + e)/(19*x^19) - (5*(9*d + 2*e))/(18*x^18) - (15*(8*d + 3*e))/(17*x^17) - (15*(7*d + 4*e))/(8*x^16) - (14*(6*d + 5*e))/(5*x^15) - (3*(5*d + 6*e))/x^14 - (30*(4*d + 7*e))/(13*x^13) - (5*(3*d + 8*e))/(4* x^12) - (5*(2*d + 9*e))/(11*x^11) - (d + 10*e)/(10*x^10) - e/(9*x^9)
Time = 0.51 (sec) , antiderivative size = 151, 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.
| \(\displaystyle \int \frac {\left (x^2+2 x+1\right )^5 (d+e x)}{x^{21}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \int \frac {(x+1)^{10} (d+e x)}{x^{21}}dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left (\frac {10 d+e}{x^{20}}+\frac {5 (9 d+2 e)}{x^{19}}+\frac {15 (8 d+3 e)}{x^{18}}+\frac {30 (7 d+4 e)}{x^{17}}+\frac {42 (6 d+5 e)}{x^{16}}+\frac {42 (5 d+6 e)}{x^{15}}+\frac {30 (4 d+7 e)}{x^{14}}+\frac {15 (3 d+8 e)}{x^{13}}+\frac {5 (2 d+9 e)}{x^{12}}+\frac {d+10 e}{x^{11}}+\frac {d}{x^{21}}+\frac {e}{x^{10}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {10 d+e}{19 x^{19}}-\frac {5 (9 d+2 e)}{18 x^{18}}-\frac {15 (8 d+3 e)}{17 x^{17}}-\frac {15 (7 d+4 e)}{8 x^{16}}-\frac {14 (6 d+5 e)}{5 x^{15}}-\frac {3 (5 d+6 e)}{x^{14}}-\frac {30 (4 d+7 e)}{13 x^{13}}-\frac {5 (3 d+8 e)}{4 x^{12}}-\frac {5 (2 d+9 e)}{11 x^{11}}-\frac {d+10 e}{10 x^{10}}-\frac {d}{20 x^{20}}-\frac {e}{9 x^9}\) |
Input:
Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^21,x]
Output:
-1/20*d/x^20 - (10*d + e)/(19*x^19) - (5*(9*d + 2*e))/(18*x^18) - (15*(8*d + 3*e))/(17*x^17) - (15*(7*d + 4*e))/(8*x^16) - (14*(6*d + 5*e))/(5*x^15) - (3*(5*d + 6*e))/x^14 - (30*(4*d + 7*e))/(13*x^13) - (5*(3*d + 8*e))/(4* x^12) - (5*(2*d + 9*e))/(11*x^11) - (d + 10*e)/(10*x^10) - e/(9*x^9)
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 = 0.75 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.81
| method | result | size |
| norman | \(\frac {-\frac {d}{20}+\left (-\frac {10 d}{19}-\frac {e}{19}\right ) x +\left (-\frac {5 d}{2}-\frac {5 e}{9}\right ) x^{2}+\left (-\frac {120 d}{17}-\frac {45 e}{17}\right ) x^{3}+\left (-\frac {105 d}{8}-\frac {15 e}{2}\right ) x^{4}+\left (-\frac {84 d}{5}-14 e \right ) x^{5}+\left (-15 d -18 e \right ) x^{6}+\left (-\frac {120 d}{13}-\frac {210 e}{13}\right ) x^{7}+\left (-\frac {15 d}{4}-10 e \right ) x^{8}+\left (-\frac {10 d}{11}-\frac {45 e}{11}\right ) x^{9}+\left (-\frac {d}{10}-e \right ) x^{10}-\frac {e \,x^{11}}{9}}{x^{20}}\) | \(123\) |
| risch | \(\frac {-\frac {d}{20}+\left (-\frac {10 d}{19}-\frac {e}{19}\right ) x +\left (-\frac {5 d}{2}-\frac {5 e}{9}\right ) x^{2}+\left (-\frac {120 d}{17}-\frac {45 e}{17}\right ) x^{3}+\left (-\frac {105 d}{8}-\frac {15 e}{2}\right ) x^{4}+\left (-\frac {84 d}{5}-14 e \right ) x^{5}+\left (-15 d -18 e \right ) x^{6}+\left (-\frac {120 d}{13}-\frac {210 e}{13}\right ) x^{7}+\left (-\frac {15 d}{4}-10 e \right ) x^{8}+\left (-\frac {10 d}{11}-\frac {45 e}{11}\right ) x^{9}+\left (-\frac {d}{10}-e \right ) x^{10}-\frac {e \,x^{11}}{9}}{x^{20}}\) | \(123\) |
| default | \(-\frac {45 d +10 e}{18 x^{18}}-\frac {d}{20 x^{20}}-\frac {120 d +210 e}{13 x^{13}}-\frac {210 d +252 e}{14 x^{14}}-\frac {e}{9 x^{9}}-\frac {d +10 e}{10 x^{10}}-\frac {120 d +45 e}{17 x^{17}}-\frac {252 d +210 e}{15 x^{15}}-\frac {10 d +45 e}{11 x^{11}}-\frac {10 d +e}{19 x^{19}}-\frac {210 d +120 e}{16 x^{16}}-\frac {45 d +120 e}{12 x^{12}}\) | \(130\) |
| gosper | \(-\frac {1847560 e \,x^{11}+1662804 d \,x^{10}+16628040 e \,x^{10}+15116400 d \,x^{9}+68023800 e \,x^{9}+62355150 d \,x^{8}+166280400 e \,x^{8}+153489600 d \,x^{7}+268606800 e \,x^{7}+249420600 d \,x^{6}+299304720 e \,x^{6}+279351072 d \,x^{5}+232792560 x^{5} e +218243025 d \,x^{4}+124710300 x^{4} e +117374400 d \,x^{3}+44015400 x^{3} e +41570100 d \,x^{2}+9237800 e \,x^{2}+8751600 d x +875160 e x +831402 d}{16628040 x^{20}}\) | \(132\) |
| parallelrisch | \(\frac {-1847560 e \,x^{11}-1662804 d \,x^{10}-16628040 e \,x^{10}-15116400 d \,x^{9}-68023800 e \,x^{9}-62355150 d \,x^{8}-166280400 e \,x^{8}-153489600 d \,x^{7}-268606800 e \,x^{7}-249420600 d \,x^{6}-299304720 e \,x^{6}-279351072 d \,x^{5}-232792560 x^{5} e -218243025 d \,x^{4}-124710300 x^{4} e -117374400 d \,x^{3}-44015400 x^{3} e -41570100 d \,x^{2}-9237800 e \,x^{2}-8751600 d x -875160 e x -831402 d}{16628040 x^{20}}\) | \(132\) |
| orering | \(-\frac {\left (1847560 e \,x^{11}+1662804 d \,x^{10}+16628040 e \,x^{10}+15116400 d \,x^{9}+68023800 e \,x^{9}+62355150 d \,x^{8}+166280400 e \,x^{8}+153489600 d \,x^{7}+268606800 e \,x^{7}+249420600 d \,x^{6}+299304720 e \,x^{6}+279351072 d \,x^{5}+232792560 x^{5} e +218243025 d \,x^{4}+124710300 x^{4} e +117374400 d \,x^{3}+44015400 x^{3} e +41570100 d \,x^{2}+9237800 e \,x^{2}+8751600 d x +875160 e x +831402 d \right ) \left (x^{2}+2 x +1\right )^{5}}{16628040 x^{20} \left (x +1\right )^{10}}\) | \(147\) |
Input:
int((e*x+d)*(x^2+2*x+1)^5/x^21,x,method=_RETURNVERBOSE)
Output:
(-1/20*d+(-10/19*d-1/19*e)*x+(-5/2*d-5/9*e)*x^2+(-120/17*d-45/17*e)*x^3+(- 105/8*d-15/2*e)*x^4+(-84/5*d-14*e)*x^5+(-15*d-18*e)*x^6+(-120/13*d-210/13* e)*x^7+(-15/4*d-10*e)*x^8+(-10/11*d-45/11*e)*x^9+(-1/10*d-e)*x^10-1/9*e*x^ 11)/x^20
Time = 0.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.85 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=-\frac {1847560 \, e x^{11} + 1662804 \, {\left (d + 10 \, e\right )} x^{10} + 7558200 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 20785050 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 38372400 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 49884120 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 46558512 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 31177575 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 14671800 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 4618900 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 875160 \, {\left (10 \, d + e\right )} x + 831402 \, d}{16628040 \, x^{20}} \] Input:
integrate((e*x+d)*(x^2+2*x+1)^5/x^21,x, algorithm="fricas")
Output:
-1/16628040*(1847560*e*x^11 + 1662804*(d + 10*e)*x^10 + 7558200*(2*d + 9*e )*x^9 + 20785050*(3*d + 8*e)*x^8 + 38372400*(4*d + 7*e)*x^7 + 49884120*(5* d + 6*e)*x^6 + 46558512*(6*d + 5*e)*x^5 + 31177575*(7*d + 4*e)*x^4 + 14671 800*(8*d + 3*e)*x^3 + 4618900*(9*d + 2*e)*x^2 + 875160*(10*d + e)*x + 8314 02*d)/x^20
Time = 19.23 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=\frac {- 831402 d - 1847560 e x^{11} + x^{10} \left (- 1662804 d - 16628040 e\right ) + x^{9} \left (- 15116400 d - 68023800 e\right ) + x^{8} \left (- 62355150 d - 166280400 e\right ) + x^{7} \left (- 153489600 d - 268606800 e\right ) + x^{6} \left (- 249420600 d - 299304720 e\right ) + x^{5} \left (- 279351072 d - 232792560 e\right ) + x^{4} \left (- 218243025 d - 124710300 e\right ) + x^{3} \left (- 117374400 d - 44015400 e\right ) + x^{2} \left (- 41570100 d - 9237800 e\right ) + x \left (- 8751600 d - 875160 e\right )}{16628040 x^{20}} \] Input:
integrate((e*x+d)*(x**2+2*x+1)**5/x**21,x)
Output:
(-831402*d - 1847560*e*x**11 + x**10*(-1662804*d - 16628040*e) + x**9*(-15 116400*d - 68023800*e) + x**8*(-62355150*d - 166280400*e) + x**7*(-1534896 00*d - 268606800*e) + x**6*(-249420600*d - 299304720*e) + x**5*(-279351072 *d - 232792560*e) + x**4*(-218243025*d - 124710300*e) + x**3*(-117374400*d - 44015400*e) + x**2*(-41570100*d - 9237800*e) + x*(-8751600*d - 875160*e ))/(16628040*x**20)
Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.85 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=-\frac {1847560 \, e x^{11} + 1662804 \, {\left (d + 10 \, e\right )} x^{10} + 7558200 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 20785050 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 38372400 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 49884120 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 46558512 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 31177575 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 14671800 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 4618900 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 875160 \, {\left (10 \, d + e\right )} x + 831402 \, d}{16628040 \, x^{20}} \] Input:
integrate((e*x+d)*(x^2+2*x+1)^5/x^21,x, algorithm="maxima")
Output:
-1/16628040*(1847560*e*x^11 + 1662804*(d + 10*e)*x^10 + 7558200*(2*d + 9*e )*x^9 + 20785050*(3*d + 8*e)*x^8 + 38372400*(4*d + 7*e)*x^7 + 49884120*(5* d + 6*e)*x^6 + 46558512*(6*d + 5*e)*x^5 + 31177575*(7*d + 4*e)*x^4 + 14671 800*(8*d + 3*e)*x^3 + 4618900*(9*d + 2*e)*x^2 + 875160*(10*d + e)*x + 8314 02*d)/x^20
Time = 0.18 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=-\frac {1847560 \, e x^{11} + 1662804 \, d x^{10} + 16628040 \, e x^{10} + 15116400 \, d x^{9} + 68023800 \, e x^{9} + 62355150 \, d x^{8} + 166280400 \, e x^{8} + 153489600 \, d x^{7} + 268606800 \, e x^{7} + 249420600 \, d x^{6} + 299304720 \, e x^{6} + 279351072 \, d x^{5} + 232792560 \, e x^{5} + 218243025 \, d x^{4} + 124710300 \, e x^{4} + 117374400 \, d x^{3} + 44015400 \, e x^{3} + 41570100 \, d x^{2} + 9237800 \, e x^{2} + 8751600 \, d x + 875160 \, e x + 831402 \, d}{16628040 \, x^{20}} \] Input:
integrate((e*x+d)*(x^2+2*x+1)^5/x^21,x, algorithm="giac")
Output:
-1/16628040*(1847560*e*x^11 + 1662804*d*x^10 + 16628040*e*x^10 + 15116400* d*x^9 + 68023800*e*x^9 + 62355150*d*x^8 + 166280400*e*x^8 + 153489600*d*x^ 7 + 268606800*e*x^7 + 249420600*d*x^6 + 299304720*e*x^6 + 279351072*d*x^5 + 232792560*e*x^5 + 218243025*d*x^4 + 124710300*e*x^4 + 117374400*d*x^3 + 44015400*e*x^3 + 41570100*d*x^2 + 9237800*e*x^2 + 8751600*d*x + 875160*e*x + 831402*d)/x^20
Time = 0.13 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.80 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=-\frac {\frac {e\,x^{11}}{9}+\left (\frac {d}{10}+e\right )\,x^{10}+\left (\frac {10\,d}{11}+\frac {45\,e}{11}\right )\,x^9+\left (\frac {15\,d}{4}+10\,e\right )\,x^8+\left (\frac {120\,d}{13}+\frac {210\,e}{13}\right )\,x^7+\left (15\,d+18\,e\right )\,x^6+\left (\frac {84\,d}{5}+14\,e\right )\,x^5+\left (\frac {105\,d}{8}+\frac {15\,e}{2}\right )\,x^4+\left (\frac {120\,d}{17}+\frac {45\,e}{17}\right )\,x^3+\left (\frac {5\,d}{2}+\frac {5\,e}{9}\right )\,x^2+\left (\frac {10\,d}{19}+\frac {e}{19}\right )\,x+\frac {d}{20}}{x^{20}} \] Input:
int(((d + e*x)*(2*x + x^2 + 1)^5)/x^21,x)
Output:
-(d/20 + x^2*((5*d)/2 + (5*e)/9) + x^8*((15*d)/4 + 10*e) + x^6*(15*d + 18* e) + x^9*((10*d)/11 + (45*e)/11) + x^5*((84*d)/5 + 14*e) + x^4*((105*d)/8 + (15*e)/2) + x^3*((120*d)/17 + (45*e)/17) + x^7*((120*d)/13 + (210*e)/13) + (e*x^11)/9 + x*((10*d)/19 + e/19) + x^10*(d/10 + e))/x^20
Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=\frac {-1847560 e \,x^{11}-1662804 d \,x^{10}-16628040 e \,x^{10}-15116400 d \,x^{9}-68023800 e \,x^{9}-62355150 d \,x^{8}-166280400 e \,x^{8}-153489600 d \,x^{7}-268606800 e \,x^{7}-249420600 d \,x^{6}-299304720 e \,x^{6}-279351072 d \,x^{5}-232792560 e \,x^{5}-218243025 d \,x^{4}-124710300 e \,x^{4}-117374400 d \,x^{3}-44015400 e \,x^{3}-41570100 d \,x^{2}-9237800 e \,x^{2}-8751600 d x -875160 e x -831402 d}{16628040 x^{20}} \] Input:
int((e*x+d)*(x^2+2*x+1)^5/x^21,x)
Output:
( - 1662804*d*x**10 - 15116400*d*x**9 - 62355150*d*x**8 - 153489600*d*x**7 - 249420600*d*x**6 - 279351072*d*x**5 - 218243025*d*x**4 - 117374400*d*x* *3 - 41570100*d*x**2 - 8751600*d*x - 831402*d - 1847560*e*x**11 - 16628040 *e*x**10 - 68023800*e*x**9 - 166280400*e*x**8 - 268606800*e*x**7 - 2993047 20*e*x**6 - 232792560*e*x**5 - 124710300*e*x**4 - 44015400*e*x**3 - 923780 0*e*x**2 - 875160*e*x)/(16628040*x**20)