Integrand size = 19, antiderivative size = 151 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{19}} \, dx=-\frac {d}{18 x^{18}}-\frac {10 d+e}{17 x^{17}}-\frac {5 (9 d+2 e)}{16 x^{16}}-\frac {8 d+3 e}{x^{15}}-\frac {15 (7 d+4 e)}{7 x^{14}}-\frac {42 (6 d+5 e)}{13 x^{13}}-\frac {7 (5 d+6 e)}{2 x^{12}}-\frac {30 (4 d+7 e)}{11 x^{11}}-\frac {3 (3 d+8 e)}{2 x^{10}}-\frac {5 (2 d+9 e)}{9 x^9}-\frac {d+10 e}{8 x^8}-\frac {e}{7 x^7} \]
-1/18*d/x^18+1/17*(-10*d-e)/x^17-5/16*(9*d+2*e)/x^16+(-8*d-3*e)/x^15-15/7* (7*d+4*e)/x^14-42/13*(6*d+5*e)/x^13-7/2*(5*d+6*e)/x^12-30/11*(4*d+7*e)/x^1 1-3/2*(3*d+8*e)/x^10-5/9*(2*d+9*e)/x^9+1/8*(-d-10*e)/x^8-1/7*e/x^7
Time = 0.03 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{19}} \, dx=-\frac {d}{18 x^{18}}-\frac {10 d+e}{17 x^{17}}-\frac {5 (9 d+2 e)}{16 x^{16}}-\frac {8 d+3 e}{x^{15}}-\frac {15 (7 d+4 e)}{7 x^{14}}-\frac {42 (6 d+5 e)}{13 x^{13}}-\frac {7 (5 d+6 e)}{2 x^{12}}-\frac {30 (4 d+7 e)}{11 x^{11}}-\frac {3 (3 d+8 e)}{2 x^{10}}-\frac {5 (2 d+9 e)}{9 x^9}-\frac {d+10 e}{8 x^8}-\frac {e}{7 x^7} \]
-1/18*d/x^18 - (10*d + e)/(17*x^17) - (5*(9*d + 2*e))/(16*x^16) - (8*d + 3 *e)/x^15 - (15*(7*d + 4*e))/(7*x^14) - (42*(6*d + 5*e))/(13*x^13) - (7*(5* d + 6*e))/(2*x^12) - (30*(4*d + 7*e))/(11*x^11) - (3*(3*d + 8*e))/(2*x^10) - (5*(2*d + 9*e))/(9*x^9) - (d + 10*e)/(8*x^8) - e/(7*x^7)
Time = 0.30 (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.
| \(\displaystyle \int \frac {\left (x^2+2 x+1\right )^5 (d+e x)}{x^{19}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \int \frac {(x+1)^{10} (d+e x)}{x^{19}}dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left (\frac {10 d+e}{x^{18}}+\frac {5 (9 d+2 e)}{x^{17}}+\frac {15 (8 d+3 e)}{x^{16}}+\frac {30 (7 d+4 e)}{x^{15}}+\frac {42 (6 d+5 e)}{x^{14}}+\frac {42 (5 d+6 e)}{x^{13}}+\frac {30 (4 d+7 e)}{x^{12}}+\frac {15 (3 d+8 e)}{x^{11}}+\frac {5 (2 d+9 e)}{x^{10}}+\frac {d+10 e}{x^9}+\frac {d}{x^{19}}+\frac {e}{x^8}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {10 d+e}{17 x^{17}}-\frac {5 (9 d+2 e)}{16 x^{16}}-\frac {8 d+3 e}{x^{15}}-\frac {15 (7 d+4 e)}{7 x^{14}}-\frac {42 (6 d+5 e)}{13 x^{13}}-\frac {7 (5 d+6 e)}{2 x^{12}}-\frac {30 (4 d+7 e)}{11 x^{11}}-\frac {3 (3 d+8 e)}{2 x^{10}}-\frac {5 (2 d+9 e)}{9 x^9}-\frac {d+10 e}{8 x^8}-\frac {d}{18 x^{18}}-\frac {e}{7 x^7}\) |
-1/18*d/x^18 - (10*d + e)/(17*x^17) - (5*(9*d + 2*e))/(16*x^16) - (8*d + 3 *e)/x^15 - (15*(7*d + 4*e))/(7*x^14) - (42*(6*d + 5*e))/(13*x^13) - (7*(5* d + 6*e))/(2*x^12) - (30*(4*d + 7*e))/(11*x^11) - (3*(3*d + 8*e))/(2*x^10) - (5*(2*d + 9*e))/(9*x^9) - (d + 10*e)/(8*x^8) - e/(7*x^7)
3.6.85.3.1 Defintions of rubi rules used
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 = 0.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.81
| method | result | size |
| norman | \(\frac {-\frac {d}{18}+\left (-\frac {10 d}{17}-\frac {e}{17}\right ) x +\left (-\frac {45 d}{16}-\frac {5 e}{8}\right ) x^{2}+\left (-8 d -3 e \right ) x^{3}+\left (-15 d -\frac {60 e}{7}\right ) x^{4}+\left (-\frac {252 d}{13}-\frac {210 e}{13}\right ) x^{5}+\left (-\frac {35 d}{2}-21 e \right ) x^{6}+\left (-\frac {120 d}{11}-\frac {210 e}{11}\right ) x^{7}+\left (-\frac {9 d}{2}-12 e \right ) x^{8}+\left (-\frac {10 d}{9}-5 e \right ) x^{9}+\left (-\frac {d}{8}-\frac {5 e}{4}\right ) x^{10}-\frac {e \,x^{11}}{7}}{x^{18}}\) | \(123\) |
| risch | \(\frac {-\frac {d}{18}+\left (-\frac {10 d}{17}-\frac {e}{17}\right ) x +\left (-\frac {45 d}{16}-\frac {5 e}{8}\right ) x^{2}+\left (-8 d -3 e \right ) x^{3}+\left (-15 d -\frac {60 e}{7}\right ) x^{4}+\left (-\frac {252 d}{13}-\frac {210 e}{13}\right ) x^{5}+\left (-\frac {35 d}{2}-21 e \right ) x^{6}+\left (-\frac {120 d}{11}-\frac {210 e}{11}\right ) x^{7}+\left (-\frac {9 d}{2}-12 e \right ) x^{8}+\left (-\frac {10 d}{9}-5 e \right ) x^{9}+\left (-\frac {d}{8}-\frac {5 e}{4}\right ) x^{10}-\frac {e \,x^{11}}{7}}{x^{18}}\) | \(123\) |
| default | \(-\frac {d}{18 x^{18}}-\frac {210 d +120 e}{14 x^{14}}-\frac {45 d +120 e}{10 x^{10}}-\frac {252 d +210 e}{13 x^{13}}-\frac {10 d +45 e}{9 x^{9}}-\frac {10 d +e}{17 x^{17}}-\frac {e}{7 x^{7}}-\frac {45 d +10 e}{16 x^{16}}-\frac {120 d +45 e}{15 x^{15}}-\frac {120 d +210 e}{11 x^{11}}-\frac {210 d +252 e}{12 x^{12}}-\frac {d +10 e}{8 x^{8}}\) | \(130\) |
| gosper | \(-\frac {350064 e \,x^{11}+306306 d \,x^{10}+3063060 e \,x^{10}+2722720 d \,x^{9}+12252240 e \,x^{9}+11027016 d \,x^{8}+29405376 e \,x^{8}+26732160 d \,x^{7}+46781280 e \,x^{7}+42882840 d \,x^{6}+51459408 e \,x^{6}+47500992 d \,x^{5}+39584160 e \,x^{5}+36756720 d \,x^{4}+21003840 e \,x^{4}+19603584 d \,x^{3}+7351344 e \,x^{3}+6891885 d \,x^{2}+1531530 e \,x^{2}+1441440 d x +144144 e x +136136 d}{2450448 x^{18}}\) | \(132\) |
| parallelrisch | \(\frac {-350064 e \,x^{11}-306306 d \,x^{10}-3063060 e \,x^{10}-2722720 d \,x^{9}-12252240 e \,x^{9}-11027016 d \,x^{8}-29405376 e \,x^{8}-26732160 d \,x^{7}-46781280 e \,x^{7}-42882840 d \,x^{6}-51459408 e \,x^{6}-47500992 d \,x^{5}-39584160 e \,x^{5}-36756720 d \,x^{4}-21003840 e \,x^{4}-19603584 d \,x^{3}-7351344 e \,x^{3}-6891885 d \,x^{2}-1531530 e \,x^{2}-1441440 d x -144144 e x -136136 d}{2450448 x^{18}}\) | \(132\) |
(-1/18*d+(-10/17*d-1/17*e)*x+(-45/16*d-5/8*e)*x^2+(-8*d-3*e)*x^3+(-15*d-60 /7*e)*x^4+(-252/13*d-210/13*e)*x^5+(-35/2*d-21*e)*x^6+(-120/11*d-210/11*e) *x^7+(-9/2*d-12*e)*x^8+(-10/9*d-5*e)*x^9+(-1/8*d-5/4*e)*x^10-1/7*e*x^11)/x ^18
Time = 0.34 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.85 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{19}} \, dx=-\frac {350064 \, e x^{11} + 306306 \, {\left (d + 10 \, e\right )} x^{10} + 1361360 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 3675672 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 6683040 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 8576568 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 7916832 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 5250960 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 2450448 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 765765 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 144144 \, {\left (10 \, d + e\right )} x + 136136 \, d}{2450448 \, x^{18}} \]
-1/2450448*(350064*e*x^11 + 306306*(d + 10*e)*x^10 + 1361360*(2*d + 9*e)*x ^9 + 3675672*(3*d + 8*e)*x^8 + 6683040*(4*d + 7*e)*x^7 + 8576568*(5*d + 6* e)*x^6 + 7916832*(6*d + 5*e)*x^5 + 5250960*(7*d + 4*e)*x^4 + 2450448*(8*d + 3*e)*x^3 + 765765*(9*d + 2*e)*x^2 + 144144*(10*d + e)*x + 136136*d)/x^18
Time = 15.59 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{19}} \, dx=\frac {- 136136 d - 350064 e x^{11} + x^{10} \left (- 306306 d - 3063060 e\right ) + x^{9} \left (- 2722720 d - 12252240 e\right ) + x^{8} \left (- 11027016 d - 29405376 e\right ) + x^{7} \left (- 26732160 d - 46781280 e\right ) + x^{6} \left (- 42882840 d - 51459408 e\right ) + x^{5} \left (- 47500992 d - 39584160 e\right ) + x^{4} \left (- 36756720 d - 21003840 e\right ) + x^{3} \left (- 19603584 d - 7351344 e\right ) + x^{2} \left (- 6891885 d - 1531530 e\right ) + x \left (- 1441440 d - 144144 e\right )}{2450448 x^{18}} \]
(-136136*d - 350064*e*x**11 + x**10*(-306306*d - 3063060*e) + x**9*(-27227 20*d - 12252240*e) + x**8*(-11027016*d - 29405376*e) + x**7*(-26732160*d - 46781280*e) + x**6*(-42882840*d - 51459408*e) + x**5*(-47500992*d - 39584 160*e) + x**4*(-36756720*d - 21003840*e) + x**3*(-19603584*d - 7351344*e) + x**2*(-6891885*d - 1531530*e) + x*(-1441440*d - 144144*e))/(2450448*x**1 8)
Time = 0.20 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.85 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{19}} \, dx=-\frac {350064 \, e x^{11} + 306306 \, {\left (d + 10 \, e\right )} x^{10} + 1361360 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 3675672 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 6683040 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 8576568 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 7916832 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 5250960 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 2450448 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 765765 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 144144 \, {\left (10 \, d + e\right )} x + 136136 \, d}{2450448 \, x^{18}} \]
-1/2450448*(350064*e*x^11 + 306306*(d + 10*e)*x^10 + 1361360*(2*d + 9*e)*x ^9 + 3675672*(3*d + 8*e)*x^8 + 6683040*(4*d + 7*e)*x^7 + 8576568*(5*d + 6* e)*x^6 + 7916832*(6*d + 5*e)*x^5 + 5250960*(7*d + 4*e)*x^4 + 2450448*(8*d + 3*e)*x^3 + 765765*(9*d + 2*e)*x^2 + 144144*(10*d + e)*x + 136136*d)/x^18
Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{19}} \, dx=-\frac {350064 \, e x^{11} + 306306 \, d x^{10} + 3063060 \, e x^{10} + 2722720 \, d x^{9} + 12252240 \, e x^{9} + 11027016 \, d x^{8} + 29405376 \, e x^{8} + 26732160 \, d x^{7} + 46781280 \, e x^{7} + 42882840 \, d x^{6} + 51459408 \, e x^{6} + 47500992 \, d x^{5} + 39584160 \, e x^{5} + 36756720 \, d x^{4} + 21003840 \, e x^{4} + 19603584 \, d x^{3} + 7351344 \, e x^{3} + 6891885 \, d x^{2} + 1531530 \, e x^{2} + 1441440 \, d x + 144144 \, e x + 136136 \, d}{2450448 \, x^{18}} \]
-1/2450448*(350064*e*x^11 + 306306*d*x^10 + 3063060*e*x^10 + 2722720*d*x^9 + 12252240*e*x^9 + 11027016*d*x^8 + 29405376*e*x^8 + 26732160*d*x^7 + 467 81280*e*x^7 + 42882840*d*x^6 + 51459408*e*x^6 + 47500992*d*x^5 + 39584160* e*x^5 + 36756720*d*x^4 + 21003840*e*x^4 + 19603584*d*x^3 + 7351344*e*x^3 + 6891885*d*x^2 + 1531530*e*x^2 + 1441440*d*x + 144144*e*x + 136136*d)/x^18
Time = 0.13 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.81 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{19}} \, dx=-\frac {\frac {e\,x^{11}}{7}+\left (\frac {d}{8}+\frac {5\,e}{4}\right )\,x^{10}+\left (\frac {10\,d}{9}+5\,e\right )\,x^9+\left (\frac {9\,d}{2}+12\,e\right )\,x^8+\left (\frac {120\,d}{11}+\frac {210\,e}{11}\right )\,x^7+\left (\frac {35\,d}{2}+21\,e\right )\,x^6+\left (\frac {252\,d}{13}+\frac {210\,e}{13}\right )\,x^5+\left (15\,d+\frac {60\,e}{7}\right )\,x^4+\left (8\,d+3\,e\right )\,x^3+\left (\frac {45\,d}{16}+\frac {5\,e}{8}\right )\,x^2+\left (\frac {10\,d}{17}+\frac {e}{17}\right )\,x+\frac {d}{18}}{x^{18}} \]