Integrand size = 19, antiderivative size = 109 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{17}} \, dx=-\frac {d (1+x)^{11}}{16 x^{16}}+\frac {(5 d-16 e) (1+x)^{11}}{240 x^{15}}-\frac {(5 d-16 e) (1+x)^{11}}{840 x^{14}}+\frac {(5 d-16 e) (1+x)^{11}}{3640 x^{13}}-\frac {(5 d-16 e) (1+x)^{11}}{21840 x^{12}}+\frac {(5 d-16 e) (1+x)^{11}}{240240 x^{11}} \]
-1/16*d*(1+x)^11/x^16+1/240*(5*d-16*e)*(1+x)^11/x^15-1/840*(5*d-16*e)*(1+x )^11/x^14+1/3640*(5*d-16*e)*(1+x)^11/x^13-1/21840*(5*d-16*e)*(1+x)^11/x^12 +1/240240*(5*d-16*e)*(1+x)^11/x^11
Time = 0.03 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.40 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{17}} \, dx=-\frac {d}{16 x^{16}}-\frac {10 d+e}{15 x^{15}}-\frac {5 (9 d+2 e)}{14 x^{14}}-\frac {15 (8 d+3 e)}{13 x^{13}}-\frac {5 (7 d+4 e)}{2 x^{12}}-\frac {42 (6 d+5 e)}{11 x^{11}}-\frac {21 (5 d+6 e)}{5 x^{10}}-\frac {10 (4 d+7 e)}{3 x^9}-\frac {15 (3 d+8 e)}{8 x^8}-\frac {5 (2 d+9 e)}{7 x^7}-\frac {d+10 e}{6 x^6}-\frac {e}{5 x^5} \]
-1/16*d/x^16 - (10*d + e)/(15*x^15) - (5*(9*d + 2*e))/(14*x^14) - (15*(8*d + 3*e))/(13*x^13) - (5*(7*d + 4*e))/(2*x^12) - (42*(6*d + 5*e))/(11*x^11) - (21*(5*d + 6*e))/(5*x^10) - (10*(4*d + 7*e))/(3*x^9) - (15*(3*d + 8*e)) /(8*x^8) - (5*(2*d + 9*e))/(7*x^7) - (d + 10*e)/(6*x^6) - e/(5*x^5)
Time = 0.19 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {1184, 87, 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.
| \(\displaystyle \int \frac {\left (x^2+2 x+1\right )^5 (d+e x)}{x^{17}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \int \frac {(x+1)^{10} (d+e x)}{x^{17}}dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {1}{16} (5 d-16 e) \int \frac {(x+1)^{10}}{x^{16}}dx-\frac {d (x+1)^{11}}{16 x^{16}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {1}{16} (5 d-16 e) \left (-\frac {4}{15} \int \frac {(x+1)^{10}}{x^{15}}dx-\frac {(x+1)^{11}}{15 x^{15}}\right )-\frac {d (x+1)^{11}}{16 x^{16}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {1}{16} (5 d-16 e) \left (-\frac {4}{15} \left (-\frac {3}{14} \int \frac {(x+1)^{10}}{x^{14}}dx-\frac {(x+1)^{11}}{14 x^{14}}\right )-\frac {(x+1)^{11}}{15 x^{15}}\right )-\frac {d (x+1)^{11}}{16 x^{16}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {1}{16} (5 d-16 e) \left (-\frac {4}{15} \left (-\frac {3}{14} \left (-\frac {2}{13} \int \frac {(x+1)^{10}}{x^{13}}dx-\frac {(x+1)^{11}}{13 x^{13}}\right )-\frac {(x+1)^{11}}{14 x^{14}}\right )-\frac {(x+1)^{11}}{15 x^{15}}\right )-\frac {d (x+1)^{11}}{16 x^{16}}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {1}{16} (5 d-16 e) \left (-\frac {4}{15} \left (-\frac {3}{14} \left (-\frac {2}{13} \left (-\frac {1}{12} \int \frac {(x+1)^{10}}{x^{12}}dx-\frac {(x+1)^{11}}{12 x^{12}}\right )-\frac {(x+1)^{11}}{13 x^{13}}\right )-\frac {(x+1)^{11}}{14 x^{14}}\right )-\frac {(x+1)^{11}}{15 x^{15}}\right )-\frac {d (x+1)^{11}}{16 x^{16}}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {1}{16} \left (-\frac {(x+1)^{11}}{15 x^{15}}-\frac {4}{15} \left (-\frac {(x+1)^{11}}{14 x^{14}}-\frac {3}{14} \left (-\frac {(x+1)^{11}}{13 x^{13}}-\frac {2}{13} \left (\frac {(x+1)^{11}}{132 x^{11}}-\frac {(x+1)^{11}}{12 x^{12}}\right )\right )\right )\right ) (5 d-16 e)-\frac {d (x+1)^{11}}{16 x^{16}}\) |
-1/16*(d*(1 + x)^11)/x^16 - ((5*d - 16*e)*(-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
3.6.83.3.1 Defintions of rubi rules used
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 = 0.09 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.13
| method | result | size |
| norman | \(\frac {-\frac {d}{16}+\left (-\frac {2 d}{3}-\frac {e}{15}\right ) x +\left (-\frac {45 d}{14}-\frac {5 e}{7}\right ) x^{2}+\left (-\frac {120 d}{13}-\frac {45 e}{13}\right ) x^{3}+\left (-\frac {35 d}{2}-10 e \right ) x^{4}+\left (-\frac {252 d}{11}-\frac {210 e}{11}\right ) x^{5}+\left (-21 d -\frac {126 e}{5}\right ) x^{6}+\left (-\frac {40 d}{3}-\frac {70 e}{3}\right ) x^{7}+\left (-\frac {45 d}{8}-15 e \right ) x^{8}+\left (-\frac {10 d}{7}-\frac {45 e}{7}\right ) x^{9}+\left (-\frac {d}{6}-\frac {5 e}{3}\right ) x^{10}-\frac {e \,x^{11}}{5}}{x^{16}}\) | \(123\) |
| risch | \(\frac {-\frac {d}{16}+\left (-\frac {2 d}{3}-\frac {e}{15}\right ) x +\left (-\frac {45 d}{14}-\frac {5 e}{7}\right ) x^{2}+\left (-\frac {120 d}{13}-\frac {45 e}{13}\right ) x^{3}+\left (-\frac {35 d}{2}-10 e \right ) x^{4}+\left (-\frac {252 d}{11}-\frac {210 e}{11}\right ) x^{5}+\left (-21 d -\frac {126 e}{5}\right ) x^{6}+\left (-\frac {40 d}{3}-\frac {70 e}{3}\right ) x^{7}+\left (-\frac {45 d}{8}-15 e \right ) x^{8}+\left (-\frac {10 d}{7}-\frac {45 e}{7}\right ) x^{9}+\left (-\frac {d}{6}-\frac {5 e}{3}\right ) x^{10}-\frac {e \,x^{11}}{5}}{x^{16}}\) | \(123\) |
| default | \(-\frac {d +10 e}{6 x^{6}}-\frac {45 d +10 e}{14 x^{14}}-\frac {210 d +252 e}{10 x^{10}}-\frac {e}{5 x^{5}}-\frac {120 d +45 e}{13 x^{13}}-\frac {120 d +210 e}{9 x^{9}}-\frac {10 d +45 e}{7 x^{7}}-\frac {d}{16 x^{16}}-\frac {10 d +e}{15 x^{15}}-\frac {252 d +210 e}{11 x^{11}}-\frac {210 d +120 e}{12 x^{12}}-\frac {45 d +120 e}{8 x^{8}}\) | \(130\) |
| gosper | \(-\frac {48048 e \,x^{11}+40040 d \,x^{10}+400400 e \,x^{10}+343200 d \,x^{9}+1544400 e \,x^{9}+1351350 d \,x^{8}+3603600 e \,x^{8}+3203200 d \,x^{7}+5605600 e \,x^{7}+5045040 d \,x^{6}+6054048 e \,x^{6}+5503680 d \,x^{5}+4586400 e \,x^{5}+4204200 d \,x^{4}+2402400 e \,x^{4}+2217600 d \,x^{3}+831600 e \,x^{3}+772200 d \,x^{2}+171600 e \,x^{2}+160160 d x +16016 e x +15015 d}{240240 x^{16}}\) | \(132\) |
| parallelrisch | \(\frac {-48048 e \,x^{11}-40040 d \,x^{10}-400400 e \,x^{10}-343200 d \,x^{9}-1544400 e \,x^{9}-1351350 d \,x^{8}-3603600 e \,x^{8}-3203200 d \,x^{7}-5605600 e \,x^{7}-5045040 d \,x^{6}-6054048 e \,x^{6}-5503680 d \,x^{5}-4586400 e \,x^{5}-4204200 d \,x^{4}-2402400 e \,x^{4}-2217600 d \,x^{3}-831600 e \,x^{3}-772200 d \,x^{2}-171600 e \,x^{2}-160160 d x -16016 e x -15015 d}{240240 x^{16}}\) | \(132\) |
(-1/16*d+(-2/3*d-1/15*e)*x+(-45/14*d-5/7*e)*x^2+(-120/13*d-45/13*e)*x^3+(- 35/2*d-10*e)*x^4+(-252/11*d-210/11*e)*x^5+(-21*d-126/5*e)*x^6+(-40/3*d-70/ 3*e)*x^7+(-45/8*d-15*e)*x^8+(-10/7*d-45/7*e)*x^9+(-1/6*d-5/3*e)*x^10-1/5*e *x^11)/x^16
Time = 0.38 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.18 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{17}} \, dx=-\frac {48048 \, e x^{11} + 40040 \, {\left (d + 10 \, e\right )} x^{10} + 171600 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 450450 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 800800 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 1009008 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 917280 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 600600 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 277200 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 85800 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 16016 \, {\left (10 \, d + e\right )} x + 15015 \, d}{240240 \, x^{16}} \]
-1/240240*(48048*e*x^11 + 40040*(d + 10*e)*x^10 + 171600*(2*d + 9*e)*x^9 + 450450*(3*d + 8*e)*x^8 + 800800*(4*d + 7*e)*x^7 + 1009008*(5*d + 6*e)*x^6 + 917280*(6*d + 5*e)*x^5 + 600600*(7*d + 4*e)*x^4 + 277200*(8*d + 3*e)*x^ 3 + 85800*(9*d + 2*e)*x^2 + 16016*(10*d + e)*x + 15015*d)/x^16
Time = 12.95 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.20 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{17}} \, dx=\frac {- 15015 d - 48048 e x^{11} + x^{10} \left (- 40040 d - 400400 e\right ) + x^{9} \left (- 343200 d - 1544400 e\right ) + x^{8} \left (- 1351350 d - 3603600 e\right ) + x^{7} \left (- 3203200 d - 5605600 e\right ) + x^{6} \left (- 5045040 d - 6054048 e\right ) + x^{5} \left (- 5503680 d - 4586400 e\right ) + x^{4} \left (- 4204200 d - 2402400 e\right ) + x^{3} \left (- 2217600 d - 831600 e\right ) + x^{2} \left (- 772200 d - 171600 e\right ) + x \left (- 160160 d - 16016 e\right )}{240240 x^{16}} \]
(-15015*d - 48048*e*x**11 + x**10*(-40040*d - 400400*e) + x**9*(-343200*d - 1544400*e) + x**8*(-1351350*d - 3603600*e) + x**7*(-3203200*d - 5605600* e) + x**6*(-5045040*d - 6054048*e) + x**5*(-5503680*d - 4586400*e) + x**4* (-4204200*d - 2402400*e) + x**3*(-2217600*d - 831600*e) + x**2*(-772200*d - 171600*e) + x*(-160160*d - 16016*e))/(240240*x**16)
Time = 0.20 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.18 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{17}} \, dx=-\frac {48048 \, e x^{11} + 40040 \, {\left (d + 10 \, e\right )} x^{10} + 171600 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 450450 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 800800 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 1009008 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 917280 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 600600 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 277200 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 85800 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 16016 \, {\left (10 \, d + e\right )} x + 15015 \, d}{240240 \, x^{16}} \]
-1/240240*(48048*e*x^11 + 40040*(d + 10*e)*x^10 + 171600*(2*d + 9*e)*x^9 + 450450*(3*d + 8*e)*x^8 + 800800*(4*d + 7*e)*x^7 + 1009008*(5*d + 6*e)*x^6 + 917280*(6*d + 5*e)*x^5 + 600600*(7*d + 4*e)*x^4 + 277200*(8*d + 3*e)*x^ 3 + 85800*(9*d + 2*e)*x^2 + 16016*(10*d + e)*x + 15015*d)/x^16
Time = 0.27 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.20 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{17}} \, dx=-\frac {48048 \, e x^{11} + 40040 \, d x^{10} + 400400 \, e x^{10} + 343200 \, d x^{9} + 1544400 \, e x^{9} + 1351350 \, d x^{8} + 3603600 \, e x^{8} + 3203200 \, d x^{7} + 5605600 \, e x^{7} + 5045040 \, d x^{6} + 6054048 \, e x^{6} + 5503680 \, d x^{5} + 4586400 \, e x^{5} + 4204200 \, d x^{4} + 2402400 \, e x^{4} + 2217600 \, d x^{3} + 831600 \, e x^{3} + 772200 \, d x^{2} + 171600 \, e x^{2} + 160160 \, d x + 16016 \, e x + 15015 \, d}{240240 \, x^{16}} \]
-1/240240*(48048*e*x^11 + 40040*d*x^10 + 400400*e*x^10 + 343200*d*x^9 + 15 44400*e*x^9 + 1351350*d*x^8 + 3603600*e*x^8 + 3203200*d*x^7 + 5605600*e*x^ 7 + 5045040*d*x^6 + 6054048*e*x^6 + 5503680*d*x^5 + 4586400*e*x^5 + 420420 0*d*x^4 + 2402400*e*x^4 + 2217600*d*x^3 + 831600*e*x^3 + 772200*d*x^2 + 17 1600*e*x^2 + 160160*d*x + 16016*e*x + 15015*d)/x^16
Time = 0.13 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.13 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{17}} \, dx=-\frac {\frac {e\,x^{11}}{5}+\left (\frac {d}{6}+\frac {5\,e}{3}\right )\,x^{10}+\left (\frac {10\,d}{7}+\frac {45\,e}{7}\right )\,x^9+\left (\frac {45\,d}{8}+15\,e\right )\,x^8+\left (\frac {40\,d}{3}+\frac {70\,e}{3}\right )\,x^7+\left (21\,d+\frac {126\,e}{5}\right )\,x^6+\left (\frac {252\,d}{11}+\frac {210\,e}{11}\right )\,x^5+\left (\frac {35\,d}{2}+10\,e\right )\,x^4+\left (\frac {120\,d}{13}+\frac {45\,e}{13}\right )\,x^3+\left (\frac {45\,d}{14}+\frac {5\,e}{7}\right )\,x^2+\left (\frac {2\,d}{3}+\frac {e}{15}\right )\,x+\frac {d}{16}}{x^{16}} \]