Integrand size = 19, antiderivative size = 71 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{15}} \, dx=-\frac {d (1+x)^{11}}{14 x^{14}}+\frac {(3 d-14 e) (1+x)^{11}}{182 x^{13}}-\frac {(3 d-14 e) (1+x)^{11}}{1092 x^{12}}+\frac {(3 d-14 e) (1+x)^{11}}{12012 x^{11}} \] Output:
-1/14*d*(1+x)^11/x^14+1/182*(3*d-14*e)*(1+x)^11/x^13-1/1092*(3*d-14*e)*(1+ x)^11/x^12+1/12012*(3*d-14*e)*(1+x)^11/x^11
Leaf count is larger than twice the leaf count of optimal. \(149\) vs. \(2(71)=142\).
Time = 0.04 (sec) , antiderivative size = 149, normalized size of antiderivative = 2.10 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{15}} \, dx=-\frac {d}{14 x^{14}}-\frac {10 d+e}{13 x^{13}}-\frac {5 (9 d+2 e)}{12 x^{12}}-\frac {15 (8 d+3 e)}{11 x^{11}}-\frac {3 (7 d+4 e)}{x^{10}}-\frac {14 (6 d+5 e)}{3 x^9}-\frac {21 (5 d+6 e)}{4 x^8}-\frac {30 (4 d+7 e)}{7 x^7}-\frac {5 (3 d+8 e)}{2 x^6}-\frac {2 d+9 e}{x^5}-\frac {d+10 e}{4 x^4}-\frac {e}{3 x^3} \] Input:
Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^15,x]
Output:
-1/14*d/x^14 - (10*d + e)/(13*x^13) - (5*(9*d + 2*e))/(12*x^12) - (15*(8*d + 3*e))/(11*x^11) - (3*(7*d + 4*e))/x^10 - (14*(6*d + 5*e))/(3*x^9) - (21 *(5*d + 6*e))/(4*x^8) - (30*(4*d + 7*e))/(7*x^7) - (5*(3*d + 8*e))/(2*x^6) - (2*d + 9*e)/x^5 - (d + 10*e)/(4*x^4) - e/(3*x^3)
Time = 0.30 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {1184, 87, 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.
\(\displaystyle \int \frac {\left (x^2+2 x+1\right )^5 (d+e x)}{x^{15}} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int \frac {(x+1)^{10} (d+e x)}{x^{15}}dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {1}{14} (3 d-14 e) \int \frac {(x+1)^{10}}{x^{14}}dx-\frac {d (x+1)^{11}}{14 x^{14}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {1}{14} (3 d-14 e) \left (-\frac {2}{13} \int \frac {(x+1)^{10}}{x^{13}}dx-\frac {(x+1)^{11}}{13 x^{13}}\right )-\frac {d (x+1)^{11}}{14 x^{14}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {1}{14} (3 d-14 e) \left (-\frac {2}{13} \left (-\frac {1}{12} \int \frac {(x+1)^{10}}{x^{12}}dx-\frac {(x+1)^{11}}{12 x^{12}}\right )-\frac {(x+1)^{11}}{13 x^{13}}\right )-\frac {d (x+1)^{11}}{14 x^{14}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {1}{14} \left (-\frac {(x+1)^{11}}{13 x^{13}}-\frac {2}{13} \left (\frac {(x+1)^{11}}{132 x^{11}}-\frac {(x+1)^{11}}{12 x^{12}}\right )\right ) (3 d-14 e)-\frac {d (x+1)^{11}}{14 x^{14}}\) |
Input:
Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^15,x]
Output:
-1/14*(d*(1 + x)^11)/x^14 - ((3*d - 14*e)*(-1/13*(1 + x)^11/x^13 - (2*(-1/ 12*(1 + x)^11/x^12 + (1 + x)^11/(132*x^11)))/13))/14
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 = 0.77 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.73
method | result | size |
norman | \(\frac {-\frac {d}{14}+\left (-\frac {10 d}{13}-\frac {e}{13}\right ) x +\left (-\frac {15 d}{4}-\frac {5 e}{6}\right ) x^{2}+\left (-\frac {120 d}{11}-\frac {45 e}{11}\right ) x^{3}+\left (-21 d -12 e \right ) x^{4}+\left (-28 d -\frac {70 e}{3}\right ) x^{5}+\left (-\frac {105 d}{4}-\frac {63 e}{2}\right ) x^{6}+\left (-\frac {120 d}{7}-30 e \right ) x^{7}+\left (-\frac {15 d}{2}-20 e \right ) x^{8}+\left (-2 d -9 e \right ) x^{9}+\left (-\frac {d}{4}-\frac {5 e}{2}\right ) x^{10}-\frac {e \,x^{11}}{3}}{x^{14}}\) | \(123\) |
risch | \(\frac {-\frac {d}{14}+\left (-\frac {10 d}{13}-\frac {e}{13}\right ) x +\left (-\frac {15 d}{4}-\frac {5 e}{6}\right ) x^{2}+\left (-\frac {120 d}{11}-\frac {45 e}{11}\right ) x^{3}+\left (-21 d -12 e \right ) x^{4}+\left (-28 d -\frac {70 e}{3}\right ) x^{5}+\left (-\frac {105 d}{4}-\frac {63 e}{2}\right ) x^{6}+\left (-\frac {120 d}{7}-30 e \right ) x^{7}+\left (-\frac {15 d}{2}-20 e \right ) x^{8}+\left (-2 d -9 e \right ) x^{9}+\left (-\frac {d}{4}-\frac {5 e}{2}\right ) x^{10}-\frac {e \,x^{11}}{3}}{x^{14}}\) | \(123\) |
default | \(-\frac {10 d +45 e}{5 x^{5}}-\frac {45 d +120 e}{6 x^{6}}-\frac {e}{3 x^{3}}-\frac {10 d +e}{13 x^{13}}-\frac {210 d +252 e}{8 x^{8}}-\frac {d +10 e}{4 x^{4}}-\frac {d}{14 x^{14}}-\frac {252 d +210 e}{9 x^{9}}-\frac {120 d +210 e}{7 x^{7}}-\frac {210 d +120 e}{10 x^{10}}-\frac {120 d +45 e}{11 x^{11}}-\frac {45 d +10 e}{12 x^{12}}\) | \(130\) |
gosper | \(-\frac {4004 e \,x^{11}+3003 d \,x^{10}+30030 e \,x^{10}+24024 d \,x^{9}+108108 e \,x^{9}+90090 d \,x^{8}+240240 e \,x^{8}+205920 d \,x^{7}+360360 e \,x^{7}+315315 d \,x^{6}+378378 e \,x^{6}+336336 d \,x^{5}+280280 x^{5} e +252252 d \,x^{4}+144144 x^{4} e +131040 d \,x^{3}+49140 x^{3} e +45045 d \,x^{2}+10010 e \,x^{2}+9240 d x +924 e x +858 d}{12012 x^{14}}\) | \(132\) |
parallelrisch | \(\frac {-4004 e \,x^{11}-3003 d \,x^{10}-30030 e \,x^{10}-24024 d \,x^{9}-108108 e \,x^{9}-90090 d \,x^{8}-240240 e \,x^{8}-205920 d \,x^{7}-360360 e \,x^{7}-315315 d \,x^{6}-378378 e \,x^{6}-336336 d \,x^{5}-280280 x^{5} e -252252 d \,x^{4}-144144 x^{4} e -131040 d \,x^{3}-49140 x^{3} e -45045 d \,x^{2}-10010 e \,x^{2}-9240 d x -924 e x -858 d}{12012 x^{14}}\) | \(132\) |
orering | \(-\frac {\left (4004 e \,x^{11}+3003 d \,x^{10}+30030 e \,x^{10}+24024 d \,x^{9}+108108 e \,x^{9}+90090 d \,x^{8}+240240 e \,x^{8}+205920 d \,x^{7}+360360 e \,x^{7}+315315 d \,x^{6}+378378 e \,x^{6}+336336 d \,x^{5}+280280 x^{5} e +252252 d \,x^{4}+144144 x^{4} e +131040 d \,x^{3}+49140 x^{3} e +45045 d \,x^{2}+10010 e \,x^{2}+9240 d x +924 e x +858 d \right ) \left (x^{2}+2 x +1\right )^{5}}{12012 x^{14} \left (x +1\right )^{10}}\) | \(147\) |
Input:
int((e*x+d)*(x^2+2*x+1)^5/x^15,x,method=_RETURNVERBOSE)
Output:
(-1/14*d+(-10/13*d-1/13*e)*x+(-15/4*d-5/6*e)*x^2+(-120/11*d-45/11*e)*x^3+( -21*d-12*e)*x^4+(-28*d-70/3*e)*x^5+(-105/4*d-63/2*e)*x^6+(-120/7*d-30*e)*x ^7+(-15/2*d-20*e)*x^8+(-2*d-9*e)*x^9+(-1/4*d-5/2*e)*x^10-1/3*e*x^11)/x^14
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (63) = 126\).
Time = 0.07 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.82 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{15}} \, dx=-\frac {4004 \, e x^{11} + 3003 \, {\left (d + 10 \, e\right )} x^{10} + 12012 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 30030 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 51480 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 63063 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 56056 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 36036 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 16380 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 5005 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 924 \, {\left (10 \, d + e\right )} x + 858 \, d}{12012 \, x^{14}} \] Input:
integrate((e*x+d)*(x^2+2*x+1)^5/x^15,x, algorithm="fricas")
Output:
-1/12012*(4004*e*x^11 + 3003*(d + 10*e)*x^10 + 12012*(2*d + 9*e)*x^9 + 300 30*(3*d + 8*e)*x^8 + 51480*(4*d + 7*e)*x^7 + 63063*(5*d + 6*e)*x^6 + 56056 *(6*d + 5*e)*x^5 + 36036*(7*d + 4*e)*x^4 + 16380*(8*d + 3*e)*x^3 + 5005*(9 *d + 2*e)*x^2 + 924*(10*d + e)*x + 858*d)/x^14
Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (58) = 116\).
Time = 9.49 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.85 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{15}} \, dx=\frac {- 858 d - 4004 e x^{11} + x^{10} \left (- 3003 d - 30030 e\right ) + x^{9} \left (- 24024 d - 108108 e\right ) + x^{8} \left (- 90090 d - 240240 e\right ) + x^{7} \left (- 205920 d - 360360 e\right ) + x^{6} \left (- 315315 d - 378378 e\right ) + x^{5} \left (- 336336 d - 280280 e\right ) + x^{4} \left (- 252252 d - 144144 e\right ) + x^{3} \left (- 131040 d - 49140 e\right ) + x^{2} \left (- 45045 d - 10010 e\right ) + x \left (- 9240 d - 924 e\right )}{12012 x^{14}} \] Input:
integrate((e*x+d)*(x**2+2*x+1)**5/x**15,x)
Output:
(-858*d - 4004*e*x**11 + x**10*(-3003*d - 30030*e) + x**9*(-24024*d - 1081 08*e) + x**8*(-90090*d - 240240*e) + x**7*(-205920*d - 360360*e) + x**6*(- 315315*d - 378378*e) + x**5*(-336336*d - 280280*e) + x**4*(-252252*d - 144 144*e) + x**3*(-131040*d - 49140*e) + x**2*(-45045*d - 10010*e) + x*(-9240 *d - 924*e))/(12012*x**14)
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (63) = 126\).
Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.82 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{15}} \, dx=-\frac {4004 \, e x^{11} + 3003 \, {\left (d + 10 \, e\right )} x^{10} + 12012 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 30030 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 51480 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 63063 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 56056 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 36036 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 16380 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 5005 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 924 \, {\left (10 \, d + e\right )} x + 858 \, d}{12012 \, x^{14}} \] Input:
integrate((e*x+d)*(x^2+2*x+1)^5/x^15,x, algorithm="maxima")
Output:
-1/12012*(4004*e*x^11 + 3003*(d + 10*e)*x^10 + 12012*(2*d + 9*e)*x^9 + 300 30*(3*d + 8*e)*x^8 + 51480*(4*d + 7*e)*x^7 + 63063*(5*d + 6*e)*x^6 + 56056 *(6*d + 5*e)*x^5 + 36036*(7*d + 4*e)*x^4 + 16380*(8*d + 3*e)*x^3 + 5005*(9 *d + 2*e)*x^2 + 924*(10*d + e)*x + 858*d)/x^14
Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (63) = 126\).
Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.85 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{15}} \, dx=-\frac {4004 \, e x^{11} + 3003 \, d x^{10} + 30030 \, e x^{10} + 24024 \, d x^{9} + 108108 \, e x^{9} + 90090 \, d x^{8} + 240240 \, e x^{8} + 205920 \, d x^{7} + 360360 \, e x^{7} + 315315 \, d x^{6} + 378378 \, e x^{6} + 336336 \, d x^{5} + 280280 \, e x^{5} + 252252 \, d x^{4} + 144144 \, e x^{4} + 131040 \, d x^{3} + 49140 \, e x^{3} + 45045 \, d x^{2} + 10010 \, e x^{2} + 9240 \, d x + 924 \, e x + 858 \, d}{12012 \, x^{14}} \] Input:
integrate((e*x+d)*(x^2+2*x+1)^5/x^15,x, algorithm="giac")
Output:
-1/12012*(4004*e*x^11 + 3003*d*x^10 + 30030*e*x^10 + 24024*d*x^9 + 108108* e*x^9 + 90090*d*x^8 + 240240*e*x^8 + 205920*d*x^7 + 360360*e*x^7 + 315315* d*x^6 + 378378*e*x^6 + 336336*d*x^5 + 280280*e*x^5 + 252252*d*x^4 + 144144 *e*x^4 + 131040*d*x^3 + 49140*e*x^3 + 45045*d*x^2 + 10010*e*x^2 + 9240*d*x + 924*e*x + 858*d)/x^14
Time = 10.63 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.73 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{15}} \, dx=-\frac {\frac {e\,x^{11}}{3}+\left (\frac {d}{4}+\frac {5\,e}{2}\right )\,x^{10}+\left (2\,d+9\,e\right )\,x^9+\left (\frac {15\,d}{2}+20\,e\right )\,x^8+\left (\frac {120\,d}{7}+30\,e\right )\,x^7+\left (\frac {105\,d}{4}+\frac {63\,e}{2}\right )\,x^6+\left (28\,d+\frac {70\,e}{3}\right )\,x^5+\left (21\,d+12\,e\right )\,x^4+\left (\frac {120\,d}{11}+\frac {45\,e}{11}\right )\,x^3+\left (\frac {15\,d}{4}+\frac {5\,e}{6}\right )\,x^2+\left (\frac {10\,d}{13}+\frac {e}{13}\right )\,x+\frac {d}{14}}{x^{14}} \] Input:
int(((d + e*x)*(2*x + x^2 + 1)^5)/x^15,x)
Output:
-(d/14 + x^9*(2*d + 9*e) + x^10*(d/4 + (5*e)/2) + x^2*((15*d)/4 + (5*e)/6) + x^4*(21*d + 12*e) + x^8*((15*d)/2 + 20*e) + x^5*(28*d + (70*e)/3) + x^7 *((120*d)/7 + 30*e) + x^6*((105*d)/4 + (63*e)/2) + x^3*((120*d)/11 + (45*e )/11) + (e*x^11)/3 + x*((10*d)/13 + e/13))/x^14
Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.85 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{15}} \, dx=\frac {-4004 e \,x^{11}-3003 d \,x^{10}-30030 e \,x^{10}-24024 d \,x^{9}-108108 e \,x^{9}-90090 d \,x^{8}-240240 e \,x^{8}-205920 d \,x^{7}-360360 e \,x^{7}-315315 d \,x^{6}-378378 e \,x^{6}-336336 d \,x^{5}-280280 e \,x^{5}-252252 d \,x^{4}-144144 e \,x^{4}-131040 d \,x^{3}-49140 e \,x^{3}-45045 d \,x^{2}-10010 e \,x^{2}-9240 d x -924 e x -858 d}{12012 x^{14}} \] Input:
int((e*x+d)*(x^2+2*x+1)^5/x^15,x)
Output:
( - 3003*d*x**10 - 24024*d*x**9 - 90090*d*x**8 - 205920*d*x**7 - 315315*d* x**6 - 336336*d*x**5 - 252252*d*x**4 - 131040*d*x**3 - 45045*d*x**2 - 9240 *d*x - 858*d - 4004*e*x**11 - 30030*e*x**10 - 108108*e*x**9 - 240240*e*x** 8 - 360360*e*x**7 - 378378*e*x**6 - 280280*e*x**5 - 144144*e*x**4 - 49140* e*x**3 - 10010*e*x**2 - 924*e*x)/(12012*x**14)