\[ \boxed { \left ( {{\rm e}^{x}}+2\,x \right ) {\it d4y} \left ( x \right ) +4\, \left ( {{\rm e}^{x}}+2 \right ) {\frac {{\rm d}^{3}}{{\rm d}{x}^{3}}}y \left ( x \right ) +6\,{{\rm e}^{x}}{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) +4\,{{\rm e}^{x}}{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +y \left ( x \right ) {{\rm e}^{x}}-{x}^{-5}=0} \]
Mathematica: cpu = 0.054507 (sec), leaf count = 59 \[ \text {DSolve}\left [-\frac {1}{x^5}+\left (2 x+e^x\right ) y^{(4)}(x)+4 \left (e^x+2\right ) y^{(3)}(x)+6 e^x y''(x)+4 e^x y'(x)+e^x y(x)=0,y(x),x\right ] \]
Maple: cpu = 0.016 (sec), leaf count = 65 \[ \left \{ y \left ( x \right ) ={\frac {{\it \_C4}}{{{\rm e}^{x}}+2\,x}}+ {\frac {{\it \_C1}\,{x}^{3}}{{{\rm e}^{x}}+2\,x}}+{\frac {{\it \_C2}\, {x}^{2}}{{{\rm e}^{x}}+2\,x}}+{\frac {{\it \_C3}\,x}{{{\rm e}^{x}}+2\, x}}+{\frac {1}{ \left ( 24\,{{\rm e}^{x}}+48\,x \right ) x}} \right \} \]