\[ \boxed { {\frac {{\rm d}^{5}}{{\rm d}{x}^{5}}}y \left ( x \right ) +a{\frac {{\rm d}^{4}}{{\rm d}{x}^{4}}}y \left ( x \right ) -f=0} \]
Mathematica: cpu = 1218.103679 (sec), leaf count = 117 \[ \left \{\left \{y(x)\to (\text {Integrate$\grave { }\$\$$a$\$$3645023}-1) (\text {Integrate$\grave { }\$\$$a$\$$3818055}-1) (\text {Integrate$\grave { }\$\$$a$\$$3863942}-1) (x-1) e^{-a \text {Integrate$\grave { }\$\$$a$\$$3873056}} \left (\text {Integrate}\left [e^{a K[1]} f(K[1]),\{K[1],1,\text {Integrate$\grave { }\$\$$a$\$$3873056}\},\text {Assumptions}\to (\Im (\text {Integrate$\grave { }\$\$$a$\$$3645023})\neq 0\lor 1<\text {Integrate$\grave { }\$\$$a$\$$3818055}<\text {Integrate$\grave { }\$\$$a$\$$3645023}\lor \text {Integrate$\grave { }\$\$$a$\$$3645023}<\text {Integrate$\grave { }\$\$$a$\$$3818055}<1)\land (\Im (\text {Integrate$\grave { }\$\$$a$\$$3818055})\neq 0\lor 1<\text {Integrate$\grave { }\$\$$a$\$$3863942}<\text {Integrate$\grave { }\$\$$a$\$$3818055}\lor \text {Integrate$\grave { }\$\$$a$\$$3818055}<\text {Integrate$\grave { }\$\$$a$\$$3863942}<1)\land (\Im (\text {Integrate$\grave { }\$\$$a$\$$3863942})\neq 0\lor 1<\text {Integrate$\grave { }\$\$$a$\$$3873056}<\text {Integrate$\grave { }\$\$$a$\$$3863942}\lor \text {Integrate$\grave { }\$\$$a$\$$3863942}<\text {Integrate$\grave { }\$\$$a$\$$3873056}<1)\land (\Im (x)\neq 0\lor 1<\text {Integrate$\grave { }\$\$$a$\$$3645023}<x\lor x<\text {Integrate$\grave { }\$\$$a$\$$3645023}<1)\right ]+c_1\right )+c_5 x^3+c_4 x^2+c_3 x+c_2\right \}\right \} \]
Maple: cpu = 0.015 (sec), leaf count = 40 \[ \left \{ y \left ( x \right ) ={\frac {{{\rm e}^{-ax}}{\it \_C1}}{{a}^{4 }}}+{\frac {f{x}^{4}}{24\,a}}+{\frac {{\it \_C2}\,{x}^{3}}{6}}+{\frac {{\it \_C3}\,{x}^{2}}{2}}+{\it \_C4}\,x+{\it \_C5} \right \} \]