OrePolynomial[{{4*(73872*t^3 - 247487643*t^4 - 51281208432*t^5 + 
     2031521448960*t^6 - 29222960627712*t^7 + 25907242729472*t^8), {5}}, 
  {432*(5130*t^2 - 15026211*t^3 - 2173403100*t^4 + 179528539648*t^5 - 
     3822412103680*t^6 + 3238405341184*t^7), {4}}, 
  {3601260*t - 7991255151*t^2 - 38440990800*t^3 + 169972269308928*t^4 - 
    6553924203970560*t^5 + 5200878977941504*t^6, {3}}, 
  {14*(69255 - 39306546*t + 54806431680*t^2 + 4013271664128*t^3 - 
     554096158310400*t^4 + 399943059636224*t^5), {2}}, 
  {140*(315495 + 31347216*t - 88612337664*t^2 - 12973663322112*t^3 + 
     8096013352960*t^4), {1}}}, OreAlgebraObject[{Der[t]}, Together, 
  Together[#1 + #2] & , Together[#1*#2] & , None], DegreeLexicographic]
