Distribute operator on functions

Question

I wrote this code:

 L = 1;
u[x_] := Subscript[u, 0][x] + p*Subscript[u, 1][x]
Distribute[
 Refine[(1/Gamma[L - α])*
   Integrate[(x - τ)^(L - α - 1)*
     D[u[τ], {τ, L}], {τ, 0, x}], 
  Assumptions -> {L - 1 < α < L, x > 0}]]

Output is:

 Integrate[(x - τ)^-α (Derivative[1][Subscript[u, 
     0]][τ] + p Derivative[1][Subscript[u, 1]][τ]), {τ,
   0, x}, Assumptions -> x > 0 && 0 < α < 1]/Gamma[
 1 - α]

I need to this output:

  Integrate[(x - τ)^-α (Derivative[1][Subscript[u, 
     0]][τ]), {τ, 0, x}, 
  Assumptions -> x > 0 && 0 < α < 1]/Gamma[1 - α] + 
 Integrate[(x - τ)^-α (p Derivative[1][Subscript[u, 
      1]][τ]), {τ, 0, x}, 
  Assumptions -> x > 0 && 0 < α < 1]/Gamma[1 - α]

Answer 1

Maybe something like this would be useful.

L = 1;
u[x_] := Subscript[u, 0][x] + p*Subscript[u, 1][x]


Distribute[ Integrate[
  Distribute[(1/Gamma[L-α]) (x-τ)^(L-α-1)*D[u[τ],{τ,L}]],
  {τ, 0, x}
]]