How I can get faster code for computing a fractional integral?

Question

I use of this code Fractional Integral and wrote the following code:

INT[α_, f_, x_, opts___] := 
  Integrate[(x - t)^(-α - 1) (f /. x -> t), {t, 0, x}, opts]/Gamma[-α]
INT[mu_?Positive, f_, x_, opts___] := 
  Module[{m = Ceiling[mu]},
    D[INT[-(m - mu), f, x, opts], {x, m}]]

I run for:

u1[x] = 
  INT[-α, 0.9 - 0.1 x, x, Assumptions -> {x > 0 && Re[β] > -1 && Re[α] > 0}]

u2[x] = 
  INT[-α, 8. u1[x] - 2. x u1[x], x, 
    Assumptions -> {x > 0 && Re[β] > -1 && Re[α] > 0}]

But Mathematica needs a lot of time to make the calculations. Any suggestion code for get faster calculation?

Answer 1

I have two suggestions to speed up the code:

1. Always Expand the input expression f
2. Set option GenerateConditions→False to Integrate to bypass convergence test routines.

This Expands the input and passes the result to the integrators internalINT:

INT[α_, f_, x_, opts___] :=
  Module[{expr = Expand[f]},
    internalINT[α, f, x, opts]
  ]

Of the following, the first two definitions for internalINT is your code copied here. The third definition Maps the internalINT over addition (Plus):

SetOptions[Integrate, GenerateConditions -> False];

internalINT[α_, f_, x_, opts___] := 
  Integrate[(x - t)^(-α - 1) (f /. x -> t), {t, 0, x}, opts]/Gamma[-α]

internalINT[mu_?Positive, f_, x_, opts___] := 
  Module[{m = Ceiling[mu]}, D[internalINT[-(m - mu), f, x, opts], {x, m}]]

internalINT[α_, f_Plus, x_, opts___] := Map[internalINT[α, #, x, opts] &, Expand@f]