AsymptoticIntegrate compute a series solution of integral.

Notation "{b, 0, 1}" mean a compute series at b=0 with order 1.

"b->0" mean b tending to zero or b close to zero.

I have only, if: b->0

AsymptoticIntegrate[E^(-(a/x) - x)/(b + c/x), {x, 0, Infinity}, {b, 0, 1}, Assumptions -> a > 0]

(*(2 a BesselK[2, 2 Sqrt[a]])/c - (2 a^(3/2) b BesselK[3, 2 Sqrt[a]])/c^2*)

If: b->Infinity

 AsymptoticIntegrate[E^(-(a/x) - x)/(b + c/x), {x, 0, Infinity}, {b, Infinity, 1}, Assumptions -> a > 0]

 (*(2 Sqrt[a] BesselK[1, 2 Sqrt[a]])/b*)

if:a ->0

 AsymptoticIntegrate[E^(-(a/x) - x)/(b + c/x), {x, 0, Infinity}, {a, 0, 1}]
 (*-((a E^(c/b) (Gamma[0, c/b] + Log[b/c] + Log[c/b]))/b) + (
  b - c E^(c/b) Gamma[0, c/b] - c E^(c/b) Log[b/c] - 
   c E^(c/b) Log[c/b])/b^2*)

if: c->0

 AsymptoticIntegrate[E^(-(a/x) - x)/(b + c/x), {x, 0, Infinity}, {c, 0, 1}]
 (*-((2 c BesselK[0, 2 Sqrt[a]])/b^2) + (2 Sqrt[a] BesselK[1, 2 Sqrt[a]])/b*)

if: c->Infinity

 AsymptoticIntegrate[ E^(-(a/x) - x)/(b + c/x), {x, 0, Infinity}, {c, Infinity, 1}]
 (*(2 a BesselK[2, 2 Sqrt[a]])/c*)

Regards.