I opened a book about MellinTransform and tried a few examples:
MellinTransform[(Sqrt[x^2 + a^2] + a)^\[Rho]/Sqrt[x^2 + a^2]*
BesselJ[v, b*x], x, s, Method -> "Integrate",
Assumptions -> {b > 0,
Re[a] > 0, -Re[v] < Re[s] < 5/2 - Re[\[Rho]]}](*Can't Solve !*)
MellinTransform[
Exp[-\[Sigma] Sqrt[x] - \[Omega]*x] Hypergeometric1F1[a,
b, \[Omega]*x], x, s, Method -> "Integrate",
Assumptions -> {a > 0, b > 0, Re[\[Sigma]] > 0,
Re[s] > 0}](*Can't Solve !*)
MellinTransform[
1/(x + \[Sigma])^\[Rho] Hypergeometric2F1[a, b,
c, -\[Omega]*x], x, s, Method -> "Integrate",
Assumptions -> {Re[\[Rho]] > 0, Re[\[Sigma]] > 0,
Re[s] > 0}](*Can't Solve !*)