Integrate[(E^(I*k0*x)*(1 - E^(-(c*x)))^5*BesselJ[0, k*x])/(k0^4*x^3), {x, 0, Infinity}, Assumptions -> n == 0 && q == 4 && κ == 5 && k > k0 && c >= 0 && k >= 0 && k0 >= 0 && n >= 0]
Integrate((Power(E,I*k0*x)*Power(1 - Power(E,-(c*x)),5)*BesselJ(0,k*x))/(Power(k0,4)*Power(x,3)),List(x,0,DirectedInfinity(1)),Rule(Assumptions,n == 0 && q == 4 && κ == 5 && k > k0 && c >= 0 && k >= 0 && k0 >= 0 && n >= 0))

                              -5 c x + I k0 x        c x 5                                2  2               4  4               6  6               8  8      Pi                                                   2  2              4  4              6  6               8  8
                             E                (-1 + E   )  ((-418854310875 + 29682132480 k  x  - 3901685760 k  x  + 1258291200 k  x  - 2147483648 k  x ) Cos[-- + k x] + 4 Sqrt[2] k x (13043905875 - 1229437440 k  x  + 240844800 k  x  - 150994944 k  x  + 2147483648 k  x ) (Cos[k x] + Sin[k x]))
                                                                                                                                                             4
Integrate::idiv: Integral of ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ does not converge on {0, Infinity}.
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                                                                                                                                              8589934592 k     k0  Sqrt[2 Pi] x
Series[Integrate[(E^(I*k0*x)*(1 - E^(-(c*x)))^5*BesselJ[0, k*x])/(k0^4*x^3), {x, 0, Infinity}, Assumptions -> n == 0 && q == 4 && κ == 5 && k > k0 && c >= 0 && k >= 0 && k0 >= 0 && n >= 0], {k, Infinity, 10}]
Integrate[(E^(I*k0*x)*(1 - E^(-(c*x)))^5*BesselJ[0, k*x])/(k0^4*x^3), {x, 0, Infinity}, Assumptions -> n == 0 && q == 4 && κ == 5]