gfortran compiler Error: result of exponentiation exceeds the range of INTEGER(4)

The problem arises as one part of your computation is done using integer arithmetic of type integer(4). That type has an upper limit of 2^31-1 = 2147483647 whereas your intermediate result 5^15 = 30517578125 is slightly larger (thanks to @evets comment).

As pointed out in your question: you save the result in a real variable. Therefor, you could just compute that exponentiation using real data types: 5.0**15. Your formula will end up like the following

dFeV(x)= R1 * (5.0**15) * (a**2) * exp(-(VmigFe)/kbt)

Note that integer(4) need not be the same implementation for every processor (thanks @IanBush). Which just means that for some specific machines the upper limit might be different from 2^31-1 = 2147483647.