[gnramires]

Correction: I've got the integral wrong, of course :)

I've got an expression (after correcting my rusty maths) of

i = k*(n-1+k) / ( (1-r)*(n-1+k)^(p-1) + r*k^(p-1) ) ^ (1/(p-1)) - k

I think there could be precision problems (with p<2 specially)