  We study free infinite divisibility (FID) for a class which is called
generalized power distributions with free Poisson term by using a complex
analytic technique and a calculation for the free cumulants and Hankel
determinants. In particular, our main result implies that (i) if $X$ follows
the free Generalized Inverse Gaussian distribution, then $X^r$ follows an FID
distribution when $|r|\ge1$, (ii) if $S$ follows the standard semicircle law
and $u\ge 2$, then $(S+u)^r$ follows an FID distribution when $r\le -1$, and
(iii) if $B_p$ follows the beta distribution with parameters $p$ and $3/2$,
then (a) $B_p^r$ follows an FID distribution when $|r|\ge 1$ and $0<p\le 1/2$,
and (b) $B_p^r$ follows an FID distribution when $r\le -1$ and $p>1/2$.
