Abstract: We study free infinite divisibility (FID) for a class of generalized power distributions with free Poisson term by using complex analytic methods and free cumulants. In particular, we prove that (i) if $X$ follows the free generalized inverse Gaussian distribution, then the distribution of $X^r$ is FID when $|r|\ge 1$; (ii) if $S$ follows the standard semicircle law and $u>2$, then the distribution of $(S+u{)}^r$ is FID when $r\le -1$; (iii) if $B_p$ follows the beta distribution with parameters $p$ and $3/2$, then (iii-a) the distribution of $B_p^r$ is FID when $|r|\ge 1$ and $0<p\le 1/2$; (iii-b) the distribution of $B_p^r$ is FID when $r\le -1$ and $p>1/2$.