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 \(|r|\ge1\); (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 \(p>1/2\).