There exists a long list of papers on the univalence of the integral operator \[ F_{\alpha,\beta}(z)= \int^z_0 (f'(t))^\alpha\Biggl({f(t)\over t}\Biggr)^\beta\,dt,\qquad \alpha,\beta\text{ -- real numbers}, \] defined on the class \(S\) or on one of its subclasses. Nevertheless there are many iteresting questions to answer. The present authors study univalence of functions that belong to the minimal invariant family \({\mathcal M}_{\alpha,\beta}(S^c)\) that includes the set of functions \(F_{\alpha,\beta}(z)\) generated by functions \(f\in S^c\) (univalent and convex on the unit disk). They determined the order of \({\mathcal M}_{\alpha,\beta}(S^c)\), gave an estimation to the radius of univalence and described the set \({\mathcal A}\) such that if \((\alpha,\beta)\in{\mathcal A}\) then \(f\in{\mathcal M}_{\alpha,\beta}(S^c)\) is univalent and close-to-convex in the unit disk.