LetA ⊂ Bbe an extension of domains andXbe an indeterminate overB. In this paper, we study the elasticity of atomic domains of the formA+XB[X] whereA ⊂ Bis an integral extension. First, we study the case whereA = Zand, show that, whateverB, this elasticity is infinite. Finally, we investigate the case whereAis a discrete valuation ring with quotient fieldKandBits integral closure in a finite extension ofK.