The aim of the present paper is to discuss a number of interesting classes of Eulerian integrals and the theorem based upon the fractional calculus associated with general class of polynomials given by Srivastava [4, P.1, Eq.(1)], generalized polynomials given by Srivastava [8, P.185, Eq.(7)] and the multivariable H-function given by Srivastava and Panda [13, P.271, eq.(4.1)]. The results derived here are of a very general nature and hence encompass several cases of interest hitherto scattered in the literature.