New fractional integral unifying six existing fractional integrals

In this paper we introduce a new fractional integral that generalizes six existing fractional integrals, namely, Riemann-Liouville, Hadamard, Erdu0027elyi-Kober, Katugampola, Weyl and Liouville fractional integrals in to one form. Such a generalization takes the form [ left({}^{rho}mathcal{I}^{alpha, beta}_{a+;eta, kappa}fright)(x)=frac{rho^{1-beta}x^{kappa}}{Gamma(alpha)}int_a^x frac{tau^{rho eta +rho-1}}{(x^rho-tau^rho)^{1-alpha}}f(tau)text{d}tau, quad 0leq a u003c x u003c b leq infty. ] A similar generalization is not possible with the Erdu0027elyi-Kober operator though there is a close resemblance with the operator in question. We also give semigroup, boundedness, shift and integration-by-parts formulas for completeness.