An open type mixed quadrature rule is constructed blending the anti-Gauss 3-point rule with Steffensen's 4-point rule. The analytical convergence of the mixed rule is studied. An adaptive integration scheme is designed based on the mixed quadrature rule. A comparative study of the mixed quadrature rule and the Gauss‒Laguerre quadrature rule is given by evaluating several improper integrals of the form $\int\limits_{0}^{\infty}e^{-x}f(x)dx$. The advantage of implementing mixed quadrature rule in developing an efficient adaptive integration scheme is shown by evaluating some improper integrals.