The use of non-standard abscissae in the infinite range work had an obvious application to classical Laguerre quadrature. In this paper these ideas are pursued to allow modified Laguerre quadrature to operate accurately on non-exponential decreasing integrands. The evaluation of the relevant monomials now requires exponential integrals, and the functions ci(x) and si(x) for which simple fast algorithms are available. A simple ‘black-box’ algorithm for combining a conventional Clenshaw-Curtis or Gauss-Legendre rule with this modified Gauss-Laguerre algorithm is provided. The work gives a fresh aspect to a classical quadarature rule which arises in areas such as statistics and wave theory.