We present an effective numerical technique for computing the electrical image coefficients for rounded rectangular pipes with perfectly conducting walls. The method of moments is used to solve the involved boundary-value problem in integral form, by means of(a rapidly converging representation of) the rectangular-domain Green’s function, together with a set of piecewise parabolic subdomain basis functions, yielding high speed and accuracy, with minimum storage budget (no prior meshing required). As a distinctive feature of the proposed method no numerical differentiation is required, resulting into far better accuracy as compared, e.g., to finite-element and finite-difference methods. Application to some simplified cross section geometries relevant to LHC (square with rounded corners, stadium and cut-circle)are presented. As a check of accuracy of the proposed approach, a comparison with available exact (analytical) results for the circular pipe (hardest possible benchmark), shows an excellent agreement.