[en] Scaling relations for the FEL gain, including the e-beam quality effects, have been usefully exploited to design FEL devices. We propose further extension of the above formulae including high gain, inhomogeneous broadening and saturation effects. A crucial role to get these relations is the use of approximant methods generalizing the Pade procedure. We derive gain equations containing the corrections due to energy spread, emittances and field intensity. It is shown that these equations can be exploited to open-quotes simulateclose quotes the FEL evolution with an almost negligible computational effort. Comments on the role of the saturation intensity and its dependence on the e-beam quality, high gain corrections etc. are also presented