Complicated expansions of solutions to an ordinary differential equation

We consider an ordinary differential equation of a very general form. Let its truncated equation, corresponding to a vertex or to a nonhorisontal edge of the polygon of the initial equation, have a solution in the form of product of a power of the independent variable and of a series of powers of its multiple logarithms. We show, that under absence of critical numbers such a nonpower asymptotics of a solution to the initial equation can be prolonged as a power-logarithmic expansion of a solution to the initial equation. It is a series of powers of the independent variable, coefficients of which are series of powers of its multiple logarithms. We give examples of the calculations. The main attention is given to explanations of the computational algorithms.