Based on the simplified Weiss-Tabor-Carnevale (WTC) algorithm and symbolic computation, Painlevé properties and analytic solutions of the variable coefficient nonlinear Schrdinger (NLS) equation are investigated, which involves four arbitrary functions of space-time. Among the four variable coefficients of the equation, the first two are two-order dispersion of longitudinal distance and nonlinear coefficient respectively, and the last two are the real and imaginary parts of the fiber loss factor. Relationship among the four variable coefficients are derived with WTC method when the equation is Painlevé integrable. Three special forms of rational function solutions are derived with Painlevé truncation method, and the partial solutions of the equation are obtained by using the variable separation method. The obtained results are the extension of existing conclusions.