A formula for evaluation of the distribution of a linear combination of independent inverted gamma random variables by one-dimensional numerical integration is presented. The formula is direct application of the inversion formula given by [J.Gil-Pelaez: Note on the inversion theorem. Biometrika 38 (1951), 481-482.]. This method is applied to computation of the generalized $p$-values used for exact significance testing and interval estimation of the parameter of interest in the Behrens-Fisher problem and for variance components in balanced mixed linear model.