The distribution of linear combinations of independent chi-square random variables is intimately related with the distribution of quadratic forms in normal random variables. As such, this distribution has been studied by many authors. However, there is still some room left for improvement, since while some simpler approximations do not yield sufficiently good results, other approximations which show a better performance are sometimes too complicated to be implemented in practical terms. In this paper the exact distribution of linear combinations of independent chi-square random variables is obtained, for some particular cases, in closed finite highly manageable forms, while for more general cases near-exact approximations are obtained, which are able to yield very manageable and well-performing approximations. Numerical studies compare the performance of these near-exact distributions with other existing approximations and distributions and show how sharp are the approximations provided by these near-exact distributions. A useful subproduct that is obtained is closed form expressions for the distribution of quadratic forms and for some instances of ratios of quadratic forms, useful in anova and other linear or mixed-linear models where heterocedasticity is present or assumed. Solutions for the problem of the distribution of the statistic associated with the Behrens-Fisher problem are then in turn obtained as a much useful subproduct of the distribution of ratios of quadratic forms. Modules programmed in Mathematica®, Maxima and R for the implementation of the distributions developed are made available at the site https://sites.google.com/site/lincombchisquares .