Yuri N. KOSOVTSOV
It is known (due to Mordukhai-Boltovski, Ritt, Prelle, Singer, Christopher and others) that if given rational ODE has a Liouvillian first integral then the corresponding integrating factor of the ODE must be of a very special form  $\mu=\prod_i P_i^{a_i} \exp(\prod_j Q_j^{b_j})$, where $P_i$'s and $Q_j$'s are irreducible polynomials, $a_i$'s and $b_j$'s are constants. These results lead to a partial algorithm for finding Liouvillian first integrals. However there are two main complications on the way to obtaining polynomials in the integrating factor form. One have to find first of all upper bound for the degrees of polynomials $P_i$ and $Q_j$ (this problem is unsolved yet) and then the set of coefficients for each of the polynomials by computation-intensive method of undetermined parameters. As a result this approach was implemented in CAS only for first and relatively simple second order ODEs. We propose an algebraic method for finding polynomials of the integrating factors forms of rational ODEs of any order $\frac{d^n y}{dx^n}= A(x,y,y^{(1)},...,y^{(n-1)}) / B(x,y,y^{(1)},...,y^{(n-1)})$, where $A$ and $B$ are polynomials, based on examination of the resultants of  $A$ and $B$ (or $A$ and $P_i$ and the like). If one of the polynomials $P_i$ or $Q_j$ is known, we are able to obtain by elementary computations all of the possible polynomials as candidates to the integrating factor form. We consider ways for obtaining such initial polynomial. If both of $A$ and $B$ of given ODE's are not constant polynomials, the method can determine in finite terms the explicit expression of an integrating factor if the ODE permits integrating factors of above mentioned form and then the Liouvillian first integral. The tests of procedures, based on proposed method, implemented in Maple in the case of  rational integrating factors (see http://www.maplesoft.com/applications/app_center_view.aspx?AID=1309