Let \(k\) be a numerical field of characteristic zero. The authors exhibit new algorithms connected with a problem of numerical integration of systems of first order algebraic differential equations over \(k\) of the form (1) \(f(x, \dot{x}) =0\), where \(f:k^{2n}\rightarrow k^{n}\) is a rational mapping, when a Jacobian determinant of the system is zero. In such cases the application of the implicit function theorem is impossible and local passage from a system (1) to its explicit form (2) \(\dot{x}=g(x)\) needs the additional discrete information (differentiation index, differential Hilbert function, algebraic parametric set). In order to obtain this information, one may consider a system (3) \(f(x, \dot{x}) =y\) (generic perturbation of (1)) and fulfill the process of completion, with the help of algorithms which have exponential complexity. In this paper the authors offer an alternative approach based on the use of differentially algebraic technics [see \textit{E. R. Kolchin}, Differential algebra and algebraic group, New York: Academic Press (1973; Zbl 0264.12102), Chs. 3-4] and numeric-symbolic algorithms [see \textit{G. Reid, C. Smith, J. Verschelde}, ''Geometric completion of differential systems using numeric-symbolic continuation'', SIGSAM Bull. 36, No. 2, 1--17 (2002; Zbl 1054.65090)] which have polynomial complexity. Reviewer's remark: The results of the paper are obtained assuming that the components of the map \(f\) are differentially algebraically independent over \(k\) (i.e. do not satisfy any differential equation over \(k\)). However it is not clear as far as realization of this condition for rational functions is easily checked.