In this work we provide an effective method to prove the formal integrability of the resonant saddles. The method is based on the use of a blow-up and the resolution of a recurrence differential equation using induction. Using the method some open integrability problems for certain resonant saddles are solved.The authors are grateful to the referee for his/her valuable instructions and suggestions to improve this paper. The first author acknowledges the financial support of the Slovenian Research Agency (core funding No. P1-0306). The second author is partially supported by a MINECO/ FEDER grant number MTM2017-84383-P and an AGAUR (Generalitat de Catalunya) grant number 2017SGR-1276.