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We perform a combined study of the two hadronic decays $D^+\to K^-\pi^+\pi^+$and $D_s^+\to K^+K^+\pi^-$ using a detailed analysis of the semileptonic decays$D^+\to K^-\pi^+ \ell^+\nu_\ell$ ($\ell=e, \mu$) thanks to the high-statisticsdataset provided by the BESIII Collaboration. We propose simple and suitableamplitude parametrizations of the studied reactions that shall be of interestto experimentalists for upcoming analyses. These new parametrizations are basedon the na\"ive factorization hypothesis and the description of the resultingmatrix elements in terms of well-known hadronic form factors, with specialemphasis on the $K\pi$ scalar and vector cases. Such form factors account fortwo-body final state interactions which fulfill analyticity, unitarity andchiral symmetry constraints. As a result of our study, we find that the$P$-wave contribution fits nicely within the na\"ive-factorization approach,whereas the $S$-wave contribution requires complex Wilson coefficients thathint for possibly genuine three-body non-factorizable effects. Our hypothesisis further supported by the examination of $D_s^+\to K^+K^+\pi^-$ decays, wherewe achieve a description in overall good agreement with data.

A combined study of hadronic $D^+\to K^-\pi^+\pi^+$ and $D_s^+\to  K^+K^+\pi^-$ decays by means of the analysis of semileptonic $D^+\to K^-\pi^+  \ell^+\nu_\ell$ decays