  We theoretically analyze the intensity of the resonant inelastic X-ray
scattering (RIXS) based on Ansatz that the electron is fractionalized into two
components. By fitting the angle resolved photoemission spectroscopy (ARPES)
data of a cuprate high-$T_{\rm c}$ superconductor
Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ and the accompanied machine learning results
obtained before by Yamaji {\it et al.} (arXiv:1903.08060), we determine the
paprameters of the two-component fermion model which describes the electron
fractionalization. The RIXS spectra are predicted by using this description. We
find that the intensity is enhanced in the superconducting phase relative to
the normal or pseudogap phase. Since the enhancement is unusual, we propose
this can be used as a critical test whether the unconventional electron
fractionalization captures the essence of the cuprate superconductivity. This
type of combined analyses using multiple independent spectroscopy data, with
the help of theoretical and machine learning insights, regarded as {\it
integrated spectroscopy} opens a new route to study difficult open issues in
strongly correlated electron systems.
