We present a detailed study of the incorporation of gallium and indium into (GaIn)As grown by low-pressure metal-organic vapor-phase epitaxy (MOVPE) using the group-V precursors arsine (AsH3), tertiarybutylarsine (TBAs) and diethyltertiarybutylarsine (DEtBAs). We have grown strain-compensated multilayer structures of (GaIn)As layers alternately with AsH3 and one of the organic sources DEtBAs and TBAs, In order to clarify the origin of the changed solid composition using organic sources, we analyzed X-ray diffraction data by dividing each (GaIn)As layer into a GaAs sublayer and an InAs sublayer. Comparing simulated and measured X-ray diffraction spectra, we find that the use of organic precursors in MOVPE growth increases the In incorporation rate by increasing the InAs growth rate. This effect is even higher for DEtBAs than for TBAs. An As-source-dependent supply of active hydrogen, controlling the In incorporation, as well as vapor phase processes and surface reactions are discussed.