The design of large-scale, high-numerical-aperture, and broadband achromatism is a big challenge in metalens research. In fact, many colorful imaging systems have RGB color filters, which means the achromatism only for RGB lights would be sufficient. Avoiding broadband achromatism is expected to greatly improve the working efficiency of metalenses. Nevertheless, a proper bandpass filter is necessary under a white light illumination in the metalens integrated imaging system. Here we propose a bandpass-filter-integrated multiwavelength achromatic metalens ($\mathrm{NA}=0.2$), which is designed using a searching optimization algorithm to achieve the achromatism of RGB lights with high efficiencies. The bandpass filter is implemented by composite DBRs and defect layers, by which three desired wavelengths are selected out. The simulations and experiments on the filter-integrated metalens definitely show a good RGB achromatism. Further imaging experiments demonstrate a higher signal-to-noise ratio and resolution compared with the one without the filter. Our approach provides not only an RGB achromatic meta-imaging device but also a new route to access a highly efficient spectrum tailoring metasystem by incorporating bandpass filter designs.