Radar communication integration is an effective solution to spectrum competition and electromagnetic interference. The integrated waveform determines the architecture and performance of the integrated system. This paper proposes a radar communication integrated waveform based on multi-frequency-band Chirp-BOK to solve the low communication rate issue in Chirp Binary Orthogonal Keying (Chirp-BOK). Furthermore, it designs modulation and demodulation methods based on Inverse Fast Fourier Transform(IFFT) and Fast Fourier Transform(FFT) to reduce system complexity. To tackle the high Bit Error Rate(BER) issue, an optimization method using a double-window approach to reduce the judgment frequency is proposed. The ambiguity function and Doppler performance of the multi-frequency-band Chirp-BOK are analyzed. Simulation results demonstrate that the multi-frequency-band Chirp-BOK not only enhances the communication rate but also reduces the BER, ensuring detection resolution and excellent Doppler tolerance.