We present a real-time and fully integrated quantum random number generator (QRNG) by measuring laser phase fluctuations. The QRNG scheme based on laser phase fluctuations is featured for its capability of generating ultra high-speed random numbers. However, the speed bottleneck of a practical QRNG lies on the limited speed of randomness extraction. To close the gap between the fast randomness generation and the slow post-processing, we propose a pipeline extraction algorithm based on Toeplitz matrix hashing and implement it in a high-speed field-programmable gate array. Further, all the QRNG components are integrated into a module, including a compact and actively stabilized interferometer, high-speed data acquisition, and real-time data post-processing and transmission. The final generation rate of the QRNG module with real-time extraction can reach 3.2 Gbps.