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We present a comprehensive analysis of practical p-n-p Ge/Ge_1-xSn_x/Ge heterojunction phototransistors (HPTs) for design optimization for efficient infrared detection. Our design includes a Ge_1-xSn_x narrow-bandgap semiconductor as the active layer in the base layer, enabling extension of the photodetection range from near-infrared to mid-infrared to perform wide-range infrared detection. We calculate the current gain, signal-to-noise ratio (SNR), and optical responsivity and investigate their dependences on the structural parameters to optimize the proposed Ge_1-xSn_x p-n-p HPTs. The results show that the SNR is strongly dependent on the operation frequency and that the introduction of Sn into the base layer can improve the SNR in the high-frequency region. In addition, the current gain strongly depends on the Sn content in the Ge_1-xSn_x base layer, and a Sn content of 6%-9% maximizes the optical responsivity achievable in the infrared range. These results provide useful guidelines for designing and optimizing practical p-n-p Ge_1-xSn_x HPTs for high-performance infrared photodetection.