To enable the performance optimization of application mapping on modern field-programmable gate arrays (FPGAs), certain critical path portions of the designs might be prearranged into many multi-cell macros during synthesis. These movable macros with constraints of shape and resources lead to challenging mixed-size placement for FPGA designs which cannot be addressed by previous works of analytical placers. In this work, we propose AMF-Placer, an open-source Analytical Mixed-size FPGA placer supporting mixed-size placement on FPGA, with an interface to Xilinx Vivado. To speed up the convergence and improve the quality of the placement, AMF-Placer is equipped with a series of new techniques for wirelength optimization, cell spreading, packing, and legalization. Based on a set of the latest large open-source benchmarks from various domains for Xilinx Ultrascale FPGAs, experimental results indicate that AMF-Placer can improve HPWL by 20.4%-89.3% and reduce runtime by 8.0%-84.2%, compared to the baseline. Furthermore, utilizing the parallelism of the proposed algorithms, with 8 threads, the placement procedure can be accelerated by 2.41x on average.