This paper proposes a flexible beamforming approach for integrated sensing and communication (ISAC) systems with movable antennas (MAs). The system integrates a bistatic radar with a multi-user multiple-input-single-output (MU-MISO) system, where the base station (BS) is equipped with MAs. This allows for array response reconfiguration by adjusting antenna positions, enabling a joint beamforming and antenna position optimization problem to maximize communication rate and sensing mutual information (MI). The fractional programming (FP) method is used to transform the non-convex objective function, and the beamforming matrix and antenna positions are alternatively updated. Karush–Kuhn–Tucker (KKT) conditions are used to derive the closed-form solution for the beamforming matrix, while an efficient search-based projected gradient ascent (SPGA) method is proposed for updating the antenna positions. Simulation results show that MAs significantly enhance ISAC performance, achieving a 59.8% performance gain compared to fixed uniform arrays. The system model considers a dual functional radar and communication (DFRC) BS serving K users and sensing one target. The communication channel between the transmitter (Tx) and user k is expressed as a function of antenna positions. The communication and sensing signal models are derived, with the communication rate and sensing MI as performance metrics. The problem formulation aims to maximize the sum of communication rate and MI by optimizing beamforming and antenna positions. A fractional programming-based alternating optimization (AO) algorithm is proposed to find the optimal solution, with the beamforming and antenna positions solved using direct gradient ascent (DGA) and SPGA methods, respectively. The proposed SPGA-based flexible beamforming algorithm outperforms the DGA-based method, achieving significant performance gains in high SNR settings and with a large feasible moving region. The algorithm is shown to be effective for ISAC systems, with the weighting factor carefully selected to meet specific sensing and communication requirements. The results demonstrate that flexible beamforming with MAs can enhance ISAC performance, and the system with only 4 MAs can outperform the system with 8 fixed antennas, providing insights for reducing hardware costs in engineering applications.This paper proposes a flexible beamforming approach for integrated sensing and communication (ISAC) systems with movable antennas (MAs). The system integrates a bistatic radar with a multi-user multiple-input-single-output (MU-MISO) system, where the base station (BS) is equipped with MAs. This allows for array response reconfiguration by adjusting antenna positions, enabling a joint beamforming and antenna position optimization problem to maximize communication rate and sensing mutual information (MI). The fractional programming (FP) method is used to transform the non-convex objective function, and the beamforming matrix and antenna positions are alternatively updated. Karush–Kuhn–Tucker (KKT) conditions are used to derive the closed-form solution for the beamforming matrix, while an efficient search-based projected gradient ascent (SPGA) method is proposed for updating the antenna positions. Simulation results show that MAs significantly enhance ISAC performance, achieving a 59.8% performance gain compared to fixed uniform arrays. The system model considers a dual functional radar and communication (DFRC) BS serving K users and sensing one target. The communication channel between the transmitter (Tx) and user k is expressed as a function of antenna positions. The communication and sensing signal models are derived, with the communication rate and sensing MI as performance metrics. The problem formulation aims to maximize the sum of communication rate and MI by optimizing beamforming and antenna positions. A fractional programming-based alternating optimization (AO) algorithm is proposed to find the optimal solution, with the beamforming and antenna positions solved using direct gradient ascent (DGA) and SPGA methods, respectively. The proposed SPGA-based flexible beamforming algorithm outperforms the DGA-based method, achieving significant performance gains in high SNR settings and with a large feasible moving region. The algorithm is shown to be effective for ISAC systems, with the weighting factor carefully selected to meet specific sensing and communication requirements. The results demonstrate that flexible beamforming with MAs can enhance ISAC performance, and the system with only 4 MAs can outperform the system with 8 fixed antennas, providing insights for reducing hardware costs in engineering applications.