The known dynamic regressor extension and mixing method (DREM) is combined with the proposed filter of a new type, which uses the integration operation with forgetting, and the recursive least-squares method to develop the new I-DREM model reference adaptive control (MRAC) system. It does not require a priori knowledge of the sign or the elements values of the control input matrix of the plant. It also provides the exponential convergence of the adaptation process (with the automatically adjustable adaptation rate) without the regressor persistent excitation. Such control system allows to solve three actual problems of the adaptive control: 1) to provide the exponential convergence of the controller parameter error under the condition of the regressor initial excitation, 2) to make such convergence monotonic, 3) to calculate the adaptation rate online according to the current regressor value. Some numerical experiments are conducted to demonstrate the effectiveness of the proposed method.