This paper investigates the visual inertial structure from motion problem with special focus on its observability properties. The considered system is a sensor assembling consisting of a monocular camera and inertial sensors (i.e., three orthogonal accelerometers and three orthogonal gyroscopes). The considered state contains the vehicle speed and attitude, the biases on the inertial measurements, the position of the features observed by the camera, the parameters characterizing the extrinsic camera-inertial sensors calibration and the magnitude of the gravity. In the case of a single feature, this state consists of $24$ components. The observability analysis is carried out by using the method of {\it continuous symmetries}, which has recently been introduced. Due to the system complexity, a direct derivation of the system continuous symmetries would require an excessive computational cost. For this reason, new theoretical results are derived in order to significantly reduce the load of symbolic computation. The paper contribution is therefore twofold. From one side it provides new theoretical results, which allow significantly reducing the symbolic computational load necessary to investigate the observability properties of any estimation problem by making the use of the concept of continuous symmetry. From the other side, it provides a new result in the framework of the observability of the visual-inertial structure from motion. Specifically, it is proven that, the information contained in the data provided by a monocular camera which observes a single point-feature and by an Inertial Measurement Unit (IMU), allows estimating the absolute scale, the speed in the local frame, the absolute roll and pitch angles, the biases which affect the accelerometer's and the gyroscope's measurements, the magnitude of the gravitational acceleration and the extrinsic camera-IMU calibration.