We present a method that enables performing x-ray phase-contrast imaging (XPCI) computed tomography with a laboratory setup using a single image per projection angle, eliminating the need to move optical elements during acquisition. Theoretical derivation of the method is presented, and its validity conditions are provided. The object is assumed to be quasihomogeneous, i.e., to feature a ratio between the refractive index and the linear attenuation coefficient that is approximately constant across the field of view. The method is experimentally demonstrated on a plastics phantom and on biological samples using a continuous rotation acquisition scheme achieving scan times of a few minutes. Moreover, we show that such acquisition times can be further reduced with the use of a high-efficiency photon-counting detector. Thanks to its ability to substantially simplify the image-acquisition procedure and greatly reduce collection times, we believe this method represents a very important step towards the application of XPCI to real-world problems.