Comparative analysis between a network and a random graph model can uncover network properties that significantly deviate from those in random networks. The standard random graph model used for comparison uniformly samples random graphs with the same degrees as the network data, often achieved through edge-swap algorithms. However, for hypergraphs, fewer such methodologies are available. This study introduces the Hypercurveball algorithm, designed to sample random, potentially directed, hypergraphs with fixed degrees. Minor adjustments enable the sampling of hypergraphs without degenerate hyperedges, self-loops, or multi-hyperedges. For most of these algorithms, we prove whether they sample uniformly or with bias. We experimentally show that the Hypercurveball algorithm can be significantly faster or slower than the standard hyperedge-shuffling algorithm, which is the hyperedge-equivalent of the edge-swap algorithm. We present criteria on the hypergraph degree sequence that indicate when the Hypercurveball algorithm is more efficient than the standard hyperedge-shuffling method. Finally, our experimental results suggest polynomial scaling of the mixing time for both the Hypercurveball and hyperedge-shuffling algorithms.