Fetal cerebral brain magnetic resonance imaging (MRI) is critical for the detection of abnormal brain development before birth. A key image processing step is the reconstruction of a 3D high resolution volume from the acquired series of 2D slices. Several types of MR sequences are commonly acquired during a scanning session, but current reconstruction methods consider each sequence (or contrast) separately. Multi-contrast techniques have been proposed but they do not compensate for potential movement during the acquisition, which occurs almost systematically in the context of fetal MRI. In this work, we introduce a new method for the joint reconstruction of multiple 3D volumes from different contrasts. Our method combines the redundant and complementary information across several stacks of 2D slices from different acquisition sequences via an implicit neural representation, and includes a slice motion correction module. Our results on both simulations and real data acquired in clinical routine demonstrates the relevance and efficiency of the proposed method.