Knowledge base completion (KBC) has benefitted greatly by learning explainable rules in an human-interpretable dialect such as first-order logic. Rule-based KBC has so far, mainly focussed on learning one of two types of rules: conjunction-of-disjunctions and disjunction-of-conjunctions. We qualitatively show, via examples, that one of these has an advantage over the other when it comes to achieving high quality KBC. To the best of our knowledge, we are the first to propose learning both kinds of rules within a common framework. To this end, we propose to utilize logical neural networks (LNN), a powerful neuro-symbolic AI framework that can express both kinds of rules and learn these end-to-end using gradient-based optimization. Our in-depth experiments show that our LNN-based approach to learning rules for KBC leads to roughly 10% relative improvements, if not more, over SotA rule-based KBC methods. Moreover, by showing how to combine our proposed methods with knowledge graph embeddings we further achieve an additional 7.5% relative improvement.