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The nonequilibrium dynamics of disordered many-body quantum systems after a quantum quench unveils important insights about the competition between interactions and disorder, yielding, in particular, an interesting perspective toward the understanding of many-body localization. Still, the experimentally relevant effect of bond randomness in long-range interacting spin chains on their dynamical properties have so far not been investigated. In this Letter, we examine the entanglement entropy growth after a global quench in a quantum spin chain with randomly placed spins and long-range tunable interactions decaying with distance with power α. Using a dynamical version of the strong disorder renormalization group we find for α>αc that the entanglement entropy grows logarithmically with time and becomes smaller with larger α as S(t)=Spln(t)/(2α). Here, Sp=2ln2−1. We present results of numerical exact diagonalization calculations for system sizes up to N∼16 spins, in good agreement with the analytical results for sufficiently large α>αc≈1.8. For α<αc, we find that the entanglement entropy grows as a power law with time, S(t)∼tγ(α) with 0<γ(α)<1 a decaying function of the interaction exponent α.