Partial pluricomplex energy and integrability exponents of plurisubharmonic functions

We give a sufficient condition on the Monge-Amp\`ere mass of a plurisubharmonic function $u$ for $\exp (- 2 u)$ to be locally integrable. This gives a pluripotential theoretic proof of a theorem by J-P. Demailly.
Comment: extended version with new results and more applications