We have derived the fourth-order generalized density matrix expansion and used it to construct various exchange-energy functionals. The fourth-order terms depend on several quantities containing invariants of the second-order derivative matrices of the orbitals and the electron density. The impact of these variables on the accuracy of exchange functionals has never been studied before and we here demonstrate their importance. The new exchange functionals show excellent accuracy (as compared to Hartree-Fock exchange energies) surpassing those of our previously developed exchange functionals.