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Abstract: A generalization of Hodges's method of obtaining the gradient expansion is derived and then employed to determine the sixth-order term. The formula for it is as follows: ${T}_{6}[\ensuremath{\rho}]=\frac{{(3{\ensuremath{\pi}}^{2})}^{\ensuremath{-}413}}{45360\ensuremath{\int}{\ensuremath{\rho}}^{\frac{\ensuremath{-}1}{3}}[13{(\frac{\ensuremath{\nabla}{\ensuremath{\nabla}}^{2}\ensuremath{\rho}}{\ensuremath{\rho}})}^{2}+\frac{2575}{144{(\frac{{\ensuremath{\nabla}}^{2}\ensuremath{\rho}}{\ensuremath{\rho}})}^{3}}+\frac{249}{16{(\frac{\ensuremath{\nabla}\ensuremath{\rho}}{\ensuremath{\rho}})}^{2}(\frac{{\ensuremath{\nabla}}^{4}\ensuremath{\rho}}{\ensuremath{\rho}})}+\frac{1499}{18{(\frac{\ensuremath{\nabla}\ensuremath{\rho}}{\ensuremath{\rho}})}^{2}{(\frac{{\ensuremath{\nabla}}^{2}\ensuremath{\rho}}{\ensuremath{\rho}})}^{2}}\ensuremath{-}\frac{1307}{36{(\frac{\ensuremath{\nabla}\ensuremath{\rho}}{\ensuremath{\rho}})}^{2}(\frac{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\nabla}}\ensuremath{\rho}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\nabla}}{\ensuremath{\nabla}}^{2}\ensuremath{\rho}}{{\ensuremath{\rho}}^{2}})}+\frac{343}{18{(\frac{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\nabla}}\ensuremath{\rho}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\nabla}}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\nabla}}\ensuremath{\rho}}{{\ensuremath{\rho}}^{2}})}^{2}+\frac{8341}{72(\frac{{\ensuremath{\nabla}}^{2}\ensuremath{\rho}}{\ensuremath{\rho}}){(\frac{\ensuremath{\nabla}\ensuremath{\rho}}{\ensuremath{\rho}})}^{4}}\ensuremath{-}\frac{1600495}{2592{(\frac{\ensuremath{\nabla}\ensuremath{\rho}}{\ensuremath{\rho}})}^{6}}]{d}^{3}r}}$. For atomic densities, ${T}_{6}[\ensuremath{\rho}]$ is divergent near the nucleus and at large distances.