  Recently a Mellin-space formula was conjectured for the form of correlation
functions of $1/2$ BPS operators in planar $\mathcal{N}=4$ SYM in the strong 't
Hooft coupling limit. In this work we report on the computation of two
previously unknown four-point functions of operators with weights $\langle 2345
\rangle$ and $\langle 3456\rangle$, from the effective type-IIB supergravity
action using AdS/CFT. These correlators are novel: they are the first
correlators with all-different weights and in particular $\langle 3456\rangle$
is the first next-next-next-to-extremal correlator to ever have been computed.
We also present simplifications of the known algorithm, without which these
computations could not have been executed without considerable computer power.
The main simplifications we found are present in the computation of the
exchange Lagrangian and in the computation of $a$ tensors. After bringing our
results in the appropriate form we successfully corroborate the recently
conjectured formula.
