  We derive analytic results for the symbol of certain two-loop Feynman
integrals relevant for seven- and eight-point two-loop scattering amplitudes in
planar $\mathcal{N}=4$ super-Yang--Mills theory. We use a bootstrap inspired
strategy, combined with a set of second-order partial differential equations
that provide powerful constraints on the symbol ansatz. When the complete
symbol alphabet is not available, we adopt a hybrid approach. Instead of the
full function, we bootstrap a certain discontinuity for which the alphabet is
known. Then we write a one-fold dispersion integral to recover the complete
result. At six and seven points, we find that the individual Feynman integrals
live in the same space of functions as the amplitude, which is described by the
9- and 42-letter cluster alphabets respectively. Starting at eight points
however, the symbol alphabet of the MHV amplitude is insufficient for
individual integrals. In particular, some of the integrals require algebraic
letters involving four-mass box square-root singularities. We point out that
these algebraic letters are relevant at the amplitude level directly starting
with N$^2$MHV amplitudes even at one loop.
