  Superfield methods can be used to determine the precise way the self-dual
five-form couples to the metric in the first non-trivial $\alpha'$ corrections
to type IIB supergravity. We explicitly compute the exact tensor structure of
these terms. This requires extensive use of computing algorithms to reduce the
complicated expressions that appear to a surprisingly simple form. Along the
way we show a new method of computing Schouten identities. With this result we
clarify under which conditions one may neglect the five-form higher derivative
terms. We comment on corrections to the thermodynamics of charged black holes.
