Formula for Integration (by Integration by Parts) For a Product of 3 Functions

Question

Given general formulas $u$, $v$, and $w$, what is the general formula for integration by parts to calculate $\int uvw \mathbb{d}x$?

Answer 1

Here is my take on it: $$\int uvw=u(v \int w- \int v'( \int w))- \int u'(v \int w- \int v'( \int w))$$

Answer 2

One general idea with products of three functions is to use the product rule in the form $$ (u v w)' = u' v w + u v' w + uv w' $$ and the get partial integration in the form $$ \int u' v w = uvw - \int u v' w - \int uv w' $$ and then the solution of your problem could be straightforward, but tedious.