Various (Poly)logarithmic Integrals and Multiple Zeta Values

We show how integrals over [0,1] of products of the form
log^n(x)log^m(1-x)Li_p(x)Li_q(1-x)/x can be evaluated in
terms of multiple zeta values.  We also show how
integrals over [0,1] of log^n(x)log^m(1-x)x^i, i nonnegative,
can be written as rational polynomials in ordinary zeta
values and a kind of generalized binomial coefficients.