How to evaluate the closed form of the following integrals

\[\int\limits_0^1 {\frac{{{{\log }^{{p_1}}}\left( {1 - x} \right){{\log }^{{p_2}}}\left( x \right){{\log }^{{p_3}}}\left( {1 + x} \right)}}{x}} dx,\]

where $p_1,p_2,p_3$ are nonnegative integers and ${p_1} + {p_3} \ge 1$.

It can be expressed in terms of multiple zeta values?