If
\[
\int_{2}^{3}
\frac{3\log x}
{3\log x+\log(125-75x+15x^2-x^3)}
\,dx
=
k,
\]
then
\[
4k^2+2k+1=
\]
Show Hint
If an integral contains expressions like
\[
f(x)
\quad \text{and} \quad
f(a+b-x),
\]
always test the substitution \(x\to a+b-x\). Many definite integrals collapse instantly using symmetry.