$\int \frac{dx}{x[10+7\log x+{\left(\log x\right)}^2]}=$

$\log x=t$

$\frac{1}{x}dx=dt$

Given integral

$=\int \frac{dt}{10+7t+t^2}$

$\Rightarrow \int \frac{dt}{{(t+\frac{7}{2})}^2-\frac{9}{4}}$

$=\int \frac{1}{{(t+\frac{7}{2})}^2-{\left(\frac{3}{2}\right)}^2}dt$

$=\frac{1}{2\left(\frac{3}{2}\right)}\log \left|\frac{t+\frac{7}{2}-\frac{3}{2}}{t+\frac{7}{2}+\frac{3}{2}}\right|+c$

$=\frac{1}{3}\log \left|\frac{t+2}{t+5}\right|+c$

$=\frac{1}{3}\log \left|\frac{\log x+2}{\log x+5}\right|+c$