Convergence of an improprer integral

Question

For which $\alpha \in \mathbb{R}$ does

$$ \int_1^{+\infty} \frac{\log x^2 +3}{x^\alpha(12\log^4x + 7 \log^2x + 1)} \,dx$$

converge?

In particular, how do I use the function $g(x) = 1/x^\alpha$ to study it?

Thanks