Does the integral $\int \frac{1}{(ax+b)(cx+d)} dx$ converge?

Question

Let $a, b, c, d \in \mathbb{R}$. I was wondering does the integreal $\int_A^{\infty} \frac{1}{(ax+b)(cx+d)} dx$ converge? where the integrand is well defined for $x\geq A$?

I think it should because $1/x^2$ does... but I wasn't sure. Any info would be appreciated!

Answer 1

Yes, it converges, since$$\left\lvert\frac1{(ax+b)(cx+d)}\right\rvert\leqslant\frac2{\lvert ac\rvert x^2}$$if $x$ is large enough and because $\int_A^\infty\frac{\mathrm dx}{x^2}$ converges.