Suppose $f(x,y)\geq 0$ is a measurable function on $\mathbb{R}\times\mathbb{R}$, and $$\int_\mathbb{R}\left(\int_\mathbb{R}f(x,y)d\mu(x)\right)d\mu(y)<\infty$$
Do we have that $\dfrac{xyf(x,y)}{x^2+y^2}$ is integrable on $\mathbb{R}\times\mathbb{R}$.
By Fubini, we have that $$\int_{\mathbb{R}\times\mathbb{R}}f(x,y)d(\mu\times\mu)<\infty$$but I don't know about the factor $\dfrac{xy}{x^2+y^2}$.
It would help to show $\dfrac{xy}{x^2+y^2}$ is bounded --- say $\left|\dfrac{xy}{x^2+y^2}\right| \le C$ for all $x,y$. Then $$ \left|f(x,y)\dfrac{xy}{x^2+y^2}\right| \le C|f(x,y)|, $$ so the integral of that can easily be shown to be finite.
The function $\dfrac{xy}{x^2+y^2}$ is homogeneous of degree $0$, i.e. multiplying $(x,y)$ by a constant has the effect of multiplying the whole fraction by $1$. Therefore it's enough to show that the function is bounded on the unit circle $x^2+y^2=1$. It's a continuous function on a compact set (the circle is compact), so it's bounded.