Let
$\begin{aligned} I & =\int \frac{2 x^{12}+5 x^9}{\left(1+x^3+x^5\right)^3} d x \\ & =\int \frac{2 x^{12}+5 x^9}{\left[x^5\left(1+\frac{1}{x^2}+\frac{1}{x^5}\right)\right]^3} d x \\ & =\int \frac{2 x^{12}+5 x^9}{x^{15}\left(1+\frac{1}{x^2}+\frac{1}{x^5}\right)^3} d x \\ & =\int \frac{2 x^{-3}+5 x^{-6}}{\left(1+\frac{1}{x^2}+\frac{1}{x^5}\right)^3} d x\end{aligned}$
$\begin{aligned} & \text { Put, } 1+\frac{1}{x^2}+\frac{1}{x^5}=t \Rightarrow\left(-\frac{2}{x^3}-\frac{5}{x^6}\right) d x=d t \\ & \therefore I=-\int \frac{d t}{t^3}=\frac{1}{2 t^2}+C=\frac{1}{2\left(1+\frac{1}{x^2}+\frac{1}{x^5}\right)^2}+C \\ & \quad=\int \frac{x^{10}}{2\left(x^5+x^3+1\right)^2}+C \\ & \therefore m=10, l=2, r=2 \\ & \therefore \quad \frac{m-l}{r}=\frac{10-2}{2}=4\end{aligned}$