$f=\frac{x+3}{x\left(4x+1\right)}$
$x\neq -\frac{1}{4}\text{ and }x\neq 0$

$\left\{\begin{matrix}x=\frac{\sqrt{f^{2}+46f+1}-f+1}{8f}\text{; }x=\frac{-\sqrt{f^{2}+46f+1}-f+1}{8f}\text{, }&f\neq 0\\x=-3\text{, }&f=0\end{matrix}\right.$

$\left\{\begin{matrix}x=\frac{\sqrt{f^{2}+46f+1}-f+1}{8f}\text{; }x=\frac{-\sqrt{f^{2}+46f+1}-f+1}{8f}\text{, }&f\leq -4\sqrt{33}-23\text{ or }\left(f\neq 0\text{ and }f\geq 4\sqrt{33}-23\right)\\x=-3\text{, }&f=0\end{matrix}\right.$