Clearly, $x=0$ is not a solution. Dividing both numerator and denominator by $x$ leads $$\frac{2}{2x - 5 +\frac{3}{x}} +\frac{13}{2x +1 +\frac{3}{x}}=6$$ Letting $y=2x+\frac{3}{x}$, then $$\frac{2}{y-5}+\frac{13}{y+1}=6\implies y=1, \frac{11}{2}$$ Setting this back: $$2x+\frac{3}{x}=1, \frac{11}{2}\implies x=\boxed{\frac{3}{4}, 2}$$