Points on hyperelliptic curves: $y^2=5(x^2-3)(x^2+2)(x^2-11/5)$

You can apply the so-called Elliptic Chabauty over the biquadratic field $K:=\mathbb{Q}(\sqrt{3},\sqrt{11/5})$ (also equal to the field adjoining a root of $ 25x^4 - 260x^2 + 16$). Over this field there are two possible 2-coverings (one corresponding to the points with coordinate $x=1$, the other with coordinate $x=-1$). Both curves have a map to a genus one curve given by an equation $$ \delta_{\pm} y^2=(x-\sqrt{3})(x-\sqrt{11/5})(x^2-2)$$ with $$\delta_{\pm}=-(\pm 1-\sqrt{3})(\pm 1-\sqrt{11/5})$$ (one for every sign). Both cases we get an elliptic curve with rank 2, so we can apply Elliptic Chabauty MAGMA function, which answers that the only points with rational $x$-coordinate are the ones with $x=\pm 1$.

You can do (attempt) etale descent over one of the quadratic number fields defined by any of the factors. See section 8.3 of my paper Elliptic curves over $\mathbb Q$ and 2-adic images of Galois with Jeremy Rouse for an example of how to do this (there is code available on the arXiv too).