Counting points on hyperelliptic curves of type $y^2=x^{2g+1}+ax^{g+1}+bx$

In this work, we investigate hyperelliptic curves of type shown in the title over the finite field $\mathbb F_q$, $q=p^n$, $p>2$. For the case of $g=3$ or $4$, $p\nmid4g$ and $b$ is a $4g$-root, we provide efficient methods to compute the number of points in the Jacobian of the curve.