Rational solutions to the Variants of Erd\H{o}s- Selfridge superelliptic curves

For the superelliptic curves of the form $$(x+1)\cdots (x+i-1)(x+i+1)\cdots (x+k)=y^{\mathrm{\ell }}$$ with $x,y\in \mathbb{Q}$, $y\mathrm{\ne }0$, $k\ge 3$, $1\leq i\leq k$,$\ell \geq 2,$ a prime, Das, Laishram, Saradha, and Edis showed that the superelliptic curve has no rational points for $\mathrm{\ell }\ge e^{3^k}$. In fact, the double exponential bound, obtained in these papers is far from reality. In this paper, we study the superelliptic curves for small values of $k$. In particular, we explicitly solve the above equation for $4\le k\le 8.$