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$$\int_{l}^{M}(j-1)^{\beta -2}\sqrt{j} \text{ d}j \quad \beta \in \mathbb{R}$$

I have not tried anything useful. I have tried various substitutions and have come to a dead end each time.

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    Please show us what particular substitution you tried ? – 2017-02-09
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    No, this is an incomplete Beta integral, it has no closed form. (And when the bounds are $l=0,M=1$, you will need the Gamma function.) – 2017-02-09

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As indicated by Yves Daoust, we must use the incomplete beta function

\begin{equation} \mathrm{B}_{x}(a,b) = \int\limits_{0}^{x} z^{a-1}(1-z)^{b-1} dz \end{equation}

If we split the integral, we have \begin{align} \int\limits_{l}^{M} (j-1)^{\beta - 2} j^{1/2} dj &= (-1)^{\beta - 2} \int\limits_{l}^{M} (1-j)^{\beta - 2} j^{1/2} dj \\ &= (-1)^{\beta - 2} \left( \int\limits_{0}^{M} (1-j)^{\beta - 2} j^{1/2} dj - \int\limits_{0}^{l} (1-j)^{\beta - 2} j^{1/2} dj \right) \\ &= (-1)^{\beta - 2} \big[ \mathrm{B}_{M}(3/2,\beta - 1) - \mathrm{B}_{l}(3/2,\beta - 1) \big] \end{align}