Let us take an element at a distance $x$ from point mass as shown in figure below, 

Force on $m$ due to element is given by, $dF=\frac{Gmdm}{x^2}$
$\Rightarrow dF=\frac{Gm\lambda dx}{x^2}$
$\Rightarrow {\int }_0^{F}dF=\frac{GmM}{L}{\int }_r^{r+L}\frac{dx}{x^2}$
$\Rightarrow F=\frac{GmM}{L}{\left[-\frac{1}{x}\right]}_r^{r+L}$
$\Rightarrow F=\frac{GmM}{L}\left[\frac{1}{r}-\frac{1}{r+L}\right]$
$\Rightarrow F=\frac{GmM}{L}\left[\frac{r+L-r}{r\left(r+L\right)}\right]$
$\Rightarrow F=\frac{GmM}{r\left(r+L\right)}$