Generally, signs are a matter of convention and of definition. Here they phrased the question as "find the work done to move the charge". Hence, this implies that if an external source supplies energy and the charge gains energy, the work is positive. This is why they start the calculation off with a negative sign in the equation for work W.
In this case, the charge is positive and the fields is directed outward. Hence the integral itself is positive (as you say) and the charge naturally wants to move outward to a lower energy state. So, the charge is able to do work on something else, while it is moving outward.
However, if ρL is negative, the opposite is true, and work is done on the positive charge Q, when it moves outward.
If the question was "find the work done by the charge when it moves outward" the negative sign would not have been placed in the equation for work W.