I want to integrate the following function w.r.t. $s$ from 0 to $v$. The function is:
$$\frac{1}{s^{\frac{3}{2}}}\frac{1}{(v-s)^{\frac{3}{2}}}e^{-\frac{y^2}{2s}}e^{-\frac{(y-a)^2}{2(v-s)}}$$
From my opinion, I think it is integrable since the singularity occurs at 0 and $v$ only but the function converges to 0 at both end. However, when I enter it into mathematica and matlab, it does not give me the result. Even worse, mathematica say it is not integrable. Is it possible to integrate? What's wrong with my argument? Also, could someone show me the answer
$$\int_{0}^{v}\frac{1}{s^{\frac{3}{2}}}\frac{1}{(v-s)^{\frac{3}{2}}}e^{-\frac{y^2}{2s}}e^{-\frac{(y-a)^2}{2(v-s)}}\; ds$$ if it is integrable?
Indeed, I need to show if it is $\frac{2(2y-a)}{v\sqrt{2\pi v}}e^{-\frac{(2y-a)^2}{2v}}\frac{\pi}{4y(y-a)}$, but since I do not know how to do it, I check if it is integrable but the result is not my expectation using mathematica or matlab.