Let $X$ and $Y$ be two random variables with joint PDF $$f_{XY}(x,y)=\frac{1}{\Gamma(a)\Gamma(b)}x^{a-1}(y-x)^{b-1}e^{-y},\quad0<x<y,\quad\text{Bivariate Gamma Distribution}$$ where $a$ and $b$ are positive parameters. Find the margianl PDF's of $X$ and $Y$ respectively.
My Solution: For the marginal PDF of $X$ I integrated the joint PDF from $y_{1}=0$ to $y_{2}=\infty$ and I got $$f_{X}\left(x\right)=\int_{0}^{\infty}f\left(x,y\right)dy=\int_{0}^{\infty}\frac{1}{\Gamma\left(a\right)\Gamma\left(b\right)}x^{a-1}\left(y-x\right)^{b-1}e^{-y}dy=\frac{x^{a-1}}{\Gamma\left(a\right)\Gamma\left(b\right)}\int_{0}^{\infty}\left(y-x\right)^{b-1}e^{-y}dy.$$ We substitute $u=y-x\Leftrightarrow y=u+x$ with $dy=du$ $$ \int_{-x}^{\infty}u^{b-1}e^{-u-x}du=e^{-x}\int_{-x}^{\infty}u^{b-1}e^{-u}du=e^{-x}\Gamma\left(b,-x\right) $$ where $\Gamma\left(a,x\right)$ is the upper invomplete Gamma function. As a result the marginal PDF of $X$ is $$ f_{X}\left(x\right)=\frac{x^{a-1}e^{-x}}{\Gamma\left(a\right)\Gamma\left(b\right)}\Gamma\left(b,-x\right).$$ which must be clearly wrong as a result, since the incomplete Gamma function can't take a negative $x$ ($x>0\Leftrightarrow -x<0$) as an input.
For the marginal PDF of $Y$ I integrated the joint PDF from $x_{1}=0$ to $x_{2}=y$ and made it this far $$ f_{Y}\left(y\right)=\int_{0}^{y}f\left(x,y\right)dx=\int_{0}^{y}\frac{1}{\Gamma\left(a\right)\Gamma\left(b\right)}x^{a-1}\left(y-x\right)^{b-1}e^{-y}dx=\frac{e^{-y}}{\Gamma\left(a\right)\Gamma\left(b\right)}\int_{0}^{y}x^{a-1}\left(y-x\right)^{b-1}dx $$ (which isn't far enough) and I can't continue.
I'm stuck with this problem for days, and I can't seem to find a solution. What did I do wrong when computing the first integer, and how can I compute the second one?