I spent so much time doing this integral yet I can't seem to get the right answer. Please help me find where I made a mistake. Thanks!

The joint pdf of $\displaystyle X$ and $\displaystyle Y$ is given by:

$\displaystyle f_{X,Y}(x,y) = \frac{16}{15}(x^{2} + \frac{xy}{2}), 0<x<1, 0<y<x+1$

Verify that this is a legitimate joint pdf. So I try to integrate the pdf and see if it equals one.

$\displaystyle \frac{16}{15}\int_0^1 \int_0^{x+1}(x^{2} + \frac{xy}{2})dydx = \frac{16}{15}\int_0^1 [x^{2}y + \frac{xy^{2}}{4}]_0^{x+1}dx =$$\displaystyle \frac{16}{15}\int_0^1 (x^3 + x^2 + \frac{x^3 + 2x^2 + x}{4})dx = $$\displaystyle \frac{16}{15}[\frac{x^4}{4} + \frac{x^3}{3} + \frac{x^4}{16} + \frac{x^2}{4} +$$\displaystyle \frac{x^2}{8}]_0^1=\frac{49}{45}$

As you can see, the integral of the pdf does not equal one. But it has to since it asked me to verify that it is legitimate! Where is my mistake? Thanks.