You made the right thing when you split the integral into two.
Alright,
$\displaystyle \int xe^{x^2}dx=\frac{e^{x^2}}{2}+C$
Now to the first integral:
$\displaystyle \int{xe^{\sqrt{x}}dx$
First... substitute $\displaystyle \sqrt{x}=t$ and you will get:
$\displaystyle 2\int {e^t}{ t^3}dt$
Now you can do integration by part 3 times... or:
Find the coefficients of:
$\displaystyle \int {e^t}{ t^3}dt=Ae^t {t^3}+Be^t {t^2}+Ce^t {t}+De^t$
(by differentiation)