Originally Posted by

**ThePerfectHacker** This is not elementary.

Loiuville shown that if $\displaystyle f(x),g(x)$ are rational function then the integral $\displaystyle \int f(x) \exp (g(x)) dx$ is elementary if and only if, there exists a rational function $\displaystyle h(x)$ such as $\displaystyle f(x)=h'(x)+h(x)g(x)$.

In this case, let us we have that.

$\displaystyle f(x)=1$ and $\displaystyle g(x)=x^2$ solve the differencial equation,

$\displaystyle 1=y'+x^2y$.

This equation has no rational solutions.