Find $\int e^{x^4}(x+x^3+2x^5)e^{x^2}dx$ Find $\int e^{x^4}(x+x^3+2x^5)e^{x^2}dx$

$$I=\int e^{x^4}(x+x^3+2x^5)e^{x^2}dx$$
$$I=\int e^{x^4}(1+x^2+2x^4)e^{x^2}xdx$$Let $x^2=t$
$I=\int e^{t^2}(1+t+2t^2)e^t\frac{dt}{2}=\frac{1}{2}\int e^{t^2+t}(1+t+2t^2)dt$
I am stuck here.

Assuming the answer is given by an elementary function, it has to be a function of the form $p(x) e^{x^2+x^4}$ with $p(x)$ being a quadratic polynomial. We have
$$\frac{d}{dx} x^2 e^{x^2+x^4} = 2(x+x^3+2x^5)e^{x^2+x^4} $$
hence such inspired guess immediately leads to
$$ \int (x+x^3+2x^5) e^{x^2+x^4}\,dx = C+\frac{x^2}{2}e^{x^2+x^4}.$$
