Integral of $e^{x^3+x^2-1}(3x^4+2x^3+2x)$ I understand that this looks exactly like a "Do my Homework" kinda question but trust me, I've spent hours(I won't go into detail as it's off topic).
Note: I'm a High School student, our teacher gave this question as a challenge.
I'm struggling with

$$\int e^{x^3+x^2-1}(3x^4+2x^3+2x)\ dx$$ 

My Progress: I tried finding integral by parts, and found that$$\int e^{x^3+x^2-1}\ dx$$
was the only trouble maker(for now).
So, I tried finding it's integral then gave up and tried using a calculator to see what I missed.
The website said: That it's antiderivative is not elementary. I didn't even know what that means.
New Approach: Now, I went on to trying to plot the graph of $$ e^{x^3+x^2-1}$$  to see if I could related it to $$\int_{-a}^x e^{x^3+x^2-1}\ dx$$ and say whether $$\int e^{x^3+x^2-1}\ dx$$ exists or not.
 I was hoping for some discontinuity in the graph of the definite integral but I didn't seem to find any.
Note: I drew the graph of the definite integral through observation and intuition, I don't think there was any other method.
So is there anyway of helping me?

Assume the solution to be of the form $\exp(x^3+x^2-1)Q(x)$ and differentiate this expression to get $\exp(x^3+x^2-1)\left[(3x^2+2x)Q(x)+Q'(x)\right]$. Compare this with $\exp(x^3+x^2-1)\left[3x^4+2x^3+2x\right]$ to obtain the ODE:
$$(3x^2+2x)Q(x)+Q'(x)=3x^4+2x^3+2x.$$
The particular solution is $x^2$, you could also solve this ODE with separation of variables and variation of parameters to get $$Q(x)=x^2+c_1\exp(-x^3-x^2).$$ Setting $c_1=0$ will result in the particular solution $x^2$.
So, the integral is given by $\exp(x^3+x^2-1)x^2+c$
