Evaluate $\int_{0}^{1}2^{x^2+x}\mathrm dx$ 
Question : Evaluate - $$\int_{0}^{1}2^{x^2+x}\mathrm dx$$

My Attempt : First I tried to evaluate the indefinite integral of $2^{x^2+x}$ in order to put the limits $0$ and $1$ later on, but couldn't integrate it. Then I checked on WA and came to know that it's elementary integral doesn't exist. Now I moved one to using properties of definite integration such as $$\int_a^b f(x) \mathrm dx=\int_a^b f(a+b-x) \mathrm dx$$
But it couldn't help either. Can you please give me hint to proceed on this question?
P.S. - This is a high school level problem and therefore its solution shouldn't involve any special functions, such as Gaussian Integral etc.
Edit :  I asked my teacher this question and basically this was an approximation based question. This was a MCQ type question which has an option "None of the above" and it was the correct answer, since the other options were made in such a way that can be rejected by bounding this integral between 2 functions. For example we can use $$2^{x^2+x}<2^{2x} ~; ~x\in (0,1)$$ and thus can be sure that this integral is less than $3/\ln(4)$.
Thanks all for devoting your time in my question!

Hint:
Using $u=\frac{2x+1}{2}$ yields an imaginary error function.
