What is the closed form of $\int \exp ( \sqrt{x}-x^2)\ dx$?

Question

My attempt to get the closed form of $\int \exp ( \sqrt{x}-x^2)\ dx$ using integration by part to get something related to error function is failed, I believe that function has a closed form because we can write its form into product of two simple function has closed form such that $\int \exp ( \sqrt{x}-x^2)=\int (exp (-x^2)\times \int(\exp(\sqrt{x})$. We have simply $\int e^\sqrt{x}dx=2\int e^u u\ du=2e^uu-2e^u+C=2e^\sqrt{x}(\sqrt{x}-1)+C $ and the first is error function but if we use integration by part with those simple two function it's hard to get that then why ?

Answer 1

The expression $$\int \exp ( \sqrt{x}-x^2)=\int (exp (-x^2)\times \int(\exp(\sqrt{x})$$ is not true.

In general integral of product is not product of integrals.

Also there is no reason to expect a closed form for your integral $$\int \exp ( \sqrt{x}-x^2)dx$$ simply because it is the integral of a product.