Is there any hope to find solution of the integral $\int e^{-a \left(\frac{b x^3}{2}-\frac{x^2}{4}\right)} dx $?

Question

Is there any hope to find a closed-form solution to this integral

$$\int e^{-a \left(\frac{b x^3}{2}-\frac{x^2}{4}\right)} dx=? \qquad\textstyle{with}\qquad a,b>0 \qquad\textstyle{and}\qquad a,b\in\mathbb{R}$$

The range of $x$ is $-\frac c2\leq x\leq 0$ where $c$ is a real positive and finite. I would appreciate any hints or comments on how I can find its solution.

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P.S. Mathematica does not answer.

 Integrate[Exp[(-a (-(x^2/4) + (x^3 b)/2))], x, 
  Assumptions -> {a > 0 && b > 0  && a \[Element] Reals && b \[Element] Reals}]