calculus - Closed form solution for this integral?

I'm hitting a road block in finding an expression (closed form preferably) for the following integral:

\begin{equation}

\int^{+\infty}_0 x^b \left ( 1-\frac{x}{u} \right )^c \exp(-a x^3) dx
\end{equation}

where $a,b$ are positive constants; $b>1$ is an odd multiple of $0.5$, while $c$ is a positive or negative odd multiple of $0.5$; $u$ is a (positive) parameter.

Things I have considered or tried:

• look up in tables (Gradshsteyn and Ryzhik): there are very few explicit results for integrals involving $\exp(-a x^3)$ (or for the other factors after transforming via $y=x^3$). Also, tabulated results involving $\exp(-a x^p)$ for more general $p$ do not include the other factors $x^b (1-x/u)^c$. One exception is (3.478.3):
  \begin{equation}
  
  \int^{u}_0 x^b (1-ux)^c \exp(-a x^3) dx,
  \end{equation}
  but the limits of integration do not match with my case;




• there is a closed form solution (3.478.1) for the simpler integral
  \begin{equation}
  \int^{+\infty}_0 x^{d-1} \exp(-a x^3) dx = \frac{a^{-d/3}}{3} \Gamma(d/3).
  \end{equation}
  (NB: there is also an expression for the indefinite integral.)
  A binomial expansion of $[1-(x/u)]^n$ for integer $n$ would produce a solution in series form. However, in my case, the exponents $b$ and $c$ are strictly half-integer. For the same reason, integration by parts does not lead to a simpler integral without the factor $[1-(x/u)]^c$;




• Wolfram Math online did not produce a result;





• the integral is an intermediate step in a longer analysis, so numerical solution (with given values for the parameter) is not practical.

Grateful for any pointers or solution.