I want to evaluate the following integral: \begin{equation} f_{abcd}(t) = \int_{-\infty}^{\infty}d\lambda\int_0^{t-\lambda} d\tau \frac{e^{i a \tau}}{ (b+i \tau)^{5/2} } \int_0^{t-\lambda} d\tau \left( \frac{e^{i c \tau}}{ (d+i \tau)^{5/2} } \right)^*, \end{equation} where $a,b,c$ and $d$ are real positive constants; $t$ is a real positive variable.
We can rewrite this as: \begin{equation} f_{abcd}(t) = \int_{-\infty}^{\infty}d\lambda \:g_{ab}(t-\lambda) \:\left(g_{cd}(t-\lambda)\right)^*. \end{equation} where we have defined \begin{equation} g_{ab}(t) = \int_{0}^t d\tau \, \frac{e^{i a \tau}}{ (b+i \tau)^{5/2} } \end{equation}
Now the integral $g_{ab}(t)$ can be evaluated and expressed as function of the complex error function erfi.
The problem then is that I don't know how to evaluate the integral $\int d\lambda$ to obtain $f(t)$. I am not even sure it has a definite form.
I tried to play with the boundaries of integration, or write the integral as sum of separate integrals, but without success. Any idea is welcome!