Prove that $$\int_0^r\frac{2x}{r^2}\exp\left({-\frac{πvx^\frac{2d}{c}}{t^{\frac2c}}}\right) \,\mathrm dx=\frac cd\frac{t^{\frac2d}}{r^2}\left(\frac1{πv}\right)^{\frac cd}γ\left(\frac cd,\ πv\left(\frac{r^d}t\right)^{\frac2c}\right).$$ Hint: on changing variables $$y=\frac{πvx^{\frac{2d}{c}}}{t^{\frac{2}{c}}}.$$
Note: $$ \gamma \left ( a,b \right )=\int_{0}^{b}x^{a-1}e^{-x} \,\mathrm{d}x$$ is the incomplete gamma function.
I am asking here as I have not been able to get through with its step-by-step proof. However I have verified the answer is correct using integral calculator
(Mis)using this integral calculator was a bad idea.
Your integral can be written as
$$2pq\int xe^{-px^2}dx=-qe^{-px^2}+C.$$