Difficult integral #3

$\begin{aligned} I & = \int_0^\infty e^{-x^\frac 25} dx & \small \blue{\text{Let }t = x^\frac 25 \implies dt = \frac 25 x^{-\frac 35} dx} \\ & = \frac 52 \int_0^\infty t^\frac 32 e^{-t} dt & \small \blue{\text{Gamma function }\Gamma (s) = \int_0^\infty t^{s-1}e^{-t} dt} \\ & = \frac 52 \Gamma \left(\frac 52\right) & \small \blue{\text{Note that }\Gamma (s+1) = s\Gamma(s)} \\ & = \frac 52 \cdot \frac 32 \Gamma \left(\frac 32\right) \\ & = \frac 52 \cdot \frac 32 \cdot \frac 12 \Gamma \left(\frac 12\right) & \small \blue{\text{and }\Gamma \left(\frac 12\right) = \sqrt \pi} \\ & = \frac {15\sqrt \pi}8 \end{aligned}$

Therefore $a+b = 15+8 = \boxed{23}$ .