<s>[INST] Is there a solution possible for the following integral? I want to solve the following integral
$$\int_0^{\infty}e^{-a(1+x^{m})^{\frac{2}{m}}}x\,{\rm d}x$$
where $a$ is real number but is not equal to $0$ and $m>2$. Any help or upper/lower bounds on this integral will be very helpful. Thank you.
 [/INST] Using CAS like Mathematica and MellinTransform can be solve it for m even integers and $a>0$.
$$\int_0^{\infty } \exp \left(-a \left(1+x^m\right)^{2/m}\right) \, dx=\mathcal{M}_a\left[\int_0^{\infty } \exp \left(-a
   \left(1+x^m\right)^{2/m}\right) \, dx\right](s)=\int_0^{\infty } \mathcal{M}_a\left[\exp \left(-a \left(1+x^m\right)^{2/m}\right)\right](s) \,
   dx=\mathcal{M}_s^{-1}\left[\int_0^{\infty } \left(1+x^m\right)^{-\frac{2 s}{m}} \Gamma (s) \,
   dx\right](a)=\mathcal{M}_s^{-1}\left[\frac{\Gamma \left(1+\frac{1}{m}\right) \Gamma (s) \Gamma \left(\frac{-1+2 s}{m}\right)}{\Gamma
   \left(\frac{2 s}{m}\right)}\right](a)$$
for m=3 and odd m Mathematica can't inverse Mellin transform.
for m=4
$$\mathcal{M}_s^{-1}\left[\frac{\Gamma \left(\frac{5}{4}\right) \Gamma (s) \Gamma \left(\frac{1}{4} (-1+2 s)\right)}{\Gamma
   \left(\frac{s}{2}\right)}\right](a)=\frac{2^{3/4} \sqrt[4]{a} K_{\frac{3}{4}}(a) \Gamma \left(\frac{5}{4}\right)}{\sqrt{\pi }}$$
where $K_{\frac{3}{4}}(a)$ is  modified Bessel function of the second kind.
for m=6
$\mathcal{M}_s^{-1}\left[\frac{\Gamma \left(\frac{7}{6}\right) \Gamma (s) \Gamma \left(\frac{1}{6} (-1+2 s)\right)}{\Gamma
   \left(\frac{s}{3}\right)}\right](a)=\Gamma \left(\frac{7}{6}\right) \left(\frac{3 \sqrt{\pi } \,
   _0F_2\left(;\frac{1}{6},\frac{1}{2};-\frac{a^3}{27}\right)}{\sqrt{a} \Gamma \left(\frac{1}{6}\right)}-\frac{2 a \sqrt{\pi } \,
   _0F_2\left(;\frac{2}{3},\frac{3}{2};-\frac{a^3}{27}\right)}{3 \Gamma \left(\frac{2}{3}\right)}+\frac{2 a^2 \pi  \,
   _0F_2\left(;\frac{4}{3},\frac{11}{6};-\frac{a^3}{27}\right)}{9 \Gamma \left(\frac{4}{3}\right) \Gamma \left(\frac{11}{6}\right)}\right)$
where $\, _0F_2\left(;\frac{1}{6},\frac{1}{2};-\frac{a^3}{27}\right)$ is the generalized hypergeometric function.
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