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Evaluate intergal $$ I(\mu, \alpha, \lambda)=\int_{0}^{\infty} y^\mu e^{-\alpha (y+y^{-1})y^{\lambda}}dy. $$ It comes from phisics.

WolframAlpha gives partial answer $$ I(\mu, \alpha, 1)=\frac{1}{2} e^{-\alpha} \alpha^{-\frac{\mu}{2}-\frac{1}{2}} \Gamma \left( \frac{\mu+1}{2} \right) . $$ It is possible to find it in close form?

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    without making any serious attempt to prove that, i suppose that the cases $\lambda=-1,0,1,3$ are the only ones which can be developed in a rather nice closed form (including Bessel functions) – 2017-02-03
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    @tired. I totally agree with you. But, even for $\lambda=3$, I arrive to some confluent hypergeometric function. – 2017-02-03
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    @ClaudeLeibovici i think at least for $\mu\in N$ they should be reduceable to Besselfunctions..but for nonintegers i think you are right – 2017-02-03

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