Whether the following integral expression can be calculated in closed-form?
$y=\int_0^\infty r \exp(-a_1r^{b_1}-a_2r^{b_2})\mathrm{d}r$, where $a_i,b_i>0,i=1,2$.
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With $b_2 \neq b_1$, I would be surprized by a closed form. – Claude Leibovici Jul 06 '17 at 06:15
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@ClaudeLeibovici Why, there might be a closed form (admittedly, not in elementary functions) if $b_1=1,;b_2=2$ or if $b_1=2,;b_2=4$, but hardly anywhere else. – Ivan Neretin Jul 06 '17 at 07:32
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@IvanNeretin. You are totally right ! It seems that the cases you gave are the only which do not lead to hypergeometric and Airy functions. – Claude Leibovici Jul 06 '17 at 08:10
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After Ivan Neretin's comment, I used a CAS to compute $$f(b,d)=\int_0^\infty r \,e^{-a r^b-c r^d}\,dr$$ The simplest is, for sure $$f(1,2)=\frac{1}{2 c}-\frac{\sqrt{\pi } a e^{\frac{a^2}{4 c}} \text{erfc}\left(\frac{a}{2 \sqrt{c}}\right)}{4 c^{3/2}}$$ For $f(1,n)$ with $n>2$, the CAS was able to generate expressions involving hypergeometric functions (this was checked up to $n=10$).
As Ivan Neretin commented, a simple one is $$f(2,4)=\frac{\sqrt{\pi } e^{\frac{a^2}{4 c}} \text{erfc}\left(\frac{a}{2 \sqrt{c}}\right)}{4 \sqrt{c}}$$ For the other $f(2,n)$, hypergeometric functions appear again and again (as well as Airy functions too); this was also checked up to $n=10$.
The cases corresponding to $b \geq 3$ seem to only lead to long linear combinations of hypergeometric functions
Claude Leibovici
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