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I need some some function f(r) with asymptotics

as x -> 0, f-> 1*x+a*x^3+b*x^5+c*x^7+..., a,b,c>0

as x -> infinity, f-> x*Log[x]

I construct one like A/B*Log[1+x], where A=1+x/2+x^3/12, B=1+d*x^2.

But I think it is too complex, such that I cannot continue my successive latter works.

Can somebody give me a more simple such functions?

Suggestion or hints are also welcome.

Thank you very much!

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# Who can give me some function with asymptotics ?

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