<s>[INST] Convergence of $\int_0^{\infty}\frac{\ln (1 + x^2)}{\sqrt {2x^5 + x^6}}\,dx $ 
Find out if the following integral diverges or converges:
$$
\int_0^\infty \frac{\ln (1 + x^2)}{\sqrt {2x^5 + x^6}}\,dx 
$$ 


First I split the integral as $\displaystyle \int_0^1 \frac{\ln (1 + x^2)}{\sqrt {2x^5 + x^6}}\,dx + \int_1^\infty \frac{\ln (1 + x^2)}{\sqrt {2x^5 + x^6}}\,dx $.


*

*For $\int_0^{1}\frac{\ln (1 + x^2)}{\sqrt {2x^5 + x^6}}\,dx $ , I prove $\ln(1+x^2)< x^2$, using that I can prove $\int_0^1\frac{\ln (1 + x^2)}{\sqrt {2x^5 + x^6}}\,dx $ converges, but it take too long so is there shorter way to do this problem?  

*For $\int_1^{\infty}\frac{\ln (1 + x^2)}{\sqrt {2x^5 + x^6}}\,dx $ , I have no idea how to do this problem.
 [/INST] For small $x$, $$\frac{\log(1+x^2)}{x^2\sqrt{2x+x^2}}=\frac1{\sqrt{2x}}+o(x^{-1/2}).$$
For large $x$, $$\frac{\log(1+x^2)}{x^2\sqrt{2x+x^2}}=\frac{\log(x)}{x^3}+o\left(\frac{\log(x)}{x^3}\right)=o\left(x^{-2}\right).$$
In both cases, the integrals of the asymptotic expressions do converge.
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