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.

Your approach on $[0,1]$ seems fine. For the other part, you can verify that:
$$\frac{\ln (1 + x^2)}{\sqrt {2x^5 + x^6}} \le \frac{\ln (1 + x^2)}{\sqrt {x^6}}=\frac{\ln (1 + x^2)}{x^3} \quad \mbox{and} \quad \int_1^{\infty}\frac{\ln (1 + x^2)}{x^3}\,\mbox{d}x = \ln 2$$
