<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] Just for the fun of it.
The convergence issues having been clearly explained and one answer for the value showing that a CAS is able to find a (nasty) antiderivative, the result is in fact
$$\int_0^{\infty}\frac{\log (1 + x^2)}{\sqrt {2x^5 + x^6}}\,dx=\frac{1}{3} \log \left(\frac{1}{4} \left(-1+i+\sqrt{-1-2 i}\right)^{-\sqrt{4-2 i}}\,\,
   \left(-1-i+\sqrt{-1+2 i}\right)^{-\sqrt{4+2 i}}\right)$$
</s>