How to take $\int_1^{+\infty} \frac{dx}{x\sqrt{1+x^5+x^{10}}}$?

I have the integral:

$$\int_1^{+\infty} \frac{dx}{x\sqrt{1+x^5+x^{10}}}$$

I have no clue how to approach it. i was thinking of $x = \frac{1}{t}$ but it does not seem to be correct, or I do something wrong

• Have you tried $x^5=t$? then $5dx=x^{-4} dt$ and you'll get something like $\frac{dt}{t\sqrt{1+t+t^2}}$. – Mikhail Tikhonov May 21 '17 at 12:02

Sun $x=1/t$ and get

$$\int_0^1 dt \frac{t^4}{\sqrt{1+t^5+t^{10}}} = \frac15 \int_0^1 \frac{du}{\sqrt{1+u+u^2}}$$

You should be able to do the latter integral. The result is $\frac15 \log{(1+2/\sqrt{3})}$.

• Excellent $(+1)$ – tired May 21 '17 at 12:06
• I don't know what I am doing wrong since I get $\frac{1}{5} \log \left(1+\frac{2}{\sqrt{3}}\right)$ – Claude Leibovici May 21 '17 at 14:01
• @ClaudeLeibovici: you did nothing wrong - I did. Messed up the integration limits. – Ron Gordon May 21 '17 at 14:33
• This was just a very minor detail ! The solution you provided is really nice. Cheers. – Claude Leibovici May 21 '17 at 14:45


The change of variables in \eqref{1} is Euler First Substitution.