Struggling to Solve \int\sqrt{\frac{x}{x^2+2}}dx

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Homework Statement



Find \int\sqrt{\frac{x}{x^2+2}}dx

Homework Equations



N/A

The Attempt at a Solution


Don't know where to start.
I tried a few substitution but doesn't yield anything.
It seems to be short, so any small hint will be great.
Thanks!
 
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The answer isn't pretty at all. Try it out at integrator.wolfram.com There's an imaginary number in it, along with "elliptical integrals", whatever that may mean. Probably there isn't any antiderivative which you can express in elementary form.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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