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How can I find closed-form expression of the following double integral $$\int_0^{\frac{\pi}4} \int_0^\infty \frac{dr \, d\phi}{ u_1^2 + u_2^2 r + 2 u_1 u_2 \sqrt{r} \cos \phi}?$$ Please help me as soon as you can.

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Are you sure it is convergent? The integral with respect to $r$ diverges at infinity. – Sasha Jun 29 '12 at 12:44
Are you missing a factor of like $1/r$ or $1/r^2$? – KennyTM Jun 29 '12 at 17:49
1  
Hence the answer is: there is a closed-form expression, which is $+\infty$. – Did Jul 1 '12 at 20:36

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