# Help solving this Linear First order ODE

I'm trying to solve this ODE :- $x (dy/dx) + y \log(x) = e^x x^{(1-1/2 \log(x))}$

I divided the equation throughout by $x$, obtaining $(dy/dx) + y\log(x)/x = e^x x^{(\log(x^{-1/2}))}$.

Then, I obtained the Integrating factor as $e^{((\log (x)^2)/2)}$.

Then, $y e^{((\log(x)^2)/2)} = \int e^{(x + ((\log(x)^2)/2))} x^{(\log(x^{-1/2}))} dx$.

I'm not sure how to proceed from here. It seems the integral on the right has to be evaluated by parts but it is quite tedious and tends to get messy. Is there a simpler way to evaluate the integral? Or is there any other approach to this differential equation?

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$$x^{\log{x^{-1/2}}} = e^{-(1/2) \log^2{x}}$$