$$I = \int_{0}^\infty \dfrac{e^{-\sqrt{Ax}} \ln \left(1+x\right)}{\sqrt{x}} dx , \quad A>0.$$
Any hint for solving this integral, I think Gauss quadrature cannot be applied to this. Matlab is giving numerical answers to this, that means its solvable. Any help would be highly appreciated.
Your question is equivalent to finding $$ g(B) = \int_{0}^{+\infty} \log(1+x^2)\,e^{- B x}\,dx $$ for $B>0$, i.e. to finding the Laplace transform of $\log(1+x^2)$. By integration by parts we have $$ g(B) = \frac{2}{B}\int_{0}^{+\infty}\frac{x}{1+x^2}\,e^{-B x}\,dx $$ and by partial fraction decomposition that equals $$ \frac{2}{B}\left[\frac{\pi}{2}\sin(B)-\sin(B)\text{Si}(B)-\cos(B)\text{Ci}(B)\right] $$ where $\text{Si}$ and $\text{Ci}$ are the sine/cosine integrals. A numerical evaluation is indeed simpler from the previous integral representation, since both $\frac{x}{1+x^2}$ and $e^{-Bx}$ are smooth and decreasing functions. For instance, by the Cauchy-Schwarz inequality
$$ g(B)\leq\frac{2}{B}\sqrt{\int_{0}^{+\infty}\frac{x\,dx}{(1+x^2)^2}\int_{0}^{+\infty}x\,e^{-2B x}\,dx}=\frac{1}{B^2\sqrt{2}}. $$