<s>[INST] Integration involving rational function and exponentials I hope I can find closed form solution for two following definite integrals. Unfortunately I don't have Mathematica and I can't find similar integrals in Tables. Can any one help me please? 
\begin{equation}
\int_0^\infty \frac{e^{-x}}{x^{1/2}+a x^{3/2}} dx
\end{equation}
\begin{equation}
\int_0^\infty \frac{xe^{-x}}{x^{1/2}(1+bx)^2} dx
\end{equation}
 [/INST] The second integral is calculated in the same fashion:
\begin{eqnarray}
&&\int\limits_0^\infty \frac{x e^{-x}}{\sqrt{x} (1+b x)^2} dx \underbrace{=}_{z=\sqrt{x}}\\
&&\int\limits_0^\infty \frac{z^2}{(1+b z^2)^2} 2 e^{-z^2} dz=\\
&&\int\limits_0^\infty \left(\frac{1}{b} \frac{1}{1+b z^2} - \frac{1}{b} \frac{1}{1+b z^2)^2} \right) 2 e^{-z^2} dz=\\
&&\frac{2}{b^{3/2}} \left( \int\limits_0^\infty \frac{e^{-\frac{z^2}{b}}}{1+z^2}dz - \int\limits_0^\infty \frac{e^{-\frac{z^2}{b}}}{(1+z^2)^2}dz\right)=\\
&&\frac{2}{b^{3/2}} \left( \frac{\pi}{2} \exp(\frac{1}{b}) erfc(\frac{1}{\sqrt{b}}) - \int\limits_0^\infty \frac{e^{-\frac{z^2}{b}}}{(1+z^2)^2}dz\right)=\\
&&\frac{2}{b^{3/2}} \left( \frac{\pi}{2} \exp(\frac{1}{b}) erfc(\frac{1}{\sqrt{b}}) - \int\limits_0^\infty \frac{1}{b} \left( \frac{z}{1+z^2} + \arctan(z)\right) z e^{-\frac{z^2}{b}} dz\right)=\\
&& \frac{\pi}{b^{3/2}}\exp(\frac{1}{b})erfc(\frac{1}{\sqrt{b}}) - \frac{2}{b^{5/2}} \int\limits_0^\infty \left( 1-\frac{1}{1+z^2} + z \arctan(z) \right) e^{-\frac{z^2}{b}} dz=\\
&&\frac{\pi}{b^{3/2}}\exp(\frac{1}{b})erfc(\frac{1}{\sqrt{b}}) - \frac{2}{b^{5/2}}\left( \sqrt{b} \frac{\sqrt{\pi}}{2} - \frac{\pi}{2} \exp(\frac{1}{b}) erfc(\frac{1}{\sqrt{b}}) +   \int\limits_0^\infty   z e^{-\frac{z^2}{b}} \arctan(z)   dz \right)=\\
&&\frac{\pi}{b^{3/2}}\exp(\frac{1}{b})erfc(\frac{1}{\sqrt{b}}) - \frac{2}{b^{5/2}}\left( \sqrt{b} \frac{\sqrt{\pi}}{2} - \frac{\pi}{2} \exp(\frac{1}{b}) erfc(\frac{1}{\sqrt{b}}) +  \frac{b}{2} \int\limits_0^\infty   e^{-\frac{z^2}{b}} \frac{1}{1+z^2}  dz \right)=\\
&&\frac{\pi}{b^{3/2}}\exp(\frac{1}{b})erfc(\frac{1}{\sqrt{b}}) - \frac{2}{b^{5/2}}\left( \sqrt{b} \frac{\sqrt{\pi}}{2} - \frac{\pi}{2} \exp(\frac{1}{b}) erfc(\frac{1}{\sqrt{b}}) +  \frac{b}{2} \frac{\pi}{2} \exp(\frac{1}{b}) erfc(\frac{1}{\sqrt{b}}) \right)=\\
&& -\frac{\sqrt{\pi}}{b^2} +  \frac{\pi}{2 b^{3/2}} \exp(\frac{1}{b}) erfc(\frac{1}{\sqrt{b}}) \left( 1+ \frac{2}{b} \right)
\end{eqnarray}
</s>