I have tried much with the following two integrals, even with Mathematica and Matlab but in vain
$$\int \frac{x^{4}-3x^{3}+x^{2}+2x-2}{\sqrt{x^{2}+2x+3}}dx$$ and $$\int \frac{x^{\frac{2}{5}}}{\sqrt[3]{1+4x^{\frac{5}{2}}}}dx$$
Any help with that is very appreciated.
Mathematica 12.0 gives (up to the integration constant) the first integral as $$\frac1{12}\left((3x^3-19x^2+40x+18)\sqrt{x^2+2x+3}-162\sinh^{-1}\frac{x+1}{\sqrt2}\right)$$ and the second as $$\frac57x^{7/5}{_2F_1}\left(\frac13,\frac{14}{25};\frac{39}{25};-4x^{5/2}\right)$$