Easy looking Integrals...can be toooo tough!

$\displaystyle\int_0^{2} \left(\sqrt{1+x^3}+\sqrt[3]{x^2+2x}\right) \ dx$

If the above integral can be expressed as $2_2 F_1\left(\frac{-1}{a},\frac{1}{b};\frac{4}{3};c\right)+\frac{18}{5}-\dfrac{3\Gamma({\frac{1}{\lambda}})\Gamma({\frac{\mu}{3}})}{10\sqrt[\nu]{\pi}}$

Evaluate $\lfloor{a+b+c+\mu+\nu+\lambda \rfloor}$

$\rightarrow$ $_2 F_1(a',b';c';d')$ is the Hypergeometric function