smc

Lets set our original expression equal to $x$. So $\sqrt[3]{5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}}=x$. Cubing this gives us $x^3={\left(\sqrt[3]{5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}}\right)}^3=5+2\sqrt{13}+5-2\sqrt{13}+3\left(\sqrt[3]{5+2\sqrt{13}}\ast \sqrt[3]{5-2\sqrt{13}}\right)\left(\sqrt[3]{5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}}\right)=10-9x$ So we have $x^3+9x-10=0$. We can easily see that 1 is a root of this polynomial. By synthetic division, the new polynomial is $x^2+x+10$, which has no real roots. Thus $\sqrt[3]{5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}}=1$. Since 1 is not A-D, our answer is $\boxed{\text{none of these}}$.