Need solve systems or not?

Answer is $9$ .

Appying AM-GM inequality, we have: $\sqrt{ab}\le\dfrac{a+b}{2}=\dfrac{25}{8}$

Since $(1+\sqrt{a})(1+\sqrt{b})=\dfrac{15}{2}$ , we have $1+\sqrt{a}+\sqrt{b}+\sqrt{ab}=\dfrac{15}{2}$

Or $\sqrt{a}+\sqrt{b}=\dfrac{13}{2}-\sqrt{ab}\qquad(1)$ .

Since $a+b=\dfrac{25}{4}$ , we get $\left(\sqrt{a}+\sqrt{b}\right)^2-2\sqrt{ab}=\dfrac{25}{4}\qquad(2)$ .

From $(1)$ and $(2)$ , we get: $\left(\dfrac{13}{2}-\sqrt{ab}\right)^2-2\sqrt{ab}=\dfrac{25}{4}$

$\Leftrightarrow ab-15\sqrt{ab}+36=0$

$\Leftrightarrow (\sqrt{ab}-3)(\sqrt{ab}-12)=0$

$\Leftrightarrow \sqrt{ab}=3$ , since $\sqrt{ab}\le\dfrac{25}{8}$

Thus, $ab=\boxed{9}$ .