$$x+|x|= \begin{cases}2 x, & x \geq 0 \\ 0, & x<0\end{cases}$$

$\Rightarrow \frac{1}{\sqrt{x+|x|}}$, domain is $x>0$, as $2 x \neq 0$

Similarly,

$$\begin{aligned} &\frac{1}{\sqrt{3 x+10-x^2}} \text { is defined when } 3 x+10-x^2>0\\ &\begin{aligned} \Rightarrow & x^2-3 x-10<0 \\ & (x-5)(x+2)<0 \\ \Rightarrow & x \in(-2,5) \\ \Rightarrow & \text { Domain will be }(0, \infty) \cap(-2,5)=(0,5) \\ \Rightarrow & (1+a)^2+b^2=1+25=26 \end{aligned} \end{aligned}$$