Given:

$f^1\left(x\right)=\frac{3x+2}{2x+3}$

Now,

$f^2\left(x\right)=f^{1}o{f}^1\left(x\right)$

$\Rightarrow f^2\left(x\right)=\frac{3\left(\frac{3x+2}{2x+3}\right)+2}{2\left(\frac{3x+2}{2x+3}\right)+3}$

$\Rightarrow f^2\left(x\right)=\frac{9x+6+4x+6}{6x+4+6x+9}$

$\Rightarrow f^2\left(x\right)=\frac{13x+12}{12x+13}$

$\Rightarrow f^3\left(x\right)=\frac{3\left(\frac{13x+12}{12x+13}\right)+2}{2\left(\frac{13x+12}{12x+13}\right)+3}$

$\Rightarrow f^3\left(x\right)=\frac{63x+62}{62x+63}$

$\Rightarrow f^4\left(x\right)=\frac{3\left(\frac{63x+62}{62x+63}\right)+2}{2\left(\frac{63x+62}{62x+63}\right)+3}$

$\Rightarrow f^4\left(x\right)=\frac{313x+312}{312x+313}$

$\therefore f^5\left(x\right)=\frac{1563x+1562}{1562x+1563}=\frac{ax+b}{bx+a}$

$\therefore a+b=1563+1562=3125$