The mutual inductance between the rectangular loop and the long straight wire as shown in figure is M.                

Through an elementary strip of area$\text{da}=\text{a}\left(\text{d}x\right)$ ,

Flux linked due to magnetic field produced by current i, sent through

straight wire is $\text{d}\varphi =\text{BdA}\cos \text{O}=\frac{{\mu }_0{i}_1}{2\pi x}\text{a}\left(\text{d}x\right)$  

Total flux through the given rectangular loop

$\varphi ={\displaystyle \int d\varphi =\frac{{\mu }_0{i}_1\text{a}}{2\pi }{\displaystyle {\int }_{\text{c}}^{\text{c}+\text{b}}\frac{dx}{x}=\frac{{\mu }_0{i}_1\text{a}}{2\pi }}}ln\left[\frac{\text{c}+\text{b}}{\text{c}}\right]$

$\text{M}=\frac{\varphi }{i_1}=\frac{{\mu }_0\text{a}}{2\pi }ln\left[1+\frac{\text{b}}{\text{c}}\right]$