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

Consider the rectangular loop of distance c apart from long straight wire, then net flux across current carrying straight wire, i.e.,
 
   $\varphi =\underset{c}{\overset{b+c}{\int }}BdA=\underset{c}{\overset{b+c}{\int }}\frac{{\mu }_0{i}_1}{2\pi x}adx$
  =  $\frac{{\mu }_{0}Ia}{2\pi }\ln \frac{b+c}{c}$