Consider a small strip of length a and width dx at a distance dx from the wire 

$\text{Magnetic field at a distance x from the wire is } B = \frac{\mu_0 I}{2\pi x}$

$\text{ Flux passing through this small portion is } d\phi = Badx = \frac{\mu_0 Ia}{2\pi x}dx$

$\text{Total Flux = }\frac{\mu_0 Ia}{2\pi} ln\frac{b+c}{c}$

$\text{Mutual Inductance M }= \frac{\phi}{I} = \frac{\mu_0 a}{2\pi} ln(1+\frac{b}{c})$