Scale factor in standard cosmology

In standard cosmology the density $rho$ is given by:

$rho=rho_0left(frac{a_0}{a}right)^{3(1+w)}$

where $w=P/rho=$ const depends on the particle content ($P$ is the pressure). The Friedmann equation is

$left(frac{a’}{a}right)^2=frac{8pi G}{3}rho$,

where the prime is a cosmic time detivative and I have neglected curvature and the cosmological constant.

Now inserting the first equation into the Friedmann equation I get:

$left(frac{a’}{a}right)^2sim left(frac{a_0}{a}right)^{3(1+w)}Rightarrow a’sim a^{-3/2(1+w)+1}$

integrating this gives

$asim tau^{-3/2(1+w)+2}$

where $tau$ is the cosmic time. However in my book ( The Cosmic Microwave Background by Ruth Durrer, page 6, equation 1.25 https://www.cambridge.org/core/books/cosmic-microwave-background/10D066B56BBBA899F3B89A29E0B3B78B ) they get

$asim tau^{2/3(1+w)}$

Can you spot my mistake?