  Compact binaries are an important class of gravitational-wave (GW) sources
that can be detected by current and future GW observatories. They provide a
testbed for general relativity (GR) in the highly dynamical strong-field
regime. Here, we use GWs from inspiraling binary neutron stars and binary black
holes to investigate dipolar gravitational radiation (DGR) and varying
gravitational constant predicted by some alternative theories to GR, such as
the scalar-tensor gravity. Within the parametrized post-Einsteinian framework,
we introduce the parametrization of these two effects simultaneously into
compact binaries' inspiral waveform and perform the Fisher-information-matrix
analysis to estimate their simultaneous bounds. In general, the space-based GW
detectors can give a tighter limit than ground-based ones. The tightest
constraints can reach $\sigma_B<3\times10^{-11}$ for the DGR parameter $B$ and
$\sigma_{\dot{G}}/G < 7\times10^{-9} \, {\rm yr}^{-1} $ for the varying $G$,
when the time to coalescence of the GW event is close to the lifetime of
space-based detectors. In addition, we analyze the correlation between these
two effects and highlight the importance of considering both effects in order
to arrive at more realistic results.
