In[1]:= expr = Product[(x - i)^(-2/5), {i, 0, 3, 1}]
Out[1]= 1/((-3 + x)^(2/5) (-2 + x)^(2/5) (-1 + x)^(2/5) x^(2/5))
In[9]:= NIntegrate[expr, {x, 0, 10}]
During evaluation of In[9]:= NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small. >>
During evaluation of In[9]:= NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in x near {x} = {2.01156}. NIntegrate obtained -0.982375-1.58796 I and 0.08710236099655635` for the integral and error estimates. >>
Out[9]= -0.982375 - 1.58796 I
In[15]:= int[x_] =
NDSolveValue[{y'[x] == expr, y[-1] == 0}, y[x], {x, -1, 10}]
Out[15]= InterpolatingFunction[{{-1., 10.}}, <>][x]
In[16]:= int[10] - int[0]
Out[16]= -0.975968 - 1.57868 I