Calculating the integral.

The interval $[0,T]$ is divided into $n$ subintervals $[ih, (i + 1)h], i = 0,1,2, ldots ,n -1,$ of equal lengths $h$ where $h=T/n$.

The first $n+1$ hat functions $ psi_{n}(t)$ are given as follows:
begin{aligned}
&psi_{0}(t)=left{begin{array}{ll}
frac{h-t}{h}, & 0 leqslant t<h \
0, & text { otherwise }
end{array}right.\
&psi_{i}(t)=left{begin{array}{ll}
frac{t-(i-1) h}{h}, & (i-1) h leqslant t<i h \
frac{(i+1) h-t}{h}, & i h leqslant t<(i+1) h, quad i=1,2, ldots, n-1 \
0, & text { otherwise }
end{array}right.\
&psi_{n}(t)=left{begin{array}{ll}
frac{t-(T-h)}{h}, & T-h leqslant t leqslant T \
0, & text { otherwise }
end{array}right.
end{aligned}
Question: How to find an explicit formula for calculating following integral:
begin{equation}
g_{ij}=frac{(rho+1)^{1-alpha}}{Gamma(alpha)} int_{0}^{jh}left((jh)^{rho+1}-tau^{rho+1}right)^{alpha-1} tau^{rho} psi_i(tau) ,d tau
end{equation}
where
$alpha$ and $rho neq-1$ are real numbers and $Gamma$ is gamma function.

For example; if $rho=0$ in the formula above, then
$$
g_{0 j}=left{begin{array}{ll}
0, & j=0 \
frac{h^{alpha}}{Gamma(alpha+2)}left((j-1)^{alpha+1}+j^{alpha}(1-j+alpha)right), & j=1,2,3, ldots, n
end{array}right.
$$
and for $ i=1,2,3, ldots, n, j=0,1,2, ldots, n,$
$$
begin{aligned}
& \
& g_{i j}=left{begin{array}{ll}
0, & & i>j, \
frac{h^{alpha}}{Gamma(alpha+2)}, & & i=j, \
frac{h^{alpha}}{Gamma(alpha+2)}left((j-i+1)^{alpha+1}-2(j-i)^{alpha+1}+(j-i-1)^{alpha+1}right), & &i<j
end{array}right.
end{aligned}
$$

Hope to see ideas, suggestions, comments.

Best regards.