CoCalc -- 3523.sagews

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var('k_i b_i x_i x_i_1 x a')
show(integrate((k_i*x + b_i)*(a-x)/(a^(3/2)), (x, x_i, x_i_1)))

<html><div class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\frac{2 \, k_{i} x_{i}^{3} - 2 \, k_{i} x_{i_{1}}^{3} - 3 \, {\left(a k_{i} - b_{i}\right)} x_{i}^{2} + 3 \, {\left(a k_{i} - b_{i}\right)} x_{i_{1}}^{2} - 6 \, a b_{i} x_{i} + 6 \, a b_{i} x_{i_{1

6 \, a^{\left(\frac{3}{2}\right)}} }}}

show(_.expand())

<html><div class="math">\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{k_{i} x_{i}^{2}}{2 \, \sqrt{a}} + \frac{k_{i} x_{i_{1}}^{2}}{2 \, \sqrt{a}} + \frac{k_{i} x_{i}^{3}}{3 \, a^{\left(\frac{3}{2}\right)}} - \frac{k_{i} x_{i_{1}}^{3}}{3 \, a^{\left(\frac{3}{2}\right)}} - \frac{b_{i} x_{i}}{\sqrt{a}} + \frac{b_{i} x_{i_{1

\sqrt{a}} + \frac{b_{i} x_{i}^{2}}{2 \, a^{\left(\frac{3}{2}\right)}} - \frac{b_{i} x_{i_{1}}^{2}}{2 \, a^{\left(\frac{3}{2}\right)}} }}}

show(_.factor())

\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{{\left(x_{i} - x_{i_{1}}\right)} {\left(3 \, a k_{i} x_{i} + 3 \, a k_{i} x_{i_{1}} - 2 \, k_{i} x_{i}^{2} - 2 \, k_{i} x_{i} x_{i_{1}} - 2 \, k_{i} x_{i_{1}}^{2} + 6 \, a b_{i} - 3 \, b_{i} x_{i} - 3 \, b_{i} x_{i_{1}}\right)}}{6 \, a^{\left(\frac{3}{2}\right)}}