16.2 Problem number 94

$$\int x^2{(a+b{\sinh }^{-1}(c+dx))}^{n}dx$$

Optimal antiderivative $$\frac{3^{-1-n}{(a+b\mathrm{arcsinh}(dx+c))}^n\mathrm{\Gamma }(1+n,-\frac{3(a+b\mathrm{arcsinh}(dx+c))}{b}){\mathrm{e}}^{-\frac{3a}{b}}{\left(\frac{-a-b\mathrm{arcsinh}(dx+c)}{b}\right)}^{-n}}{8d^3}-\frac{2^{-2-n}c{(a+b\mathrm{arcsinh}(dx+c))}^n\mathrm{\Gamma }(1+n,-\frac{2(a+b\mathrm{arcsinh}(dx+c))}{b}){\mathrm{e}}^{-\frac{2a}{b}}{\left(\frac{-a-b\mathrm{arcsinh}(dx+c)}{b}\right)}^{-n}}{d^3}-\frac{{(a+b\mathrm{arcsinh}(dx+c))}^n\mathrm{\Gamma }(1+n,\frac{-a-b\mathrm{arcsinh}(dx+c)}{b}){\mathrm{e}}^{-\frac{a}{b}}{\left(\frac{-a-b\mathrm{arcsinh}(dx+c)}{b}\right)}^{-n}}{8d^3}+\frac{c^2{(a+b\mathrm{arcsinh}(dx+c))}^n\mathrm{\Gamma }(1+n,\frac{-a-b\mathrm{arcsinh}(dx+c)}{b}){\mathrm{e}}^{-\frac{a}{b}}{\left(\frac{-a-b\mathrm{arcsinh}(dx+c)}{b}\right)}^{-n}}{2d^3}+\frac{{\mathrm{e}}^{\frac{a}{b}}{(a+b\mathrm{arcsinh}(dx+c))}^n\mathrm{\Gamma }(1+n,\frac{a+b\mathrm{arcsinh}(dx+c)}{b}){\left(\frac{a+b\mathrm{arcsinh}(dx+c)}{b}\right)}^{-n}}{8d^3}-\frac{c^2{\mathrm{e}}^{\frac{a}{b}}{(a+b\mathrm{arcsinh}(dx+c))}^n\mathrm{\Gamma }(1+n,\frac{a+b\mathrm{arcsinh}(dx+c)}{b}){\left(\frac{a+b\mathrm{arcsinh}(dx+c)}{b}\right)}^{-n}}{2d^3}-\frac{2^{-2-n}c{\mathrm{e}}^{\frac{2a}{b}}{(a+b\mathrm{arcsinh}(dx+c))}^n\mathrm{\Gamma }(1+n,\frac{2a+2b\mathrm{arcsinh}(dx+c)}{b}){\left(\frac{a+b\mathrm{arcsinh}(dx+c)}{b}\right)}^{-n}}{d^3}-\frac{3^{-1-n}{\mathrm{e}}^{\frac{3a}{b}}{(a+b\mathrm{arcsinh}(dx+c))}^n\mathrm{\Gamma }(1+n,\frac{3a+3b\mathrm{arcsinh}(dx+c)}{b}){\left(\frac{a+b\mathrm{arcsinh}(dx+c)}{b}\right)}^{-n}}{8d^3}$$

command

Integrate[x^2*(a + b*ArcSinh[c + d*x])^n,x]

Mathematica 13.1 output

$$\int x^2{(a+b{\sinh }^{-1}(c+dx))}^{n}dx$$

Mathematica 12.3 output

$$\frac{2^{-n-3}{3}^{-n-1}{e}^{-\frac{3a}{b}}{(a+b{\sinh }^{-1}(c+dx))}^n{(-\frac{{(a+b{\sinh }^{-1}(c+dx))}^2}{b^2})}^{-n}((4c^2-1)2^n{3}^{n+1}{e}^{\frac{2a}{b}}{(\frac{a}{b}+{\sinh }^{-1}(c+dx))}^n\mathrm{\Gamma }(n+1,-\frac{a+b{\sinh }^{-1}(c+dx)}{b})-(4c^2-1)2^n{3}^{n+1}{e}^{\frac{4a}{b}}{(-\frac{a+b{\sinh }^{-1}(c+dx)}{b})}^n\mathrm{\Gamma }(n+1,\frac{a}{b}+{\sinh }^{-1}(c+dx))+2^n{(\frac{a}{b}+{\sinh }^{-1}(c+dx))}^n\mathrm{\Gamma }(n+1,-\frac{3(a+b{\sinh }^{-1}(c+dx))}{b})-2c{3}^{n+1}{e}^{a/b}{(\frac{a}{b}+{\sinh }^{-1}(c+dx))}^n\mathrm{\Gamma }(n+1,-\frac{2(a+b{\sinh }^{-1}(c+dx))}{b})-e^{\frac{5a}{b}}{(-\frac{a+b{\sinh }^{-1}(c+dx)}{b})}^n(2c{3}^{n+1}\mathrm{\Gamma }(n+1,\frac{2(a+b{\sinh }^{-1}(c+dx))}{b})+2^n{e}^{a/b}\mathrm{\Gamma }(n+1,\frac{3(a+b{\sinh }^{-1}(c+dx))}{b})))}{d^3}$$