\[ \int x^2 \left (a+b \sinh ^{-1}(c+d x)\right )^n \, dx \]
Optimal antiderivative \[ \frac {3^{-1-n} \left (a +b \arcsinh \! \left (d x +c \right )\right )^{n} \Gamma \! \left (1+n , -\frac {3 \left (a +b \arcsinh \! \left (d x +c \right )\right )}{b}\right ) {\mathrm e}^{-\frac {3 a}{b}} \left (\frac {-a -b \arcsinh \left (d x +c \right )}{b}\right )^{-n}}{8 d^{3}}-\frac {2^{-2-n} c \left (a +b \arcsinh \! \left (d x +c \right )\right )^{n} \Gamma \! \left (1+n , -\frac {2 \left (a +b \arcsinh \! \left (d x +c \right )\right )}{b}\right ) {\mathrm e}^{-\frac {2 a}{b}} \left (\frac {-a -b \arcsinh \left (d x +c \right )}{b}\right )^{-n}}{d^{3}}-\frac {\left (a +b \arcsinh \! \left (d x +c \right )\right )^{n} \Gamma \! \left (1+n , \frac {-a -b \arcsinh \! \left (d x +c \right )}{b}\right ) {\mathrm e}^{-\frac {a}{b}} \left (\frac {-a -b \arcsinh \left (d x +c \right )}{b}\right )^{-n}}{8 d^{3}}+\frac {c^{2} \left (a +b \arcsinh \! \left (d x +c \right )\right )^{n} \Gamma \! \left (1+n , \frac {-a -b \arcsinh \! \left (d x +c \right )}{b}\right ) {\mathrm e}^{-\frac {a}{b}} \left (\frac {-a -b \arcsinh \left (d x +c \right )}{b}\right )^{-n}}{2 d^{3}}+\frac {{\mathrm e}^{\frac {a}{b}} \left (a +b \arcsinh \! \left (d x +c \right )\right )^{n} \Gamma \! \left (1+n , \frac {a +b \arcsinh \! \left (d x +c \right )}{b}\right ) \left (\frac {a +b \arcsinh \left (d x +c \right )}{b}\right )^{-n}}{8 d^{3}}-\frac {c^{2} {\mathrm e}^{\frac {a}{b}} \left (a +b \arcsinh \! \left (d x +c \right )\right )^{n} \Gamma \! \left (1+n , \frac {a +b \arcsinh \! \left (d x +c \right )}{b}\right ) \left (\frac {a +b \arcsinh \left (d x +c \right )}{b}\right )^{-n}}{2 d^{3}}-\frac {2^{-2-n} c \,{\mathrm e}^{\frac {2 a}{b}} \left (a +b \arcsinh \! \left (d x +c \right )\right )^{n} \Gamma \! \left (1+n , \frac {2 a +2 b \arcsinh \! \left (d x +c \right )}{b}\right ) \left (\frac {a +b \arcsinh \left (d x +c \right )}{b}\right )^{-n}}{d^{3}}-\frac {3^{-1-n} {\mathrm e}^{\frac {3 a}{b}} \left (a +b \arcsinh \! \left (d x +c \right )\right )^{n} \Gamma \! \left (1+n , \frac {3 a +3 b \arcsinh \! \left (d x +c \right )}{b}\right ) \left (\frac {a +b \arcsinh \left (d x +c \right )}{b}\right )^{-n}}{8 d^{3}} \]
command
Integrate[x^2*(a + b*ArcSinh[c + d*x])^n,x]
Mathematica 13.1 output
\[ \int x^2 \left (a+b \sinh ^{-1}(c+d x)\right )^n \, dx \]
Mathematica 12.3 output
\[ \frac {2^{-n-3} 3^{-n-1} e^{-\frac {3 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{b^2}\right )^{-n} \left (\left (4 c^2-1\right ) 2^n 3^{n+1} e^{\frac {2 a}{b}} \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )^n \Gamma \left (n+1,-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )-\left (4 c^2-1\right ) 2^n 3^{n+1} e^{\frac {4 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )^n \Gamma \left (n+1,\frac {a}{b}+\sinh ^{-1}(c+d x)\right )+2^n \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )^n \Gamma \left (n+1,-\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )-2 c 3^{n+1} e^{a/b} \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )^n \Gamma \left (n+1,-\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )-e^{\frac {5 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )^n \left (2 c 3^{n+1} \Gamma \left (n+1,\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )+2^n e^{a/b} \Gamma \left (n+1,\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )\right )\right )}{d^3} \]