\[ \int x^2 (a+b \text {ArcSin}(c+d x))^n \, dx \]
Optimal antiderivative \[ -\frac {\mathrm {I} \left (a +b \arcsin \! \left (d x +c \right )\right )^{n} \Gamma \! \left (1+n , \frac {\mathrm {-I} \left (a +b \arcsin \! \left (d x +c \right )\right )}{b}\right ) {\mathrm e}^{\frac {\mathrm {-I} a}{b}} \left (\frac {\mathrm {-I} \left (a +b \arcsin \left (d x +c \right )\right )}{b}\right )^{-n}}{8 d^{3}}-\frac {\mathrm {I} c^{2} \left (a +b \arcsin \! \left (d x +c \right )\right )^{n} \Gamma \! \left (1+n , \frac {\mathrm {-I} \left (a +b \arcsin \! \left (d x +c \right )\right )}{b}\right ) {\mathrm e}^{\frac {\mathrm {-I} a}{b}} \left (\frac {\mathrm {-I} \left (a +b \arcsin \left (d x +c \right )\right )}{b}\right )^{-n}}{2 d^{3}}+\frac {\mathrm {I} \,{\mathrm e}^{\frac {\mathrm {I} a}{b}} \left (a +b \arcsin \! \left (d x +c \right )\right )^{n} \Gamma \! \left (1+n , \frac {\mathrm {I} \left (a +b \arcsin \! \left (d x +c \right )\right )}{b}\right ) \left (\frac {\mathrm {I} \left (a +b \arcsin \left (d x +c \right )\right )}{b}\right )^{-n}}{8 d^{3}}+\frac {\mathrm {I} c^{2} {\mathrm e}^{\frac {\mathrm {I} a}{b}} \left (a +b \arcsin \! \left (d x +c \right )\right )^{n} \Gamma \! \left (1+n , \frac {\mathrm {I} \left (a +b \arcsin \! \left (d x +c \right )\right )}{b}\right ) \left (\frac {\mathrm {I} \left (a +b \arcsin \left (d x +c \right )\right )}{b}\right )^{-n}}{2 d^{3}}+\frac {2^{-2-n} c \left (a +b \arcsin \! \left (d x +c \right )\right )^{n} \Gamma \! \left (1+n , \frac {-2 \,\mathrm {I} \left (a +b \arcsin \! \left (d x +c \right )\right )}{b}\right ) {\mathrm e}^{\frac {-2 \,\mathrm {I} a}{b}} \left (\frac {\mathrm {-I} \left (a +b \arcsin \left (d x +c \right )\right )}{b}\right )^{-n}}{d^{3}}+\frac {2^{-2-n} c \,{\mathrm e}^{\frac {2 \,\mathrm {I} a}{b}} \left (a +b \arcsin \! \left (d x +c \right )\right )^{n} \Gamma \! \left (1+n , \frac {2 \,\mathrm {I} \left (a +b \arcsin \! \left (d x +c \right )\right )}{b}\right ) \left (\frac {\mathrm {I} \left (a +b \arcsin \left (d x +c \right )\right )}{b}\right )^{-n}}{d^{3}}+\frac {\mathrm {I} \,3^{-1-n} \left (a +b \arcsin \! \left (d x +c \right )\right )^{n} \Gamma \! \left (1+n , \frac {-3 \,\mathrm {I} \left (a +b \arcsin \! \left (d x +c \right )\right )}{b}\right ) {\mathrm e}^{\frac {-3 \,\mathrm {I} a}{b}} \left (\frac {\mathrm {-I} \left (a +b \arcsin \left (d x +c \right )\right )}{b}\right )^{-n}}{8 d^{3}}-\frac {\mathrm {I} \,3^{-1-n} {\mathrm e}^{\frac {3 \,\mathrm {I} a}{b}} \left (a +b \arcsin \! \left (d x +c \right )\right )^{n} \Gamma \! \left (1+n , \frac {3 \,\mathrm {I} \left (a +b \arcsin \! \left (d x +c \right )\right )}{b}\right ) \left (\frac {\mathrm {I} \left (a +b \arcsin \left (d x +c \right )\right )}{b}\right )^{-n}}{8 d^{3}} \]
command
Integrate[x^2*(a + b*ArcSin[c + d*x])^n,x]
Mathematica 13.1 output
\[ \int x^2 (a+b \text {ArcSin}(c+d x))^n \, dx \]
Mathematica 12.3 output
\[ \frac {2^{-n-3} 3^{-n-1} e^{-\frac {3 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^n \left (\frac {\left (a+b \sin ^{-1}(c+d x)\right )^2}{b^2}\right )^{-n} \left (i \left (4 c^2+1\right ) 2^n 3^{n+1} e^{\frac {4 i a}{b}} \left (-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^n \Gamma \left (n+1,\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-i \left (4 c^2+1\right ) 2^n 3^{n+1} e^{\frac {2 i a}{b}} \left (\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^n \Gamma \left (n+1,-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+2 c 3^{n+1} e^{\frac {5 i a}{b}} \left (-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^n \Gamma \left (n+1,\frac {2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-i 2^n e^{\frac {6 i a}{b}} \left (-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^n \Gamma \left (n+1,\frac {3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+2 c 3^{n+1} e^{\frac {i a}{b}} \left (\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^n \Gamma \left (n+1,-\frac {2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+i 2^n \left (\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )^n \Gamma \left (n+1,-\frac {3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )}{d^3} \]