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