\[ \int (f x)^m \left (d+e x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx \]
Optimal antiderivative \[ \frac {d \left (f x \right )^{1+m} \left (a +b \,\mathrm {arccosh}\! \left (c x \right )\right )}{f \left (1+m \right )}+\frac {e \left (f x \right )^{3+m} \left (a +b \,\mathrm {arccosh}\! \left (c x \right )\right )}{f^{3} \left (3+m \right )}-\frac {b e \left (f x \right )^{2+m} \sqrt {c x -1}\, \sqrt {c x +1}}{c \,f^{2} \left (3+m \right )^{2}}-\frac {b \left (e \left (1+m \right ) \left (2+m \right )+c^{2} d \left (3+m \right )^{2}\right ) \left (f x \right )^{2+m} \hypergeom \! \left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [2+\frac {m}{2}\right ], c^{2} x^{2}\right ) \sqrt {-c^{2} x^{2}+1}}{c \,f^{2} \left (1+m \right ) \left (2+m \right ) \left (3+m \right )^{2} \sqrt {c x -1}\, \sqrt {c x +1}} \]
command
Integrate[(f*x)^m*(d + e*x^2)*(a + b*ArcCosh[c*x]),x]
Mathematica 13.1 output
\[ \int (f x)^m \left (d+e x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx \]
Mathematica 12.3 output
\[ x (f x)^m \left (\frac {\frac {\left (d (m+3)+e (m+1) x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{m+1}-\frac {b c e x^3 \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {m+4}{2};\frac {m+6}{2};c^2 x^2\right )}{(m+4) \sqrt {c x-1} \sqrt {c x+1}}}{m+3}-\frac {b c d x \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{\left (m^2+3 m+2\right ) \sqrt {c x-1} \sqrt {c x+1}}\right ) \]