\[ \int (f x)^m \left (d+e x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx \]
Optimal antiderivative \[ \frac {d^{2} \left (f x \right )^{1+m} \left (a +b \,\mathrm {arccosh}\! \left (c x \right )\right )}{f \left (1+m \right )}+\frac {2 d e \left (f x \right )^{3+m} \left (a +b \,\mathrm {arccosh}\! \left (c x \right )\right )}{f^{3} \left (3+m \right )}+\frac {e^{2} \left (f x \right )^{5+m} \left (a +b \,\mathrm {arccosh}\! \left (c x \right )\right )}{f^{5} \left (5+m \right )}+\frac {b e \left (2 c^{2} d \left (5+m \right )^{2}+e \left (m^{2}+7 m +12\right )\right ) \left (f x \right )^{2+m} \left (-c^{2} x^{2}+1\right )}{c^{3} f^{2} \left (3+m \right )^{2} \left (5+m \right )^{2} \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {b \,e^{2} \left (f x \right )^{4+m} \left (-c^{2} x^{2}+1\right )}{c \,f^{4} \left (5+m \right )^{2} \sqrt {c x -1}\, \sqrt {c x +1}}-\frac {b \left (\frac {c^{4} d^{2} \left (3+m \right ) \left (5+m \right )}{1+m}+\frac {e \left (2+m \right ) \left (2 c^{2} d \left (5+m \right )^{2}+e \left (m^{2}+7 m +12\right )\right )}{\left (3+m \right ) \left (5+m \right )}\right ) \left (f x \right )^{2+m} \hypergeom \! \left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [2+\frac {m}{2}\right ], c^{2} x^{2}\right ) \sqrt {-c^{2} x^{2}+1}}{c^{3} f^{2} \left (2+m \right ) \left (3+m \right ) \left (5+m \right ) \sqrt {c x -1}\, \sqrt {c x +1}} \]
command
Integrate[(f*x)^m*(d + e*x^2)^2*(a + b*ArcCosh[c*x]),x]
Mathematica 13.1 output
\[ \int (f x)^m \left (d+e x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx \]
Mathematica 12.3 output
\[ x (f x)^m \left (\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )}{m+1}+\frac {2 d e x^2 \left (a+b \cosh ^{-1}(c x)\right )}{m+3}+\frac {e^2 x^4 \left (a+b \cosh ^{-1}(c x)\right )}{m+5}-\frac {b c d^2 x \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{\left (m^2+3 m+2\right ) \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c d e x^3 \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {m+4}{2};\frac {m+6}{2};c^2 x^2\right )}{\left (m^2+7 m+12\right ) \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c e^2 x^5 \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {m+6}{2};\frac {m+8}{2};c^2 x^2\right )}{(m+5) (m+6) \sqrt {c x-1} \sqrt {c x+1}}\right ) \]