\[ \int \frac {a+b \cosh ^{-1}(c+d x)}{(c e+d e x)^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {4 b \EllipticF \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}, i\right ) \sqrt {-d x -c +1}}{d \,e^{\frac {3}{2}} \sqrt {d x +c -1}}-\frac {2 \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}{d e \sqrt {e \left (d x +c \right )}} \]
command
integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (\sqrt {{\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )} b d^{2} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) + \sqrt {{\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )} a d^{2} - 2 \, \sqrt {d^{3} \cosh \left (1\right ) + d^{3} \sinh \left (1\right )} {\left (b d x + b c\right )} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right )}}{{\left (d^{4} x + c d^{3}\right )} \cosh \left (1\right )^{2} + 2 \, {\left (d^{4} x + c d^{3}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (d^{4} x + c d^{3}\right )} \sinh \left (1\right )^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {d e x + c e} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}}{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}, x\right ) \]