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