\[ \int \frac {a+b \text {ArcCos}(c x)}{(d x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (a +b \arccos \left (c x \right )\right )}{3 d \left (d x \right )^{\frac {3}{2}}}+\frac {4 b \,c^{\frac {3}{2}} \EllipticE \left (\frac {\sqrt {c}\, \sqrt {d x}}{\sqrt {d}}, i\right )}{3 d^{\frac {5}{2}}}-\frac {4 b \,c^{\frac {3}{2}} \EllipticF \left (\frac {\sqrt {c}\, \sqrt {d x}}{\sqrt {d}}, i\right )}{3 d^{\frac {5}{2}}}+\frac {4 b c \sqrt {-c^{2} x^{2}+1}}{3 d^{2} \sqrt {d x}} \]
command
integrate((a+b*arccos(c*x))/(d*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (2 \, \sqrt {-c^{2} d} b c x^{2} {\rm weierstrassZeta}\left (\frac {4}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{c^{2}}, 0, x\right )\right ) + {\left (2 \, \sqrt {-c^{2} x^{2} + 1} b c x - b \arccos \left (c x\right ) - a\right )} \sqrt {d x}\right )}}{3 \, d^{3} x^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {d x} {\left (b \arccos \left (c x\right ) + a\right )}}{d^{3} x^{3}}, x\right ) \]