\[ \int (d x)^{5/2} (a+b \text {ArcCos}(c x)) \, dx \]
Optimal antiderivative \[ \frac {2 \left (d x \right )^{\frac {7}{2}} \left (a +b \arccos \left (c x \right )\right )}{7 d}+\frac {20 b \,d^{\frac {5}{2}} \EllipticF \left (\frac {\sqrt {c}\, \sqrt {d x}}{\sqrt {d}}, i\right )}{147 c^{\frac {7}{2}}}-\frac {4 b \left (d x \right )^{\frac {5}{2}} \sqrt {-c^{2} x^{2}+1}}{49 c}-\frac {20 b \,d^{2} \sqrt {d x}\, \sqrt {-c^{2} x^{2}+1}}{147 c^{3}} \]
command
integrate((d*x)^(5/2)*(a+b*arccos(c*x)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (10 \, \sqrt {-c^{2} d} b d^{2} {\rm weierstrassPInverse}\left (\frac {4}{c^{2}}, 0, x\right ) - {\left (21 \, b c^{5} d^{2} x^{3} \arccos \left (c x\right ) + 21 \, a c^{5} d^{2} x^{3} - 2 \, {\left (3 \, b c^{4} d^{2} x^{2} + 5 \, b c^{2} d^{2}\right )} \sqrt {-c^{2} x^{2} + 1}\right )} \sqrt {d x}\right )}}{147 \, c^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b d^{2} x^{2} \arccos \left (c x\right ) + a d^{2} x^{2}\right )} \sqrt {d x}, x\right ) \]