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