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