\[ \int \frac {\text {ArcTan}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{\sqrt {x}}+\frac {2 \sqrt {\frac {\cos \left (4 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {-e}\, \left (\sqrt {d}+x \sqrt {e}\right ) \sqrt {\frac {e \,x^{2}+d}{\left (\sqrt {d}+x \sqrt {e}\right )^{2}}}}{\cos \left (2 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right ) d^{\frac {1}{4}} e^{\frac {1}{4}} \sqrt {e \,x^{2}+d}} \]
command
integrate(arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))/x^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {4 i \, x {\rm weierstrassPInverse}\left (-4 \, d e^{\left (-1\right )}, 0, x\right ) - i \, \sqrt {x} \log \left (\frac {2 \, x^{2} e + 2 \, \sqrt {x^{2} e + d} x e^{\frac {1}{2}} + d}{d}\right )}{x} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{x^{\frac {3}{2}}}, x\right ) \]