\[ \int x^{3/2} \text {ArcTan}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx \]
Optimal antiderivative \[ \frac {2 x^{\frac {5}{2}} \arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{5}+\frac {4 x^{\frac {3}{2}} \sqrt {e \,x^{2}+d}}{25 \sqrt {-e}}+\frac {12 d \sqrt {-e}\, \sqrt {x}\, \sqrt {e \,x^{2}+d}}{25 e^{\frac {3}{2}} \left (\sqrt {d}+x \sqrt {e}\right )}-\frac {12 d^{\frac {5}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {-e}\, \left (\sqrt {d}+x \sqrt {e}\right ) \sqrt {\frac {e \,x^{2}+d}{\left (\sqrt {d}+x \sqrt {e}\right )^{2}}}}{25 \cos \left (2 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right ) e^{\frac {7}{4}} \sqrt {e \,x^{2}+d}}+\frac {6 d^{\frac {5}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {-e}\, \left (\sqrt {d}+x \sqrt {e}\right ) \sqrt {\frac {e \,x^{2}+d}{\left (\sqrt {d}+x \sqrt {e}\right )^{2}}}}{25 \cos \left (2 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right ) e^{\frac {7}{4}} \sqrt {e \,x^{2}+d}} \]
command
integrate(x^(3/2)*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {1}{25} \, {\left (5 i \, x^{\frac {5}{2}} e \log \left (\frac {2 \, x^{2} e + 2 \, \sqrt {x^{2} e + d} x e^{\frac {1}{2}} + d}{d}\right ) - 4 i \, \sqrt {x^{2} e + d} x^{\frac {3}{2}} e^{\frac {1}{2}} - 12 i \, d {\rm weierstrassZeta}\left (-4 \, d e^{\left (-1\right )}, 0, {\rm weierstrassPInverse}\left (-4 \, d e^{\left (-1\right )}, 0, x\right )\right )\right )} e^{\left (-1\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (x^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ), x\right ) \]