\[ \int x^{7/2} \text {ArcTan}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx \]
Optimal antiderivative \[ \frac {2 x^{\frac {9}{2}} \arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{9}+\frac {28 d \,x^{\frac {3}{2}} \sqrt {e \,x^{2}+d}}{405 \left (-e \right )^{\frac {3}{2}}}+\frac {4 x^{\frac {7}{2}} \sqrt {e \,x^{2}+d}}{81 \sqrt {-e}}-\frac {28 d^{2} \sqrt {-e}\, \sqrt {x}\, \sqrt {e \,x^{2}+d}}{135 e^{\frac {5}{2}} \left (\sqrt {d}+x \sqrt {e}\right )}+\frac {28 d^{\frac {9}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {-e}\, \left (\sqrt {d}+x \sqrt {e}\right ) \sqrt {\frac {e \,x^{2}+d}{\left (\sqrt {d}+x \sqrt {e}\right )^{2}}}}{135 \cos \left (2 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right ) e^{\frac {11}{4}} \sqrt {e \,x^{2}+d}}-\frac {14 d^{\frac {9}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {-e}\, \left (\sqrt {d}+x \sqrt {e}\right ) \sqrt {\frac {e \,x^{2}+d}{\left (\sqrt {d}+x \sqrt {e}\right )^{2}}}}{135 \cos \left (2 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right ) e^{\frac {11}{4}} \sqrt {e \,x^{2}+d}} \]
command
integrate(x^(7/2)*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
\[ \frac {1}{405} \, {\left (45 i \, x^{\frac {9}{2}} e^{2} \log \left (\frac {2 \, x^{2} e + 2 \, \sqrt {x^{2} e + d} x e^{\frac {1}{2}} + d}{d}\right ) - 4 \, {\left (5 i \, x^{3} e - 7 i \, d x\right )} \sqrt {x^{2} e + d} \sqrt {x} e^{\frac {1}{2}} + 84 i \, d^{2} {\rm weierstrassZeta}\left (-4 \, d e^{\left (-1\right )}, 0, {\rm weierstrassPInverse}\left (-4 \, d e^{\left (-1\right )}, 0, x\right )\right )\right )} e^{\left (-2\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (x^{\frac {7}{2}} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ), x\right ) \]