81.2 Problem number 19

\[ \int x^{5/2} \text {ArcTan}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx \]

Optimal antiderivative \[ \frac {2 x^{\frac {7}{2}} \arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{7}+\frac {4 x^{\frac {5}{2}} \sqrt {e \,x^{2}+d}}{49 \sqrt {-e}}+\frac {20 d \sqrt {x}\, \sqrt {e \,x^{2}+d}}{147 \left (-e \right )^{\frac {3}{2}}}-\frac {10 d^{\frac {7}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {-e}\, \left (\sqrt {d}+x \sqrt {e}\right ) \sqrt {\frac {e \,x^{2}+d}{\left (\sqrt {d}+x \sqrt {e}\right )^{2}}}}{147 \cos \left (2 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right ) e^{\frac {9}{4}} \sqrt {e \,x^{2}+d}} \]

command

integrate(x^(5/2)*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

\[ \frac {1}{147} \, {\left (21 i \, x^{\frac {7}{2}} e^{2} \log \left (\frac {2 \, x^{2} e + 2 \, \sqrt {x^{2} e + d} x e^{\frac {1}{2}} + d}{d}\right ) - 4 \, \sqrt {x^{2} e + d} {\left (3 i \, x^{2} e - 5 i \, d\right )} \sqrt {x} e^{\frac {1}{2}} - 20 i \, d^{2} {\rm weierstrassPInverse}\left (-4 \, d e^{\left (-1\right )}, 0, x\right )\right )} e^{\left (-2\right )} \]

Fricas 1.3.7 via sagemath 9.3 output

\[ {\rm integral}\left (x^{\frac {5}{2}} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ), x\right ) \]