\[ \int x^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx \]
Optimal antiderivative \[ \frac {2 x^{\frac {5}{2}} \arctanh \! \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{5}-\frac {4 x^{\frac {3}{2}} \sqrt {e \,x^{2}+d}}{25 \sqrt {e}}+\frac {12 d \sqrt {x}\, \sqrt {e \,x^{2}+d}}{25 e \left (\sqrt {d}+x \sqrt {e}\right )}-\frac {12 d^{\frac {5}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \! \left (\sin \! \left (2 \arctan \! \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {d}+x \sqrt {e}\right ) \sqrt {\frac {e \,x^{2}+d}{\left (\sqrt {d}+x \sqrt {e}\right )^{2}}}}{25 \cos \! \left (2 \arctan \! \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right ) e^{\frac {5}{4}} \sqrt {e \,x^{2}+d}}+\frac {6 d^{\frac {5}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \! \left (\sin \! \left (2 \arctan \! \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {d}+x \sqrt {e}\right ) \sqrt {\frac {e \,x^{2}+d}{\left (\sqrt {d}+x \sqrt {e}\right )^{2}}}}{25 \cos \! \left (2 \arctan \! \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right ) e^{\frac {5}{4}} \sqrt {e \,x^{2}+d}} \]
command
integrate(x^(3/2)*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ +\infty \]