\[ \int \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx \]
Optimal antiderivative \[ \frac {2 x^{\frac {3}{2}} \arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{3}-\frac {4 \sqrt {x}\, \sqrt {e \,x^{2}+d}}{9 \sqrt {e}}+\frac {2 d^{\frac {3}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {d}+x \sqrt {e}\right ) \sqrt {\frac {e \,x^{2}+d}{\left (\sqrt {d}+x \sqrt {e}\right )^{2}}}}{9 \cos \left (2 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right ) e^{\frac {3}{4}} \sqrt {e \,x^{2}+d}} \]
command
integrate(x^(1/2)*arctanh(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
\[ \frac {3 \, {\left (x \cosh \left (\frac {1}{2}\right )^{2} + 2 \, x \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right ) + x \sinh \left (\frac {1}{2}\right )^{2}\right )} \sqrt {x} \log \left (\frac {2 \, x^{2} \cosh \left (\frac {1}{2}\right )^{2} + 4 \, x^{2} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right ) + 2 \, x^{2} \sinh \left (\frac {1}{2}\right )^{2} + 2 \, {\left (x \cosh \left (\frac {1}{2}\right ) + x \sinh \left (\frac {1}{2}\right )\right )} \sqrt {\frac {{\left (x^{2} + d\right )} \cosh \left (\frac {1}{2}\right ) + {\left (x^{2} - d\right )} \sinh \left (\frac {1}{2}\right )}{\cosh \left (\frac {1}{2}\right ) - \sinh \left (\frac {1}{2}\right )}} + d}{d}\right ) - 4 \, \sqrt {x} \sqrt {\frac {{\left (x^{2} + d\right )} \cosh \left (\frac {1}{2}\right ) + {\left (x^{2} - d\right )} \sinh \left (\frac {1}{2}\right )}{\cosh \left (\frac {1}{2}\right ) - \sinh \left (\frac {1}{2}\right )}} {\left (\cosh \left (\frac {1}{2}\right ) + \sinh \left (\frac {1}{2}\right )\right )} + 4 \, d {\rm weierstrassPInverse}\left (-\frac {4 \, d}{\cosh \left (\frac {1}{2}\right )^{2} + 2 \, \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right ) + \sinh \left (\frac {1}{2}\right )^{2}}, 0, x\right )}{9 \, {\left (\cosh \left (\frac {1}{2}\right )^{2} + 2 \, \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right ) + \sinh \left (\frac {1}{2}\right )^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {x} \operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right ), x\right ) \]