\[ \int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x^{15/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{13 x^{\frac {13}{2}}}+\frac {36 e^{\frac {3}{2}} \sqrt {e \,x^{2}+d}}{1001 d^{2} x^{\frac {7}{2}}}-\frac {60 e^{\frac {5}{2}} \sqrt {e \,x^{2}+d}}{1001 d^{3} x^{\frac {3}{2}}}-\frac {4 \sqrt {e}\, \sqrt {e \,x^{2}+d}}{143 d \,x^{\frac {11}{2}}}-\frac {30 e^{\frac {13}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {d}+x \sqrt {e}\right ) \sqrt {\frac {e \,x^{2}+d}{\left (\sqrt {d}+x \sqrt {e}\right )^{2}}}}{1001 \cos \left (2 \arctan \left (\frac {e^{\frac {1}{4}} \sqrt {x}}{d^{\frac {1}{4}}}\right )\right ) d^{\frac {13}{4}} \sqrt {e \,x^{2}+d}} \]
command
integrate(arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^(15/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {77 \, d^{3} \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 \, {\left (15 \, x^{5} \cosh \left (\frac {1}{2}\right )^{5} + 75 \, x^{5} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right )^{4} + 15 \, x^{5} \sinh \left (\frac {1}{2}\right )^{5} - 9 \, d x^{3} \cosh \left (\frac {1}{2}\right )^{3} + 7 \, d^{2} x \cosh \left (\frac {1}{2}\right ) + 3 \, {\left (50 \, x^{5} \cosh \left (\frac {1}{2}\right )^{2} - 3 \, d x^{3}\right )} \sinh \left (\frac {1}{2}\right )^{3} + 3 \, {\left (50 \, x^{5} \cosh \left (\frac {1}{2}\right )^{3} - 9 \, d x^{3} \cosh \left (\frac {1}{2}\right )\right )} \sinh \left (\frac {1}{2}\right )^{2} + {\left (75 \, x^{5} \cosh \left (\frac {1}{2}\right )^{4} - 27 \, d x^{3} \cosh \left (\frac {1}{2}\right )^{2} + 7 \, d^{2} x\right )} \sinh \left (\frac {1}{2}\right )\right )} \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 )}} + 60 \, {\left (x^{7} \cosh \left (\frac {1}{2}\right )^{6} + 6 \, x^{7} \cosh \left (\frac {1}{2}\right )^{5} \sinh \left (\frac {1}{2}\right ) + 15 \, x^{7} \cosh \left (\frac {1}{2}\right )^{4} \sinh \left (\frac {1}{2}\right )^{2} + 20 \, x^{7} \cosh \left (\frac {1}{2}\right )^{3} \sinh \left (\frac {1}{2}\right )^{3} + 15 \, x^{7} \cosh \left (\frac {1}{2}\right )^{2} \sinh \left (\frac {1}{2}\right )^{4} + 6 \, x^{7} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right )^{5} + x^{7} \sinh \left (\frac {1}{2}\right )^{6}\right )} {\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 )}{1001 \, d^{3} x^{7}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right )}{x^{\frac {15}{2}}}, x\right ) \]