\[ \int \frac {(d \sec (e+f x))^{5/2}}{(b \tan (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 d^{2} \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticF \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\right ) \sqrt {d \sec \left (f x +e \right )}\, \left (\sqrt {\sin }\left (f x +e \right )\right )}{3 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) b^{2} f \sqrt {b \tan \left (f x +e \right )}}-\frac {2 d^{2} \sqrt {d \sec \left (f x +e \right )}}{3 b f \left (b \tan \left (f x +e \right )\right )^{\frac {3}{2}}} \]
command
integrate((d*sec(f*x+e))^(5/2)/(b*tan(f*x+e))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, d^{2} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{2} + {\left (d^{2} \cos \left (f x + e\right )^{2} - d^{2}\right )} \sqrt {-2 i \, b d} {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + {\left (d^{2} \cos \left (f x + e\right )^{2} - d^{2}\right )} \sqrt {2 i \, b d} {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}{3 \, {\left (b^{3} f \cos \left (f x + e\right )^{2} - b^{3} f\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {d \sec \left (f x + e\right )} \sqrt {b \tan \left (f x + e\right )} d^{2} \sec \left (f x + e\right )^{2}}{b^{3} \tan \left (f x + e\right )^{3}}, x\right ) \]