\[ \int \frac {1}{(d \sec (e+f x))^{3/2} (b \tan (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {8 \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticF \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\right ) \sqrt {d \sec \left (f x +e \right )}\, \left (\sqrt {\sin }\left (f x +e \right )\right )}{3 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) b^{2} d^{2} f \sqrt {b \tan \left (f x +e \right )}}-\frac {4 \sqrt {b \tan \left (f x +e \right )}}{3 b^{3} f \left (d \sec \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {2}{3 b f \left (d \sec \left (f x +e \right )\right )^{\frac {3}{2}} \left (b \tan \left (f x +e \right )\right )^{\frac {3}{2}}} \]
command
integrate(1/(d*sec(f*x+e))^(3/2)/(b*tan(f*x+e))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (2 \, \sqrt {-2 i \, b d} {\left (\cos \left (f x + e\right )^{2} - 1\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 2 \, \sqrt {2 i \, b d} {\left (\cos \left (f x + e\right )^{2} - 1\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) + {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{\cos \left (f x + e\right )}}\right )}}{3 \, {\left (b^{3} d^{2} f \cos \left (f x + e\right )^{2} - b^{3} d^{2} f\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {d \sec \left (f x + e\right )} \sqrt {b \tan \left (f x + e\right )}}{b^{3} d^{2} \sec \left (f x + e\right )^{2} \tan \left (f x + e\right )^{3}}, x\right ) \]