\[ \int (d \sec (e+f x))^{5/2} (b \tan (e+f x))^{3/2} \, dx \]
Optimal antiderivative \[ \frac {b^{2} d^{2} \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticF \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\right ) \sqrt {d \sec \left (f x +e \right )}\, \left (\sqrt {\sin }\left (f x +e \right )\right )}{6 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) f \sqrt {b \tan \left (f x +e \right )}}+\frac {b \left (d \sec \left (f x +e \right )\right )^{\frac {5}{2}} \sqrt {b \tan \left (f x +e \right )}}{3 f}-\frac {b \,d^{2} \sqrt {d \sec \left (f x +e \right )}\, \sqrt {b \tan \left (f x +e \right )}}{6 f} \]
command
integrate((d*sec(f*x+e))^(5/2)*(b*tan(f*x+e))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {\sqrt {-2 i \, b d} b d^{2} \cos \left (f x + e\right )^{2} {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + \sqrt {2 i \, b d} b d^{2} \cos \left (f x + e\right )^{2} {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) + 2 \, {\left (b d^{2} \cos \left (f x + e\right )^{2} - 2 \, b d^{2}\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{12 \, f \cos \left (f x + e\right )^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {d \sec \left (f x + e\right )} \sqrt {b \tan \left (f x + e\right )} b d^{2} \sec \left (f x + e\right )^{2} \tan \left (f x + e\right ), x\right ) \]