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