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