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