\[ \int (d \sec (e+f x))^{5/2} (a+b \tan (e+f x))^3 \, dx \]
Optimal antiderivative \[ \frac {a \left (7 a^{2}-6 b^{2}\right ) d^{2} \sqrt {2}\, \sqrt {\frac {1+\sqrt {1+\tan ^{2}\left (f x +e \right )}}{\sqrt {1+\tan ^{2}\left (f x +e \right )}}}\, \EllipticF \left (\sin \left (\frac {\arctan \left (\tan \left (f x +e \right )\right )}{2}\right ), \sqrt {2}\right ) \sqrt {d \sec \left (f x +e \right )}}{21 \cos \left (\frac {\arctan \left (\tan \left (f x +e \right )\right )}{2}\right ) f \left (\sec ^{2}\left (f x +e \right )\right )^{\frac {1}{4}}}+\frac {2 a \left (7 a^{2}-6 b^{2}\right ) d^{2} \sqrt {d \sec \left (f x +e \right )}\, \tan \left (f x +e \right )}{21 f}+\frac {2 b \,d^{2} \left (\sec ^{2}\left (f x +e \right )\right ) \sqrt {d \sec \left (f x +e \right )}\, \left (a +b \tan \left (f x +e \right )\right )^{2}}{9 f}+\frac {2 b \,d^{2} \left (\sec ^{2}\left (f x +e \right )\right ) \sqrt {d \sec \left (f x +e \right )}\, \left (154 a^{2}-28 b^{2}+65 a b \tan \left (f x +e \right )\right )}{315 f} \]
command
integrate((d*sec(f*x+e))^(5/2)*(a+b*tan(f*x+e))^3,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-15 i \, \sqrt {2} {\left (7 \, a^{3} - 6 \, a b^{2}\right )} d^{\frac {5}{2}} \cos \left (f x + e\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 15 i \, \sqrt {2} {\left (7 \, a^{3} - 6 \, a b^{2}\right )} d^{\frac {5}{2}} \cos \left (f x + e\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) + 2 \, {\left (35 \, b^{3} d^{2} + 63 \, {\left (3 \, a^{2} b - b^{3}\right )} d^{2} \cos \left (f x + e\right )^{2} + 15 \, {\left (9 \, a b^{2} d^{2} \cos \left (f x + e\right ) + {\left (7 \, a^{3} - 6 \, a b^{2}\right )} d^{2} \cos \left (f x + e\right )^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{315 \, f \cos \left (f x + e\right )^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b^{3} d^{2} \sec \left (f x + e\right )^{2} \tan \left (f x + e\right )^{3} + 3 \, a b^{2} d^{2} \sec \left (f x + e\right )^{2} \tan \left (f x + e\right )^{2} + 3 \, a^{2} b d^{2} \sec \left (f x + e\right )^{2} \tan \left (f x + e\right ) + a^{3} d^{2} \sec \left (f x + e\right )^{2}\right )} \sqrt {d \sec \left (f x + e\right )}, x\right ) \]