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