\[ \int \frac {(a+b \sin (e+f x))^2}{\sqrt {c+d \sin (e+f x)}} \, dx \]
Optimal antiderivative \[ -\frac {2 b^{2} \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}}{3 d f}+\frac {4 b \left (-3 a d +b c \right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticE \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {c +d \sin \left (f x +e \right )}}{3 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) d^{2} f \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}-\frac {2 \left (\left (3 a^{2}+b^{2}\right ) d^{2}+2 b c \left (-3 a d +b c \right )\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticF \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}{3 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) d^{2} f \sqrt {c +d \sin \left (f x +e \right )}} \]
command
integrate((a+b*sin(f*x+e))^2/(c+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 {6 \, \sqrt {d \sin \left (f x + e\right ) + c} b^{2} d^{2} \cos \left (f x + e\right ) - \sqrt {2} {\left (4 \, b^{2} c^{2} - 12 \, a b c d + 3 \, {\left (3 \, a^{2} + b^{2}\right )} d^{2}\right )} \sqrt {i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) - \sqrt {2} {\left (4 \, b^{2} c^{2} - 12 \, a b c d + 3 \, {\left (3 \, a^{2} + b^{2}\right )} d^{2}\right )} \sqrt {-i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right ) + 6 \, \sqrt {2} {\left (-i \, b^{2} c d + 3 i \, a b d^{2}\right )} \sqrt {i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) + 6 \, \sqrt {2} {\left (i \, b^{2} c d - 3 i \, a b d^{2}\right )} \sqrt {-i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right )}{9 \, d^{3} f} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {b^{2} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2}}{\sqrt {d \sin \left (f x + e\right ) + c}}, x\right ) \]