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