\[ \int \frac {a+b \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-a d +b c \right ) \cos \left (f x +e \right )}{3 \left (c^{2}-d^{2}\right ) f \left (c +d \sin \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {2 \left (4 a c d -b \left (c^{2}+3 d^{2}\right )\right ) \cos \left (f x +e \right )}{3 \left (c^{2}-d^{2}\right )^{2} f \sqrt {c +d \sin \left (f x +e \right )}}-\frac {2 \left (4 a c d -b \left (c^{2}+3 d^{2}\right )\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticE \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {c +d \sin \left (f x +e \right )}}{3 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) d \left (c^{2}-d^{2}\right )^{2} f \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}-\frac {2 \left (-a d +b c \right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticF \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}{3 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) d \left (c^{2}-d^{2}\right ) f \sqrt {c +d \sin \left (f x +e \right )}} \]
command
integrate((a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (\sqrt {2} {\left (2 \, b c^{3} d^{2} + a c^{2} d^{3} - 6 \, b c d^{4} + 3 \, a d^{5}\right )} \cos \left (f x + e\right )^{2} - 2 \, \sqrt {2} {\left (2 \, b c^{4} d + a c^{3} d^{2} - 6 \, b c^{2} d^{3} + 3 \, a c d^{4}\right )} \sin \left (f x + e\right ) - \sqrt {2} {\left (2 \, b c^{5} + a c^{4} d - 4 \, b c^{3} d^{2} + 4 \, a c^{2} d^{3} - 6 \, b c d^{4} + 3 \, a d^{5}\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 (2 \, b c^{3} d^{2} + a c^{2} d^{3} - 6 \, b c d^{4} + 3 \, a d^{5}\right )} \cos \left (f x + e\right )^{2} - 2 \, \sqrt {2} {\left (2 \, b c^{4} d + a c^{3} d^{2} - 6 \, b c^{2} d^{3} + 3 \, a c d^{4}\right )} \sin \left (f x + e\right ) - \sqrt {2} {\left (2 \, b c^{5} + a c^{4} d - 4 \, b c^{3} d^{2} + 4 \, a c^{2} d^{3} - 6 \, b c d^{4} + 3 \, a d^{5}\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 (i \, b c^{2} d^{3} - 4 i \, a c d^{4} + 3 i \, b d^{5}\right )} \cos \left (f x + e\right )^{2} + 2 \, \sqrt {2} {\left (-i \, b c^{3} d^{2} + 4 i \, a c^{2} d^{3} - 3 i \, b c d^{4}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (-i \, b c^{4} d + 4 i \, a c^{3} d^{2} - 4 i \, b c^{2} d^{3} + 4 i \, a c d^{4} - 3 i \, b d^{5}\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 (-i \, b c^{2} d^{3} + 4 i \, a c d^{4} - 3 i \, b d^{5}\right )} \cos \left (f x + e\right )^{2} + 2 \, \sqrt {2} {\left (i \, b c^{3} d^{2} - 4 i \, a c^{2} d^{3} + 3 i \, b c d^{4}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (i \, b c^{4} d - 4 i \, a c^{3} d^{2} + 4 i \, b c^{2} d^{3} - 4 i \, a c d^{4} + 3 i \, b d^{5}\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 ) + 6 \, {\left ({\left (b c^{2} d^{3} - 4 \, a c d^{4} + 3 \, b d^{5}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (2 \, b c^{3} d^{2} - 5 \, a c^{2} d^{3} + 2 \, b c d^{4} + a d^{5}\right )} \cos \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{9 \, {\left ({\left (c^{4} d^{4} - 2 \, c^{2} d^{6} + d^{8}\right )} f \cos \left (f x + e\right )^{2} - 2 \, {\left (c^{5} d^{3} - 2 \, c^{3} d^{5} + c d^{7}\right )} f \sin \left (f x + e\right ) - {\left (c^{6} d^{2} - c^{4} d^{4} - c^{2} d^{6} + d^{8}\right )} f\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (b \sin \left (f x + e\right ) + a\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{3 \, c d^{2} \cos \left (f x + e\right )^{2} - c^{3} - 3 \, c d^{2} + {\left (d^{3} \cos \left (f x + e\right )^{2} - 3 \, c^{2} d - d^{3}\right )} \sin \left (f x + e\right )}, x\right ) \]