\[ \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-a d +b c \right )^{2} \cos \left (f x +e \right )}{3 d \left (c^{2}-d^{2}\right ) f \left (c +d \sin \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {4 \left (-a d +b c \right ) \left (2 a c d +b \left (c^{2}-3 d^{2}\right )\right ) \cos \left (f x +e \right )}{3 d \left (c^{2}-d^{2}\right )^{2} f \sqrt {c +d \sin \left (f x +e \right )}}+\frac {4 \left (-a d +b c \right ) \left (2 a c d +b \left (c^{2}-3 d^{2}\right )\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticE \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {c +d \sin \left (f x +e \right )}}{3 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) d^{2} \left (c^{2}-d^{2}\right )^{2} f \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}-\frac {2 \left (2 a b c d -a^{2} d^{2}+b^{2} \left (2 c^{2}-3 d^{2}\right )\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticF \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}{3 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) d^{2} \left (c^{2}-d^{2}\right ) f \sqrt {c +d \sin \left (f x +e \right )}} \]
command
integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (\sqrt {2} {\left (4 \, b^{2} c^{4} d^{2} + 4 \, a b c^{3} d^{3} - 12 \, a b c d^{5} + {\left (a^{2} - 9 \, b^{2}\right )} c^{2} d^{4} + 3 \, {\left (a^{2} + 3 \, b^{2}\right )} d^{6}\right )} \cos \left (f x + e\right )^{2} - 2 \, \sqrt {2} {\left (4 \, b^{2} c^{5} d + 4 \, a b c^{4} d^{2} - 12 \, a b c^{2} d^{4} + {\left (a^{2} - 9 \, b^{2}\right )} c^{3} d^{3} + 3 \, {\left (a^{2} + 3 \, b^{2}\right )} c d^{5}\right )} \sin \left (f x + e\right ) - \sqrt {2} {\left (4 \, b^{2} c^{6} + 4 \, a b c^{5} d - 8 \, a b c^{3} d^{3} + 4 \, a^{2} c^{2} d^{4} - 12 \, a b c d^{5} + {\left (a^{2} - 5 \, b^{2}\right )} c^{4} d^{2} + 3 \, {\left (a^{2} + 3 \, b^{2}\right )} d^{6}\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^{4} d^{2} + 4 \, a b c^{3} d^{3} - 12 \, a b c d^{5} + {\left (a^{2} - 9 \, b^{2}\right )} c^{2} d^{4} + 3 \, {\left (a^{2} + 3 \, b^{2}\right )} d^{6}\right )} \cos \left (f x + e\right )^{2} - 2 \, \sqrt {2} {\left (4 \, b^{2} c^{5} d + 4 \, a b c^{4} d^{2} - 12 \, a b c^{2} d^{4} + {\left (a^{2} - 9 \, b^{2}\right )} c^{3} d^{3} + 3 \, {\left (a^{2} + 3 \, b^{2}\right )} c d^{5}\right )} \sin \left (f x + e\right ) - \sqrt {2} {\left (4 \, b^{2} c^{6} + 4 \, a b c^{5} d - 8 \, a b c^{3} d^{3} + 4 \, a^{2} c^{2} d^{4} - 12 \, a b c d^{5} + {\left (a^{2} - 5 \, b^{2}\right )} c^{4} d^{2} + 3 \, {\left (a^{2} + 3 \, b^{2}\right )} d^{6}\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 ) + 6 \, {\left (\sqrt {2} {\left (i \, b^{2} c^{3} d^{3} + i \, a b c^{2} d^{4} + 3 i \, a b d^{6} - i \, {\left (2 \, a^{2} + 3 \, b^{2}\right )} c d^{5}\right )} \cos \left (f x + e\right )^{2} + 2 \, \sqrt {2} {\left (-i \, b^{2} c^{4} d^{2} - i \, a b c^{3} d^{3} - 3 i \, a b c d^{5} + i \, {\left (2 \, a^{2} + 3 \, b^{2}\right )} c^{2} d^{4}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (-i \, b^{2} c^{5} d - i \, a b c^{4} d^{2} - 4 i \, a b c^{2} d^{4} - 3 i \, a b d^{6} + 2 i \, {\left (a^{2} + b^{2}\right )} c^{3} d^{3} + i \, {\left (2 \, a^{2} + 3 \, b^{2}\right )} c 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 (\sqrt {2} {\left (-i \, b^{2} c^{3} d^{3} - i \, a b c^{2} d^{4} - 3 i \, a b d^{6} + i \, {\left (2 \, a^{2} + 3 \, b^{2}\right )} c d^{5}\right )} \cos \left (f x + e\right )^{2} + 2 \, \sqrt {2} {\left (i \, b^{2} c^{4} d^{2} + i \, a b c^{3} d^{3} + 3 i \, a b c d^{5} - i \, {\left (2 \, a^{2} + 3 \, b^{2}\right )} c^{2} d^{4}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (i \, b^{2} c^{5} d + i \, a b c^{4} d^{2} + 4 i \, a b c^{2} d^{4} + 3 i \, a b d^{6} - 2 i \, {\left (a^{2} + b^{2}\right )} c^{3} d^{3} - i \, {\left (2 \, a^{2} + 3 \, b^{2}\right )} c 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 (2 \, {\left (b^{2} c^{3} d^{3} + a b c^{2} d^{4} + 3 \, a b d^{6} - {\left (2 \, a^{2} + 3 \, b^{2}\right )} c d^{5}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (b^{2} c^{4} d^{2} + 4 \, a b c^{3} d^{3} + 4 \, a b c d^{5} + a^{2} d^{6} - 5 \, {\left (a^{2} + b^{2}\right )} c^{2} d^{4}\right )} \cos \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{9 \, {\left ({\left (c^{4} d^{5} - 2 \, c^{2} d^{7} + d^{9}\right )} f \cos \left (f x + e\right )^{2} - 2 \, {\left (c^{5} d^{4} - 2 \, c^{3} d^{6} + c d^{8}\right )} f \sin \left (f x + e\right ) - {\left (c^{6} d^{3} - c^{4} d^{5} - c^{2} d^{7} + d^{9}\right )} f\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b^{2} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2}\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{3 \, c d^{2} \cos \left (f x + e\right )^{2} - c^{3} - 3 \, c d^{2} + {\left (d^{3} \cos \left (f x + e\right )^{2} - 3 \, c^{2} d - d^{3}\right )} \sin \left (f x + e\right )}, x\right ) \]