\[ \int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (7 \left (9 a^{2}+7 b^{2}\right ) d^{2}-10 b c \left (-9 a d +b c \right )\right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{315 d f}+\frac {4 b \left (-9 a d +b c \right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{\frac {5}{2}}}{63 d f}-\frac {2 b^{2} \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{\frac {7}{2}}}{9 d f}-\frac {4 \left (84 a^{2} c \,d^{2}+15 a b d \left (3 c^{2}+5 d^{2}\right )-b^{2} \left (5 c^{3}-57 c \,d^{2}\right )\right ) \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}}{315 d f}-\frac {2 \left (21 a^{2} d^{2} \left (23 c^{2}+9 d^{2}\right )+30 a b d \left (3 c^{3}+29 c \,d^{2}\right )-b^{2} \left (10 c^{4}-279 c^{2} d^{2}-147 d^{4}\right )\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticE \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {c +d \sin \left (f x +e \right )}}{315 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) d^{2} f \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}-\frac {4 \left (c^{2}-d^{2}\right ) \left (-84 a^{2} c \,d^{2}-45 a b \,c^{2} d -75 a b \,d^{3}+5 b^{2} c^{3}-57 b^{2} c \,d^{2}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticF \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}{315 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) d^{2} f \sqrt {c +d \sin \left (f x +e \right )}} \]
command
integrate((a+b*sin(f*x+e))^2*(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (20 \, b^{2} c^{5} - 180 \, a b c^{4} d + 690 \, a b c^{2} d^{3} + 450 \, a b d^{5} - 3 \, {\left (7 \, a^{2} + 31 \, b^{2}\right )} c^{3} d^{2} + 3 \, {\left (231 \, a^{2} + 163 \, b^{2}\right )} c d^{4}\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 (20 \, b^{2} c^{5} - 180 \, a b c^{4} d + 690 \, a b c^{2} d^{3} + 450 \, a b d^{5} - 3 \, {\left (7 \, a^{2} + 31 \, b^{2}\right )} c^{3} d^{2} + 3 \, {\left (231 \, a^{2} + 163 \, b^{2}\right )} c d^{4}\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 \, \sqrt {2} {\left (-10 i \, b^{2} c^{4} d + 90 i \, a b c^{3} d^{2} + 870 i \, a b c d^{4} + 3 i \, {\left (161 \, a^{2} + 93 \, b^{2}\right )} c^{2} d^{3} + 21 i \, {\left (9 \, a^{2} + 7 \, b^{2}\right )} d^{5}\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 \, \sqrt {2} {\left (10 i \, b^{2} c^{4} d - 90 i \, a b c^{3} d^{2} - 870 i \, a b c d^{4} - 3 i \, {\left (161 \, a^{2} + 93 \, b^{2}\right )} c^{2} d^{3} - 21 i \, {\left (9 \, a^{2} + 7 \, b^{2}\right )} d^{5}\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 (5 \, {\left (19 \, b^{2} c d^{4} + 18 \, a b d^{5}\right )} \cos \left (f x + e\right )^{3} - {\left (5 \, b^{2} c^{3} d^{2} + 270 \, a b c^{2} d^{3} + 240 \, a b d^{5} + 3 \, {\left (77 \, a^{2} + 86 \, b^{2}\right )} c d^{4}\right )} \cos \left (f x + e\right ) + {\left (35 \, b^{2} d^{5} \cos \left (f x + e\right )^{3} - 3 \, {\left (25 \, b^{2} c^{2} d^{3} + 90 \, a b c d^{4} + 7 \, {\left (3 \, a^{2} + 4 \, b^{2}\right )} d^{5}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{945 \, d^{3} f} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b^{2} d^{2} \cos \left (f x + e\right )^{4} + 4 \, a b c d + {\left (a^{2} + b^{2}\right )} c^{2} + {\left (a^{2} + b^{2}\right )} d^{2} - {\left (b^{2} c^{2} + 4 \, a b c d + {\left (a^{2} + 2 \, b^{2}\right )} d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (a b c^{2} + a b d^{2} + {\left (a^{2} + b^{2}\right )} c d - {\left (b^{2} c d + a b d^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}, x\right ) \]