\[ \int (a+b \cosh (x)+c \sinh (x))^{5/2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (c \cosh \left (x \right )+b \sinh \left (x \right )\right ) \left (a +b \cosh \left (x \right )+c \sinh \left (x \right )\right )^{\frac {3}{2}}}{5}+\frac {16 \left (a c \cosh \left (x \right )+a b \sinh \left (x \right )\right ) \sqrt {a +b \cosh \left (x \right )+c \sinh \left (x \right )}}{15}-\frac {2 i \left (23 a^{2}+9 b^{2}-9 c^{2}\right ) \sqrt {\frac {\cos \left (i x -\arctan \left (b , -i c \right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {i x}{2}-\frac {\arctan \left (b , -i c \right )}{2}\right ), \sqrt {2}\, \sqrt {\frac {\sqrt {b^{2}-c^{2}}}{a +\sqrt {b^{2}-c^{2}}}}\right ) \sqrt {a +b \cosh \left (x \right )+c \sinh \left (x \right )}}{15 \cos \left (\frac {i x}{2}-\frac {\arctan \left (b , -i c \right )}{2}\right ) \sqrt {\frac {a +b \cosh \left (x \right )+c \sinh \left (x \right )}{a +\sqrt {b^{2}-c^{2}}}}}+\frac {16 i a \left (a^{2}-b^{2}+c^{2}\right ) \sqrt {\frac {\cos \left (i x -\arctan \left (b , -i c \right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {i x}{2}-\frac {\arctan \left (b , -i c \right )}{2}\right ), \sqrt {2}\, \sqrt {\frac {\sqrt {b^{2}-c^{2}}}{a +\sqrt {b^{2}-c^{2}}}}\right ) \sqrt {\frac {a +b \cosh \left (x \right )+c \sinh \left (x \right )}{a +\sqrt {b^{2}-c^{2}}}}}{15 \cos \left (\frac {i x}{2}-\frac {\arctan \left (b , -i c \right )}{2}\right ) \sqrt {a +b \cosh \left (x \right )+c \sinh \left (x \right )}} \]
command
integrate((a+b*cosh(x)+c*sinh(x))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {4 \, {\left (\sqrt {2} {\left (a^{3} - 33 \, a b^{2} + 33 \, a c^{2}\right )} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (a^{3} - 33 \, a b^{2} + 33 \, a c^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} {\left (a^{3} - 33 \, a b^{2} + 33 \, a c^{2}\right )} \sinh \left (x\right )^{2}\right )} \sqrt {b + c} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2} + 3 \, c^{2}\right )}}{3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2} + 9 \, a c^{2}\right )}}{27 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )}}, \frac {3 \, {\left (b + c\right )} \cosh \left (x\right ) + 3 \, {\left (b + c\right )} \sinh \left (x\right ) + 2 \, a}{3 \, {\left (b + c\right )}}\right ) + 12 \, {\left (\sqrt {2} {\left (23 \, a^{2} b + 9 \, b^{3} - 9 \, b c^{2} - 9 \, c^{3} + {\left (23 \, a^{2} + 9 \, b^{2}\right )} c\right )} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (23 \, a^{2} b + 9 \, b^{3} - 9 \, b c^{2} - 9 \, c^{3} + {\left (23 \, a^{2} + 9 \, b^{2}\right )} c\right )} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} {\left (23 \, a^{2} b + 9 \, b^{3} - 9 \, b c^{2} - 9 \, c^{3} + {\left (23 \, a^{2} + 9 \, b^{2}\right )} c\right )} \sinh \left (x\right )^{2}\right )} \sqrt {b + c} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2} + 3 \, c^{2}\right )}}{3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2} + 9 \, a c^{2}\right )}}{27 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2} + 3 \, c^{2}\right )}}{3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2} + 9 \, a c^{2}\right )}}{27 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )}}, \frac {3 \, {\left (b + c\right )} \cosh \left (x\right ) + 3 \, {\left (b + c\right )} \sinh \left (x\right ) + 2 \, a}{3 \, {\left (b + c\right )}}\right )\right ) - 3 \, {\left (3 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )^{4} + 3 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \sinh \left (x\right )^{4} + 22 \, {\left (a b^{2} + 2 \, a b c + a c^{2}\right )} \cosh \left (x\right )^{3} + 2 \, {\left (11 \, a b^{2} + 22 \, a b c + 11 \, a c^{2} + 6 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 3 \, b^{3} + 3 \, b^{2} c + 3 \, b c^{2} - 3 \, c^{3} - 4 \, {\left (23 \, a^{2} b + 9 \, b^{3} - 9 \, b c^{2} - 9 \, c^{3} + {\left (23 \, a^{2} + 9 \, b^{2}\right )} c\right )} \cosh \left (x\right )^{2} - 2 \, {\left (46 \, a^{2} b + 18 \, b^{3} - 18 \, b c^{2} - 18 \, c^{3} - 9 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (23 \, a^{2} + 9 \, b^{2}\right )} c - 33 \, {\left (a b^{2} + 2 \, a b c + a c^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} - 22 \, {\left (a b^{2} - a c^{2}\right )} \cosh \left (x\right ) + 2 \, {\left (6 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )} \cosh \left (x\right )^{3} - 11 \, a b^{2} + 11 \, a c^{2} + 33 \, {\left (a b^{2} + 2 \, a b c + a c^{2}\right )} \cosh \left (x\right )^{2} - 4 \, {\left (23 \, a^{2} b + 9 \, b^{3} - 9 \, b c^{2} - 9 \, c^{3} + {\left (23 \, a^{2} + 9 \, b^{2}\right )} c\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {b \cosh \left (x\right ) + c \sinh \left (x\right ) + a}}{90 \, {\left ({\left (b + c\right )} \cosh \left (x\right )^{2} + 2 \, {\left (b + c\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (b + c\right )} \sinh \left (x\right )^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b^{2} \cosh \left (x\right )^{2} + c^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2} + 2 \, {\left (b c \cosh \left (x\right ) + a c\right )} \sinh \left (x\right )\right )} \sqrt {b \cosh \left (x\right ) + c \sinh \left (x\right ) + a}, x\right ) \]