\[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (c \cosh \left (x \right )+b \sinh \left (x \right )\right )}{3 \left (a^{2}-b^{2}+c^{2}\right ) \left (a +b \cosh \left (x \right )+c \sinh \left (x \right )\right )^{\frac {3}{2}}}-\frac {8 \left (a c \cosh \left (x \right )+a b \sinh \left (x \right )\right )}{3 \left (a^{2}-b^{2}+c^{2}\right )^{2} \sqrt {a +b \cosh \left (x \right )+c \sinh \left (x \right )}}-\frac {8 i a \sqrt {\frac {\cos \left (i x -\arctan \left (b , -i c \right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {i x}{2}-\frac {\arctan \left (b , -i c \right )}{2}\right ), \sqrt {2}\, \sqrt {\frac {\sqrt {b^{2}-c^{2}}}{a +\sqrt {b^{2}-c^{2}}}}\right ) \sqrt {a +b \cosh \left (x \right )+c \sinh \left (x \right )}}{3 \cos \left (\frac {i x}{2}-\frac {\arctan \left (b , -i c \right )}{2}\right ) \left (a^{2}-b^{2}+c^{2}\right )^{2} \sqrt {\frac {a +b \cosh \left (x \right )+c \sinh \left (x \right )}{a +\sqrt {b^{2}-c^{2}}}}}+\frac {2 i \sqrt {\frac {\cos \left (i x -\arctan \left (b , -i c \right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {i x}{2}-\frac {\arctan \left (b , -i c \right )}{2}\right ), \sqrt {2}\, \sqrt {\frac {\sqrt {b^{2}-c^{2}}}{a +\sqrt {b^{2}-c^{2}}}}\right ) \sqrt {\frac {a +b \cosh \left (x \right )+c \sinh \left (x \right )}{a +\sqrt {b^{2}-c^{2}}}}}{3 \cos \left (\frac {i x}{2}-\frac {\arctan \left (b , -i c \right )}{2}\right ) \left (a^{2}-b^{2}+c^{2}\right ) \sqrt {a +b \cosh \left (x \right )+c \sinh \left (x \right )}} \]
command
integrate(1/(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
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ {\rm integral}\left (\frac {\sqrt {b \cosh \left (x\right ) + c \sinh \left (x\right ) + a}}{b^{3} \cosh \left (x\right )^{3} + c^{3} \sinh \left (x\right )^{3} + 3 \, a b^{2} \cosh \left (x\right )^{2} + 3 \, a^{2} b \cosh \left (x\right ) + a^{3} + 3 \, {\left (b c^{2} \cosh \left (x\right ) + a c^{2}\right )} \sinh \left (x\right )^{2} + 3 \, {\left (b^{2} c \cosh \left (x\right )^{2} + 2 \, a b c \cosh \left (x\right ) + a^{2} c\right )} \sinh \left (x\right )}, x\right ) \]