\[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx \]
Optimal antiderivative \[ \frac {26 a^{2} \sqrt {\frac {\cos \left (x \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {x}{2}\right ), \sqrt {2}\right ) \sqrt {a \left (\cos ^{3}\left (x \right )\right )}}{77 \cos \left (\frac {x}{2}\right ) \cos \left (x \right )^{\frac {3}{2}}}+\frac {78 a^{2} \cos \left (x \right ) \sin \left (x \right ) \sqrt {a \left (\cos ^{3}\left (x \right )\right )}}{385}+\frac {26 a^{2} \left (\cos ^{3}\left (x \right )\right ) \sin \left (x \right ) \sqrt {a \left (\cos ^{3}\left (x \right )\right )}}{165}+\frac {2 a^{2} \left (\cos ^{5}\left (x \right )\right ) \sin \left (x \right ) \sqrt {a \left (\cos ^{3}\left (x \right )\right )}}{15}+\frac {26 a^{2} \sqrt {a \left (\cos ^{3}\left (x \right )\right )}\, \tan \left (x \right )}{77} \]
command
integrate((a*cos(x)^3)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {195 i \, \sqrt {2} a^{\frac {5}{2}} \cos \left (x\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) - 195 i \, \sqrt {2} a^{\frac {5}{2}} \cos \left (x\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 2 \, {\left (77 \, a^{2} \cos \left (x\right )^{6} + 91 \, a^{2} \cos \left (x\right )^{4} + 117 \, a^{2} \cos \left (x\right )^{2} + 195 \, a^{2}\right )} \sqrt {a \cos \left (x\right )^{3}} \sin \left (x\right )}{1155 \, \cos \left (x\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {a \cos \left (x\right )^{3}} a^{2} \cos \left (x\right )^{6}, x\right ) \]