\[ \int \frac {\text {ArcTan}(a x)^{5/2}}{c+a^2 c x^2} \, dx \]
Optimal antiderivative \[ \frac {2 \arctan \left (a x \right )^{\frac {7}{2}}}{7 a c} \]
command
integrate(atan(a*x)**(5/2)/(a**2*c*x**2+c),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {2 \operatorname {atan}^{\frac {7}{2}}{\left (a x \right )}}{7 a c} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \begin {cases} \frac {2 \operatorname {atan}^{\frac {7}{2}}{\left (a x \right )}}{7 a c} & \text {for}\: c \neq 0 \\\tilde {\infty } \int \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx & \text {otherwise} \end {cases} \]________________________________________________________________________________________