\[ \int \frac {1}{(a+a x)^{9/2} (c-c x)^{9/2}} \, dx \]
Optimal antiderivative \[ \frac {x}{7 a c \left (a x +a \right )^{\frac {7}{2}} \left (-c x +c \right )^{\frac {7}{2}}}+\frac {6 x}{35 a^{2} c^{2} \left (a x +a \right )^{\frac {5}{2}} \left (-c x +c \right )^{\frac {5}{2}}}+\frac {8 x}{35 a^{3} c^{3} \left (a x +a \right )^{\frac {3}{2}} \left (-c x +c \right )^{\frac {3}{2}}}+\frac {16 x}{35 a^{4} c^{4} \sqrt {a x +a}\, \sqrt {-c x +c}} \]
command
integrate(1/(a*x+a)**(9/2)/(-c*x+c)**(9/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {4 i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {9}{4}, \frac {11}{4}, 1 & \frac {1}{2}, \frac {9}{2}, 5 \\\frac {9}{4}, \frac {11}{4}, 4, \frac {9}{2}, 5 & 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{105 \pi ^{\frac {3}{2}} a^{\frac {9}{2}} c^{\frac {9}{2}}} + \frac {4 {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {1}{2}, \frac {7}{4}, \frac {9}{4}, 1 & \\\frac {7}{4}, \frac {9}{4} & - \frac {1}{2}, 0, 4, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )}}{105 \pi ^{\frac {3}{2}} a^{\frac {9}{2}} c^{\frac {9}{2}}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________