\[ \int (1-x)^{7/2} (1+x)^{5/2} \, dx \]
Optimal antiderivative \[ \frac {5 \left (1-x \right )^{\frac {3}{2}} x \left (1+x \right )^{\frac {3}{2}}}{24}+\frac {\left (1-x \right )^{\frac {5}{2}} x \left (1+x \right )^{\frac {5}{2}}}{6}+\frac {\left (1-x \right )^{\frac {7}{2}} \left (1+x \right )^{\frac {7}{2}}}{7}+\frac {5 \arcsin \! \left (x \right )}{16}+\frac {5 x \sqrt {1-x}\, \sqrt {1+x}}{16} \]
command
integrate((1-x)**(7/2)*(1+x)**(5/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \begin {cases} - \frac {5 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{8} - \frac {i \left (x + 1\right )^{\frac {15}{2}}}{7 \sqrt {x - 1}} + \frac {55 i \left (x + 1\right )^{\frac {13}{2}}}{42 \sqrt {x - 1}} - \frac {193 i \left (x + 1\right )^{\frac {11}{2}}}{42 \sqrt {x - 1}} + \frac {1237 i \left (x + 1\right )^{\frac {9}{2}}}{168 \sqrt {x - 1}} - \frac {769 i \left (x + 1\right )^{\frac {7}{2}}}{168 \sqrt {x - 1}} - \frac {i \left (x + 1\right )^{\frac {5}{2}}}{48 \sqrt {x - 1}} - \frac {5 i \left (x + 1\right )^{\frac {3}{2}}}{48 \sqrt {x - 1}} + \frac {5 i \sqrt {x + 1}}{8 \sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\\frac {5 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{8} + \frac {\left (x + 1\right )^{\frac {15}{2}}}{7 \sqrt {1 - x}} - \frac {55 \left (x + 1\right )^{\frac {13}{2}}}{42 \sqrt {1 - x}} + \frac {193 \left (x + 1\right )^{\frac {11}{2}}}{42 \sqrt {1 - x}} - \frac {1237 \left (x + 1\right )^{\frac {9}{2}}}{168 \sqrt {1 - x}} + \frac {769 \left (x + 1\right )^{\frac {7}{2}}}{168 \sqrt {1 - x}} + \frac {\left (x + 1\right )^{\frac {5}{2}}}{48 \sqrt {1 - x}} + \frac {5 \left (x + 1\right )^{\frac {3}{2}}}{48 \sqrt {1 - x}} - \frac {5 \sqrt {x + 1}}{8 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]