4.3 Problem number 1089

\[ \int (1-x)^{9/2} (1+x)^{5/2} \, dx \]

Optimal antiderivative \[ \frac {15 \left (1-x \right )^{\frac {3}{2}} x \left (1+x \right )^{\frac {3}{2}}}{64}+\frac {3 \left (1-x \right )^{\frac {5}{2}} x \left (1+x \right )^{\frac {5}{2}}}{16}+\frac {9 \left (1-x \right )^{\frac {7}{2}} \left (1+x \right )^{\frac {7}{2}}}{56}+\frac {\left (1-x \right )^{\frac {9}{2}} \left (1+x \right )^{\frac {7}{2}}}{8}+\frac {45 \arcsin \! \left (x \right )}{128}+\frac {45 x \sqrt {1-x}\, \sqrt {1+x}}{128} \]

command

integrate((1-x)**(9/2)*(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

\[ \begin {cases} - \frac {45 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{64} + \frac {i \left (x + 1\right )^{\frac {17}{2}}}{8 \sqrt {x - 1}} - \frac {79 i \left (x + 1\right )^{\frac {15}{2}}}{56 \sqrt {x - 1}} + \frac {725 i \left (x + 1\right )^{\frac {13}{2}}}{112 \sqrt {x - 1}} - \frac {1699 i \left (x + 1\right )^{\frac {11}{2}}}{112 \sqrt {x - 1}} + \frac {8191 i \left (x + 1\right )^{\frac {9}{2}}}{448 \sqrt {x - 1}} - \frac {4099 i \left (x + 1\right )^{\frac {7}{2}}}{448 \sqrt {x - 1}} - \frac {3 i \left (x + 1\right )^{\frac {5}{2}}}{128 \sqrt {x - 1}} - \frac {15 i \left (x + 1\right )^{\frac {3}{2}}}{128 \sqrt {x - 1}} + \frac {45 i \sqrt {x + 1}}{64 \sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\\frac {45 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{64} - \frac {\left (x + 1\right )^{\frac {17}{2}}}{8 \sqrt {1 - x}} + \frac {79 \left (x + 1\right )^{\frac {15}{2}}}{56 \sqrt {1 - x}} - \frac {725 \left (x + 1\right )^{\frac {13}{2}}}{112 \sqrt {1 - x}} + \frac {1699 \left (x + 1\right )^{\frac {11}{2}}}{112 \sqrt {1 - x}} - \frac {8191 \left (x + 1\right )^{\frac {9}{2}}}{448 \sqrt {1 - x}} + \frac {4099 \left (x + 1\right )^{\frac {7}{2}}}{448 \sqrt {1 - x}} + \frac {3 \left (x + 1\right )^{\frac {5}{2}}}{128 \sqrt {1 - x}} + \frac {15 \left (x + 1\right )^{\frac {3}{2}}}{128 \sqrt {1 - x}} - \frac {45 \sqrt {x + 1}}{64 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]