\[ \int \frac {x^4 (1+x)^{3/2}}{\sqrt {1-x}} \, dx \]
Optimal antiderivative \[ \frac {11 \arcsin \left (x \right )}{16}-\frac {11 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{48}-\frac {x^{2} \left (1+x \right )^{\frac {5}{2}} \sqrt {1-x}}{15}-\frac {x^{3} \left (1+x \right )^{\frac {5}{2}} \sqrt {1-x}}{6}-\frac {\left (1+x \right )^{\frac {5}{2}} \left (18+19 x \right ) \sqrt {1-x}}{120}-\frac {11 \sqrt {1-x}\, \sqrt {1+x}}{16} \]
command
integrate(x**4*(1+x)**(3/2)/(1-x)**(1/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ 2 \left (\begin {cases} - \frac {x \sqrt {1 - x} \sqrt {x + 1}}{4} - \sqrt {1 - x} \sqrt {x + 1} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{2} & \text {for}\: \sqrt {x + 1} > - \sqrt {2} \wedge \sqrt {x + 1} < \sqrt {2} \end {cases}\right ) - 8 \left (\begin {cases} - \frac {3 x \sqrt {1 - x} \sqrt {x + 1}}{4} + \frac {\left (1 - x\right )^{\frac {3}{2}} \left (x + 1\right )^{\frac {3}{2}}}{6} - 2 \sqrt {1 - x} \sqrt {x + 1} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{2} & \text {for}\: \sqrt {x + 1} > - \sqrt {2} \wedge \sqrt {x + 1} < \sqrt {2} \end {cases}\right ) + 12 \left (\begin {cases} - \frac {7 x \sqrt {1 - x} \sqrt {x + 1}}{4} + \frac {2 \left (1 - x\right )^{\frac {3}{2}} \left (x + 1\right )^{\frac {3}{2}}}{3} + \frac {\sqrt {1 - x} \sqrt {x + 1} \left (- 5 x - 2 \left (x + 1\right )^{3} + 6 \left (x + 1\right )^{2} - 4\right )}{16} - 4 \sqrt {1 - x} \sqrt {x + 1} + \frac {35 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{8} & \text {for}\: \sqrt {x + 1} > - \sqrt {2} \wedge \sqrt {x + 1} < \sqrt {2} \end {cases}\right ) - 8 \left (\begin {cases} - \frac {15 x \sqrt {1 - x} \sqrt {x + 1}}{4} - \frac {\left (1 - x\right )^{\frac {5}{2}} \left (x + 1\right )^{\frac {5}{2}}}{10} + 2 \left (1 - x\right )^{\frac {3}{2}} \left (x + 1\right )^{\frac {3}{2}} + \frac {5 \sqrt {1 - x} \sqrt {x + 1} \left (- 5 x - 2 \left (x + 1\right )^{3} + 6 \left (x + 1\right )^{2} - 4\right )}{16} - 8 \sqrt {1 - x} \sqrt {x + 1} + \frac {63 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{8} & \text {for}\: \sqrt {x + 1} > - \sqrt {2} \wedge \sqrt {x + 1} < \sqrt {2} \end {cases}\right ) + 2 \left (\begin {cases} \frac {x^{3} \left (1 - x\right )^{\frac {3}{2}} \left (x + 1\right )^{\frac {3}{2}}}{12} - \frac {31 x \sqrt {1 - x} \sqrt {x + 1}}{4} - \frac {3 \left (1 - x\right )^{\frac {5}{2}} \left (x + 1\right )^{\frac {5}{2}}}{5} + \frac {16 \left (1 - x\right )^{\frac {3}{2}} \left (x + 1\right )^{\frac {3}{2}}}{3} + \frac {33 \sqrt {1 - x} \sqrt {x + 1} \left (- 5 x - 2 \left (x + 1\right )^{3} + 6 \left (x + 1\right )^{2} - 4\right )}{32} - 16 \sqrt {1 - x} \sqrt {x + 1} + \frac {231 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{16} & \text {for}\: \sqrt {x + 1} > - \sqrt {2} \wedge \sqrt {x + 1} < \sqrt {2} \end {cases}\right ) \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________