\[ \int (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {5}{2}}}{65}-\frac {51601293223 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{15202687500}-\frac {776112041 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{7601343750}+\frac {601 \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}}{160875}+\frac {178 \left (2+3 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}}{10725}-\frac {11725073 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{56306250}-\frac {18034 \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{625625}-\frac {776112041 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{506756250} \]
command
integrate((1-2*x)^(3/2)*(2+3*x)^(5/2)*(3+5*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {1}{506756250} \, {\left (7016625000 \, x^{5} + 12374775000 \, x^{4} + 3047388750 \, x^{3} - 5775295500 \, x^{2} - 3548873565 \, x + 325972172\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]