\[ \int (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {7}{2}}}{75}-\frac {836091184171 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{68412093750}-\frac {50299451003 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{136824187500}+\frac {2503 \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {7}{2}} \sqrt {1-2 x}}{804375}+\frac {178 \left (2+3 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {7}{2}} \sqrt {1-2 x}}{14625}-\frac {380132617 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{506756250}-\frac {57509209 \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{506756250}-\frac {199721 \left (3+5 x \right )^{\frac {7}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{12065625}-\frac {50299451003 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{9121612500} \]
command
integrate((1-2*x)^(3/2)*(2+3*x)^(5/2)*(3+5*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {1}{9121612500} \, {\left (547296750000 \, x^{6} + 1316318850000 \, x^{5} + 888419542500 \, x^{4} - 227285730000 \, x^{3} - 522917547750 \, x^{2} - 177853891770 \, x + 44426819351\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]