\[ \int (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {7}{2}}}{65}-\frac {9380126059 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{1824322500}-\frac {70536439 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{456080625}+\frac {62 \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {7}{2}} \sqrt {1-2 x}}{3575}-\frac {2133359 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{6756750}-\frac {160084 \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{3378375}-\frac {67 \left (3+5 x \right )^{\frac {7}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{160875}-\frac {70536439 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{30405375} \]
command
integrate((1-2*x)^(3/2)*(2+3*x)^(3/2)*(3+5*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {1}{60810750} \, {\left (1403325000 \, x^{5} + 2364390000 \, x^{4} + 496455750 \, x^{3} - 1110242250 \, x^{2} - 638983395 \, x + 67302101\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]