\[ \int (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx \]
Optimal antiderivative \[ \frac {106 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {7}{2}}}{4875}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}} \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {7}{2}}}{75}-\frac {1580201444291 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{410472562500}-\frac {23763809947 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{205236281250}+\frac {8038 \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {7}{2}} \sqrt {1-2 x}}{804375}-\frac {359748241 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{1520268750}-\frac {26534891 \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{760134375}+\frac {364267 \left (3+5 x \right )^{\frac {7}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{36196875}-\frac {23763809947 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{13682418750} \]
command
integrate((1-2*x)^(5/2)*(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
\[ \frac {1}{13682418750} \, {\left (547296750000 \, x^{6} + 579573225000 \, x^{5} - 352885207500 \, x^{4} - 487924998750 \, x^{3} + 59959633500 \, x^{2} + 157612390605 \, x + 9093216326\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]