\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (1-2 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {5}{2}}}{21 \left (2+3 x \right )^{\frac {7}{2}}}+\frac {74 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {5}{2}}}{63 \left (2+3 x \right )^{\frac {5}{2}}}-\frac {904798 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{107163}+\frac {270668 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{107163}-\frac {1844 \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}}{567 \left (2+3 x \right )^{\frac {3}{2}}}-\frac {62596 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{3969 \sqrt {2+3 x}}+\frac {1353340 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{35721} \]
command
integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(9/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (396900 \, x^{4} + 9846603 \, x^{3} + 17788023 \, x^{2} + 11107911 \, x + 2337569\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{35721 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32}, x\right ) \]