\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (1-2 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {5}{2}}}{33 \left (2+3 x \right )^{\frac {11}{2}}}+\frac {370 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {5}{2}}}{891 \left (2+3 x \right )^{\frac {9}{2}}}-\frac {584888452 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{173282571}-\frac {13235368 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{173282571}-\frac {55772 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{43659 \left (2+3 x \right )^{\frac {5}{2}}}+\frac {36980 \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}}{18711 \left (2+3 x \right )^{\frac {7}{2}}}-\frac {17089252 \sqrt {1-2 x}\, \sqrt {3+5 x}}{8251551 \left (2+3 x \right )^{\frac {3}{2}}}+\frac {584888452 \sqrt {1-2 x}\, \sqrt {3+5 x}}{57760857 \sqrt {2+3 x}} \]
command
integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(13/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (71063946918 \, x^{5} + 237923150688 \, x^{4} + 320012032635 \, x^{3} + 215597947743 \, x^{2} + 72620507583 \, x + 9770732477\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{57760857 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128}, x\right ) \]