\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (1-2 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {5}{2}}}{27 \left (2+3 x \right )^{\frac {9}{2}}}+\frac {370 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {5}{2}}}{567 \left (2+3 x \right )^{\frac {7}{2}}}-\frac {100444 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{2250423}-\frac {1241596 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{2250423}-\frac {13316 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{35721 \left (2+3 x \right )^{\frac {3}{2}}}+\frac {2776 \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}}{1701 \left (2+3 x \right )^{\frac {5}{2}}}-\frac {1241596 \sqrt {1-2 x}\, \sqrt {3+5 x}}{750141 \sqrt {2+3 x}} \]
command
integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(11/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (29072682 \, x^{4} + 115002639 \, x^{3} + 142557831 \, x^{2} + 71920155 \, x + 12903031\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{750141 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64}, x\right ) \]