\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (1-2 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {5}{2}}}{9 \left (2+3 x \right )^{\frac {3}{2}}}-\frac {452399 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{76545}+\frac {135334 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{76545}+\frac {370 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {5}{2}}}{27 \sqrt {2+3 x}}-\frac {31298 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{567}+\frac {5260 \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{567}+\frac {135334 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{5103} \]
command
integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (24300 \, x^{4} - 25110 \, x^{3} + 5949 \, x^{2} + 108285 \, x + 56963\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{5103 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8}, x\right ) \]