\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {25111 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{1148175}-\frac {310399 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{1148175}-\frac {2 \left (1-2 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {5}{2}}}{3 \sqrt {2+3 x}}-\frac {40 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {5}{2}} \sqrt {2+3 x}}{81}+\frac {64628 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{8505}-\frac {2108 \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{1701}-\frac {310399 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{76545} \]
command
integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (567000 \, x^{4} - 386100 \, x^{3} - 259650 \, x^{2} + 245751 \, x + 21964\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{76545 \, \sqrt {3 \, x + 2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{9 \, x^{2} + 12 \, x + 4}, x\right ) \]