\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{9/2}} \, dx \]
Optimal antiderivative \[ \frac {11 \left (3+5 x \right )^{\frac {3}{2}}}{21 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{\frac {7}{2}}}+\frac {98642 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{2470629}-\frac {65672 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{2470629}+\frac {220 \sqrt {3+5 x}}{49 \left (2+3 x \right )^{\frac {7}{2}} \sqrt {1-2 x}}-\frac {4545 \sqrt {1-2 x}\, \sqrt {3+5 x}}{2401 \left (2+3 x \right )^{\frac {7}{2}}}-\frac {11433 \sqrt {1-2 x}\, \sqrt {3+5 x}}{16807 \left (2+3 x \right )^{\frac {5}{2}}}-\frac {33778 \sqrt {1-2 x}\, \sqrt {3+5 x}}{117649 \left (2+3 x \right )^{\frac {3}{2}}}-\frac {98642 \sqrt {1-2 x}\, \sqrt {3+5 x}}{823543 \sqrt {2+3 x}} \]
command
integrate((3+5*x)^(5/2)/(1-2*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
\[ -\frac {2 \, {\left (15980004 \, x^{5} + 28748088 \, x^{4} + 7681599 \, x^{3} - 10746933 \, x^{2} - 6524789 \, x - 866085\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2470629 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1944 \, x^{8} + 3564 \, x^{7} + 378 \, x^{6} - 2583 \, x^{5} - 1050 \, x^{4} + 616 \, x^{3} + 336 \, x^{2} - 48 \, x - 32}, x\right ) \]