\[ \int \frac {1}{(1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {119732 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{2152227}-\frac {7388 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{2152227}+\frac {4 \sqrt {2+3 x}}{231 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}}}+\frac {368 \sqrt {2+3 x}}{5929 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}-\frac {18470 \sqrt {1-2 x}\, \sqrt {2+3 x}}{195657 \left (3+5 x \right )^{\frac {3}{2}}}+\frac {598660 \sqrt {1-2 x}\, \sqrt {2+3 x}}{2152227 \sqrt {3+5 x}} \]
command
integrate(1/(1-2*x)^(5/2)/(3+5*x)^(5/2)/(2+3*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (5986600 \, x^{3} - 2800980 \, x^{2} - 1822554 \, x + 881831\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2152227 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{3000 \, x^{7} + 2900 \, x^{6} - 2010 \, x^{5} - 2277 \, x^{4} + 425 \, x^{3} + 603 \, x^{2} - 27 \, x - 54}, x\right ) \]