\[ \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {4157 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{99825}-\frac {412 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{99825}+\frac {7 \left (2+3 x \right )^{\frac {3}{2}}}{11 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}-\frac {107 \sqrt {1-2 x}\, \sqrt {2+3 x}}{1815 \left (3+5 x \right )^{\frac {3}{2}}}-\frac {4157 \sqrt {1-2 x}\, \sqrt {2+3 x}}{19965 \sqrt {3+5 x}} \]
command
integrate((2+3*x)^(5/2)/(1-2*x)^(3/2)/(3+5*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (20785 \, x^{2} + 22313 \, x + 5881\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{19965 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{500 \, x^{5} + 400 \, x^{4} - 235 \, x^{3} - 207 \, x^{2} + 27 \, x + 27}, x\right ) \]