\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {17 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{1029}-\frac {146 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{343}+\frac {11 \left (3+5 x \right )^{\frac {3}{2}}}{21 \left (1-2 x \right )^{\frac {3}{2}} \sqrt {2+3 x}}-\frac {143 \sqrt {3+5 x}}{49 \sqrt {1-2 x}\, \sqrt {2+3 x}}+\frac {438 \sqrt {1-2 x}\, \sqrt {3+5 x}}{343 \sqrt {2+3 x}} \]
command
integrate((3+5*x)^(5/2)/(1-2*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
\[ \frac {{\left (5256 \, x^{2} + 3445 \, x - 72\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1029 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{72 \, x^{5} - 12 \, x^{4} - 58 \, x^{3} + 15 \, x^{2} + 12 \, x - 4}, x\right ) \]