\[ \int \frac {(2+3 x)^{3/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {49 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{3993}-\frac {8 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{3993}+\frac {7 \sqrt {2+3 x}}{33 \left (1-2 x \right )^{\frac {3}{2}} \sqrt {3+5 x}}+\frac {8 \sqrt {2+3 x}}{363 \sqrt {1-2 x}\, \sqrt {3+5 x}}-\frac {245 \sqrt {1-2 x}\, \sqrt {2+3 x}}{3993 \sqrt {3+5 x}} \]
command
integrate((2+3*x)^(3/2)/(1-2*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (490 \, x^{2} - 402 \, x - 345\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{3993 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}}{200 \, x^{5} - 60 \, x^{4} - 138 \, x^{3} + 47 \, x^{2} + 24 \, x - 9}, x\right ) \]