\[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {55019 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{30250}+\frac {823 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{15125}+\frac {7 \left (2+3 x \right )^{\frac {5}{2}}}{11 \sqrt {1-2 x}\, \sqrt {3+5 x}}-\frac {37 \left (2+3 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{605 \sqrt {3+5 x}}+\frac {2388 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{3025} \]
command
integrate((2+3*x)^(7/2)/(1-2*x)^(3/2)/(3+5*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (5445 \, x^{2} - 20897 \, x - 14494\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{3025 \, {\left (10 \, x^{2} + x - 3\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9}, x\right ) \]