\[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^{13/2}} \, dx \]
Optimal antiderivative \[ -\frac {924247516 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{2201330439}-\frac {31704544 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{2201330439}+\frac {2 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{231 \left (2+3 x \right )^{\frac {11}{2}}}+\frac {940 \sqrt {1-2 x}\, \sqrt {3+5 x}}{43659 \left (2+3 x \right )^{\frac {9}{2}}}-\frac {251590 \sqrt {1-2 x}\, \sqrt {3+5 x}}{2139291 \left (2+3 x \right )^{\frac {7}{2}}}-\frac {362666 \sqrt {1-2 x}\, \sqrt {3+5 x}}{14975037 \left (2+3 x \right )^{\frac {5}{2}}}+\frac {11460644 \sqrt {1-2 x}\, \sqrt {3+5 x}}{104825259 \left (2+3 x \right )^{\frac {3}{2}}}+\frac {924247516 \sqrt {1-2 x}\, \sqrt {3+5 x}}{733776813 \sqrt {2+3 x}} \]
command
integrate((3+5*x)^(5/2)/(2+3*x)^(13/2)/(1-2*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (112296073194 \, x^{5} + 377569336554 \, x^{4} + 507518001945 \, x^{3} + 340525216341 \, x^{2} + 113962415157 \, x + 15211411193\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{733776813 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{4374 \, x^{8} + 18225 \, x^{7} + 30618 \, x^{6} + 24948 \, x^{5} + 7560 \, x^{4} - 3024 \, x^{3} - 3360 \, x^{2} - 1088 \, x - 128}, x\right ) \]