\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{13/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{11 c d \left (2 c d x +b d \right )^{\frac {11}{2}}}-\frac {3 \sqrt {c \,x^{2}+b x +a}}{154 c^{2} d^{3} \left (2 c d x +b d \right )^{\frac {7}{2}}}+\frac {\sqrt {c \,x^{2}+b x +a}}{77 c^{2} \left (-4 a c +b^{2}\right ) d^{5} \left (2 c d x +b d \right )^{\frac {3}{2}}}+\frac {\EllipticF \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{154 c^{3} \left (-4 a c +b^{2}\right )^{\frac {3}{4}} d^{\frac {13}{2}} \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^(13/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (64 \, c^{6} x^{6} + 192 \, b c^{5} x^{5} + 240 \, b^{2} c^{4} x^{4} + 160 \, b^{3} c^{3} x^{3} + 60 \, b^{4} c^{2} x^{2} + 12 \, b^{5} c x + b^{6}\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + 2 \, {\left (32 \, c^{6} x^{4} + 64 \, b c^{5} x^{3} - b^{4} c^{2} - 2 \, a b^{2} c^{3} + 56 \, a^{2} c^{4} + 2 \, {\left (11 \, b^{2} c^{4} + 52 \, a c^{5}\right )} x^{2} - 2 \, {\left (5 \, b^{3} c^{3} - 52 \, a b c^{4}\right )} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{308 \, {\left (64 \, {\left (b^{2} c^{10} - 4 \, a c^{11}\right )} d^{7} x^{6} + 192 \, {\left (b^{3} c^{9} - 4 \, a b c^{10}\right )} d^{7} x^{5} + 240 \, {\left (b^{4} c^{8} - 4 \, a b^{2} c^{9}\right )} d^{7} x^{4} + 160 \, {\left (b^{5} c^{7} - 4 \, a b^{3} c^{8}\right )} d^{7} x^{3} + 60 \, {\left (b^{6} c^{6} - 4 \, a b^{4} c^{7}\right )} d^{7} x^{2} + 12 \, {\left (b^{7} c^{5} - 4 \, a b^{5} c^{6}\right )} d^{7} x + {\left (b^{8} c^{4} - 4 \, a b^{6} c^{5}\right )} d^{7}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {2 \, c d x + b d} {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{128 \, c^{7} d^{7} x^{7} + 448 \, b c^{6} d^{7} x^{6} + 672 \, b^{2} c^{5} d^{7} x^{5} + 560 \, b^{3} c^{4} d^{7} x^{4} + 280 \, b^{4} c^{3} d^{7} x^{3} + 84 \, b^{5} c^{2} d^{7} x^{2} + 14 \, b^{6} c d^{7} x + b^{7} d^{7}}, x\right ) \]