\[ \int \frac {1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2}{\left (-4 a c +b^{2}\right ) d \left (2 c d x +b d \right )^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}}-\frac {56 c \sqrt {c \,x^{2}+b x +a}}{5 \left (-4 a c +b^{2}\right )^{2} d \left (2 c d x +b d \right )^{\frac {5}{2}}}-\frac {168 c \sqrt {c \,x^{2}+b x +a}}{5 \left (-4 a c +b^{2}\right )^{3} d^{3} \sqrt {2 c d x +b d}}+\frac {84 \EllipticE \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{5 \left (-4 a c +b^{2}\right )^{\frac {9}{4}} d^{\frac {7}{2}} \sqrt {c \,x^{2}+b x +a}}-\frac {84 \EllipticF \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{5 \left (-4 a c +b^{2}\right )^{\frac {9}{4}} d^{\frac {7}{2}} \sqrt {c \,x^{2}+b x +a}} \]
command
integrate(1/(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (42 \, \sqrt {2} {\left (8 \, c^{4} x^{5} + 20 \, b c^{3} x^{4} + a b^{3} + 2 \, {\left (9 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{3} + {\left (7 \, b^{3} c + 12 \, a b c^{2}\right )} x^{2} + {\left (b^{4} + 6 \, a b^{2} c\right )} x\right )} \sqrt {c^{2} d} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) + {\left (336 \, c^{4} x^{4} + 672 \, b c^{3} x^{3} + 5 \, b^{4} + 72 \, a b^{2} c - 32 \, a^{2} c^{2} + 224 \, {\left (2 \, b^{2} c^{2} + a c^{3}\right )} x^{2} + 112 \, {\left (b^{3} c + 2 \, a b c^{2}\right )} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}\right )}}{5 \, {\left (8 \, {\left (b^{6} c^{4} - 12 \, a b^{4} c^{5} + 48 \, a^{2} b^{2} c^{6} - 64 \, a^{3} c^{7}\right )} d^{4} x^{5} + 20 \, {\left (b^{7} c^{3} - 12 \, a b^{5} c^{4} + 48 \, a^{2} b^{3} c^{5} - 64 \, a^{3} b c^{6}\right )} d^{4} x^{4} + 2 \, {\left (9 \, b^{8} c^{2} - 104 \, a b^{6} c^{3} + 384 \, a^{2} b^{4} c^{4} - 384 \, a^{3} b^{2} c^{5} - 256 \, a^{4} c^{6}\right )} d^{4} x^{3} + {\left (7 \, b^{9} c - 72 \, a b^{7} c^{2} + 192 \, a^{2} b^{5} c^{3} + 128 \, a^{3} b^{3} c^{4} - 768 \, a^{4} b c^{5}\right )} d^{4} x^{2} + {\left (b^{10} - 6 \, a b^{8} c - 24 \, a^{2} b^{6} c^{2} + 224 \, a^{3} b^{4} c^{3} - 384 \, a^{4} b^{2} c^{4}\right )} d^{4} x + {\left (a b^{9} - 12 \, a^{2} b^{7} c + 48 \, a^{3} b^{5} c^{2} - 64 \, a^{4} b^{3} c^{3}\right )} d^{4}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{16 \, c^{6} d^{4} x^{8} + 64 \, b c^{5} d^{4} x^{7} + 8 \, {\left (13 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{4} x^{6} + a^{2} b^{4} d^{4} + 8 \, {\left (11 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} d^{4} x^{5} + {\left (41 \, b^{4} c^{2} + 112 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{4} x^{4} + 2 \, {\left (5 \, b^{5} c + 32 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{4} x^{3} + {\left (b^{6} + 18 \, a b^{4} c + 24 \, a^{2} b^{2} c^{2}\right )} d^{4} x^{2} + 2 \, {\left (a b^{5} + 4 \, a^{2} b^{3} c\right )} d^{4} x}, x\right ) \]