\[ \int \frac {f+g x}{(d+e x)^5 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-d g +e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{9 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{5}}-\frac {2 \left (-9 b e g +10 c d g +8 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{63 e^{2} \left (-b e +2 c d \right )^{2} \left (e x +d \right )^{4}}-\frac {4 c \left (-9 b e g +10 c d g +8 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{105 e^{2} \left (-b e +2 c d \right )^{3} \left (e x +d \right )^{3}}-\frac {16 c^{2} \left (-9 b e g +10 c d g +8 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{315 e^{2} \left (-b e +2 c d \right )^{4} \left (e x +d \right )^{2}}-\frac {32 c^{3} \left (-9 b e g +10 c d g +8 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{315 e^{2} \left (-b e +2 c d \right )^{5} \left (e x +d \right )} \]
command
integrate((g*x+f)/(e*x+d)^5/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2}{315} \, {\left (\frac {16 \, {\left (10 \, \sqrt {-c} c^{4} d g + 8 \, \sqrt {-c} c^{4} f e - 9 \, b \sqrt {-c} c^{3} g e\right )} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{32 \, c^{5} d^{5} e - 80 \, b c^{4} d^{4} e^{2} + 80 \, b^{2} c^{3} d^{3} e^{3} - 40 \, b^{3} c^{2} d^{2} e^{4} + 10 \, b^{4} c d e^{5} - b^{5} e^{6}} + \frac {\frac {{\left (35 \, {\left (c - \frac {2 \, c d}{x e + d} + \frac {b e}{x e + d}\right )}^{4} \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} - 180 \, {\left (c - \frac {2 \, c d}{x e + d} + \frac {b e}{x e + d}\right )}^{3} c \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} + 378 \, {\left (c - \frac {2 \, c d}{x e + d} + \frac {b e}{x e + d}\right )}^{2} c^{2} \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} + 315 \, c^{4} \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} + 420 \, c^{3} {\left (-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}\right )}^{\frac {3}{2}}\right )} d g e^{3}}{16 \, c^{4} d^{4} e^{4} - 32 \, b c^{3} d^{3} e^{5} + 24 \, b^{2} c^{2} d^{2} e^{6} - 8 \, b^{3} c d e^{7} + b^{4} e^{8}} - \frac {{\left (35 \, {\left (c - \frac {2 \, c d}{x e + d} + \frac {b e}{x e + d}\right )}^{4} \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} - 180 \, {\left (c - \frac {2 \, c d}{x e + d} + \frac {b e}{x e + d}\right )}^{3} c \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} + 378 \, {\left (c - \frac {2 \, c d}{x e + d} + \frac {b e}{x e + d}\right )}^{2} c^{2} \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} + 315 \, c^{4} \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} + 420 \, c^{3} {\left (-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}\right )}^{\frac {3}{2}}\right )} f e^{4}}{16 \, c^{4} d^{4} e^{4} - 32 \, b c^{3} d^{3} e^{5} + 24 \, b^{2} c^{2} d^{2} e^{6} - 8 \, b^{3} c d e^{7} + b^{4} e^{8}} + \frac {9 \, {\left (5 \, {\left (c - \frac {2 \, c d}{x e + d} + \frac {b e}{x e + d}\right )}^{3} \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} - 21 \, {\left (c - \frac {2 \, c d}{x e + d} + \frac {b e}{x e + d}\right )}^{2} c \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} - 35 \, c^{3} \sqrt {-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}} - 35 \, c^{2} {\left (-c + \frac {2 \, c d}{x e + d} - \frac {b e}{x e + d}\right )}^{\frac {3}{2}}\right )} g e^{2}}{8 \, c^{3} d^{3} e^{3} - 12 \, b c^{2} d^{2} e^{4} + 6 \, b^{2} c d e^{5} - b^{3} e^{6}}}{2 \, c d \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - b e \mathrm {sgn}\left (\frac {1}{x e + d}\right )}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________