\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx \]
Optimal antiderivative \[ -\frac {5 c \left (6 b e g -13 c d g +c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{24 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {5}{2}}}+\frac {\left (6 b e g -13 c d g +c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {5}{2}}}{12 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {9}{2}}}-\frac {\left (-d g +e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {7}{2}}}{3 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {13}{2}}}+\frac {5 c^{2} \left (6 b e g -13 c d g +c e f \right ) \arctanh \left (\frac {\sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{\sqrt {-b e +2 c d}\, \sqrt {e x +d}}\right )}{8 e^{2} \sqrt {-b e +2 c d}}-\frac {5 c^{2} \left (6 b e g -13 c d g +c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{8 e^{2} \left (-b e +2 c d \right ) \sqrt {e x +d}} \]
command
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(13/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (48 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{3} g + \frac {15 \, {\left (13 \, c^{4} d g - c^{4} f e - 6 \, b c^{3} g e\right )} \arctan \left (\frac {\sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{\sqrt {-2 \, c d + b e}} + \frac {396 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{6} d^{3} g - 60 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{6} d^{2} f e - 564 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{5} d^{2} g e - 464 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{5} d^{2} g + 60 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{5} d f e^{2} + 267 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{2} c^{4} d g e^{2} + 80 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{5} d f e + 424 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{4} d g e + 141 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{4} d g - 15 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{2} c^{4} f e^{3} - 42 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{3} c^{3} g e^{3} - 40 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{4} f e^{2} - 96 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b^{2} c^{3} g e^{2} - 33 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{4} f e - 54 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{3} g e}{{\left (x e + d\right )}^{3} c^{3}}\right )} e^{\left (-2\right )}}{24 \, c} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________