\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-8 b e g +13 c d g +3 c e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {3}{2}}}{24 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {9}{2}}}-\frac {\left (-d g +e f \right ) \left (d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}\right )^{\frac {5}{2}}}{4 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {13}{2}}}-\frac {c^{3} \left (-8 b e g +13 c d g +3 c e f \right ) \arctanh \left (\frac {\sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{\sqrt {-b e +2 c d}\, \sqrt {e x +d}}\right )}{64 e^{2} \left (-b e +2 c d \right )^{\frac {5}{2}}}+\frac {c \left (-8 b e g +13 c d g +3 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{32 e^{2} \left (-b e +2 c d \right ) \left (e x +d \right )^{\frac {5}{2}}}-\frac {c^{2} \left (-8 b e g +13 c d g +3 c e f \right ) \sqrt {d \left (-b e +c d \right )-b \,e^{2} x -c \,e^{2} x^{2}}}{64 e^{2} \left (-b e +2 c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}} \]
command
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^(13/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (\frac {3 \, {\left (13 \, c^{5} d g + 3 \, c^{5} f e - 8 \, b c^{4} 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 )}{{\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} \sqrt {-2 \, c d + b e}} + \frac {312 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{8} d^{4} g + 72 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{8} d^{3} f e - 660 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{7} d^{3} g e - 572 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{7} d^{3} g - 108 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{7} d^{2} f e^{2} + 522 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{2} c^{6} d^{2} g e^{2} - 132 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{7} d^{2} f e + 924 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{6} d^{2} g e + 226 \, {\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^{6} d^{2} g + 54 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{2} c^{6} d f e^{3} - 183 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{3} c^{5} d g e^{3} + 132 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{6} d f e^{2} - 495 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b^{2} c^{5} d g e^{2} - 66 \, {\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^{6} d f e - 193 \, {\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^{5} d g e - 39 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )}^{3} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{5} d g - 9 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{3} c^{5} f e^{4} + 24 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b^{4} c^{4} g e^{4} - 33 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b^{2} c^{5} f e^{3} + 88 \, {\left (-{\left (x e + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b^{3} c^{4} g e^{3} + 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} b c^{5} f e^{2} + 40 \, {\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^{2} c^{4} g e^{2} - 9 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )}^{3} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} c^{5} f e + 24 \, {\left ({\left (x e + d\right )} c - 2 \, c d + b e\right )}^{3} \sqrt {-{\left (x e + d\right )} c + 2 \, c d - b e} b c^{4} g e}{{\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2}\right )} {\left (x e + d\right )}^{4} c^{4}}\right )} e^{\left (-2\right )}}{192 \, c} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________