\[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{13/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-A e +B d \right ) \left (b x +a \right )^{\frac {7}{2}}}{11 e \left (-a e +b d \right ) \left (e x +d \right )^{\frac {11}{2}}}+\frac {2 \left (4 A b e -11 B a e +7 B b d \right ) \left (b x +a \right )^{\frac {7}{2}}}{99 e \left (-a e +b d \right )^{2} \left (e x +d \right )^{\frac {9}{2}}}+\frac {4 b \left (4 A b e -11 B a e +7 B b d \right ) \left (b x +a \right )^{\frac {7}{2}}}{693 e \left (-a e +b d \right )^{3} \left (e x +d \right )^{\frac {7}{2}}} \]
command
integrate((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(13/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (77 \, B b^{5} d^{2} x^{4} + 11 \, {\left (19 \, B a b^{4} + 9 \, A b^{5}\right )} d^{2} x^{3} + 33 \, {\left (5 \, B a^{2} b^{3} + 9 \, A a b^{4}\right )} d^{2} x^{2} + 11 \, {\left (B a^{3} b^{2} + 27 \, A a^{2} b^{3}\right )} d^{2} x - 11 \, {\left (2 \, B a^{4} b - 9 \, A a^{3} b^{2}\right )} d^{2} + {\left (63 \, A a^{5} - 2 \, {\left (11 \, B a b^{4} - 4 \, A b^{5}\right )} x^{5} + {\left (11 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} x^{4} + 3 \, {\left (55 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + {\left (209 \, B a^{4} b + 113 \, A a^{3} b^{2}\right )} x^{2} + 7 \, {\left (11 \, B a^{5} + 23 \, A a^{4} b\right )} x\right )} e^{2} + 2 \, {\left (7 \, B b^{5} d x^{5} - 2 \, {\left (32 \, B a b^{4} - 11 \, A b^{5}\right )} d x^{4} - {\left (227 \, B a^{2} b^{3} + 11 \, A a b^{4}\right )} d x^{3} - {\left (227 \, B a^{3} b^{2} + 165 \, A a^{2} b^{3}\right )} d x^{2} - {\left (64 \, B a^{4} b + 209 \, A a^{3} b^{2}\right )} d x + 7 \, {\left (B a^{5} - 11 \, A a^{4} b\right )} d\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{693 \, {\left (b^{3} d^{9} - a^{3} x^{6} e^{9} + 3 \, {\left (a^{2} b d x^{6} - 2 \, a^{3} d x^{5}\right )} e^{8} - 3 \, {\left (a b^{2} d^{2} x^{6} - 6 \, a^{2} b d^{2} x^{5} + 5 \, a^{3} d^{2} x^{4}\right )} e^{7} + {\left (b^{3} d^{3} x^{6} - 18 \, a b^{2} d^{3} x^{5} + 45 \, a^{2} b d^{3} x^{4} - 20 \, a^{3} d^{3} x^{3}\right )} e^{6} + 3 \, {\left (2 \, b^{3} d^{4} x^{5} - 15 \, a b^{2} d^{4} x^{4} + 20 \, a^{2} b d^{4} x^{3} - 5 \, a^{3} d^{4} x^{2}\right )} e^{5} + 3 \, {\left (5 \, b^{3} d^{5} x^{4} - 20 \, a b^{2} d^{5} x^{3} + 15 \, a^{2} b d^{5} x^{2} - 2 \, a^{3} d^{5} x\right )} e^{4} + {\left (20 \, b^{3} d^{6} x^{3} - 45 \, a b^{2} d^{6} x^{2} + 18 \, a^{2} b d^{6} x - a^{3} d^{6}\right )} e^{3} + 3 \, {\left (5 \, b^{3} d^{7} x^{2} - 6 \, a b^{2} d^{7} x + a^{2} b d^{7}\right )} e^{2} + 3 \, {\left (2 \, b^{3} d^{8} x - a b^{2} d^{8}\right )} e\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]