\[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{11/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-A e +B d \right ) \left (b x +a \right )^{\frac {3}{2}}}{9 e \left (-a e +b d \right ) \left (e x +d \right )^{\frac {9}{2}}}+\frac {2 \left (2 A b e -3 B a e +B b d \right ) \left (b x +a \right )^{\frac {3}{2}}}{21 e \left (-a e +b d \right )^{2} \left (e x +d \right )^{\frac {7}{2}}}+\frac {8 b \left (2 A b e -3 B a e +B b d \right ) \left (b x +a \right )^{\frac {3}{2}}}{105 e \left (-a e +b d \right )^{3} \left (e x +d \right )^{\frac {5}{2}}}+\frac {16 b^{2} \left (2 A b e -3 B a e +B b d \right ) \left (b x +a \right )^{\frac {3}{2}}}{315 e \left (-a e +b d \right )^{4} \left (e x +d \right )^{\frac {3}{2}}} \]
command
integrate((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(11/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (63 \, B b^{4} d^{3} x^{2} + 21 \, {\left (B a b^{3} + 5 \, A b^{4}\right )} d^{3} x - 21 \, {\left (2 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} d^{3} - {\left (35 \, A a^{4} + 8 \, {\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} x^{4} - 4 \, {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} + 3 \, {\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2} + 5 \, {\left (9 \, B a^{4} + A a^{3} b\right )} x\right )} e^{3} + {\left (8 \, B b^{4} d x^{4} - 8 \, {\left (14 \, B a b^{3} - 9 \, A b^{4}\right )} d x^{3} + 3 \, {\left (19 \, B a^{2} b^{2} - 12 \, A a b^{3}\right )} d x^{2} + {\left (167 \, B a^{3} b + 27 \, A a^{2} b^{2}\right )} d x - 5 \, {\left (2 \, B a^{4} - 27 \, A a^{3} b\right )} d\right )} e^{2} + 9 \, {\left (4 \, B b^{4} d^{2} x^{3} - {\left (23 \, B a b^{3} - 14 \, A b^{4}\right )} d^{2} x^{2} - {\left (23 \, B a^{2} b^{2} + 7 \, A a b^{3}\right )} d^{2} x + {\left (4 \, B a^{3} b - 21 \, A a^{2} b^{2}\right )} d^{2}\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{315 \, {\left (b^{4} d^{9} + a^{4} x^{5} e^{9} - {\left (4 \, a^{3} b d x^{5} - 5 \, a^{4} d x^{4}\right )} e^{8} + 2 \, {\left (3 \, a^{2} b^{2} d^{2} x^{5} - 10 \, a^{3} b d^{2} x^{4} + 5 \, a^{4} d^{2} x^{3}\right )} e^{7} - 2 \, {\left (2 \, a b^{3} d^{3} x^{5} - 15 \, a^{2} b^{2} d^{3} x^{4} + 20 \, a^{3} b d^{3} x^{3} - 5 \, a^{4} d^{3} x^{2}\right )} e^{6} + {\left (b^{4} d^{4} x^{5} - 20 \, a b^{3} d^{4} x^{4} + 60 \, a^{2} b^{2} d^{4} x^{3} - 40 \, a^{3} b d^{4} x^{2} + 5 \, a^{4} d^{4} x\right )} e^{5} + {\left (5 \, b^{4} d^{5} x^{4} - 40 \, a b^{3} d^{5} x^{3} + 60 \, a^{2} b^{2} d^{5} x^{2} - 20 \, a^{3} b d^{5} x + a^{4} d^{5}\right )} e^{4} + 2 \, {\left (5 \, b^{4} d^{6} x^{3} - 20 \, a b^{3} d^{6} x^{2} + 15 \, a^{2} b^{2} d^{6} x - 2 \, a^{3} b d^{6}\right )} e^{3} + 2 \, {\left (5 \, b^{4} d^{7} x^{2} - 10 \, a b^{3} d^{7} x + 3 \, a^{2} b^{2} d^{7}\right )} e^{2} + {\left (5 \, b^{4} d^{8} x - 4 \, a b^{3} d^{8}\right )} e\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]