\[ \int \frac {(a+b x)^{3/2} (A+B x)}{(d+e x)^{13/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-A e +B d \right ) \left (b x +a \right )^{\frac {5}{2}}}{11 e \left (-a e +b d \right ) \left (e x +d \right )^{\frac {11}{2}}}+\frac {2 \left (6 A b e -11 B a e +5 B b d \right ) \left (b x +a \right )^{\frac {5}{2}}}{99 e \left (-a e +b d \right )^{2} \left (e x +d \right )^{\frac {9}{2}}}+\frac {8 b \left (6 A b e -11 B a e +5 B b d \right ) \left (b x +a \right )^{\frac {5}{2}}}{693 e \left (-a e +b d \right )^{3} \left (e x +d \right )^{\frac {7}{2}}}+\frac {16 b^{2} \left (6 A b e -11 B a e +5 B b d \right ) \left (b x +a \right )^{\frac {5}{2}}}{3465 e \left (-a e +b d \right )^{4} \left (e x +d \right )^{\frac {5}{2}}} \]
command
integrate((b*x+a)^(3/2)*(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 (495 \, B b^{5} d^{3} x^{3} + 99 \, {\left (8 \, B a b^{4} + 7 \, A b^{5}\right )} d^{3} x^{2} + 99 \, {\left (B a^{2} b^{3} + 14 \, A a b^{4}\right )} d^{3} x - 99 \, {\left (2 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} d^{3} - {\left (315 \, A a^{5} + 8 \, {\left (11 \, B a b^{4} - 6 \, A b^{5}\right )} x^{5} - 4 \, {\left (11 \, B a^{2} b^{3} - 6 \, A a b^{4}\right )} x^{4} + 3 \, {\left (11 \, B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} x^{3} + 5 \, {\left (110 \, B a^{4} b + 3 \, A a^{3} b^{2}\right )} x^{2} + 35 \, {\left (11 \, B a^{5} + 12 \, A a^{4} b\right )} x\right )} e^{3} + {\left (40 \, B b^{5} d x^{5} - 24 \, {\left (21 \, B a b^{4} - 11 \, A b^{5}\right )} d x^{4} + {\left (257 \, B a^{2} b^{3} - 132 \, A a b^{4}\right )} d x^{3} + {\left (2116 \, B a^{3} b^{2} + 99 \, A a^{2} b^{3}\right )} d x^{2} + 15 \, {\left (83 \, B a^{4} b + 110 \, A a^{3} b^{2}\right )} d x - 35 \, {\left (2 \, B a^{5} - 33 \, A a^{4} b\right )} d\right )} e^{2} + 11 \, {\left (20 \, B b^{5} d^{2} x^{4} - {\left (109 \, B a b^{4} - 54 \, A b^{5}\right )} d^{2} x^{3} - 3 \, {\left (86 \, B a^{2} b^{3} + 9 \, A a b^{4}\right )} d^{2} x^{2} - {\left (109 \, B a^{3} b^{2} + 216 \, A a^{2} b^{3}\right )} d^{2} x + 5 \, {\left (4 \, B a^{4} b - 27 \, A a^{3} b^{2}\right )} d^{2}\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{3465 \, {\left (b^{4} d^{10} + a^{4} x^{6} e^{10} - 2 \, {\left (2 \, a^{3} b d x^{6} - 3 \, a^{4} d x^{5}\right )} e^{9} + 3 \, {\left (2 \, a^{2} b^{2} d^{2} x^{6} - 8 \, a^{3} b d^{2} x^{5} + 5 \, a^{4} d^{2} x^{4}\right )} e^{8} - 4 \, {\left (a b^{3} d^{3} x^{6} - 9 \, a^{2} b^{2} d^{3} x^{5} + 15 \, a^{3} b d^{3} x^{4} - 5 \, a^{4} d^{3} x^{3}\right )} e^{7} + {\left (b^{4} d^{4} x^{6} - 24 \, a b^{3} d^{4} x^{5} + 90 \, a^{2} b^{2} d^{4} x^{4} - 80 \, a^{3} b d^{4} x^{3} + 15 \, a^{4} d^{4} x^{2}\right )} e^{6} + 6 \, {\left (b^{4} d^{5} x^{5} - 10 \, a b^{3} d^{5} x^{4} + 20 \, a^{2} b^{2} d^{5} x^{3} - 10 \, a^{3} b d^{5} x^{2} + a^{4} d^{5} x\right )} e^{5} + {\left (15 \, b^{4} d^{6} x^{4} - 80 \, a b^{3} d^{6} x^{3} + 90 \, a^{2} b^{2} d^{6} x^{2} - 24 \, a^{3} b d^{6} x + a^{4} d^{6}\right )} e^{4} + 4 \, {\left (5 \, b^{4} d^{7} x^{3} - 15 \, a b^{3} d^{7} x^{2} + 9 \, a^{2} b^{2} d^{7} x - a^{3} b d^{7}\right )} e^{3} + 3 \, {\left (5 \, b^{4} d^{8} x^{2} - 8 \, a b^{3} d^{8} x + 2 \, a^{2} b^{2} d^{8}\right )} e^{2} + 2 \, {\left (3 \, b^{4} d^{9} x - 2 \, a b^{3} d^{9}\right )} e\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]