\[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{9/2}} \, dx \]
Optimal antiderivative \[ \frac {6 d^{2} \sqrt {b x +a}}{7 \left (-a e +b d \right ) \left (e x +d \right )^{\frac {7}{2}}}+\frac {4 d \left (-14 a e +23 b d \right ) \sqrt {b x +a}}{35 \left (-a e +b d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}+\frac {16 \left (35 a^{2} e^{2}-84 a b d e +58 b^{2} d^{2}\right ) \sqrt {b x +a}}{105 \left (-a e +b d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}+\frac {32 b \left (35 a^{2} e^{2}-84 a b d e +58 b^{2} d^{2}\right ) \sqrt {b x +a}}{105 \left (-a e +b d \right )^{4} \sqrt {e x +d}} \]
command
integrate((8*e^2*x^2+20*d*e*x+15*d^2)/(e*x+d)^(9/2)/(b*x+a)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, {\left (2 \, {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (58 \, b^{10} d^{2} e^{6} - 84 \, a b^{9} d e^{7} + 35 \, a^{2} b^{8} e^{8}\right )} {\left (b x + a\right )}}{b^{6} d^{4} {\left | b \right |} e^{3} - 4 \, a b^{5} d^{3} {\left | b \right |} e^{4} + 6 \, a^{2} b^{4} d^{2} {\left | b \right |} e^{5} - 4 \, a^{3} b^{3} d {\left | b \right |} e^{6} + a^{4} b^{2} {\left | b \right |} e^{7}} + \frac {7 \, {\left (58 \, b^{11} d^{3} e^{5} - 142 \, a b^{10} d^{2} e^{6} + 119 \, a^{2} b^{9} d e^{7} - 35 \, a^{3} b^{8} e^{8}\right )}}{b^{6} d^{4} {\left | b \right |} e^{3} - 4 \, a b^{5} d^{3} {\left | b \right |} e^{4} + 6 \, a^{2} b^{4} d^{2} {\left | b \right |} e^{5} - 4 \, a^{3} b^{3} d {\left | b \right |} e^{6} + a^{4} b^{2} {\left | b \right |} e^{7}}\right )} + \frac {35 \, {\left (55 \, b^{12} d^{4} e^{4} - 188 \, a b^{11} d^{3} e^{5} + 243 \, a^{2} b^{10} d^{2} e^{6} - 142 \, a^{3} b^{9} d e^{7} + 32 \, a^{4} b^{8} e^{8}\right )}}{b^{6} d^{4} {\left | b \right |} e^{3} - 4 \, a b^{5} d^{3} {\left | b \right |} e^{4} + 6 \, a^{2} b^{4} d^{2} {\left | b \right |} e^{5} - 4 \, a^{3} b^{3} d {\left | b \right |} e^{6} + a^{4} b^{2} {\left | b \right |} e^{7}}\right )} {\left (b x + a\right )} + \frac {105 \, {\left (15 \, b^{13} d^{5} e^{3} - 65 \, a b^{12} d^{4} e^{4} + 113 \, a^{2} b^{11} d^{3} e^{5} - 99 \, a^{3} b^{10} d^{2} e^{6} + 44 \, a^{4} b^{9} d e^{7} - 8 \, a^{5} b^{8} e^{8}\right )}}{b^{6} d^{4} {\left | b \right |} e^{3} - 4 \, a b^{5} d^{3} {\left | b \right |} e^{4} + 6 \, a^{2} b^{4} d^{2} {\left | b \right |} e^{5} - 4 \, a^{3} b^{3} d {\left | b \right |} e^{6} + a^{4} b^{2} {\left | b \right |} e^{7}}\right )} \sqrt {b x + a}}{105 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {7}{2}}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________