\[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {6 d^{2} \sqrt {b x +a}}{5 \left (-a e +b d \right ) \left (e x +d \right )^{\frac {5}{2}}}+\frac {8 d \left (-5 a e +8 b d \right ) \sqrt {b x +a}}{15 \left (-a e +b d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}+\frac {16 \left (15 a^{2} e^{2}-35 a b d e +23 b^{2} d^{2}\right ) \sqrt {b x +a}}{15 \left (-a e +b d \right )^{3} \sqrt {e x +d}} \]
command
integrate((8*e^2*x^2+20*d*e*x+15*d^2)/(e*x+d)^(7/2)/(b*x+a)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (23 \, b^{8} d^{2} e^{4} - 35 \, a b^{7} d e^{5} + 15 \, a^{2} b^{6} e^{6}\right )} {\left (b x + a\right )}}{b^{5} d^{3} {\left | b \right |} e^{2} - 3 \, a b^{4} d^{2} {\left | b \right |} e^{3} + 3 \, a^{2} b^{3} d {\left | b \right |} e^{4} - a^{3} b^{2} {\left | b \right |} e^{5}} + \frac {5 \, {\left (20 \, b^{9} d^{3} e^{3} - 49 \, a b^{8} d^{2} e^{4} + 41 \, a^{2} b^{7} d e^{5} - 12 \, a^{3} b^{6} e^{6}\right )}}{b^{5} d^{3} {\left | b \right |} e^{2} - 3 \, a b^{4} d^{2} {\left | b \right |} e^{3} + 3 \, a^{2} b^{3} d {\left | b \right |} e^{4} - a^{3} b^{2} {\left | b \right |} e^{5}}\right )} + \frac {15 \, {\left (15 \, b^{10} d^{4} e^{2} - 50 \, a b^{9} d^{3} e^{3} + 63 \, a^{2} b^{8} d^{2} e^{4} - 36 \, a^{3} b^{7} d e^{5} + 8 \, a^{4} b^{6} e^{6}\right )}}{b^{5} d^{3} {\left | b \right |} e^{2} - 3 \, a b^{4} d^{2} {\left | b \right |} e^{3} + 3 \, a^{2} b^{3} d {\left | b \right |} e^{4} - a^{3} b^{2} {\left | b \right |} e^{5}}\right )} \sqrt {b x + a}}{15 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {5}{2}}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________