\[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^6} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}\, \left (105 a^{\frac {5}{2}} x^{4}+\sqrt {a}\, \left (-432 b^{4}+48 b^{3} \sqrt {a \,x^{2}+b^{2}}\right )+a^{\frac {3}{2}} \left (14 b^{2} x^{2}-70 b \,x^{2} \sqrt {a \,x^{2}+b^{2}}\right )\right )}{1920 \sqrt {a}\, b^{4} x^{5}}+\frac {7 a^{\frac {5}{2}} \arctan \left (\frac {\sqrt {a}\, x \sqrt {2}}{2 \sqrt {b}\, \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}-\frac {\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}\, \sqrt {2}}{2 \sqrt {b}}\right ) \sqrt {2}}{128 b^{\frac {9}{2}}} \]
command
integrate((b+(a*x^2+b^2)^(1/2))^(1/2)/x^6,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [\frac {105 \, \sqrt {\frac {1}{2}} a^{2} x^{5} \sqrt {-\frac {a}{b}} \log \left (-\frac {a^{2} x^{3} + 4 \, a b^{2} x - 4 \, \sqrt {a x^{2} + b^{2}} a b x - 4 \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {a x^{2} + b^{2}} b^{2} \sqrt {-\frac {a}{b}} - \sqrt {\frac {1}{2}} {\left (a b x^{2} + 2 \, b^{3}\right )} \sqrt {-\frac {a}{b}}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{x^{3}}\right ) + 2 \, {\left (105 \, a^{2} x^{4} + 14 \, a b^{2} x^{2} - 432 \, b^{4} - 2 \, {\left (35 \, a b x^{2} - 24 \, b^{3}\right )} \sqrt {a x^{2} + b^{2}}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{3840 \, b^{4} x^{5}}, -\frac {105 \, \sqrt {\frac {1}{2}} a^{2} x^{5} \sqrt {\frac {a}{b}} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {b + \sqrt {a x^{2} + b^{2}}} b \sqrt {\frac {a}{b}}}{a x}\right ) - {\left (105 \, a^{2} x^{4} + 14 \, a b^{2} x^{2} - 432 \, b^{4} - 2 \, {\left (35 \, a b x^{2} - 24 \, b^{3}\right )} \sqrt {a x^{2} + b^{2}}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{1920 \, b^{4} x^{5}}\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]