\[ \int \frac {d+c x^2}{\left (-d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
integrate((c*x^2+d)/(c*x^2-d)/(a*x+(a^2*x^2+b^2)^(1/2))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {12 \, a b^{2} \left (\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (\sqrt {a^{3} d^{6} x + \sqrt {a^{2} x^{2} + b^{2}} a^{2} d^{6} - {\left (2 \, a^{2} b^{4} c^{3} d^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - a^{2} b^{2} c d^{5} - 2 \, a^{4} d^{6}\right )} \sqrt {\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}}} {\left (b^{4} c^{4} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - a^{2} c d^{3}\right )} \sqrt {\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}} - {\left (a b^{4} c^{4} d^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - a^{3} c d^{6}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \sqrt {\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}}\right )} \left (\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}}}{a^{2} d^{7}}\right ) - 12 \, a b^{2} \left (-\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {a^{3} d^{6} x + \sqrt {a^{2} x^{2} + b^{2}} a^{2} d^{6} + {\left (2 \, a^{2} b^{4} c^{3} d^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + a^{2} b^{2} c d^{5} + 2 \, a^{4} d^{6}\right )} \sqrt {-\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}}} {\left (b^{4} c^{4} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + a^{2} c d^{3}\right )} \left (-\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {3}{4}} - {\left (a b^{4} c^{4} d^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + a^{3} c d^{6}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \left (-\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {3}{4}}}{a^{2} d^{7}}\right ) - 3 \, a b^{2} \left (\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \log \left (8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a d^{3} + 8 \, {\left (b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - a^{2} d^{3}\right )} \left (\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}}\right ) + 3 \, a b^{2} \left (\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \log \left (8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a d^{3} - 8 \, {\left (b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - a^{2} d^{3}\right )} \left (\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}}\right ) + 3 \, a b^{2} \left (-\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \log \left (8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a d^{3} + 8 \, {\left (b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + a^{2} d^{3}\right )} \left (-\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}}\right ) - 3 \, a b^{2} \left (-\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \log \left (8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a d^{3} - 8 \, {\left (b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + a^{2} d^{3}\right )} \left (-\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}}\right ) + 2 \, {\left (a^{2} x^{2} - \sqrt {a^{2} x^{2} + b^{2}} a x - b^{2}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{3 \, a b^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]