\[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
integrate((c*x^2+d)*(a*x+(a^2*x^2+b^2)^(1/2))^(1/2)/(c*x^2-d),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {12 \, a \left (\frac {b^{2} c d^{2} + 2 \, a^{2} d^{3} + 2 \, c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}}{c^{3}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {a^{3} b^{4} d^{8} x + \sqrt {a^{2} x^{2} + b^{2}} a^{2} b^{4} d^{8} + a^{2} b^{4} c d^{7} \sqrt {\frac {b^{2} c d^{2} + 2 \, a^{2} d^{3} + 2 \, c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}}{c^{3}}}} {\left (a^{2} d^{3} - c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}\right )} \left (\frac {b^{2} c d^{2} + 2 \, a^{2} d^{3} + 2 \, c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}}{c^{3}}\right )^{\frac {1}{4}} - {\left (a^{3} b^{2} d^{7} - a b^{2} c^{3} d^{4} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \left (\frac {b^{2} c d^{2} + 2 \, a^{2} d^{3} + 2 \, c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}}{c^{3}}\right )^{\frac {1}{4}}}{a^{2} b^{4} d^{7}}\right ) - 12 \, a \left (\frac {b^{2} c d^{2} + 2 \, a^{2} d^{3} - 2 \, c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}}{c^{3}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {a^{3} b^{4} d^{8} x + \sqrt {a^{2} x^{2} + b^{2}} a^{2} b^{4} d^{8} + a^{2} b^{4} c d^{7} \sqrt {\frac {b^{2} c d^{2} + 2 \, a^{2} d^{3} - 2 \, c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}}{c^{3}}}} {\left (a^{2} d^{3} + c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}\right )} \left (\frac {b^{2} c d^{2} + 2 \, a^{2} d^{3} - 2 \, c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}}{c^{3}}\right )^{\frac {1}{4}} - {\left (a^{3} b^{2} d^{7} + a b^{2} c^{3} d^{4} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \left (\frac {b^{2} c d^{2} + 2 \, a^{2} d^{3} - 2 \, c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}}{c^{3}}\right )^{\frac {1}{4}}}{a^{2} b^{4} d^{7}}\right ) - 3 \, a \left (\frac {b^{2} c d^{2} + 2 \, a^{2} d^{3} + 2 \, c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}}{c^{3}}\right )^{\frac {1}{4}} \log \left (16 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a b^{2} d^{4} + 16 \, {\left (a^{2} c d^{3} - c^{4} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}\right )} \left (\frac {b^{2} c d^{2} + 2 \, a^{2} d^{3} + 2 \, c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}}{c^{3}}\right )^{\frac {3}{4}}\right ) + 3 \, a \left (\frac {b^{2} c d^{2} + 2 \, a^{2} d^{3} + 2 \, c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}}{c^{3}}\right )^{\frac {1}{4}} \log \left (16 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a b^{2} d^{4} - 16 \, {\left (a^{2} c d^{3} - c^{4} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}\right )} \left (\frac {b^{2} c d^{2} + 2 \, a^{2} d^{3} + 2 \, c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}}{c^{3}}\right )^{\frac {3}{4}}\right ) - 3 \, a \left (\frac {b^{2} c d^{2} + 2 \, a^{2} d^{3} - 2 \, c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}}{c^{3}}\right )^{\frac {1}{4}} \log \left (16 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a b^{2} d^{4} + 16 \, {\left (a^{2} c d^{3} + c^{4} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}\right )} \left (\frac {b^{2} c d^{2} + 2 \, a^{2} d^{3} - 2 \, c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}}{c^{3}}\right )^{\frac {3}{4}}\right ) + 3 \, a \left (\frac {b^{2} c d^{2} + 2 \, a^{2} d^{3} - 2 \, c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}}{c^{3}}\right )^{\frac {1}{4}} \log \left (16 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a b^{2} d^{4} - 16 \, {\left (a^{2} c d^{3} + c^{4} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}\right )} \left (\frac {b^{2} c d^{2} + 2 \, a^{2} d^{3} - 2 \, c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{c^{6}}}}{c^{3}}\right )^{\frac {3}{4}}\right ) - 2 \, {\left (2 \, a x - \sqrt {a^{2} x^{2} + b^{2}}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{3 \, a} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]