Integrand size = 23, antiderivative size = 98 \[ \int \frac {1}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx=\frac {x}{2 \sqrt {a x^2+\sqrt {b+a^2 x^4}}}+\frac {\log \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{2 \sqrt {2} \sqrt {a}} \]
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\[ \int \frac {1}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx=\int \frac {1}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx \\ \end{align*}
Time = 0.46 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx=\frac {x}{2 \sqrt {a x^2+\sqrt {b+a^2 x^4}}}+\frac {\log \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{2 \sqrt {2} \sqrt {a}} \]
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\[\int \frac {1}{\sqrt {a \,x^{2}+\sqrt {x^{4} a^{2}+b}}}d x\]
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Time = 1.11 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.34 \[ \int \frac {1}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx=\left [\frac {\frac {\sqrt {2} b \log \left (4 \, a^{2} x^{4} + 4 \, \sqrt {a^{2} x^{4} + b} a x^{2} + 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{3} + \sqrt {2} \sqrt {a^{2} x^{4} + b} \sqrt {a} x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} + b\right )}{\sqrt {a}} - 4 \, {\left (a x^{3} - \sqrt {a^{2} x^{4} + b} x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{8 \, b}, -\frac {\sqrt {2} b \sqrt {-\frac {1}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \sqrt {-\frac {1}{a}}}{2 \, x}\right ) + 2 \, {\left (a x^{3} - \sqrt {a^{2} x^{4} + b} x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{4 \, b}\right ] \]
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\[ \int \frac {1}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx=\int \frac {1}{\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}\, dx \]
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\[ \int \frac {1}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx=\int { \frac {1}{\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}} \,d x } \]
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\[ \int \frac {1}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx=\int { \frac {1}{\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx=\int \frac {1}{\sqrt {\sqrt {a^2\,x^4+b}+a\,x^2}} \,d x \]
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