Integrand size = 37, antiderivative size = 151 \[ \int \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx=\frac {3 \sqrt {a} b x^2+2 a^{5/2} x^6+2 a^{3/2} x^4 \sqrt {b+a^2 x^4}}{8 \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}+\frac {5 b \log \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{8 \sqrt {2} \sqrt {a}} \]
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\[ \int \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx=\int \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx \\ \end{align*}
Time = 0.65 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.85 \[ \int \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx=\frac {3 b x+2 a x^3 \left (a x^2+\sqrt {b+a^2 x^4}\right )}{8 \sqrt {a x^2+\sqrt {b+a^2 x^4}}}+\frac {5 b \log \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{8 \sqrt {2} \sqrt {a}} \]
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\[\int \sqrt {a^{2} x^{4}+b}\, \sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}d x\]
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none
Time = 1.44 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.56 \[ \int \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx=\left [\frac {5 \, \sqrt {2} \sqrt {a} b \log \left (4 \, a^{2} x^{4} + 4 \, \sqrt {a^{2} x^{4} + b} a x^{2} + 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{3} + \sqrt {2} \sqrt {a^{2} x^{4} + b} \sqrt {a} x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} + b\right ) - 4 \, {\left (a^{2} x^{3} - 3 \, \sqrt {a^{2} x^{4} + b} a x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{32 \, a}, -\frac {5 \, \sqrt {2} \sqrt {-a} b \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \sqrt {-a}}{2 \, a x}\right ) + 2 \, {\left (a^{2} x^{3} - 3 \, \sqrt {a^{2} x^{4} + b} a x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{16 \, a}\right ] \]
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\[ \int \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx=\int \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \sqrt {a^{2} x^{4} + b}\, dx \]
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\[ \int \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx=\int { \sqrt {a^{2} x^{4} + b} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \,d x } \]
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\[ \int \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx=\int { \sqrt {a^{2} x^{4} + b} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \,d x } \]
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Timed out. \[ \int \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx=\int \sqrt {\sqrt {a^2\,x^4+b}+a\,x^2}\,\sqrt {a^2\,x^4+b} \,d x \]
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