Integrand size = 44, antiderivative size = 211 \[ \int \frac {\left (-b^2+a x^2\right )^2}{\left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx=-\frac {b x}{\sqrt {b^2+a x^2} \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 x \left (2 b^2+a x^2\right )}{\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}}-\frac {2 \sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {a}} \]
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\[ \int \frac {\left (-b^2+a x^2\right )^2}{\left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\int \frac {\left (-b^2+a x^2\right )^2}{\left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}}+\frac {4 b^4}{\left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {4 b^2}{\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}\right ) \, dx \\ & = -\left (\left (4 b^2\right ) \int \frac {1}{\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx\right )+\left (4 b^4\right ) \int \frac {1}{\left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx+\int \frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx \\ & = -\left (\left (4 b^2\right ) \int \left (\frac {1}{2 b \left (b-\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {1}{2 b \left (b+\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}}\right ) \, dx\right )+\left (4 b^4\right ) \int \left (-\frac {a}{4 b^2 \left (\sqrt {-a} b-a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {a}{4 b^2 \left (\sqrt {-a} b+a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {a}{2 b^2 \left (-a b^2-a^2 x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}\right ) \, dx+\int \frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx \\ & = -\left ((2 b) \int \frac {1}{\left (b-\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx\right )-(2 b) \int \frac {1}{\left (b+\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (a b^2\right ) \int \frac {1}{\left (\sqrt {-a} b-a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (a b^2\right ) \int \frac {1}{\left (\sqrt {-a} b+a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (2 a b^2\right ) \int \frac {1}{\left (-a b^2-a^2 x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx+\int \frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx \\ & = -\left ((2 b) \int \frac {1}{\left (b-\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx\right )-(2 b) \int \frac {1}{\left (b+\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (a b^2\right ) \int \frac {1}{\left (\sqrt {-a} b-a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (a b^2\right ) \int \frac {1}{\left (\sqrt {-a} b+a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (2 a b^2\right ) \int \left (-\frac {1}{2 a b \left (b-\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {1}{2 a b \left (b+\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}}\right ) \, dx+\int \frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx \\ & = b \int \frac {1}{\left (b-\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx+b \int \frac {1}{\left (b+\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-(2 b) \int \frac {1}{\left (b-\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-(2 b) \int \frac {1}{\left (b+\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (a b^2\right ) \int \frac {1}{\left (\sqrt {-a} b-a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (a b^2\right ) \int \frac {1}{\left (\sqrt {-a} b+a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx+\int \frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-b^2+a x^2\right )^2}{\left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\frac {x \left (4 b^2+2 a x^2-b \sqrt {b^2+a x^2}\right )}{\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}}-\frac {\sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \]
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\[\int \frac {\left (a \,x^{2}-b^{2}\right )^{2}}{\left (a \,x^{2}+b^{2}\right )^{2} \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}d x\]
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Timed out. \[ \int \frac {\left (-b^2+a x^2\right )^2}{\left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-b^2+a x^2\right )^2}{\left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\int \frac {\left (a x^{2} - b^{2}\right )^{2}}{\sqrt {b + \sqrt {a x^{2} + b^{2}}} \left (a x^{2} + b^{2}\right )^{2}}\, dx \]
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\[ \int \frac {\left (-b^2+a x^2\right )^2}{\left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\int { \frac {{\left (a x^{2} - b^{2}\right )}^{2}}{{\left (a x^{2} + b^{2}\right )}^{2} \sqrt {b + \sqrt {a x^{2} + b^{2}}}} \,d x } \]
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\[ \int \frac {\left (-b^2+a x^2\right )^2}{\left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\int { \frac {{\left (a x^{2} - b^{2}\right )}^{2}}{{\left (a x^{2} + b^{2}\right )}^{2} \sqrt {b + \sqrt {a x^{2} + b^{2}}}} \,d x } \]
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Timed out. \[ \int \frac {\left (-b^2+a x^2\right )^2}{\left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\int \frac {{\left (a\,x^2-b^2\right )}^2}{{\left (b^2+a\,x^2\right )}^2\,\sqrt {b+\sqrt {b^2+a\,x^2}}} \,d x \]
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