Integrand size = 33, antiderivative size = 173 \[ \int \frac {\left (b^2+a x^2\right )^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\frac {2 x \sqrt {b^2+a x^2} \left (46 b^2+15 a x^2\right )}{105 \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {2 x \left (46 b^3+3 a b x^2\right )}{105 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 \sqrt {2} b^{7/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {a}} \]
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\[ \int \frac {\left (b^2+a x^2\right )^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\int \frac {\left (b^2+a x^2\right )^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (b^2+a x^2\right )^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.74 \[ \int \frac {\left (b^2+a x^2\right )^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\frac {2 x \left (-46 b^3-3 a b x^2+46 b^2 \sqrt {b^2+a x^2}+15 a x^2 \sqrt {b^2+a x^2}\right )}{105 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {\sqrt {2} b^{7/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \]
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\[\int \frac {\left (a \,x^{2}+b^{2}\right )^{\frac {3}{2}}}{\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}d x\]
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Timed out. \[ \int \frac {\left (b^2+a x^2\right )^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (b^2+a x^2\right )^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\int \frac {\left (a x^{2} + b^{2}\right )^{\frac {3}{2}}}{\sqrt {b + \sqrt {a x^{2} + b^{2}}}}\, dx \]
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\[ \int \frac {\left (b^2+a x^2\right )^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\int { \frac {{\left (a x^{2} + b^{2}\right )}^{\frac {3}{2}}}{\sqrt {b + \sqrt {a x^{2} + b^{2}}}} \,d x } \]
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\[ \int \frac {\left (b^2+a x^2\right )^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\int { \frac {{\left (a x^{2} + b^{2}\right )}^{\frac {3}{2}}}{\sqrt {b + \sqrt {a x^{2} + b^{2}}}} \,d x } \]
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Timed out. \[ \int \frac {\left (b^2+a x^2\right )^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\int \frac {{\left (b^2+a\,x^2\right )}^{3/2}}{\sqrt {b+\sqrt {b^2+a\,x^2}}} \,d x \]
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