Integrand size = 59, antiderivative size = 167 \[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x^2 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\frac {2 \left (-1+2 a x^2\right ) \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{3 b x^2}-\frac {4 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{3 x}-\frac {\sqrt {2} a \text {arctanh}\left (\sqrt {2} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{b} \]
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\[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x^2 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x^2 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x^2 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.98 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.89 \[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x^2 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\frac {\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2 \left (-4+8 a x^2-8 b x \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}+3 i \sqrt {2} \sqrt {a} x \sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \log \left (b \left (i a x-i b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}+\sqrt {2} \sqrt {a} \sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}\right )\right )-3 i \sqrt {2} \sqrt {a} x \sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \log \left (-i a x+i b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}+\sqrt {2} \sqrt {a} \sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}\right )\right )}{6 b \left (x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )\right )^{3/2}} \]
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\[\int \frac {\sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}{x^{2} \sqrt {a \,x^{2}+b x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}}d x\]
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Time = 18.13 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x^2 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\frac {3 \, \sqrt {2} a x^{2} \log \left (4 \, a x^{2} - 4 \, b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} + 2 \, \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} {\left (2 \, \sqrt {2} b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} - \sqrt {2} {\left (2 \, a x^{2} - 1\right )}\right )} - 1\right ) + 4 \, {\left (2 \, a x^{2} - 2 \, b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} - 1\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}}}{6 \, b x^{2}} \]
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\[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x^2 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int \frac {\sqrt {\frac {a \left (a x^{2} - 1\right )}{b^{2}}}}{x^{2} \sqrt {x \left (a x + b \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}\right )}}\, dx \]
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\[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x^2 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int { \frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}}{\sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x} x^{2}} \,d x } \]
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\[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x^2 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int { \frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}}{\sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{x^2 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int \frac {\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}}{x^2\,\sqrt {a\,x^2+b\,x\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}}} \,d x \]
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