Integrand size = 26, antiderivative size = 74 \[ \int \frac {x^2}{\left (-2-b x^2\right ) \left (-1-b x^2\right )^{3/4}} \, dx=\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1-b x^2}}\right )}{\sqrt {2} b^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1-b x^2}}\right )}{\sqrt {2} b^{3/2}} \]
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Time = 0.02 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {453} \[ \int \frac {x^2}{\left (-2-b x^2\right ) \left (-1-b x^2\right )^{3/4}} \, dx=\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-b x^2-1}}\right )}{\sqrt {2} b^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-b x^2-1}}\right )}{\sqrt {2} b^{3/2}} \]
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Rule 453
Rubi steps \begin{align*} \text {integral}& = \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1-b x^2}}\right )}{\sqrt {2} b^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1-b x^2}}\right )}{\sqrt {2} b^{3/2}} \\ \end{align*}
Time = 1.77 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.88 \[ \int \frac {x^2}{\left (-2-b x^2\right ) \left (-1-b x^2\right )^{3/4}} \, dx=-\frac {-\arctan \left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1-b x^2}}\right )+\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{-1-b x^2}}\right )}{\sqrt {2} b^{3/2}} \]
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\[\int \frac {x^{2}}{\left (-b \,x^{2}-2\right ) \left (-b \,x^{2}-1\right )^{\frac {3}{4}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (57) = 114\).
Time = 0.29 (sec) , antiderivative size = 274, normalized size of antiderivative = 3.70 \[ \int \frac {x^2}{\left (-2-b x^2\right ) \left (-1-b x^2\right )^{3/4}} \, dx=\left [-\frac {2 \, \sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {2} {\left (-b x^{2} - 1\right )}^{\frac {1}{4}}}{\sqrt {b} x}\right ) - \sqrt {2} \sqrt {b} \log \left (-\frac {b^{2} x^{4} + 4 \, \sqrt {-b x^{2} - 1} b x^{2} - 4 \, b x^{2} - 2 \, \sqrt {2} {\left ({\left (-b x^{2} - 1\right )}^{\frac {1}{4}} b x^{3} + 2 \, {\left (-b x^{2} - 1\right )}^{\frac {3}{4}} x\right )} \sqrt {b} - 4}{b^{2} x^{4} + 4 \, b x^{2} + 4}\right )}{4 \, b^{2}}, \frac {2 \, \sqrt {2} \sqrt {-b} \arctan \left (\frac {\sqrt {2} {\left (-b x^{2} - 1\right )}^{\frac {1}{4}} \sqrt {-b}}{b x}\right ) - \sqrt {2} \sqrt {-b} \log \left (-\frac {b^{2} x^{4} - 4 \, \sqrt {-b x^{2} - 1} b x^{2} - 4 \, b x^{2} - 2 \, \sqrt {2} {\left ({\left (-b x^{2} - 1\right )}^{\frac {1}{4}} b x^{3} - 2 \, {\left (-b x^{2} - 1\right )}^{\frac {3}{4}} x\right )} \sqrt {-b} - 4}{b^{2} x^{4} + 4 \, b x^{2} + 4}\right )}{4 \, b^{2}}\right ] \]
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\[ \int \frac {x^2}{\left (-2-b x^2\right ) \left (-1-b x^2\right )^{3/4}} \, dx=- \int \frac {x^{2}}{b x^{2} \left (- b x^{2} - 1\right )^{\frac {3}{4}} + 2 \left (- b x^{2} - 1\right )^{\frac {3}{4}}}\, dx \]
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\[ \int \frac {x^2}{\left (-2-b x^2\right ) \left (-1-b x^2\right )^{3/4}} \, dx=\int { -\frac {x^{2}}{{\left (b x^{2} + 2\right )} {\left (-b x^{2} - 1\right )}^{\frac {3}{4}}} \,d x } \]
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\[ \int \frac {x^2}{\left (-2-b x^2\right ) \left (-1-b x^2\right )^{3/4}} \, dx=\int { -\frac {x^{2}}{{\left (b x^{2} + 2\right )} {\left (-b x^{2} - 1\right )}^{\frac {3}{4}}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\left (-2-b x^2\right ) \left (-1-b x^2\right )^{3/4}} \, dx=-\int \frac {x^2}{{\left (-b\,x^2-1\right )}^{3/4}\,\left (b\,x^2+2\right )} \,d x \]
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