Integrand size = 28, antiderivative size = 96 \[ \int \frac {x^2}{\left (-2 a+b x^2\right ) \left (-a+b x^2\right )^{3/4}} \, dx=\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a+b x^2}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a+b x^2}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/2}} \]
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Time = 0.02 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {453} \[ \int \frac {x^2}{\left (-2 a+b x^2\right ) \left (-a+b x^2\right )^{3/4}} \, dx=\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b x^2-a}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b x^2-a}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/2}} \]
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Rule 453
Rubi steps \begin{align*} \text {integral}& = \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a+b x^2}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a+b x^2}}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/2}} \\ \end{align*}
Time = 1.79 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.86 \[ \int \frac {x^2}{\left (-2 a+b x^2\right ) \left (-a+b x^2\right )^{3/4}} \, dx=\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a+b x^2}}\right )-\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a+b x^2}}{\sqrt {b} x}\right )}{\sqrt {2} \sqrt [4]{a} b^{3/2}} \]
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\[\int \frac {x^{2}}{\left (b \,x^{2}-2 a \right ) \left (b \,x^{2}-a \right )^{\frac {3}{4}}}d x\]
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.06 \[ \int \frac {x^2}{\left (-2 a+b x^2\right ) \left (-a+b x^2\right )^{3/4}} \, dx=-\frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{6}}\right )^{\frac {1}{4}} \log \left (\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} b^{2} x \left (\frac {1}{a b^{6}}\right )^{\frac {1}{4}} + {\left (b x^{2} - a\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{6}}\right )^{\frac {1}{4}} \log \left (-\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} b^{2} x \left (\frac {1}{a b^{6}}\right )^{\frac {1}{4}} - {\left (b x^{2} - a\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{6}}\right )^{\frac {1}{4}} \log \left (\frac {i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} b^{2} x \left (\frac {1}{a b^{6}}\right )^{\frac {1}{4}} + {\left (b x^{2} - a\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b^{6}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} b^{2} x \left (\frac {1}{a b^{6}}\right )^{\frac {1}{4}} + {\left (b x^{2} - a\right )}^{\frac {1}{4}}}{x}\right ) \]
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\[ \int \frac {x^2}{\left (-2 a+b x^2\right ) \left (-a+b x^2\right )^{3/4}} \, dx=\int \frac {x^{2}}{\left (- 2 a + b x^{2}\right ) \left (- a + b x^{2}\right )^{\frac {3}{4}}}\, dx \]
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\[ \int \frac {x^2}{\left (-2 a+b x^2\right ) \left (-a+b x^2\right )^{3/4}} \, dx=\int { \frac {x^{2}}{{\left (b x^{2} - a\right )}^{\frac {3}{4}} {\left (b x^{2} - 2 \, a\right )}} \,d x } \]
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\[ \int \frac {x^2}{\left (-2 a+b x^2\right ) \left (-a+b x^2\right )^{3/4}} \, dx=\int { \frac {x^{2}}{{\left (b x^{2} - a\right )}^{\frac {3}{4}} {\left (b x^{2} - 2 \, a\right )}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\left (-2 a+b x^2\right ) \left (-a+b x^2\right )^{3/4}} \, dx=-\int \frac {x^2}{{\left (b\,x^2-a\right )}^{3/4}\,\left (2\,a-b\,x^2\right )} \,d x \]
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