Integrand size = 28, antiderivative size = 98 \[ \int \frac {x^2}{\left (-2 b+a x^2\right ) \left (-b+a x^2\right )^{3/4}} \, dx=\frac {\arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}{\sqrt {a} x}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}} \]
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Time = 0.03 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.98, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {453} \[ \int \frac {x^2}{\left (-2 b+a x^2\right ) \left (-b+a x^2\right )^{3/4}} \, dx=\frac {\arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^2-b}}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}} \]
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Rule 453
Rubi steps \begin{align*} \text {integral}& = \frac {\arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}} \\ \end{align*}
Time = 1.74 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.85 \[ \int \frac {x^2}{\left (-2 b+a x^2\right ) \left (-b+a x^2\right )^{3/4}} \, dx=\frac {\arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}\right )-\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^2}}{\sqrt {a} x}\right )}{\sqrt {2} a^{3/2} \sqrt [4]{b}} \]
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\[\int \frac {x^{2}}{\left (a \,x^{2}-2 b \right ) \left (a \,x^{2}-b \right )^{\frac {3}{4}}}d x\]
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.02 \[ \int \frac {x^2}{\left (-2 b+a x^2\right ) \left (-b+a x^2\right )^{3/4}} \, dx=-\frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} \log \left (\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} x \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} + {\left (a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} \log \left (-\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} x \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} - {\left (a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} \log \left (\frac {i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} x \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} + {\left (a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} \log \left (\frac {-i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} x \left (\frac {1}{a^{6} b}\right )^{\frac {1}{4}} + {\left (a x^{2} - b\right )}^{\frac {1}{4}}}{x}\right ) \]
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\[ \int \frac {x^2}{\left (-2 b+a x^2\right ) \left (-b+a x^2\right )^{3/4}} \, dx=\int \frac {x^{2}}{\left (a x^{2} - 2 b\right ) \left (a x^{2} - b\right )^{\frac {3}{4}}}\, dx \]
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\[ \int \frac {x^2}{\left (-2 b+a x^2\right ) \left (-b+a x^2\right )^{3/4}} \, dx=\int { \frac {x^{2}}{{\left (a x^{2} - b\right )}^{\frac {3}{4}} {\left (a x^{2} - 2 \, b\right )}} \,d x } \]
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\[ \int \frac {x^2}{\left (-2 b+a x^2\right ) \left (-b+a x^2\right )^{3/4}} \, dx=\int { \frac {x^{2}}{{\left (a x^{2} - b\right )}^{\frac {3}{4}} {\left (a x^{2} - 2 \, b\right )}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\left (-2 b+a x^2\right ) \left (-b+a x^2\right )^{3/4}} \, dx=-\int \frac {x^2}{{\left (a\,x^2-b\right )}^{3/4}\,\left (2\,b-a\,x^2\right )} \,d x \]
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