Integrand size = 26, antiderivative size = 115 \[ \int \frac {x^2}{\left (a+b x^2\right )^{3/4} \left (2 a+b x^2\right )} \, dx=-\frac {\arctan \left (\frac {a^{3/4} \left (1+\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{a} b^{3/2}}+\frac {\text {arctanh}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{a} b^{3/2}} \]
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Time = 0.02 (sec) , antiderivative size = 115, 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 = {452} \[ \int \frac {x^2}{\left (a+b x^2\right )^{3/4} \left (2 a+b x^2\right )} \, dx=\frac {\text {arctanh}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{a} b^{3/2}}-\frac {\arctan \left (\frac {a^{3/4} \left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}+1\right )}{\sqrt {b} x \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{a} b^{3/2}} \]
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Rule 452
Rubi steps \begin{align*} \text {integral}& = -\frac {\tan ^{-1}\left (\frac {a^{3/4} \left (1+\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{a} b^{3/2}}+\frac {\tanh ^{-1}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{a} b^{3/2}} \\ \end{align*}
Time = 1.84 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.05 \[ \int \frac {x^2}{\left (a+b x^2\right )^{3/4} \left (2 a+b x^2\right )} \, dx=\frac {\arctan \left (\frac {b x^2-2 \sqrt {a} \sqrt {a+b x^2}}{2 \sqrt [4]{a} \sqrt {b} x \sqrt [4]{a+b x^2}}\right )-\text {arctanh}\left (\frac {2 \sqrt [4]{a} \sqrt {b} x \sqrt [4]{a+b x^2}}{b x^2+2 \sqrt {a} \sqrt {a+b x^2}}\right )}{2 \sqrt [4]{a} b^{3/2}} \]
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\[\int \frac {x^{2}}{\left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (b \,x^{2}+2 a \right )}d x\]
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.72 \[ \int \frac {x^2}{\left (a+b x^2\right )^{3/4} \left (2 a+b x^2\right )} \, dx=-\frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} \log \left (\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} b^{2} x \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} + {\left (b x^{2} + a\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} \log \left (-\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} b^{2} x \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} - {\left (b x^{2} + a\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} \log \left (\frac {i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} b^{2} x \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} + {\left (b x^{2} + a\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} b^{2} x \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} + {\left (b x^{2} + a\right )}^{\frac {1}{4}}}{x}\right ) \]
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\[ \int \frac {x^2}{\left (a+b x^2\right )^{3/4} \left (2 a+b x^2\right )} \, dx=\int \frac {x^{2}}{\left (a + b x^{2}\right )^{\frac {3}{4}} \cdot \left (2 a + b x^{2}\right )}\, dx \]
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\[ \int \frac {x^2}{\left (a+b x^2\right )^{3/4} \left (2 a+b x^2\right )} \, dx=\int { \frac {x^{2}}{{\left (b x^{2} + 2 \, a\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}}} \,d x } \]
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\[ \int \frac {x^2}{\left (a+b x^2\right )^{3/4} \left (2 a+b x^2\right )} \, dx=\int { \frac {x^{2}}{{\left (b x^{2} + 2 \, a\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\left (a+b x^2\right )^{3/4} \left (2 a+b x^2\right )} \, dx=\int \frac {x^2}{{\left (b\,x^2+a\right )}^{3/4}\,\left (b\,x^2+2\,a\right )} \,d x \]
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