Integrand size = 26, antiderivative size = 149 \[ \int \frac {a+b x}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=\frac {b \arctan \left (\frac {1-\sqrt {1-x^2}}{\sqrt {2} \sqrt [4]{1-x^2}}\right )}{\sqrt {2}}+\frac {1}{2} a \arctan \left (\frac {1-\sqrt {1-x^2}}{x \sqrt [4]{1-x^2}}\right )+\frac {b \text {arctanh}\left (\frac {1+\sqrt {1-x^2}}{\sqrt {2} \sqrt [4]{1-x^2}}\right )}{\sqrt {2}}+\frac {1}{2} a \text {arctanh}\left (\frac {1+\sqrt {1-x^2}}{x \sqrt [4]{1-x^2}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1024, 406, 450} \[ \int \frac {a+b x}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=\frac {1}{2} a \arctan \left (\frac {1-\sqrt {1-x^2}}{x \sqrt [4]{1-x^2}}\right )+\frac {1}{2} a \text {arctanh}\left (\frac {\sqrt {1-x^2}+1}{x \sqrt [4]{1-x^2}}\right )+\frac {b \arctan \left (\frac {1-\sqrt {1-x^2}}{\sqrt {2} \sqrt [4]{1-x^2}}\right )}{\sqrt {2}}+\frac {b \text {arctanh}\left (\frac {\sqrt {1-x^2}+1}{\sqrt {2} \sqrt [4]{1-x^2}}\right )}{\sqrt {2}} \]
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Rule 406
Rule 450
Rule 1024
Rubi steps \begin{align*} \text {integral}& = a \int \frac {1}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx+b \int \frac {x}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx \\ & = \frac {b \arctan \left (\frac {1-\sqrt {1-x^2}}{\sqrt {2} \sqrt [4]{1-x^2}}\right )}{\sqrt {2}}+\frac {1}{2} a \arctan \left (\frac {1-\sqrt {1-x^2}}{x \sqrt [4]{1-x^2}}\right )+\frac {b \text {arctanh}\left (\frac {1+\sqrt {1-x^2}}{\sqrt {2} \sqrt [4]{1-x^2}}\right )}{\sqrt {2}}+\frac {1}{2} a \text {arctanh}\left (\frac {1+\sqrt {1-x^2}}{x \sqrt [4]{1-x^2}}\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 10.18 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.97 \[ \int \frac {a+b x}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=\frac {1}{4} b x^2 \operatorname {AppellF1}\left (1,\frac {1}{4},1,2,x^2,\frac {x^2}{2}\right )-\frac {6 a x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},x^2,\frac {x^2}{2}\right )}{\sqrt [4]{1-x^2} \left (-2+x^2\right ) \left (6 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},x^2,\frac {x^2}{2}\right )+x^2 \left (2 \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},x^2,\frac {x^2}{2}\right )+\operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},x^2,\frac {x^2}{2}\right )\right )\right )} \]
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\[\int \frac {b x +a}{\left (-x^{2}+1\right )^{\frac {1}{4}} \left (-x^{2}+2\right )}d x\]
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Timed out. \[ \int \frac {a+b x}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {a+b x}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=- \int \frac {a}{x^{2} \sqrt [4]{1 - x^{2}} - 2 \sqrt [4]{1 - x^{2}}}\, dx - \int \frac {b x}{x^{2} \sqrt [4]{1 - x^{2}} - 2 \sqrt [4]{1 - x^{2}}}\, dx \]
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\[ \int \frac {a+b x}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=\int { -\frac {b x + a}{{\left (x^{2} - 2\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {a+b x}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=\int { -\frac {b x + a}{{\left (x^{2} - 2\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {a+b x}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=\int -\frac {a+b\,x}{{\left (1-x^2\right )}^{1/4}\,\left (x^2-2\right )} \,d x \]
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