Integrand size = 24, antiderivative size = 88 \[ \int \frac {a+b x}{\sqrt [4]{-1-x^2} \left (2+x^2\right )} \, dx=\frac {a \arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{-1-x^2}}\right )}{2 \sqrt {2}}+b \arctan \left (\sqrt [4]{-1-x^2}\right )+\frac {a \text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1-x^2}}\right )}{2 \sqrt {2}}-b \text {arctanh}\left (\sqrt [4]{-1-x^2}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1024, 407, 455, 65, 304, 209, 212} \[ \int \frac {a+b x}{\sqrt [4]{-1-x^2} \left (2+x^2\right )} \, dx=\frac {a \arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt {2}}+\frac {a \text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt {2}}+b \arctan \left (\sqrt [4]{-x^2-1}\right )-b \text {arctanh}\left (\sqrt [4]{-x^2-1}\right ) \]
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Rule 65
Rule 209
Rule 212
Rule 304
Rule 407
Rule 455
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 {a \arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{-1-x^2}}\right )}{2 \sqrt {2}}+\frac {a \text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1-x^2}}\right )}{2 \sqrt {2}}+\frac {1}{2} b \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1-x} (2+x)} \, dx,x,x^2\right ) \\ & = \frac {a \arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{-1-x^2}}\right )}{2 \sqrt {2}}+\frac {a \text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1-x^2}}\right )}{2 \sqrt {2}}-(2 b) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\sqrt [4]{-1-x^2}\right ) \\ & = \frac {a \arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{-1-x^2}}\right )}{2 \sqrt {2}}+\frac {a \text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1-x^2}}\right )}{2 \sqrt {2}}-b \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{-1-x^2}\right )+b \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{-1-x^2}\right ) \\ & = \frac {a \arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{-1-x^2}}\right )}{2 \sqrt {2}}+b \arctan \left (\sqrt [4]{-1-x^2}\right )+\frac {a \text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1-x^2}}\right )}{2 \sqrt {2}}-b \text {arctanh}\left (\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 = 162, normalized size of antiderivative = 1.84 \[ \int \frac {a+b x}{\sqrt [4]{-1-x^2} \left (2+x^2\right )} \, dx=\frac {x \left (b x \sqrt [4]{1+x^2} \operatorname {AppellF1}\left (1,\frac {1}{4},1,2,-x^2,-\frac {x^2}{2}\right )-\frac {24 a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-x^2,-\frac {x^2}{2}\right )}{\left (2+x^2\right ) \left (-6 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-x^2,-\frac {x^2}{2}\right )+x^2 \left (2 \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-x^2,-\frac {x^2}{2}\right )+\operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-x^2,-\frac {x^2}{2}\right )\right )\right )}\right )}{4 \sqrt [4]{-1-x^2}} \]
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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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\[ \int \frac {a+b x}{\sqrt [4]{-1-x^2} \left (2+x^2\right )} \, dx=\int \frac {a + b x}{\sqrt [4]{- x^{2} - 1} \left (x^{2} + 2\right )}\, dx \]
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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. \[ \int \frac {a+b x}{\sqrt [4]{-1-x^2} \left (2+x^2\right )} \, dx=\int \frac {a+b\,x}{{\left (-x^2-1\right )}^{1/4}\,\left (x^2+2\right )} \,d x \]
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