Integrand size = 27, antiderivative size = 65 \[ \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\frac {\text {RootSum}\left [2 a^2-a b-4 a \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{8 b} \]
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Leaf count is larger than twice the leaf count of optimal. \(481\) vs. \(2(65)=130\).
Time = 0.31 (sec) , antiderivative size = 481, normalized size of antiderivative = 7.40, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2081, 1284, 1443, 385, 218, 212, 209} \[ \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\frac {\sqrt {x} \sqrt [4]{a x^2+b} \arctan \left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{2\ 2^{7/8} \sqrt [8]{a} b \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2+b} \arctan \left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{2\ 2^{7/8} \sqrt [8]{a} b \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2+b} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{2\ 2^{7/8} \sqrt [8]{a} b \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2+b} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{2\ 2^{7/8} \sqrt [8]{a} b \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}} \sqrt [4]{a x^4+b x^2}} \]
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Rule 209
Rule 212
Rule 218
Rule 385
Rule 1284
Rule 1443
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt [4]{b+a x^2} \left (-2 b+a x^4\right )} \, dx}{\sqrt [4]{b x^2+a x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^4} \left (-2 b+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}} \\ & = -\frac {\left (\sqrt {a} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {2} \sqrt {a} \sqrt {b}-a x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {2} \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {a} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {2} \sqrt {a} \sqrt {b}+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {2} \sqrt {b} \sqrt [4]{b x^2+a x^4}} \\ & = -\frac {\left (\sqrt {a} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2} \sqrt {a} \sqrt {b}-\left (\sqrt {2} a^{3/2} \sqrt {b}-a b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {2} \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {a} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2} \sqrt {a} \sqrt {b}-\left (\sqrt {2} a^{3/2} \sqrt {b}+a b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {2} \sqrt {b} \sqrt [4]{b x^2+a x^4}} \\ & = -\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt [4]{a} \sqrt {\sqrt {2} \sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2\ 2^{3/4} b \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt [4]{a} \sqrt {\sqrt {2} \sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2\ 2^{3/4} b \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt [4]{a} \sqrt {\sqrt {2} \sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2\ 2^{3/4} b \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt [4]{a} \sqrt {\sqrt {2} \sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2\ 2^{3/4} b \sqrt [4]{b x^2+a x^4}} \\ & = -\frac {\sqrt {x} \sqrt [4]{b+a x^2} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [8]{a} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}} b \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [8]{a} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}} b \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [8]{a} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}} b \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [8]{a} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}} b \sqrt [4]{b x^2+a x^4}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.54 \[ \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\frac {\sqrt {x} \sqrt [4]{b+a x^2} \text {RootSum}\left [2 a^2-a b-4 a \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{8 b \sqrt [4]{x^2 \left (b+a x^2\right )}} \]
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Time = 1.33 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89
method | result | size |
pseudoelliptic | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{8}-4 \textit {\_Z}^{4} a +2 a^{2}-a b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}}{8 b}\) | \(58\) |
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Timed out. \[ \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\text {Timed out} \]
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Not integrable
Time = 3.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.37 \[ \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int \frac {1}{\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (a x^{4} - 2 b\right )}\, dx \]
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Not integrable
Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.42 \[ \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {1}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - 2 \, b\right )}} \,d x } \]
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Not integrable
Time = 2.95 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.42 \[ \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {1}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - 2 \, b\right )}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.46 \[ \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\int \frac {1}{{\left (a\,x^4+b\,x^2\right )}^{1/4}\,\left (2\,b-a\,x^4\right )} \,d x \]
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