Integrand size = 30, antiderivative size = 104 \[ \int \frac {1}{\left (b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\frac {\text {RootSum}\left [b-a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {a \log (x)-a \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}-2 \text {$\#$1}^5}\&\right ]}{2 b} \]
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Leaf count is larger than twice the leaf count of optimal. \(609\) vs. \(2(104)=208\).
Time = 0.43 (sec) , antiderivative size = 609, normalized size of antiderivative = 5.86, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2081, 1283, 1442, 385, 218, 214, 211} \[ \int \frac {1}{\left (b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2-b}}\right )}{\left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b} \sqrt [4]{a x^4-b x^2}}-\frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt [4]{a x^2-b}}\right )}{\left (\sqrt {a^2-4 b}+a\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b} \sqrt [4]{a x^4-b x^2}}+\frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2-b}}\right )}{\left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b} \sqrt [4]{a x^4-b x^2}}-\frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt [4]{a x^2-b}}\right )}{\left (\sqrt {a^2-4 b}+a\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b} \sqrt [4]{a x^4-b x^2}} \]
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Rule 211
Rule 214
Rule 218
Rule 385
Rule 1283
Rule 1442
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 (b-a x^2+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 (b-a x^4+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}} \\ & = \frac {\left (4 \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-a-\sqrt {a^2-4 b}+2 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2-4 b} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (4 \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-a+\sqrt {a^2-4 b}+2 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2-4 b} \sqrt [4]{-b x^2+a x^4}} \\ & = \frac {\left (4 \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{-a-\sqrt {a^2-4 b}-\left (a \left (-a-\sqrt {a^2-4 b}\right )+2 b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {a^2-4 b} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (4 \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{-a+\sqrt {a^2-4 b}-\left (a \left (-a+\sqrt {a^2-4 b}\right )+2 b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {a^2-4 b} \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 {a-\sqrt {a^2-4 b}}-\sqrt {a^2-a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {a-\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \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 {a-\sqrt {a^2-4 b}}+\sqrt {a^2-a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {a-\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \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 {a+\sqrt {a^2-4 b}}-\sqrt {a^2+a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {a+\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \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 {a+\sqrt {a^2-4 b}}+\sqrt {a^2+a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {a+\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \sqrt [4]{-b x^2+a x^4}} \\ & = \frac {2 \sqrt {x} \sqrt [4]{-b+a x^2} \arctan \left (\frac {\sqrt [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{-b+a x^2}}\right )}{\left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt [4]{-b x^2+a x^4}}-\frac {2 \sqrt {x} \sqrt [4]{-b+a x^2} \arctan \left (\frac {\sqrt [4]{a^2+a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{-b+a x^2}}\right )}{\left (a+\sqrt {a^2-4 b}\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{a^2+a \sqrt {a^2-4 b}-2 b} \sqrt [4]{-b x^2+a x^4}}+\frac {2 \sqrt {x} \sqrt [4]{-b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{-b+a x^2}}\right )}{\left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt [4]{-b x^2+a x^4}}-\frac {2 \sqrt {x} \sqrt [4]{-b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{a^2+a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{-b+a x^2}}\right )}{\left (a+\sqrt {a^2-4 b}\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{a^2+a \sqrt {a^2-4 b}-2 b} \sqrt [4]{-b x^2+a x^4}} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.36 \[ \int \frac {1}{\left (b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \text {RootSum}\left [b-a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {a \log (x)-2 a \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+2 \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}-2 \text {$\#$1}^5}\&\right ]}{4 b \sqrt [4]{-b x^2+a x^4}} \]
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Time = 0.33 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.65
method | result | size |
pseudoelliptic | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-a \,\textit {\_Z}^{4}+b \right )}{\sum }\frac {\left (\textit {\_R}^{4}-a \right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R} \left (2 \textit {\_R}^{4}-a \right )}}{2 b}\) | \(68\) |
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Timed out. \[ \int \frac {1}{\left (b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\text {Timed out} \]
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Not integrable
Time = 1.92 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.25 \[ \int \frac {1}{\left (b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\int \frac {1}{\sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (- a x^{2} + b + x^{4}\right )}\, dx \]
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Not integrable
Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.29 \[ \int \frac {1}{\left (b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\int { \frac {1}{{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - a x^{2} + b\right )}} \,d x } \]
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Not integrable
Time = 3.64 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.29 \[ \int \frac {1}{\left (b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\int { \frac {1}{{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - a x^{2} + b\right )}} \,d x } \]
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Not integrable
Time = 6.57 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.29 \[ \int \frac {1}{\left (b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\int \frac {1}{{\left (a\,x^4-b\,x^2\right )}^{1/4}\,\left (x^4-a\,x^2+b\right )} \,d x \]
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