Integrand size = 28, antiderivative size = 62 \[ \int \frac {1}{\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\frac {\text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{4 b} \]
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Leaf count is larger than twice the leaf count of optimal. \(413\) vs. \(2(62)=124\).
Time = 0.27 (sec) , antiderivative size = 413, normalized size of antiderivative = 6.66, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2081, 1284, 1443, 385, 218, 212, 209} \[ \int \frac {1}{\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=-\frac {\sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x^2-b}}\right )}{2 \sqrt [8]{a} b \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x^2-b}}\right )}{2 \sqrt [8]{a} b \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x^2-b}}\right )}{2 \sqrt [8]{a} b \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x^2-b}}\right )}{2 \sqrt [8]{a} b \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt [4]{a x^4-b x^2}} \]
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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 (-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 (-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 {a} \sqrt {b}-a x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\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 {a} \sqrt {b}+a x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\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 {a} \sqrt {b}-\left (a^{3/2} \sqrt {b}-a b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\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 {a} \sqrt {b}-\left (a^{3/2} \sqrt {b}+a b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\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}{1-\sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 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}{1+\sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 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}{1-\sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 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}{1+\sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 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 {a}-\sqrt {b}} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 \sqrt [8]{a} \sqrt [4]{\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 {a}+\sqrt {b}} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 \sqrt [8]{a} \sqrt [4]{\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 {a}-\sqrt {b}} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 \sqrt [8]{a} \sqrt [4]{\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 {a}+\sqrt {b}} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} b \sqrt [4]{-b x^2+a x^4}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.63 \[ \int \frac {1}{\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{4 b \sqrt [4]{-b x^2+a x^4}} \]
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Time = 1.34 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.90
method | result | size |
pseudoelliptic | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 \textit {\_Z}^{4} a +a^{2}-a b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}}{4 b}\) | \(56\) |
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Timed out. \[ \int \frac {1}{\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\text {Timed out} \]
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Not integrable
Time = 2.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.35 \[ \int \frac {1}{\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\int \frac {1}{\sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (a x^{4} - b\right )}\, dx \]
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Not integrable
Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.45 \[ \int \frac {1}{\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\int { \frac {1}{{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )}} \,d x } \]
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Not integrable
Time = 0.96 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.45 \[ \int \frac {1}{\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\int { \frac {1}{{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )}} \,d x } \]
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Not integrable
Time = 6.32 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.47 \[ \int \frac {1}{\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=-\int \frac {1}{\left (b-a\,x^4\right )\,{\left (a\,x^4-b\,x^2\right )}^{1/4}} \,d x \]
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