Integrand size = 35, antiderivative size = 101 \[ \int \frac {-b+2 a x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^8\right )} \, dx=\frac {1}{8} \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a \log (x)+a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}-\text {$\#$1}^5}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(325\) vs. \(2(101)=202\).
Time = 0.39 (sec) , antiderivative size = 325, normalized size of antiderivative = 3.22, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6857, 385, 218, 212, 209} \[ \int \frac {-b+2 a x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^8\right )} \, dx=-\frac {\left (2 \sqrt {a}-\sqrt {b}\right ) \arctan \left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x^4-b}}\right )}{4 \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {a}-\sqrt {b}}}+\frac {\left (2 \sqrt {a}+\sqrt {b}\right ) \arctan \left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x^4-b}}\right )}{4 \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {a}+\sqrt {b}}}-\frac {\left (2 \sqrt {a}-\sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x^4-b}}\right )}{4 \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {a}-\sqrt {b}}}+\frac {\left (2 \sqrt {a}+\sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x^4-b}}\right )}{4 \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {a}+\sqrt {b}}} \]
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Rule 209
Rule 212
Rule 218
Rule 385
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 a \sqrt {b}-\sqrt {a} b}{2 \sqrt {a} \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^4\right ) \sqrt [4]{-b+a x^4}}+\frac {2 a \sqrt {b}+\sqrt {a} b}{2 \sqrt {a} \sqrt {b} \left (\sqrt {b}+\sqrt {a} x^4\right ) \sqrt [4]{-b+a x^4}}\right ) \, dx \\ & = -\left (\frac {1}{2} \left (2 \sqrt {a}-\sqrt {b}\right ) \int \frac {1}{\left (\sqrt {b}-\sqrt {a} x^4\right ) \sqrt [4]{-b+a x^4}} \, dx\right )+\frac {1}{2} \left (2 \sqrt {a}+\sqrt {b}\right ) \int \frac {1}{\left (\sqrt {b}+\sqrt {a} x^4\right ) \sqrt [4]{-b+a x^4}} \, dx \\ & = -\left (\frac {1}{2} \left (2 \sqrt {a}-\sqrt {b}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}-\left (a \sqrt {b}-\sqrt {a} b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )\right )+\frac {1}{2} \left (2 \sqrt {a}+\sqrt {b}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}-\left (a \sqrt {b}+\sqrt {a} b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right ) \\ & = -\left (\frac {1}{4} \left (-1+\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )\right )-\frac {1}{4} \left (-1+\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+\frac {1}{4} \left (1+\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+\frac {1}{4} \left (1+\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right ) \\ & = \frac {\left (1-\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}}}+\frac {\left (1+\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}}}+\frac {\left (1-\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}}}+\frac {\left (1+\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}}} \\ \end{align*}
Time = 1.20 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.99 \[ \int \frac {-b+2 a x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^8\right )} \, dx=\frac {1}{8} \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {a \log (x)-a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}+\text {$\#$1}^5}\&\right ] \]
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Time = 0.18 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.62
method | result | size |
pseudoelliptic | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 a \,\textit {\_Z}^{4}+a^{2}-a b \right )}{\sum }\frac {\left (\textit {\_R}^{4}+a \right ) \ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R} \left (\textit {\_R}^{4}-a \right )}\right )}{8}\) | \(63\) |
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Timed out. \[ \int \frac {-b+2 a x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^8\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 17.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.26 \[ \int \frac {-b+2 a x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^8\right )} \, dx=\int \frac {2 a x^{4} - b}{\sqrt [4]{a x^{4} - b} \left (a x^{8} - b\right )}\, dx \]
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Not integrable
Time = 0.23 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.35 \[ \int \frac {-b+2 a x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^8\right )} \, dx=\int { \frac {2 \, a x^{4} - b}{{\left (a x^{8} - b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.35 \[ \int \frac {-b+2 a x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^8\right )} \, dx=\int { \frac {2 \, a x^{4} - b}{{\left (a x^{8} - b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]
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Not integrable
Time = 6.59 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.32 \[ \int \frac {-b+2 a x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^8\right )} \, dx=\int \frac {b-2\,a\,x^4}{{\left (a\,x^4-b\right )}^{1/4}\,\left (b-a\,x^8\right )} \,d x \]
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