Integrand size = 38, antiderivative size = 96 \[ \int \frac {-2 b+a x^4}{\sqrt [4]{b+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\frac {1}{4} \text {RootSum}\left [2 b+a \text {$\#$1}^4-\text {$\#$1}^8\&,\frac {3 a \log (x)-3 a \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )-2 \log (x) \text {$\#$1}^4+2 \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}-2 \text {$\#$1}^5}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(499\) vs. \(2(96)=192\).
Time = 0.71 (sec) , antiderivative size = 499, normalized size of antiderivative = 5.20, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {6860, 385, 218, 214, 211} \[ \int \frac {-2 b+a x^4}{\sqrt [4]{b+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=-\frac {\left (a-\frac {a^2-8 b}{\sqrt {a^2+8 b}}\right ) \arctan \left (\frac {x \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+4 b}}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^4+b}}\right )}{2 \left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+4 b}}-\frac {\left (\frac {a^2-8 b}{\sqrt {a^2+8 b}}+a\right ) \arctan \left (\frac {x \sqrt [4]{a \sqrt {a^2+8 b}+a^2+4 b}}{\sqrt [4]{\sqrt {a^2+8 b}+a} \sqrt [4]{a x^4+b}}\right )}{2 \left (\sqrt {a^2+8 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+8 b}+a^2+4 b}}-\frac {\left (a-\frac {a^2-8 b}{\sqrt {a^2+8 b}}\right ) \text {arctanh}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+4 b}}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^4+b}}\right )}{2 \left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+8 b}+a^2+4 b}}-\frac {\left (\frac {a^2-8 b}{\sqrt {a^2+8 b}}+a\right ) \text {arctanh}\left (\frac {x \sqrt [4]{a \sqrt {a^2+8 b}+a^2+4 b}}{\sqrt [4]{\sqrt {a^2+8 b}+a} \sqrt [4]{a x^4+b}}\right )}{2 \left (\sqrt {a^2+8 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+8 b}+a^2+4 b}} \]
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Rule 211
Rule 214
Rule 218
Rule 385
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a+\frac {a^2-8 b}{\sqrt {a^2+8 b}}}{\left (-a-\sqrt {a^2+8 b}+4 x^4\right ) \sqrt [4]{b+a x^4}}+\frac {a-\frac {a^2-8 b}{\sqrt {a^2+8 b}}}{\left (-a+\sqrt {a^2+8 b}+4 x^4\right ) \sqrt [4]{b+a x^4}}\right ) \, dx \\ & = \left (a-\frac {a^2-8 b}{\sqrt {a^2+8 b}}\right ) \int \frac {1}{\left (-a+\sqrt {a^2+8 b}+4 x^4\right ) \sqrt [4]{b+a x^4}} \, dx+\left (a+\frac {a^2-8 b}{\sqrt {a^2+8 b}}\right ) \int \frac {1}{\left (-a-\sqrt {a^2+8 b}+4 x^4\right ) \sqrt [4]{b+a x^4}} \, dx \\ & = \left (a-\frac {a^2-8 b}{\sqrt {a^2+8 b}}\right ) \text {Subst}\left (\int \frac {1}{-a+\sqrt {a^2+8 b}-\left (-4 b+a \left (-a+\sqrt {a^2+8 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\left (a+\frac {a^2-8 b}{\sqrt {a^2+8 b}}\right ) \text {Subst}\left (\int \frac {1}{-a-\sqrt {a^2+8 b}-\left (-4 b+a \left (-a-\sqrt {a^2+8 b}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right ) \\ & = -\frac {\left (a-\frac {a^2-8 b}{\sqrt {a^2+8 b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+8 b}}-\sqrt {a^2+4 b-a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {a-\sqrt {a^2+8 b}}}-\frac {\left (a-\frac {a^2-8 b}{\sqrt {a^2+8 b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+8 b}}+\sqrt {a^2+4 b-a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {a-\sqrt {a^2+8 b}}}-\frac {\left (a+\frac {a^2-8 b}{\sqrt {a^2+8 b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+8 b}}-\sqrt {a^2+4 b+a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {a+\sqrt {a^2+8 b}}}-\frac {\left (a+\frac {a^2-8 b}{\sqrt {a^2+8 b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+8 b}}+\sqrt {a^2+4 b+a \sqrt {a^2+8 b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {a+\sqrt {a^2+8 b}}} \\ & = -\frac {\left (a-\frac {a^2-8 b}{\sqrt {a^2+8 b}}\right ) \arctan \left (\frac {\sqrt [4]{a^2+4 b-a \sqrt {a^2+8 b}} x}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{b+a x^4}}\right )}{2 \left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2+4 b-a \sqrt {a^2+8 b}}}-\frac {\left (a+\frac {a^2-8 b}{\sqrt {a^2+8 b}}\right ) \arctan \left (\frac {\sqrt [4]{a^2+4 b+a \sqrt {a^2+8 b}} x}{\sqrt [4]{a+\sqrt {a^2+8 b}} \sqrt [4]{b+a x^4}}\right )}{2 \left (a+\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2+4 b+a \sqrt {a^2+8 b}}}-\frac {\left (a-\frac {a^2-8 b}{\sqrt {a^2+8 b}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{a^2+4 b-a \sqrt {a^2+8 b}} x}{\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt [4]{b+a x^4}}\right )}{2 \left (a-\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2+4 b-a \sqrt {a^2+8 b}}}-\frac {\left (a+\frac {a^2-8 b}{\sqrt {a^2+8 b}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{a^2+4 b+a \sqrt {a^2+8 b}} x}{\sqrt [4]{a+\sqrt {a^2+8 b}} \sqrt [4]{b+a x^4}}\right )}{2 \left (a+\sqrt {a^2+8 b}\right )^{3/4} \sqrt [4]{a^2+4 b+a \sqrt {a^2+8 b}}} \\ \end{align*}
Time = 0.77 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.01 \[ \int \frac {-2 b+a x^4}{\sqrt [4]{b+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\frac {1}{4} \text {RootSum}\left [2 b+a \text {$\#$1}^4-\text {$\#$1}^8\&,\frac {-3 a \log (x)+3 a \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^4-2 \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}+2 \text {$\#$1}^5}\&\right ] \]
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Time = 1.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.64
method | result | size |
pseudoelliptic | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4} a -2 b \right )}{\sum }\frac {\left (-2 \textit {\_R}^{4}+3 a \right ) \ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R} \left (-2 \textit {\_R}^{4}+a \right )}\right )}{4}\) | \(61\) |
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Timed out. \[ \int \frac {-2 b+a x^4}{\sqrt [4]{b+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 62.12 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.32 \[ \int \frac {-2 b+a x^4}{\sqrt [4]{b+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\int \frac {a x^{4} - 2 b}{\sqrt [4]{a x^{4} + b} \left (- a x^{4} - b + 2 x^{8}\right )}\, dx \]
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Not integrable
Time = 0.23 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.40 \[ \int \frac {-2 b+a x^4}{\sqrt [4]{b+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\int { \frac {a x^{4} - 2 \, b}{{\left (2 \, x^{8} - a x^{4} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]
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Not integrable
Time = 1.71 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.40 \[ \int \frac {-2 b+a x^4}{\sqrt [4]{b+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\int { \frac {a x^{4} - 2 \, b}{{\left (2 \, x^{8} - a x^{4} - b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]
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Not integrable
Time = 6.89 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.38 \[ \int \frac {-2 b+a x^4}{\sqrt [4]{b+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\int \frac {2\,b-a\,x^4}{{\left (a\,x^4+b\right )}^{1/4}\,\left (-2\,x^8+a\,x^4+b\right )} \,d x \]
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