Integrand size = 41, antiderivative size = 163 \[ \int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{\sqrt [4]{a}}+\frac {1}{8} \text {RootSum}\left [a^4-2 a^3 b-b^3-4 a^3 \text {$\#$1}^4+4 a^2 b \text {$\#$1}^4+6 a^2 \text {$\#$1}^8-2 a b \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-\log (x)+\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(1315\) vs. \(2(163)=326\).
Time = 1.91 (sec) , antiderivative size = 1315, normalized size of antiderivative = 8.07, number of steps used = 34, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.317, Rules used = {1607, 2081, 6860, 1284, 1531, 246, 218, 212, 209, 1443, 385, 214, 211} \[ \int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx=\frac {\sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{\sqrt [4]{a} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [4]{a \sqrt {-a-\sqrt {a^2+b}}-b} \sqrt {x}}{\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{a \sqrt {-a-\sqrt {a^2+b}}-b} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [4]{\sqrt {-a-\sqrt {a^2+b}} a+b} \sqrt {x}}{\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{\sqrt {-a-\sqrt {a^2+b}} a+b} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt [8]{\sqrt {a^2+b}-a} \sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [4]{a \sqrt {\sqrt {a^2+b}-a}-b} \sqrt {x}}{\sqrt [8]{\sqrt {a^2+b}-a} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{a \sqrt {\sqrt {a^2+b}-a}-b} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt [8]{\sqrt {a^2+b}-a} \sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [4]{\sqrt {\sqrt {a^2+b}-a} a+b} \sqrt {x}}{\sqrt [8]{\sqrt {a^2+b}-a} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{\sqrt {\sqrt {a^2+b}-a} a+b} \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{\sqrt [4]{a} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt [4]{a \sqrt {-a-\sqrt {a^2+b}}-b} \sqrt {x}}{\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{a \sqrt {-a-\sqrt {a^2+b}}-b} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {-a-\sqrt {a^2+b}} a+b} \sqrt {x}}{\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{\sqrt {-a-\sqrt {a^2+b}} a+b} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt [8]{\sqrt {a^2+b}-a} \sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt [4]{a \sqrt {\sqrt {a^2+b}-a}-b} \sqrt {x}}{\sqrt [8]{\sqrt {a^2+b}-a} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{a \sqrt {\sqrt {a^2+b}-a}-b} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt [8]{\sqrt {a^2+b}-a} \sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {\sqrt {a^2+b}-a} a+b} \sqrt {x}}{\sqrt [8]{\sqrt {a^2+b}-a} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{\sqrt {\sqrt {a^2+b}-a} a+b} \sqrt [4]{a x^4-b x^2}} \]
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Rule 209
Rule 211
Rule 212
Rule 214
Rule 218
Rule 246
Rule 385
Rule 1284
Rule 1443
Rule 1531
Rule 1607
Rule 2081
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^4 \left (a+x^4\right )}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx \\ & = \frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \int \frac {x^{7/2} \left (a+x^4\right )}{\sqrt [4]{-b+a x^2} \left (-b+2 a x^4+x^8\right )} \, dx}{\sqrt [4]{-b x^2+a x^4}} \\ & = \frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \int \left (\frac {x^{7/2}}{\sqrt [4]{-b+a x^2} \left (2 a-2 \sqrt {a^2+b}+2 x^4\right )}+\frac {x^{7/2}}{\sqrt [4]{-b+a x^2} \left (2 a+2 \sqrt {a^2+b}+2 x^4\right )}\right ) \, dx}{\sqrt [4]{-b x^2+a x^4}} \\ & = \frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \int \frac {x^{7/2}}{\sqrt [4]{-b+a x^2} \left (2 a-2 \sqrt {a^2+b}+2 x^4\right )} \, dx}{\sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \int \frac {x^{7/2}}{\sqrt [4]{-b+a x^2} \left (2 a+2 \sqrt {a^2+b}+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 {x^8}{\sqrt [4]{-b+a x^4} \left (2 a-2 \sqrt {a^2+b}+2 x^8\right )} \, dx,x,\sqrt {x}\right )}{\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 {x^8}{\sqrt [4]{-b+a x^4} \left (2 a+2 \sqrt {a^2+b}+2 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}} \\ & = 2 \frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}}-\frac {\left (2 \left (a-\sqrt {a^2+b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4} \left (2 a-2 \sqrt {a^2+b}+2 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}}-\frac {\left (2 \left (a+\sqrt {a^2+b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4} \left (2 a+2 \sqrt {a^2+b}+2 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}} \\ & = 2 \frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt [4]{-b x^2+a x^4}}+\frac {\left (\left (a-\sqrt {a^2+b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \sqrt {-a+\sqrt {a^2+b}}-2 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a+\sqrt {a^2+b}} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\left (a-\sqrt {a^2+b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \sqrt {-a+\sqrt {a^2+b}}+2 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a+\sqrt {a^2+b}} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\left (a+\sqrt {a^2+b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \sqrt {-a-\sqrt {a^2+b}}-2 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a-\sqrt {a^2+b}} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\left (a+\sqrt {a^2+b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \sqrt {-a-\sqrt {a^2+b}}+2 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a-\sqrt {a^2+b}} \sqrt [4]{-b x^2+a x^4}} \\ & = 2 \left (\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 \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 {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 \sqrt [4]{-b x^2+a x^4}}\right )+\frac {\left (\left (a-\sqrt {a^2+b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{2 \sqrt {-a+\sqrt {a^2+b}}-\left (-2 b+2 a \sqrt {-a+\sqrt {a^2+b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {-a+\sqrt {a^2+b}} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\left (a-\sqrt {a^2+b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{2 \sqrt {-a+\sqrt {a^2+b}}-\left (2 b+2 a \sqrt {-a+\sqrt {a^2+b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {-a+\sqrt {a^2+b}} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\left (a+\sqrt {a^2+b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{2 \sqrt {-a-\sqrt {a^2+b}}-\left (-2 b+2 a \sqrt {-a-\sqrt {a^2+b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {-a-\sqrt {a^2+b}} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\left (a+\sqrt {a^2+b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{2 \sqrt {-a-\sqrt {a^2+b}}-\left (2 b+2 a \sqrt {-a-\sqrt {a^2+b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {-a-\sqrt {a^2+b}} \sqrt [4]{-b x^2+a x^4}} \\ & = 2 \left (\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 \sqrt [4]{a} \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 \sqrt [4]{a} \sqrt [4]{-b x^2+a x^4}}\right )+\frac {\left (\left (a-\sqrt {a^2+b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a+\sqrt {a^2+b}}-\sqrt {-b+a \sqrt {-a+\sqrt {a^2+b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \left (-a+\sqrt {a^2+b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\left (a-\sqrt {a^2+b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a+\sqrt {a^2+b}}+\sqrt {-b+a \sqrt {-a+\sqrt {a^2+b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \left (-a+\sqrt {a^2+b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\left (a-\sqrt {a^2+b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a+\sqrt {a^2+b}}-\sqrt {b+a \sqrt {-a+\sqrt {a^2+b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \left (-a+\sqrt {a^2+b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\left (a-\sqrt {a^2+b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a+\sqrt {a^2+b}}+\sqrt {b+a \sqrt {-a+\sqrt {a^2+b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \left (-a+\sqrt {a^2+b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\left (a+\sqrt {a^2+b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a-\sqrt {a^2+b}}-\sqrt {-b+a \sqrt {-a-\sqrt {a^2+b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \left (-a-\sqrt {a^2+b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\left (a+\sqrt {a^2+b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a-\sqrt {a^2+b}}+\sqrt {-b+a \sqrt {-a-\sqrt {a^2+b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \left (-a-\sqrt {a^2+b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\left (a+\sqrt {a^2+b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a-\sqrt {a^2+b}}-\sqrt {b+a \sqrt {-a-\sqrt {a^2+b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \left (-a-\sqrt {a^2+b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\left (a+\sqrt {a^2+b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a-\sqrt {a^2+b}}+\sqrt {b+a \sqrt {-a-\sqrt {a^2+b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \left (-a-\sqrt {a^2+b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}} \\ & = -\frac {\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt {x} \sqrt [4]{-b+a x^2} \arctan \left (\frac {\sqrt [4]{-b+a \sqrt {-a-\sqrt {a^2+b}}} \sqrt {x}}{\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{-b+a \sqrt {-a-\sqrt {a^2+b}}} \sqrt [4]{-b x^2+a x^4}}-\frac {\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt {x} \sqrt [4]{-b+a x^2} \arctan \left (\frac {\sqrt [4]{b+a \sqrt {-a-\sqrt {a^2+b}}} \sqrt {x}}{\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{b+a \sqrt {-a-\sqrt {a^2+b}}} \sqrt [4]{-b x^2+a x^4}}-\frac {\sqrt [8]{-a+\sqrt {a^2+b}} \sqrt {x} \sqrt [4]{-b+a x^2} \arctan \left (\frac {\sqrt [4]{-b+a \sqrt {-a+\sqrt {a^2+b}}} \sqrt {x}}{\sqrt [8]{-a+\sqrt {a^2+b}} \sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{-b+a \sqrt {-a+\sqrt {a^2+b}}} \sqrt [4]{-b x^2+a x^4}}-\frac {\sqrt [8]{-a+\sqrt {a^2+b}} \sqrt {x} \sqrt [4]{-b+a x^2} \arctan \left (\frac {\sqrt [4]{b+a \sqrt {-a+\sqrt {a^2+b}}} \sqrt {x}}{\sqrt [8]{-a+\sqrt {a^2+b}} \sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{b+a \sqrt {-a+\sqrt {a^2+b}}} \sqrt [4]{-b x^2+a x^4}}+2 \left (\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 \sqrt [4]{a} \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 \sqrt [4]{a} \sqrt [4]{-b x^2+a x^4}}\right )-\frac {\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt {x} \sqrt [4]{-b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{-b+a \sqrt {-a-\sqrt {a^2+b}}} \sqrt {x}}{\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{-b+a \sqrt {-a-\sqrt {a^2+b}}} \sqrt [4]{-b x^2+a x^4}}-\frac {\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt {x} \sqrt [4]{-b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{b+a \sqrt {-a-\sqrt {a^2+b}}} \sqrt {x}}{\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{b+a \sqrt {-a-\sqrt {a^2+b}}} \sqrt [4]{-b x^2+a x^4}}-\frac {\sqrt [8]{-a+\sqrt {a^2+b}} \sqrt {x} \sqrt [4]{-b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{-b+a \sqrt {-a+\sqrt {a^2+b}}} \sqrt {x}}{\sqrt [8]{-a+\sqrt {a^2+b}} \sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{-b+a \sqrt {-a+\sqrt {a^2+b}}} \sqrt [4]{-b x^2+a x^4}}-\frac {\sqrt [8]{-a+\sqrt {a^2+b}} \sqrt {x} \sqrt [4]{-b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{b+a \sqrt {-a+\sqrt {a^2+b}}} \sqrt {x}}{\sqrt [8]{-a+\sqrt {a^2+b}} \sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{b+a \sqrt {-a+\sqrt {a^2+b}}} \sqrt [4]{-b x^2+a x^4}} \\ \end{align*}
\[ \int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx=\int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx \]
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Time = 0.74 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.08
method | result | size |
pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}-4 a \,\textit {\_Z}^{12}+\left (6 a^{2}-2 a b \right ) \textit {\_Z}^{8}+\left (-4 a^{3}+4 a^{2} b \right ) \textit {\_Z}^{4}+a^{4}-2 a^{3} b -b^{3}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right ) a^{\frac {1}{4}}-8 \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )+4 \ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}\right )}{8 a^{\frac {1}{4}}}\) | \(176\) |
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Timed out. \[ \int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.25 \[ \int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx=\int { \frac {x^{8} + a x^{4}}{{\left (x^{8} + 2 \, a x^{4} - b\right )} {\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}}} \,d x } \]
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Not integrable
Time = 20.01 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.02 \[ \int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx=\int { \frac {x^{8} + a x^{4}}{{\left (x^{8} + 2 \, a x^{4} - b\right )} {\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}}} \,d x } \]
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Not integrable
Time = 6.41 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.25 \[ \int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx=\int \frac {x^8+a\,x^4}{{\left (a\,x^4-b\,x^2\right )}^{1/4}\,\left (x^8+2\,a\,x^4-b\right )} \,d x \]
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