Integrand size = 42, antiderivative size = 99 \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\frac {1}{4} \text {RootSum}\left [a^4+a^3 b-2 b^3-4 a^3 \text {$\#$1}^4-2 a^2 b \text {$\#$1}^4+6 a^2 \text {$\#$1}^8+a b \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-\log (x)+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(1161\) vs. \(2(99)=198\).
Time = 1.86 (sec) , antiderivative size = 1161, normalized size of antiderivative = 11.73, number of steps used = 22, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2081, 6847, 6860, 1443, 385, 218, 214, 211} \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=-\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \arctan \left (\frac {\sqrt [4]{a \sqrt {a-\sqrt {a^2+8 b}}-2 b} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a \sqrt {a-\sqrt {a^2+8 b}}-2 b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \arctan \left (\frac {\sqrt [4]{\sqrt {a-\sqrt {a^2+8 b}} a+2 b} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{\sqrt {a-\sqrt {a^2+8 b}} a+2 b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \arctan \left (\frac {\sqrt [4]{a \sqrt {a+\sqrt {a^2+8 b}}-2 b} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a \sqrt {a+\sqrt {a^2+8 b}}-2 b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \arctan \left (\frac {\sqrt [4]{\sqrt {a+\sqrt {a^2+8 b}} a+2 b} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{\sqrt {a+\sqrt {a^2+8 b}} a+2 b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \text {arctanh}\left (\frac {\sqrt [4]{a \sqrt {a-\sqrt {a^2+8 b}}-2 b} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a \sqrt {a-\sqrt {a^2+8 b}}-2 b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {a-\sqrt {a^2+8 b}} a+2 b} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{\sqrt {a-\sqrt {a^2+8 b}} a+2 b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \text {arctanh}\left (\frac {\sqrt [4]{a \sqrt {a+\sqrt {a^2+8 b}}-2 b} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a \sqrt {a+\sqrt {a^2+8 b}}-2 b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {a+\sqrt {a^2+8 b}} a+2 b} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{\sqrt {a+\sqrt {a^2+8 b}} a+2 b} \sqrt [4]{a x^4+b x^2}} \]
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Rule 211
Rule 214
Rule 218
Rule 385
Rule 1443
Rule 2081
Rule 6847
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {2 b+a x^4}{\sqrt {x} \sqrt [4]{b+a x^2} \left (-b-a x^4+2 x^8\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 {2 b+a x^8}{\sqrt [4]{b+a x^4} \left (-b-a x^8+2 x^{16}\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 \left (\frac {a+\sqrt {a^2+8 b}}{\sqrt [4]{b+a x^4} \left (-a-\sqrt {a^2+8 b}+4 x^8\right )}+\frac {a-\sqrt {a^2+8 b}}{\sqrt [4]{b+a x^4} \left (-a+\sqrt {a^2+8 b}+4 x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}} \\ & = \frac {\left (2 \left (a-\sqrt {a^2+8 b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^4} \left (-a+\sqrt {a^2+8 b}+4 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}+\frac {\left (2 \left (a+\sqrt {a^2+8 b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^4} \left (-a-\sqrt {a^2+8 b}+4 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}} \\ & = -\frac {\left (2 \sqrt {a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \sqrt {a-\sqrt {a^2+8 b}}-4 x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (2 \sqrt {a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \sqrt {a-\sqrt {a^2+8 b}}+4 x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (2 \sqrt {a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \sqrt {a+\sqrt {a^2+8 b}}-4 x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (2 \sqrt {a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \sqrt {a+\sqrt {a^2+8 b}}+4 x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}} \\ & = -\frac {\left (2 \sqrt {a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{2 \sqrt {a-\sqrt {a^2+8 b}}-\left (-4 b+2 a \sqrt {a-\sqrt {a^2+8 b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (2 \sqrt {a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{2 \sqrt {a-\sqrt {a^2+8 b}}-\left (4 b+2 a \sqrt {a-\sqrt {a^2+8 b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (2 \sqrt {a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{2 \sqrt {a+\sqrt {a^2+8 b}}-\left (-4 b+2 a \sqrt {a+\sqrt {a^2+8 b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (2 \sqrt {a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{2 \sqrt {a+\sqrt {a^2+8 b}}-\left (4 b+2 a \sqrt {a+\sqrt {a^2+8 b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}} \\ & = -\frac {\left (\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a-\sqrt {a^2+8 b}}-\sqrt {-2 b+a \sqrt {a-\sqrt {a^2+8 b}}} 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 [4]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a-\sqrt {a^2+8 b}}+\sqrt {-2 b+a \sqrt {a-\sqrt {a^2+8 b}}} 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 [4]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a-\sqrt {a^2+8 b}}-\sqrt {2 b+a \sqrt {a-\sqrt {a^2+8 b}}} 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 [4]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a-\sqrt {a^2+8 b}}+\sqrt {2 b+a \sqrt {a-\sqrt {a^2+8 b}}} 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 [4]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a+\sqrt {a^2+8 b}}-\sqrt {-2 b+a \sqrt {a+\sqrt {a^2+8 b}}} 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 [4]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a+\sqrt {a^2+8 b}}+\sqrt {-2 b+a \sqrt {a+\sqrt {a^2+8 b}}} 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 [4]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a+\sqrt {a^2+8 b}}-\sqrt {2 b+a \sqrt {a+\sqrt {a^2+8 b}}} 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 [4]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a+\sqrt {a^2+8 b}}+\sqrt {2 b+a \sqrt {a+\sqrt {a^2+8 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{b x^2+a x^4}} \\ & = -\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \arctan \left (\frac {\sqrt [4]{-2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{-2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \arctan \left (\frac {\sqrt [4]{2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \arctan \left (\frac {\sqrt [4]{-2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{-2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \arctan \left (\frac {\sqrt [4]{2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{-2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{-2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{-2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{-2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}} \\ \end{align*}
Time = 16.30 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.17 \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\frac {\sqrt [4]{a+\frac {b}{x^2}} x \text {RootSum}\left [a^4+a^3 b-2 b^3-4 a^3 \text {$\#$1}^4-2 a^2 b \text {$\#$1}^4+6 a^2 \text {$\#$1}^8+a b \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt [4]{a+\frac {b}{x^2}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{4 \sqrt [4]{x^2 \left (b+a x^2\right )}} \]
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Time = 1.72 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.87
method | result | size |
pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}-4 a \,\textit {\_Z}^{12}+\left (6 a^{2}+a b \right ) \textit {\_Z}^{8}+\left (-4 a^{3}-2 a^{2} b \right ) \textit {\_Z}^{4}+a^{4}+a^{3} b -2 b^{3}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right )}{4}\) | \(86\) |
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Timed out. \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 71.51 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.34 \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\int \frac {a x^{4} + 2 b}{\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (- a x^{4} - b + 2 x^{8}\right )}\, dx \]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.42 \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\int { \frac {a x^{4} + 2 \, b}{{\left (2 \, x^{8} - a x^{4} - b\right )} {\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}}} \,d x } \]
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Not integrable
Time = 19.03 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.03 \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\int { \frac {a x^{4} + 2 \, b}{{\left (2 \, x^{8} - a x^{4} - b\right )} {\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}}} \,d x } \]
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Not integrable
Time = 6.71 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.40 \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\int -\frac {a\,x^4+2\,b}{{\left (a\,x^4+b\,x^2\right )}^{1/4}\,\left (-2\,x^8+a\,x^4+b\right )} \,d x \]
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