Integrand size = 28, antiderivative size = 66 \[ \int \frac {x^6}{\left (b+a x^4\right )^{3/4} \left (b^2+a^2 x^8\right )} \, dx=\frac {\text {RootSum}\left [2 a^2-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )}{a \text {$\#$1}^3-\text {$\#$1}^7}\&\right ]}{8 b} \]
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Leaf count is larger than twice the leaf count of optimal. \(325\) vs. \(2(66)=132\).
Time = 0.57 (sec) , antiderivative size = 325, normalized size of antiderivative = 4.92, number of steps used = 10, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1543, 508, 304, 211} \[ \int \frac {x^6}{\left (b+a x^4\right )^{3/4} \left (b^2+a^2 x^8\right )} \, dx=\frac {\left (-a^2\right )^{7/8} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{\sqrt {-a^2}-a} x}{\sqrt [8]{-a^2} \sqrt [4]{a x^4+b}}\right )}{4 a^{11/4} \left (\sqrt {-a^2}-a\right )^{3/4} b}-\frac {\left (-a^2\right )^{3/8} \arctan \left (\frac {\left (-a^2\right )^{3/8} \sqrt [4]{\sqrt {-a^2}-a} x}{a^{3/4} \sqrt [4]{a x^4+b}}\right )}{4 a^{7/4} \left (\sqrt {-a^2}-a\right )^{3/4} b}+\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{\sqrt {-a^2}+a} x}{\sqrt [8]{-a^2} \sqrt [4]{a x^4+b}}\right )}{4 a^{3/4} \sqrt [8]{-a^2} \left (\sqrt {-a^2}+a\right )^{3/4} b}-\frac {\sqrt [4]{a} \arctan \left (\frac {\left (-a^2\right )^{3/8} \sqrt [4]{\sqrt {-a^2}+a} x}{a^{3/4} \sqrt [4]{a x^4+b}}\right )}{4 \left (-a^2\right )^{5/8} \left (\sqrt {-a^2}+a\right )^{3/4} b} \]
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Rule 211
Rule 304
Rule 508
Rule 1543
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x^2}{2 \left (b+a x^4\right )^{3/4} \left (-\sqrt {-a^2} b+a^2 x^4\right )}+\frac {x^2}{2 \left (b+a x^4\right )^{3/4} \left (\sqrt {-a^2} b+a^2 x^4\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {x^2}{\left (b+a x^4\right )^{3/4} \left (-\sqrt {-a^2} b+a^2 x^4\right )} \, dx+\frac {1}{2} \int \frac {x^2}{\left (b+a x^4\right )^{3/4} \left (\sqrt {-a^2} b+a^2 x^4\right )} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {x^2}{-\sqrt {-a^2} b-\left (-a^2 b-a \sqrt {-a^2} b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{\sqrt {-a^2} b-\left (-a^2 b+a \sqrt {-a^2} b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right ) \\ & = \frac {\text {Subst}\left (\int \frac {1}{\sqrt [4]{-a^2}-\sqrt {a} \sqrt {-a+\sqrt {-a^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a} \sqrt {-a+\sqrt {-a^2}} b}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt [4]{-a^2}+\sqrt {a} \sqrt {-a+\sqrt {-a^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a} \sqrt {-a+\sqrt {-a^2}} b}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt [4]{-a^2}-\sqrt {a} \sqrt {a+\sqrt {-a^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a} \sqrt {a+\sqrt {-a^2}} b}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt [4]{-a^2}+\sqrt {a} \sqrt {a+\sqrt {-a^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a} \sqrt {a+\sqrt {-a^2}} b} \\ & = -\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{-a+\sqrt {-a^2}} x}{\sqrt [8]{-a^2} \sqrt [4]{b+a x^4}}\right )}{4 a^{3/4} \sqrt [8]{-a^2} \left (-a+\sqrt {-a^2}\right )^{3/4} b}+\frac {\sqrt [4]{a} \arctan \left (\frac {\left (-a^2\right )^{3/8} \sqrt [4]{-a+\sqrt {-a^2}} x}{a^{3/4} \sqrt [4]{b+a x^4}}\right )}{4 \left (-a^2\right )^{5/8} \left (-a+\sqrt {-a^2}\right )^{3/4} b}+\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{a+\sqrt {-a^2}} x}{\sqrt [8]{-a^2} \sqrt [4]{b+a x^4}}\right )}{4 a^{3/4} \sqrt [8]{-a^2} \left (a+\sqrt {-a^2}\right )^{3/4} b}-\frac {\sqrt [4]{a} \arctan \left (\frac {\left (-a^2\right )^{3/8} \sqrt [4]{a+\sqrt {-a^2}} x}{a^{3/4} \sqrt [4]{b+a x^4}}\right )}{4 \left (-a^2\right )^{5/8} \left (a+\sqrt {-a^2}\right )^{3/4} b} \\ \end{align*}
Time = 0.66 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.98 \[ \int \frac {x^6}{\left (b+a x^4\right )^{3/4} \left (b^2+a^2 x^8\right )} \, dx=\frac {\text {RootSum}\left [2 a^2-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x)-\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]}{8 b} \]
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Time = 1.39 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.86
method | result | size |
pseudoelliptic | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 \textit {\_Z}^{4} a +2 a^{2}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (\textit {\_R}^{4}-a \right )}}{8 b}\) | \(57\) |
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Timed out. \[ \int \frac {x^6}{\left (b+a x^4\right )^{3/4} \left (b^2+a^2 x^8\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 11.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.36 \[ \int \frac {x^6}{\left (b+a x^4\right )^{3/4} \left (b^2+a^2 x^8\right )} \, dx=\int \frac {x^{6}}{\left (a x^{4} + b\right )^{\frac {3}{4}} \left (a^{2} x^{8} + b^{2}\right )}\, dx \]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.42 \[ \int \frac {x^6}{\left (b+a x^4\right )^{3/4} \left (b^2+a^2 x^8\right )} \, dx=\int { \frac {x^{6}}{{\left (a^{2} x^{8} + b^{2}\right )} {\left (a x^{4} + b\right )}^{\frac {3}{4}}} \,d x } \]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.42 \[ \int \frac {x^6}{\left (b+a x^4\right )^{3/4} \left (b^2+a^2 x^8\right )} \, dx=\int { \frac {x^{6}}{{\left (a^{2} x^{8} + b^{2}\right )} {\left (a x^{4} + b\right )}^{\frac {3}{4}}} \,d x } \]
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Not integrable
Time = 5.23 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.42 \[ \int \frac {x^6}{\left (b+a x^4\right )^{3/4} \left (b^2+a^2 x^8\right )} \, dx=\int \frac {x^6}{\left (a^2\,x^8+b^2\right )\,{\left (a\,x^4+b\right )}^{3/4}} \,d x \]
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