Integrand size = 36, antiderivative size = 108 \[ \int \frac {b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\frac {1}{2} \text {RootSum}\left [2 a^2+b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {2 a \log (x)-2 a \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{3 a \text {$\#$1}-2 \text {$\#$1}^5}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(647\) vs. \(2(108)=216\).
Time = 1.29 (sec) , antiderivative size = 647, normalized size of antiderivative = 5.99, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2081, 6847, 6860, 385, 218, 214, 211} \[ \int \frac {b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\frac {\sqrt {x} \left (a-\frac {a^2-2 b}{\sqrt {a^2-4 b}}\right ) \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2+2 b}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2-b}}\right )}{\left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2+2 b} \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt {x} \left (\frac {a^2-2 b}{\sqrt {a^2-4 b}}+a\right ) \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2-4 b}+a^2+2 b}}{\sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt [4]{a x^2-b}}\right )}{\left (\sqrt {a^2-4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2-4 b}+a^2+2 b} \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt {x} \left (a-\frac {a^2-2 b}{\sqrt {a^2-4 b}}\right ) \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2+2 b}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2-b}}\right )}{\left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2+2 b} \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt {x} \left (\frac {a^2-2 b}{\sqrt {a^2-4 b}}+a\right ) \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2-4 b}+a^2+2 b}}{\sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt [4]{a x^2-b}}\right )}{\left (\sqrt {a^2-4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2-4 b}+a^2+2 b} \sqrt [4]{a x^4-b x^2}} \]
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Rule 211
Rule 214
Rule 218
Rule 385
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 {b+a x^2}{\sqrt {x} \sqrt [4]{-b+a x^2} \left (b+a x^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 {b+a x^4}{\sqrt [4]{-b+a x^4} \left (b+a x^4+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 \left (\frac {a+\frac {-a^2+2 b}{\sqrt {a^2-4 b}}}{\left (a-\sqrt {a^2-4 b}+2 x^4\right ) \sqrt [4]{-b+a x^4}}+\frac {a-\frac {-a^2+2 b}{\sqrt {a^2-4 b}}}{\left (a+\sqrt {a^2-4 b}+2 x^4\right ) \sqrt [4]{-b+a x^4}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}} \\ & = \frac {\left (2 \left (a-\frac {a^2-2 b}{\sqrt {a^2-4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (a-\sqrt {a^2-4 b}+2 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 \left (a+\frac {a^2-2 b}{\sqrt {a^2-4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (a+\sqrt {a^2-4 b}+2 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 \left (a-\frac {a^2-2 b}{\sqrt {a^2-4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{a-\sqrt {a^2-4 b}-\left (a \left (a-\sqrt {a^2-4 b}\right )+2 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 \left (a+\frac {a^2-2 b}{\sqrt {a^2-4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{a+\sqrt {a^2-4 b}-\left (a \left (a+\sqrt {a^2-4 b}\right )+2 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 (\left (a-\frac {a^2-2 b}{\sqrt {a^2-4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2-4 b}}-\sqrt {a^2-a \sqrt {a^2-4 b}+2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {a-\sqrt {a^2-4 b}} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\left (a-\frac {a^2-2 b}{\sqrt {a^2-4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2-4 b}}+\sqrt {a^2-a \sqrt {a^2-4 b}+2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {a-\sqrt {a^2-4 b}} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\left (a+\frac {a^2-2 b}{\sqrt {a^2-4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2-4 b}}-\sqrt {a^2+a \sqrt {a^2-4 b}+2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {a+\sqrt {a^2-4 b}} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\left (a+\frac {a^2-2 b}{\sqrt {a^2-4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2-4 b}}+\sqrt {a^2+a \sqrt {a^2-4 b}+2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {a+\sqrt {a^2-4 b}} \sqrt [4]{-b x^2+a x^4}} \\ & = \frac {\left (a-\frac {a^2-2 b}{\sqrt {a^2-4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2} \arctan \left (\frac {\sqrt [4]{a^2-a \sqrt {a^2-4 b}+2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{-b+a x^2}}\right )}{\left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt [4]{a^2-a \sqrt {a^2-4 b}+2 b} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (a+\frac {a^2-2 b}{\sqrt {a^2-4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2} \arctan \left (\frac {\sqrt [4]{a^2+a \sqrt {a^2-4 b}+2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{-b+a x^2}}\right )}{\left (a+\sqrt {a^2-4 b}\right )^{3/4} \sqrt [4]{a^2+a \sqrt {a^2-4 b}+2 b} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (a-\frac {a^2-2 b}{\sqrt {a^2-4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{a^2-a \sqrt {a^2-4 b}+2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{-b+a x^2}}\right )}{\left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt [4]{a^2-a \sqrt {a^2-4 b}+2 b} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (a+\frac {a^2-2 b}{\sqrt {a^2-4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{a^2+a \sqrt {a^2-4 b}+2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{-b+a x^2}}\right )}{\left (a+\sqrt {a^2-4 b}\right )^{3/4} \sqrt [4]{a^2+a \sqrt {a^2-4 b}+2 b} \sqrt [4]{-b x^2+a x^4}} \\ \end{align*}
Time = 0.86 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.34 \[ \int \frac {b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \text {RootSum}\left [2 a^2+b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {2 a \log (x)-4 a \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+2 \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4}{3 a \text {$\#$1}-2 \text {$\#$1}^5}\&\right ]}{4 \sqrt [4]{-b x^2+a x^4}} \]
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Time = 0.36 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.65
method | result | size |
pseudoelliptic | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-3 a \,\textit {\_Z}^{4}+2 a^{2}+b \right )}{\sum }\frac {\left (\textit {\_R}^{4}-2 a \right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R} \left (2 \textit {\_R}^{4}-3 a \right )}\right )}{2}\) | \(70\) |
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Timed out. \[ \int \frac {b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\text {Timed out} \]
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Not integrable
Time = 7.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.29 \[ \int \frac {b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\int \frac {a x^{2} + b}{\sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (a x^{2} + b + x^{4}\right )}\, dx \]
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Not integrable
Time = 0.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.33 \[ \int \frac {b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\int { \frac {a x^{2} + b}{{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + a x^{2} + b\right )}} \,d x } \]
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Not integrable
Time = 8.47 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.33 \[ \int \frac {b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\int { \frac {a x^{2} + b}{{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + a x^{2} + b\right )}} \,d x } \]
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Not integrable
Time = 5.74 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.33 \[ \int \frac {b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\int \frac {a\,x^2+b}{{\left (a\,x^4-b\,x^2\right )}^{1/4}\,\left (x^4+a\,x^2+b\right )} \,d x \]
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