Integrand size = 32, antiderivative size = 72 \[ \int \frac {b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\frac {1}{4} \text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}^3+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-a+\text {$\#$1}^4}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(417\) vs. \(2(72)=144\).
Time = 0.73 (sec) , antiderivative size = 417, normalized size of antiderivative = 5.79, number of steps used = 21, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2081, 1284, 1443, 399, 246, 218, 212, 209, 385} \[ \int \frac {b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\frac {\sqrt {x} \left (a-\sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt [4]{a x^2+b} \arctan \left (\frac {\sqrt {x} \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt {-a} \sqrt {b} \sqrt [4]{a x^4+b x^2}}+\frac {\sqrt {x} \left (\sqrt {-a} \sqrt {b}+a\right )^{3/4} \sqrt [4]{a x^2+b} \arctan \left (\frac {\sqrt {x} \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt {-a} \sqrt {b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt {x} \left (a-\sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt [4]{a x^2+b} \text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt {-a} \sqrt {b} \sqrt [4]{a x^4+b x^2}}+\frac {\sqrt {x} \left (\sqrt {-a} \sqrt {b}+a\right )^{3/4} \sqrt [4]{a x^2+b} \text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt {-a} \sqrt {b} \sqrt [4]{a x^4+b x^2}} \]
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Rule 209
Rule 212
Rule 218
Rule 246
Rule 385
Rule 399
Rule 1284
Rule 1443
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {\left (b+a x^2\right )^{3/4}}{\sqrt {x} \left (b+a 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 {\left (b+a x^4\right )^{3/4}}{b+a x^8} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}} \\ & = \frac {\left (\sqrt {-a} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {\left (b+a x^4\right )^{3/4}}{\sqrt {-a} \sqrt {b}-a x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (\sqrt {-a} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {\left (b+a x^4\right )^{3/4}}{\sqrt {-a} \sqrt {b}+a x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {b} \sqrt [4]{b x^2+a x^4}} \\ & = -\frac {\left (\sqrt {-a} \left (\sqrt {-a}-\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-a} \sqrt {b}+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}+\frac {\left (\sqrt {-a} \left (\sqrt {-a}+\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-a} \sqrt {b}-a x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}} \\ & = -\frac {\left (\sqrt {-a} \left (\sqrt {-a}-\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a} \sqrt {b}-\left (\sqrt {-a} a \sqrt {b}-a 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 {-a} \left (\sqrt {-a}+\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a} \sqrt {b}-\left (\sqrt {-a} a \sqrt {b}+a 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 (\sqrt {-a}-\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a+\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\left (\sqrt {-a}-\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a+\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (\left (\sqrt {-a}+\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a-\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (\left (\sqrt {-a}+\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a-\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {b} \sqrt [4]{b x^2+a x^4}} \\ & = -\frac {\left (a-\sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2} \arctan \left (\frac {\sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {-a} \sqrt {b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (a+\sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2} \arctan \left (\frac {\sqrt [4]{a+\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {-a} \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (a-\sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {-a} \sqrt {b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (a+\sqrt {-a} \sqrt {b}\right )^{3/4} \sqrt {x} \sqrt [4]{b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{a+\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {-a} \sqrt {b} \sqrt [4]{b x^2+a x^4}} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.49 \[ \int \frac {b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right ) \text {$\#$1}^3+\log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^3}{-a+\text {$\#$1}^4}\&\right ]}{4 \sqrt [4]{x^2 \left (b+a x^2\right )}} \]
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Time = 1.41 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.82
method | result | size |
pseudoelliptic | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 \textit {\_Z}^{4} a +a^{2}+a b \right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{4}-a}\right )}{4}\) | \(59\) |
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Timed out. \[ \int \frac {b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\text {Timed out} \]
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Not integrable
Time = 5.53 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.38 \[ \int \frac {b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int \frac {a x^{2} + b}{\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (a x^{4} + b\right )}\, dx \]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.44 \[ \int \frac {b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {a x^{2} + b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} + b\right )}} \,d x } \]
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Not integrable
Time = 0.86 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.44 \[ \int \frac {b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {a x^{2} + b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} + b\right )}} \,d x } \]
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Not integrable
Time = 8.39 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.44 \[ \int \frac {b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int \frac {a\,x^2+b}{\left (a\,x^4+b\right )\,{\left (a\,x^4+b\,x^2\right )}^{1/4}} \,d x \]
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