Integrand size = 37, antiderivative size = 102 \[ \int \frac {-2 b+a x^2}{\sqrt [4]{-b+a x^2} \left (-b+a x^2+x^4\right )} \, dx=-\frac {\arctan \left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b+a x^2}}{\sqrt {2}}}{x \sqrt [4]{-b+a x^2}}\right )}{\sqrt {2}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-b+a x^2}}{x^2+\sqrt {-b+a x^2}}\right )}{\sqrt {2}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 4.71 (sec) , antiderivative size = 2432, normalized size of antiderivative = 23.84, number of steps used = 18, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {1706, 408, 504, 1231, 226, 1721} \[ \int \frac {-2 b+a x^2}{\sqrt [4]{-b+a x^2} \left (-b+a x^2+x^4\right )} \, dx=-\frac {\left (a+\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}},2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right ) \left (\sqrt {2} \sqrt {b}-\sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right )^2}{4 \sqrt {2} \sqrt [4]{b} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b} \left (a^2+\sqrt {a^2+4 b} a+4 b\right ) x}-\frac {\sqrt {b} \sqrt {-a-\sqrt {a^2+4 b}} \sqrt {\frac {a x^2}{b}} \arctan \left (\frac {\sqrt {a} \sqrt {-a-\sqrt {a^2+4 b}} \sqrt [4]{a x^2-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2-\sqrt {a^2+4 b} a-2 b} \sqrt {\frac {a x^2}{b}}}\right )}{2 \sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2-\sqrt {a^2+4 b} a-2 b} x}-\frac {\sqrt {b} \sqrt {a+\sqrt {a^2+4 b}} \sqrt {\frac {a x^2}{b}} \arctan \left (\frac {\sqrt {a} \sqrt {a+\sqrt {a^2+4 b}} \sqrt [4]{a x^2-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2-\sqrt {a^2+4 b} a-2 b} \sqrt {\frac {a x^2}{b}}}\right )}{2 \sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2-\sqrt {a^2+4 b} a-2 b} x}-\frac {\sqrt {b} \sqrt {a-\sqrt {a^2+4 b}} \sqrt {\frac {a x^2}{b}} \arctan \left (\frac {\sqrt {a} \sqrt {a-\sqrt {a^2+4 b}} \sqrt [4]{a x^2-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2+\sqrt {a^2+4 b} a-2 b} \sqrt {\frac {a x^2}{b}}}\right )}{2 \sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2+\sqrt {a^2+4 b} a-2 b} x}-\frac {\sqrt {b} \sqrt {\sqrt {a^2+4 b}-a} \sqrt {\frac {a x^2}{b}} \arctan \left (\frac {\sqrt {a} \sqrt {\sqrt {a^2+4 b}-a} \sqrt [4]{a x^2-b}}{\sqrt [4]{2} \sqrt {b} \sqrt [4]{-a^2+\sqrt {a^2+4 b} a-2 b} \sqrt {\frac {a x^2}{b}}}\right )}{2 \sqrt [4]{2} \sqrt {a} \sqrt [4]{-a^2+\sqrt {a^2+4 b} a-2 b} x}-\frac {\left (a+\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}-\sqrt {2} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right ) \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{b} \left (a^2+\sqrt {a^2+4 b} a+4 b\right ) x}-\frac {\left (a+\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}+\sqrt {2} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right ) \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{b} \left (a^2+\sqrt {a^2+4 b} a+4 b\right ) x}-\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}-\sqrt {-2 a^2+2 \sqrt {a^2+4 b} a-4 b}\right ) \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{b} \left (a^2-\sqrt {a^2+4 b} a+4 b\right ) x}-\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}+\sqrt {-2 a^2+2 \sqrt {a^2+4 b} a-4 b}\right ) \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{b} \left (a^2-\sqrt {a^2+4 b} a+4 b\right ) x}+\frac {\left (a+\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right )^2 \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {2} \sqrt {b}-\sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b}},2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt [4]{b} \sqrt {-a^2-\sqrt {a^2+4 b} a-2 b} \left (a^2+\sqrt {a^2+4 b} a+4 b\right ) x}+\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}+\sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}\right )^2 \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {2} \sqrt {b}-\sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}},2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt [4]{b} \left (a^2-\sqrt {a^2+4 b} a+4 b\right ) \sqrt {-a^2+\sqrt {a^2+4 b} a-2 b} x}-\frac {\left (a-\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}-\sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}\right )^2 \sqrt {\frac {a x^2}{\left (\sqrt {b}+\sqrt {a x^2-b}\right )^2}} \left (\sqrt {b}+\sqrt {a x^2-b}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {2} \sqrt {b}+\sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}\right )^2}{4 \sqrt {2} \sqrt {b} \sqrt {-a^2+\sqrt {a^2+4 b} a-2 b}},2 \arctan \left (\frac {\sqrt [4]{a x^2-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} \sqrt [4]{b} \left (a^2-\sqrt {a^2+4 b} a+4 b\right ) \sqrt {-a^2+\sqrt {a^2+4 b} a-2 b} x} \]
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Rule 226
Rule 408
Rule 504
Rule 1231
Rule 1706
Rule 1721
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a-\sqrt {a^2+4 b}}{\left (a-\sqrt {a^2+4 b}+2 x^2\right ) \sqrt [4]{-b+a x^2}}+\frac {a+\sqrt {a^2+4 b}}{\left (a+\sqrt {a^2+4 b}+2 x^2\right ) \sqrt [4]{-b+a x^2}}\right ) \, dx \\ & = \left (a-\sqrt {a^2+4 b}\right ) \int \frac {1}{\left (a-\sqrt {a^2+4 b}+2 x^2\right ) \sqrt [4]{-b+a x^2}} \, dx+\left (a+\sqrt {a^2+4 b}\right ) \int \frac {1}{\left (a+\sqrt {a^2+4 b}+2 x^2\right ) \sqrt [4]{-b+a x^2}} \, dx \\ & = \frac {\left (2 \left (a-\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {x^2}{\left (2 b+a \left (a-\sqrt {a^2+4 b}\right )+2 x^4\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{x}+\frac {\left (2 \left (a+\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {x^2}{\left (2 b+a \left (a+\sqrt {a^2+4 b}\right )+2 x^4\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{x} \\ & = -\frac {\left (\left (a-\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}-\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} x}+\frac {\left (\left (a-\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}+\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} x}-\frac {\left (\left (a+\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}-\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} x}+\frac {\left (\left (a+\sqrt {a^2+4 b}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}+\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} x} \\ & = -\frac {\left (\left (a+\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}-\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \left (a^2+4 b+a \sqrt {a^2+4 b}\right ) x}-\frac {\left (\left (a+\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \left (a^2+4 b+a \sqrt {a^2+4 b}\right ) x}-\frac {\left (\sqrt {b} \left (a+\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}-\sqrt {2} \sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}-\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \left (a^2+4 b+a \sqrt {a^2+4 b}\right ) x}+\frac {\left (\sqrt {b} \left (a+\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}+\sqrt {2} \sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (\sqrt {-a^2-2 b-a \sqrt {a^2+4 b}}+\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \left (a^2+4 b+a \sqrt {a^2+4 b}\right ) x}-\frac {\left (\left (a-\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}-\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \left (a^2+4 b-a \sqrt {a^2+4 b}\right ) x}-\frac {\left (\left (a-\sqrt {a^2+4 b}\right ) \left (\sqrt {2} \sqrt {b}+\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \left (a^2+4 b-a \sqrt {a^2+4 b}\right ) x}-\frac {\left (\sqrt {b} \left (a-\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}-\sqrt {2} \sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}-\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \left (a^2+4 b-a \sqrt {a^2+4 b}\right ) x}+\frac {\left (\sqrt {b} \left (a-\sqrt {a^2+4 b}\right ) \left (2 \sqrt {b}+\sqrt {2} \sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}\right ) \sqrt {\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b}}}{\left (\sqrt {-a^2-2 b+a \sqrt {a^2+4 b}}+\sqrt {2} x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x^2}\right )}{\sqrt {2} \left (a^2+4 b-a \sqrt {a^2+4 b}\right ) x} \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.89 \[ \int \frac {-2 b+a x^2}{\sqrt [4]{-b+a x^2} \left (-b+a x^2+x^4\right )} \, dx=\frac {-\arctan \left (\frac {-x^2+\sqrt {-b+a x^2}}{\sqrt {2} x \sqrt [4]{-b+a x^2}}\right )+\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-b+a x^2}}{x^2+\sqrt {-b+a x^2}}\right )}{\sqrt {2}} \]
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\[\int \frac {a \,x^{2}-2 b}{\left (a \,x^{2}-b \right )^{\frac {1}{4}} \left (x^{4}+a \,x^{2}-b \right )}d x\]
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Timed out. \[ \int \frac {-2 b+a x^2}{\sqrt [4]{-b+a x^2} \left (-b+a x^2+x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {-2 b+a x^2}{\sqrt [4]{-b+a x^2} \left (-b+a x^2+x^4\right )} \, dx=\int \frac {a x^{2} - 2 b}{\sqrt [4]{a x^{2} - b} \left (a x^{2} - b + x^{4}\right )}\, dx \]
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\[ \int \frac {-2 b+a x^2}{\sqrt [4]{-b+a x^2} \left (-b+a x^2+x^4\right )} \, dx=\int { \frac {a x^{2} - 2 \, b}{{\left (x^{4} + a x^{2} - b\right )} {\left (a x^{2} - b\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {-2 b+a x^2}{\sqrt [4]{-b+a x^2} \left (-b+a x^2+x^4\right )} \, dx=\int { \frac {a x^{2} - 2 \, b}{{\left (x^{4} + a x^{2} - b\right )} {\left (a x^{2} - b\right )}^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {-2 b+a x^2}{\sqrt [4]{-b+a x^2} \left (-b+a x^2+x^4\right )} \, dx=\int -\frac {2\,b-a\,x^2}{{\left (a\,x^2-b\right )}^{1/4}\,\left (x^4+a\,x^2-b\right )} \,d x \]
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