Integrand size = 38, antiderivative size = 99 \[ \int \frac {-2 b+a x^4}{x^4 \left (b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=-\frac {4 \left (3 b+4 a x^2\right ) \left (-b x^2+a x^4\right )^{3/4}}{21 b^2 x^5}-\frac {3 a \text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{4 b} \]
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Leaf count is larger than twice the leaf count of optimal. \(474\) vs. \(2(99)=198\).
Time = 0.60 (sec) , antiderivative size = 474, normalized size of antiderivative = 4.79, number of steps used = 19, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.289, Rules used = {2081, 6857, 277, 270, 1284, 1543, 1443, 385, 218, 212, 209} \[ \int \frac {-2 b+a x^4}{x^4 \left (b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\frac {3 a \sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt {x} \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}{\sqrt [4]{a x^2-b}}\right )}{2 b \sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt [4]{a x^4-b x^2}}+\frac {3 a \sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt {x} \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}{\sqrt [4]{a x^2-b}}\right )}{2 b \sqrt [4]{\sqrt {-a} \sqrt {b}+a} \sqrt [4]{a x^4-b x^2}}+\frac {3 a \sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}{\sqrt [4]{a x^2-b}}\right )}{2 b \sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt [4]{a x^4-b x^2}}+\frac {3 a \sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}{\sqrt [4]{a x^2-b}}\right )}{2 b \sqrt [4]{\sqrt {-a} \sqrt {b}+a} \sqrt [4]{a x^4-b x^2}}+\frac {16 a \left (b-a x^2\right )}{21 b^2 x \sqrt [4]{a x^4-b x^2}}+\frac {4 \left (b-a x^2\right )}{7 b x^3 \sqrt [4]{a x^4-b x^2}} \]
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Rule 209
Rule 212
Rule 218
Rule 270
Rule 277
Rule 385
Rule 1284
Rule 1443
Rule 1543
Rule 2081
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \int \frac {-2 b+a x^4}{x^{9/2} \sqrt [4]{-b+a x^2} \left (b+a x^4\right )} \, dx}{\sqrt [4]{-b x^2+a x^4}} \\ & = \frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \int \left (\frac {1}{x^{9/2} \sqrt [4]{-b+a x^2}}-\frac {3 b}{x^{9/2} \sqrt [4]{-b+a x^2} \left (b+a x^4\right )}\right ) \, dx}{\sqrt [4]{-b x^2+a x^4}} \\ & = \frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \int \frac {1}{x^{9/2} \sqrt [4]{-b+a x^2}} \, dx}{\sqrt [4]{-b x^2+a x^4}}-\frac {\left (3 b \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \int \frac {1}{x^{9/2} \sqrt [4]{-b+a x^2} \left (b+a x^4\right )} \, dx}{\sqrt [4]{-b x^2+a x^4}} \\ & = -\frac {2 \left (b-a x^2\right )}{7 b x^3 \sqrt [4]{-b x^2+a x^4}}+\frac {\left (4 a \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \int \frac {1}{x^{5/2} \sqrt [4]{-b+a x^2}} \, dx}{7 b \sqrt [4]{-b x^2+a x^4}}-\frac {\left (6 b \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{x^8 \sqrt [4]{-b+a x^4} \left (b+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}} \\ & = -\frac {2 \left (b-a x^2\right )}{7 b x^3 \sqrt [4]{-b x^2+a x^4}}-\frac {8 a \left (b-a x^2\right )}{21 b^2 x \sqrt [4]{-b x^2+a x^4}}-\frac {\left (6 b \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{b x^8 \sqrt [4]{-b+a x^4}}-\frac {a}{b \sqrt [4]{-b+a x^4} \left (b+a x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}} \\ & = -\frac {2 \left (b-a x^2\right )}{7 b x^3 \sqrt [4]{-b x^2+a x^4}}-\frac {8 a \left (b-a x^2\right )}{21 b^2 x \sqrt [4]{-b x^2+a x^4}}-\frac {\left (6 \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{x^8 \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}}+\frac {\left (6 a \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}} \\ & = \frac {4 \left (b-a x^2\right )}{7 b x^3 \sqrt [4]{-b x^2+a x^4}}-\frac {8 a \left (b-a x^2\right )}{21 b^2 x \sqrt [4]{-b x^2+a x^4}}-\frac {\left (24 a \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{x^4 \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{7 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (3 \sqrt {-a} a \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 {b} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (3 \sqrt {-a} a \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 {b} \sqrt [4]{-b x^2+a x^4}} \\ & = \frac {4 \left (b-a x^2\right )}{7 b x^3 \sqrt [4]{-b x^2+a x^4}}+\frac {16 a \left (b-a x^2\right )}{21 b^2 x \sqrt [4]{-b x^2+a x^4}}+\frac {\left (3 \sqrt {-a} a \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 {b} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (3 \sqrt {-a} a \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 {b} \sqrt [4]{-b x^2+a x^4}} \\ & = \frac {4 \left (b-a x^2\right )}{7 b x^3 \sqrt [4]{-b x^2+a x^4}}+\frac {16 a \left (b-a x^2\right )}{21 b^2 x \sqrt [4]{-b x^2+a x^4}}+\frac {\left (3 a \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 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (3 a \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 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (3 a \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 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (3 a \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 b \sqrt [4]{-b x^2+a x^4}} \\ & = \frac {4 \left (b-a x^2\right )}{7 b x^3 \sqrt [4]{-b x^2+a x^4}}+\frac {16 a \left (b-a x^2\right )}{21 b^2 x \sqrt [4]{-b x^2+a x^4}}+\frac {3 a \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 [4]{a-\sqrt {-a} \sqrt {b}} b \sqrt [4]{-b x^2+a x^4}}+\frac {3 a \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 [4]{a+\sqrt {-a} \sqrt {b}} b \sqrt [4]{-b x^2+a x^4}}+\frac {3 a \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 [4]{a-\sqrt {-a} \sqrt {b}} b \sqrt [4]{-b x^2+a x^4}}+\frac {3 a \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 [4]{a+\sqrt {-a} \sqrt {b}} b \sqrt [4]{-b x^2+a x^4}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.31 \[ \int \frac {-2 b+a x^4}{x^4 \left (b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\frac {16 \left (3 b^2+a b x^2-4 a^2 x^4\right )-63 a b x^{7/2} \sqrt [4]{-b+a x^2} \text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{84 b^2 x^3 \sqrt [4]{-b x^2+a x^4}} \]
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Time = 0.00 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.06
method | result | size |
pseudoelliptic | \(\frac {-63 a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 \textit {\_Z}^{4} a +a^{2}+a b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right ) b \,x^{5}-64 a \left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {3}{4}} x^{2}-48 b \left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {3}{4}}}{84 b^{2} x^{5}}\) | \(105\) |
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Timed out. \[ \int \frac {-2 b+a x^4}{x^4 \left (b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\text {Timed out} \]
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Not integrable
Time = 19.81 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.32 \[ \int \frac {-2 b+a x^4}{x^4 \left (b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\int \frac {a x^{4} - 2 b}{x^{4} \sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (a x^{4} + b\right )}\, dx \]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.38 \[ \int \frac {-2 b+a x^4}{x^4 \left (b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\int { \frac {a x^{4} - 2 \, b}{{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} + b\right )} x^{4}} \,d x } \]
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Not integrable
Time = 0.96 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.38 \[ \int \frac {-2 b+a x^4}{x^4 \left (b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\int { \frac {a x^{4} - 2 \, b}{{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} + b\right )} x^{4}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.40 \[ \int \frac {-2 b+a x^4}{x^4 \left (b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx=\int -\frac {2\,b-a\,x^4}{x^4\,\left (a\,x^4+b\right )\,{\left (a\,x^4-b\,x^2\right )}^{1/4}} \,d x \]
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