Integrand size = 40, antiderivative size = 238 \[ \int \frac {\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+x^8\right )}{-b+a x^4} \, dx=\frac {\sqrt [4]{b x^2+a x^4} \left (-96 a^3 x+96 a b x-7 b^2 x+4 a b x^3+32 a^2 x^5\right )}{192 a^3}+\frac {\left (32 a^3 b-32 a b^2-7 b^3\right ) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{128 a^{15/4}}+\frac {\left (-32 a^3 b+32 a b^2+7 b^3\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{128 a^{15/4}}+\frac {\left (-2 a^2 b+b^2\right ) \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-a+\text {$\#$1}^4}\&\right ]}{4 a^2} \]
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Leaf count is larger than twice the leaf count of optimal. \(533\) vs. \(2(238)=476\).
Time = 0.75 (sec) , antiderivative size = 533, normalized size of antiderivative = 2.24, number of steps used = 24, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.325, Rules used = {2081, 6857, 285, 335, 338, 304, 209, 212, 327, 1284, 1543, 525, 524} \[ \int \frac {\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+x^8\right )}{-b+a x^4} \, dx=-\frac {7 b^3 \sqrt [4]{a x^4+b x^2} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{128 a^{15/4} \sqrt {x} \sqrt [4]{a x^2+b}}+\frac {7 b^3 \sqrt [4]{a x^4+b x^2} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{128 a^{15/4} \sqrt {x} \sqrt [4]{a x^2+b}}-\frac {7 b^2 x \sqrt [4]{a x^4+b x^2}}{192 a^3}+\frac {x \left (2-\frac {b}{a^2}\right ) \sqrt [4]{a x^4+b x^2} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 \sqrt [4]{\frac {a x^2}{b}+1}}+\frac {x \left (2-\frac {b}{a^2}\right ) \sqrt [4]{a x^4+b x^2} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt {a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 \sqrt [4]{\frac {a x^2}{b}+1}}-\frac {1}{2} x \left (1-\frac {b}{a^2}\right ) \sqrt [4]{a x^4+b x^2}+\frac {b x^3 \sqrt [4]{a x^4+b x^2}}{48 a^2}+\frac {b \left (a^2-b\right ) \sqrt [4]{a x^4+b x^2} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{4 a^{11/4} \sqrt {x} \sqrt [4]{a x^2+b}}-\frac {b \left (a^2-b\right ) \sqrt [4]{a x^4+b x^2} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{4 a^{11/4} \sqrt {x} \sqrt [4]{a x^2+b}}+\frac {x^5 \sqrt [4]{a x^4+b x^2}}{6 a} \]
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Rule 209
Rule 212
Rule 285
Rule 304
Rule 327
Rule 335
Rule 338
Rule 524
Rule 525
Rule 1284
Rule 1543
Rule 2081
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{b x^2+a x^4} \int \frac {\sqrt {x} \sqrt [4]{b+a x^2} \left (-b-a x^4+x^8\right )}{-b+a x^4} \, dx}{\sqrt {x} \sqrt [4]{b+a x^2}} \\ & = \frac {\sqrt [4]{b x^2+a x^4} \int \left (\left (-1+\frac {b}{a^2}\right ) \sqrt {x} \sqrt [4]{b+a x^2}+\frac {x^{9/2} \sqrt [4]{b+a x^2}}{a}+\frac {\left (-2 a^2 b+b^2\right ) \sqrt {x} \sqrt [4]{b+a x^2}}{a^2 \left (-b+a x^4\right )}\right ) \, dx}{\sqrt {x} \sqrt [4]{b+a x^2}} \\ & = \frac {\sqrt [4]{b x^2+a x^4} \int x^{9/2} \sqrt [4]{b+a x^2} \, dx}{a \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (\left (-1+\frac {b}{a^2}\right ) \sqrt [4]{b x^2+a x^4}\right ) \int \sqrt {x} \sqrt [4]{b+a x^2} \, dx}{\sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (\left (-2 a^2 b+b^2\right ) \sqrt [4]{b x^2+a x^4}\right ) \int \frac {\sqrt {x} \sqrt [4]{b+a x^2}}{-b+a x^4} \, dx}{a^2 \sqrt {x} \sqrt [4]{b+a x^2}} \\ & = -\frac {1}{2} \left (1-\frac {b}{a^2}\right ) x \sqrt [4]{b x^2+a x^4}+\frac {x^5 \sqrt [4]{b x^2+a x^4}}{6 a}+\frac {\left (b \sqrt [4]{b x^2+a x^4}\right ) \int \frac {x^{9/2}}{\left (b+a x^2\right )^{3/4}} \, dx}{12 a \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (b \left (-1+\frac {b}{a^2}\right ) \sqrt [4]{b x^2+a x^4}\right ) \int \frac {\sqrt {x}}{\left (b+a x^2\right )^{3/4}} \, dx}{4 \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (2 \left (-2 a^2 b+b^2\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{b+a x^4}}{-b+a x^8} \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {x} \sqrt [4]{b+a x^2}} \\ & = -\frac {1}{2} \left (1-\frac {b}{a^2}\right ) x \sqrt [4]{b x^2+a x^4}+\frac {b x^3 \sqrt [4]{b x^2+a x^4}}{48 a^2}+\frac {x^5 \sqrt [4]{b x^2+a x^4}}{6 a}-\frac {\left (7 b^2 \sqrt [4]{b x^2+a x^4}\right ) \int \frac {x^{5/2}}{\left (b+a x^2\right )^{3/4}} \, dx}{96 a^2 \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (b \left (-1+\frac {b}{a^2}\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (2 \left (-2 a^2 b+b^2\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \left (-\frac {\sqrt {a} x^2 \sqrt [4]{b+a x^4}}{2 \sqrt {b} \left (\sqrt {a} \sqrt {b}-a x^4\right )}-\frac {\sqrt {a} x^2 \sqrt [4]{b+a x^4}}{2 \sqrt {b} \left (\sqrt {a} \sqrt {b}+a x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {x} \sqrt [4]{b+a x^2}} \\ & = -\frac {7 b^2 x \sqrt [4]{b x^2+a x^4}}{192 a^3}-\frac {1}{2} \left (1-\frac {b}{a^2}\right ) x \sqrt [4]{b x^2+a x^4}+\frac {b x^3 \sqrt [4]{b x^2+a x^4}}{48 a^2}+\frac {x^5 \sqrt [4]{b x^2+a x^4}}{6 a}+\frac {\left (7 b^3 \sqrt [4]{b x^2+a x^4}\right ) \int \frac {\sqrt {x}}{\left (b+a x^2\right )^{3/4}} \, dx}{128 a^3 \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (b \left (-1+\frac {b}{a^2}\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (\left (-2 a^2 b+b^2\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{b+a x^4}}{\sqrt {a} \sqrt {b}-a x^4} \, dx,x,\sqrt {x}\right )}{a^{3/2} \sqrt {b} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (\left (-2 a^2 b+b^2\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{b+a x^4}}{\sqrt {a} \sqrt {b}+a x^4} \, dx,x,\sqrt {x}\right )}{a^{3/2} \sqrt {b} \sqrt {x} \sqrt [4]{b+a x^2}} \\ & = -\frac {7 b^2 x \sqrt [4]{b x^2+a x^4}}{192 a^3}-\frac {1}{2} \left (1-\frac {b}{a^2}\right ) x \sqrt [4]{b x^2+a x^4}+\frac {b x^3 \sqrt [4]{b x^2+a x^4}}{48 a^2}+\frac {x^5 \sqrt [4]{b x^2+a x^4}}{6 a}+\frac {\left (7 b^3 \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{64 a^3 \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (b \left (-1+\frac {b}{a^2}\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 \sqrt {a} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (b \left (-1+\frac {b}{a^2}\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 \sqrt {a} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (\left (-2 a^2 b+b^2\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1+\frac {a x^4}{b}}}{\sqrt {a} \sqrt {b}-a x^4} \, dx,x,\sqrt {x}\right )}{a^{3/2} \sqrt {b} \sqrt {x} \sqrt [4]{1+\frac {a x^2}{b}}}-\frac {\left (\left (-2 a^2 b+b^2\right ) \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1+\frac {a x^4}{b}}}{\sqrt {a} \sqrt {b}+a x^4} \, dx,x,\sqrt {x}\right )}{a^{3/2} \sqrt {b} \sqrt {x} \sqrt [4]{1+\frac {a x^2}{b}}} \\ & = -\frac {7 b^2 x \sqrt [4]{b x^2+a x^4}}{192 a^3}-\frac {1}{2} \left (1-\frac {b}{a^2}\right ) x \sqrt [4]{b x^2+a x^4}+\frac {b x^3 \sqrt [4]{b x^2+a x^4}}{48 a^2}+\frac {x^5 \sqrt [4]{b x^2+a x^4}}{6 a}+\frac {\left (2 a^2-b\right ) x \sqrt [4]{b x^2+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 a^2 \sqrt [4]{1+\frac {a x^2}{b}}}+\frac {\left (2 a^2-b\right ) x \sqrt [4]{b x^2+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt {a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 a^2 \sqrt [4]{1+\frac {a x^2}{b}}}+\frac {b \left (1-\frac {b}{a^2}\right ) \sqrt [4]{b x^2+a x^4} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {b \left (1-\frac {b}{a^2}\right ) \sqrt [4]{b x^2+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (7 b^3 \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{64 a^3 \sqrt {x} \sqrt [4]{b+a x^2}} \\ & = -\frac {7 b^2 x \sqrt [4]{b x^2+a x^4}}{192 a^3}-\frac {1}{2} \left (1-\frac {b}{a^2}\right ) x \sqrt [4]{b x^2+a x^4}+\frac {b x^3 \sqrt [4]{b x^2+a x^4}}{48 a^2}+\frac {x^5 \sqrt [4]{b x^2+a x^4}}{6 a}+\frac {\left (2 a^2-b\right ) x \sqrt [4]{b x^2+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 a^2 \sqrt [4]{1+\frac {a x^2}{b}}}+\frac {\left (2 a^2-b\right ) x \sqrt [4]{b x^2+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt {a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 a^2 \sqrt [4]{1+\frac {a x^2}{b}}}+\frac {b \left (1-\frac {b}{a^2}\right ) \sqrt [4]{b x^2+a x^4} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {b \left (1-\frac {b}{a^2}\right ) \sqrt [4]{b x^2+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (7 b^3 \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{128 a^{7/2} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (7 b^3 \sqrt [4]{b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{128 a^{7/2} \sqrt {x} \sqrt [4]{b+a x^2}} \\ & = -\frac {7 b^2 x \sqrt [4]{b x^2+a x^4}}{192 a^3}-\frac {1}{2} \left (1-\frac {b}{a^2}\right ) x \sqrt [4]{b x^2+a x^4}+\frac {b x^3 \sqrt [4]{b x^2+a x^4}}{48 a^2}+\frac {x^5 \sqrt [4]{b x^2+a x^4}}{6 a}+\frac {\left (2 a^2-b\right ) x \sqrt [4]{b x^2+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 a^2 \sqrt [4]{1+\frac {a x^2}{b}}}+\frac {\left (2 a^2-b\right ) x \sqrt [4]{b x^2+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt {a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 a^2 \sqrt [4]{1+\frac {a x^2}{b}}}-\frac {7 b^3 \sqrt [4]{b x^2+a x^4} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{128 a^{15/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {b \left (1-\frac {b}{a^2}\right ) \sqrt [4]{b x^2+a x^4} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {7 b^3 \sqrt [4]{b x^2+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{128 a^{15/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {b \left (1-\frac {b}{a^2}\right ) \sqrt [4]{b x^2+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}} \\ \end{align*}
Time = 1.14 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+x^8\right )}{-b+a x^4} \, dx=\frac {x^{3/2} \left (b+a x^2\right )^{3/4} \left (2 a^{3/4} x^{3/2} \sqrt [4]{b+a x^2} \left (-96 a^3-7 b^2+32 a^2 x^4+4 a b \left (24+x^2\right )\right )-3 b \left (-32 a^3+32 a b+7 b^2\right ) \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )+3 b \left (-32 a^3+32 a b+7 b^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )-96 a^{7/4} \left (2 a^2-b\right ) b \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right ) \text {$\#$1}+\log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}}{-a+\text {$\#$1}^4}\&\right ]\right )}{384 a^{15/4} \left (x^2 \left (b+a x^2\right )\right )^{3/4}} \]
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Time = 0.76 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.95
method | result | size |
pseudoelliptic | \(-\frac {\left (-a^{\frac {7}{4}} b^{2}+2 a^{\frac {15}{4}} b \right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 a \,\textit {\_Z}^{4}+a^{2}-a b \right )}{\sum }\frac {\textit {\_R} \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{4}-a}\right )+\frac {b \left (a^{3}-a b -\frac {7}{32} b^{2}\right ) \ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}\right )}{2}+b \left (a^{3}-a b -\frac {7}{32} b^{2}\right ) \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )-\frac {2 x \left (b \left (\frac {x^{2}}{8}+3\right ) a^{\frac {7}{4}}+a^{\frac {11}{4}} x^{4}-\frac {7 b^{2} a^{\frac {3}{4}}}{32}-3 a^{\frac {15}{4}}\right ) \left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{3}}{4 a^{\frac {15}{4}}}\) | \(226\) |
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Timed out. \[ \int \frac {\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+x^8\right )}{-b+a x^4} \, dx=\text {Timed out} \]
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Not integrable
Time = 53.36 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.13 \[ \int \frac {\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+x^8\right )}{-b+a x^4} \, dx=\int \frac {\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (- a x^{4} - b + x^{8}\right )}{a x^{4} - b}\, dx \]
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Not integrable
Time = 0.21 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.17 \[ \int \frac {\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+x^8\right )}{-b+a x^4} \, dx=\int { \frac {{\left (x^{8} - a x^{4} - b\right )} {\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}}}{a x^{4} - b} \,d x } \]
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Timed out. \[ \int \frac {\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+x^8\right )}{-b+a x^4} \, dx=\text {Timed out} \]
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Not integrable
Time = 6.48 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.16 \[ \int \frac {\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+x^8\right )}{-b+a x^4} \, dx=\int \frac {{\left (a\,x^4+b\,x^2\right )}^{1/4}\,\left (-x^8+a\,x^4+b\right )}{b-a\,x^4} \,d x \]
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