Integrand size = 41, antiderivative size = 346 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b-a x^2+x^4} \, dx=\frac {1}{2} a x \sqrt [4]{-b x^2+a x^4}+\frac {1}{4} \left (-4 a^{9/4}+\sqrt [4]{a} b\right ) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )+\frac {1}{4} \left (4 a^{9/4}-\sqrt [4]{a} b\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )+\frac {1}{2} \text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 a^4 \log (x)+a^2 b \log (x)+2 a^4 \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )-a^2 b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )+a^3 \log (x) \text {$\#$1}^4+b \log (x) \text {$\#$1}^4-a b \log (x) \text {$\#$1}^4-a^3 \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+a b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{3 a \text {$\#$1}^3-2 \text {$\#$1}^7}\&\right ] \]
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Time = 1.17 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.17, number of steps used = 17, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.268, Rules used = {2081, 6860, 285, 335, 338, 304, 209, 212, 477, 525, 524} \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b-a x^2+x^4} \, dx=-\frac {2 x \left (a^3-a^2 \sqrt {a^2+4 b}+2 a b-2 b\right ) \sqrt [4]{a x^4-b x^2} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {2 x^2}{a-\sqrt {a^2+4 b}},\frac {a x^2}{b}\right )}{3 \left (-a \sqrt {a^2+4 b}+a^2+4 b\right ) \sqrt [4]{1-\frac {a x^2}{b}}}-\frac {2 x \left (a^2+\frac {a^3+2 a b-2 b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{a x^4-b x^2} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {2 x^2}{a+\sqrt {a^2+4 b}},\frac {a x^2}{b}\right )}{3 \left (\sqrt {a^2+4 b}+a\right ) \sqrt [4]{1-\frac {a x^2}{b}}}+\frac {\sqrt [4]{a} b \sqrt [4]{a x^4-b x^2} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \sqrt {x} \sqrt [4]{a x^2-b}}-\frac {\sqrt [4]{a} b \sqrt [4]{a x^4-b x^2} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \sqrt {x} \sqrt [4]{a x^2-b}}+\frac {1}{2} a x \sqrt [4]{a x^4-b x^2} \]
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Rule 209
Rule 212
Rule 285
Rule 304
Rule 335
Rule 338
Rule 477
Rule 524
Rule 525
Rule 2081
Rule 6860
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\right )}{-b-a x^2+x^4} \, dx}{\sqrt {x} \sqrt [4]{-b+a x^2}} \\ & = \frac {\sqrt [4]{-b x^2+a x^4} \int \left (a \sqrt {x} \sqrt [4]{-b+a x^2}-\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \left (b-a b-a^2 x^2\right )}{-b-a x^2+x^4}\right ) \, dx}{\sqrt {x} \sqrt [4]{-b+a x^2}} \\ & = -\frac {\sqrt [4]{-b x^2+a x^4} \int \frac {\sqrt {x} \sqrt [4]{-b+a x^2} \left (b-a b-a^2 x^2\right )}{-b-a x^2+x^4} \, dx}{\sqrt {x} \sqrt [4]{-b+a x^2}}+\frac {\left (a \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 {1}{2} a x \sqrt [4]{-b x^2+a x^4}-\frac {\sqrt [4]{-b x^2+a x^4} \int \left (\frac {\left (-a^2+\frac {-a^3+2 b-2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}}{-a-\sqrt {a^2+4 b}+2 x^2}+\frac {\left (-a^2-\frac {-a^3+2 b-2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}}{-a+\sqrt {a^2+4 b}+2 x^2}\right ) \, dx}{\sqrt {x} \sqrt [4]{-b+a x^2}}-\frac {\left (a b \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 {1}{2} a x \sqrt [4]{-b x^2+a x^4}-\frac {\left (a b \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 (\left (-a^2-\frac {a^3-2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{-b x^2+a x^4}\right ) \int \frac {\sqrt {x} \sqrt [4]{-b+a x^2}}{-a-\sqrt {a^2+4 b}+2 x^2} \, dx}{\sqrt {x} \sqrt [4]{-b+a x^2}}-\frac {\left (\left (-a^2+\frac {a^3-2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{-b x^2+a x^4}\right ) \int \frac {\sqrt {x} \sqrt [4]{-b+a x^2}}{-a+\sqrt {a^2+4 b}+2 x^2} \, dx}{\sqrt {x} \sqrt [4]{-b+a x^2}} \\ & = \frac {1}{2} a x \sqrt [4]{-b x^2+a x^4}-\frac {\left (a b \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 (2 \left (-a^2-\frac {a^3-2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{-b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{-b+a x^4}}{-a-\sqrt {a^2+4 b}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{-b+a x^2}}-\frac {\left (2 \left (-a^2+\frac {a^3-2 b+2 a b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{-b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{-b+a x^4}}{-a+\sqrt {a^2+4 b}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{-b+a x^2}} \\ & = \frac {1}{2} a x \sqrt [4]{-b x^2+a x^4}-\frac {\left (\sqrt {a} b \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 {x} \sqrt [4]{-b+a x^2}}+\frac {\left (\sqrt {a} b \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 {x} \sqrt [4]{-b+a x^2}}-\frac {\left (2 \left (-a^2-\frac {a^3-2 b+2 a b}{\sqrt {a^2+4 b}}\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}}}{-a-\sqrt {a^2+4 b}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1-\frac {a x^2}{b}}}-\frac {\left (2 \left (-a^2+\frac {a^3-2 b+2 a b}{\sqrt {a^2+4 b}}\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}}}{-a+\sqrt {a^2+4 b}+2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1-\frac {a x^2}{b}}} \\ & = \frac {1}{2} a x \sqrt [4]{-b x^2+a x^4}-\frac {2 \left (a^2-\frac {a^3-2 b+2 a b}{\sqrt {a^2+4 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 {2 x^2}{a-\sqrt {a^2+4 b}},\frac {a x^2}{b}\right )}{3 \left (a-\sqrt {a^2+4 b}\right ) \sqrt [4]{1-\frac {a x^2}{b}}}-\frac {2 \left (a^2+\frac {a^3-2 b+2 a b}{\sqrt {a^2+4 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 {2 x^2}{a+\sqrt {a^2+4 b}},\frac {a x^2}{b}\right )}{3 \left (a+\sqrt {a^2+4 b}\right ) \sqrt [4]{1-\frac {a x^2}{b}}}+\frac {\sqrt [4]{a} b \sqrt [4]{-b x^2+a x^4} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \sqrt {x} \sqrt [4]{-b+a x^2}}-\frac {\sqrt [4]{a} b \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 \sqrt {x} \sqrt [4]{-b+a x^2}} \\ \end{align*}
Time = 1.08 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.10 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b-a x^2+x^4} \, dx=\frac {\sqrt [4]{-b x^2+a x^4} \left (2 a x^{3/2} \sqrt [4]{-b+a x^2}+\sqrt [4]{a} \left (-4 a^2+b\right ) \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )+\sqrt [4]{a} \left (4 a^2-b\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )-\text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {2 a^4 \log (x)-a^2 b \log (x)-4 a^4 \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right )+2 a^2 b \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right )-a^3 \log (x) \text {$\#$1}^4-b \log (x) \text {$\#$1}^4+a b \log (x) \text {$\#$1}^4+2 a^3 \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4+2 b \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4-2 a b \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4}{3 a \text {$\#$1}^3-2 \text {$\#$1}^7}\&\right ]\right )}{4 \sqrt {x} \sqrt [4]{-b+a x^2}} \]
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Time = 0.41 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.62
method | result | size |
pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-3 a \,\textit {\_Z}^{4}+2 a^{2}-b \right )}{\sum }\frac {\left (\left (-a^{3}+a b -b \right ) \textit {\_R}^{4}+2 a^{4}-a^{2} b \right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{x}\right )}{-2 \textit {\_R}^{7}+3 \textit {\_R}^{3} a}\right )}{2}+\frac {\left (-a^{\frac {1}{4}} b +4 a^{\frac {9}{4}}\right ) \ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}\right )}{8}+\frac {\left (-2 a^{\frac {1}{4}} b +8 a^{\frac {9}{4}}\right ) \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{8}+\frac {x a \left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{2}\) | \(215\) |
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Timed out. \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b-a x^2+x^4} \, dx=\text {Timed out} \]
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Not integrable
Time = 12.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.09 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b-a x^2+x^4} \, dx=\int \frac {\sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (a x^{4} - b\right )}{- a x^{2} - b + x^{4}}\, dx \]
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Not integrable
Time = 0.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.12 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b-a x^2+x^4} \, dx=\int { \frac {{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )}}{x^{4} - a x^{2} - b} \,d x } \]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 44.18 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.73 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b-a x^2+x^4} \, dx=\frac {1}{2} \, {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}} a x^{2} + \frac {1}{8} \, \sqrt {2} {\left (4 \, \left (-a\right )^{\frac {1}{4}} a^{2} - \left (-a\right )^{\frac {1}{4}} b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + \frac {1}{8} \, \sqrt {2} {\left (4 \, \left (-a\right )^{\frac {1}{4}} a^{2} - \left (-a\right )^{\frac {1}{4}} b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + \frac {1}{16} \, \sqrt {2} {\left (4 \, \left (-a\right )^{\frac {1}{4}} a^{2} - \left (-a\right )^{\frac {1}{4}} b\right )} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x^{2}}}\right ) - \frac {1}{16} \, \sqrt {2} {\left (4 \, \left (-a\right )^{\frac {1}{4}} a^{2} - \left (-a\right )^{\frac {1}{4}} b\right )} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x^{2}}}\right ) \]
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Not integrable
Time = 7.13 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.11 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{-b x^2+a x^4}}{-b-a x^2+x^4} \, dx=\int \frac {\left (b-a\,x^4\right )\,{\left (a\,x^4-b\,x^2\right )}^{1/4}}{-x^4+a\,x^2+b} \,d x \]
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