Integrand size = 35, antiderivative size = 174 \[ \int \frac {\left (-b+a x^2\right ) \sqrt [4]{-b x^2+a x^4}}{b+a x^2} \, dx=\frac {1}{2} x \sqrt [4]{-b x^2+a x^4}+\frac {9 b \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{4 a^{3/4}}-\frac {2 \sqrt [4]{2} b \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{a^{3/4}}-\frac {9 b \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{4 a^{3/4}}+\frac {2 \sqrt [4]{2} b \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{a^{3/4}} \]
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Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.15 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.37, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2081, 477, 525, 524} \[ \int \frac {\left (-b+a x^2\right ) \sqrt [4]{-b x^2+a x^4}}{b+a x^2} \, dx=-\frac {2 x \sqrt [4]{a x^4-b x^2} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {5}{4},\frac {7}{4},-\frac {a x^2}{b},\frac {a x^2}{b}\right )}{3 \sqrt [4]{1-\frac {a x^2}{b}}} \]
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Rule 477
Rule 524
Rule 525
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{-b x^2+a x^4} \int \frac {\sqrt {x} \left (-b+a x^2\right )^{5/4}}{b+a x^2} \, dx}{\sqrt {x} \sqrt [4]{-b+a x^2}} \\ & = \frac {\left (2 \sqrt [4]{-b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \left (-b+a x^4\right )^{5/4}}{b+a x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{-b+a x^2}} \\ & = -\frac {\left (2 b \sqrt [4]{-b x^2+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \left (1-\frac {a x^4}{b}\right )^{5/4}}{b+a x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1-\frac {a x^2}{b}}} \\ & = -\frac {2 x \sqrt [4]{-b x^2+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {5}{4},\frac {7}{4},-\frac {a x^2}{b},\frac {a x^2}{b}\right )}{3 \sqrt [4]{1-\frac {a x^2}{b}}} \\ \end{align*}
Time = 0.66 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.16 \[ \int \frac {\left (-b+a x^2\right ) \sqrt [4]{-b x^2+a x^4}}{b+a x^2} \, dx=\frac {\sqrt [4]{-b x^2+a x^4} \left (2 a^{3/4} x^{3/2} \sqrt [4]{-b+a x^2}+9 b \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )-8 \sqrt [4]{2} b \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )-9 b \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )+8 \sqrt [4]{2} b \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )\right )}{4 a^{3/4} \sqrt {x} \sqrt [4]{-b+a x^2}} \]
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Time = 0.49 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.14
method | result | size |
pseudoelliptic | \(\frac {\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}} x \,a^{\frac {3}{4}}+4 \,2^{\frac {1}{4}} \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x \,a^{\frac {1}{4}}}\right ) b +2 \,2^{\frac {1}{4}} \ln \left (\frac {x 2^{\frac {1}{4}} a^{\frac {1}{4}}+\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{-x 2^{\frac {1}{4}} a^{\frac {1}{4}}+\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}\right ) b -\frac {9 b \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{2}-\frac {9 b \ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}\right )}{4}}{2 a^{\frac {3}{4}}}\) | \(199\) |
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Timed out. \[ \int \frac {\left (-b+a x^2\right ) \sqrt [4]{-b x^2+a x^4}}{b+a x^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-b+a x^2\right ) \sqrt [4]{-b x^2+a x^4}}{b+a x^2} \, dx=\int \frac {\sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (a x^{2} - b\right )}{a x^{2} + b}\, dx \]
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\[ \int \frac {\left (-b+a x^2\right ) \sqrt [4]{-b x^2+a x^4}}{b+a x^2} \, dx=\int { \frac {{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{2} - b\right )}}{a x^{2} + b} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (134) = 268\).
Time = 0.31 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.32 \[ \int \frac {\left (-b+a x^2\right ) \sqrt [4]{-b x^2+a x^4}}{b+a x^2} \, dx=\frac {1}{2} \, {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}} x^{2} + \frac {2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} b \arctan \left (\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \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 )}{a} + \frac {2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} b \arctan \left (-\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \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 )}{a} + \frac {2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} b \log \left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a - \frac {b}{x^{2}}}\right )}{2 \, a} - \frac {2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} b \log \left (-2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a - \frac {b}{x^{2}}}\right )}{2 \, a} + \frac {9 \, \sqrt {2} b \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 )}{8 \, \left (-a\right )^{\frac {3}{4}}} + \frac {9 \, \sqrt {2} b \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 )}{8 \, \left (-a\right )^{\frac {3}{4}}} + \frac {9 \, \sqrt {2} b \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 )}{16 \, \left (-a\right )^{\frac {3}{4}}} + \frac {9 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b \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 )}{16 \, a} \]
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Timed out. \[ \int \frac {\left (-b+a x^2\right ) \sqrt [4]{-b x^2+a x^4}}{b+a x^2} \, dx=\int -\frac {\left (b-a\,x^2\right )\,{\left (a\,x^4-b\,x^2\right )}^{1/4}}{a\,x^2+b} \,d x \]
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