Integrand size = 37, antiderivative size = 218 \[ \int \frac {\left (-4 b+a x^4\right ) \sqrt [4]{-b+a x^4}}{x^6 \left (-8 b+a x^8\right )} \, dx=\frac {\sqrt [4]{-b+a x^4} \left (-4 b+9 a x^4\right )}{40 b x^5}+\frac {a \text {RootSum}\left [8 a^2-a b-16 a \text {$\#$1}^4+8 \text {$\#$1}^8\&,\frac {-8 a^2 \log (x)+a b \log (x)+8 a^2 \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-a b \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+8 a \log (x) \text {$\#$1}^4+4 b \log (x) \text {$\#$1}^4-8 a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-4 b \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-\text {$\#$1}^7}\&\right ]}{512 b} \]
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Leaf count is larger than twice the leaf count of optimal. \(928\) vs. \(2(218)=436\).
Time = 2.33 (sec) , antiderivative size = 928, normalized size of antiderivative = 4.26, number of steps used = 43, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {6857, 270, 283, 338, 304, 209, 212, 1543, 525, 524, 1533, 508} \[ \int \frac {\left (-4 b+a x^4\right ) \sqrt [4]{-b+a x^4}}{x^6 \left (-8 b+a x^8\right )} \, dx=\frac {a \sqrt [4]{a x^4-b} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right ) x^3}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {a \sqrt [4]{a x^4-b} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right ) x^3}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {a^{9/8} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}-\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{32\ 2^{3/8} \left (2 \sqrt {2} \sqrt {a}-\sqrt {b}\right )^{3/4} \sqrt {b}}+\frac {a^{13/8} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}-\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{8\ 2^{7/8} \left (2 \sqrt {2} \sqrt {a}-\sqrt {b}\right )^{3/4} b}+\frac {a^{9/8} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}+\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{32\ 2^{3/8} \left (2 \sqrt {2} \sqrt {a}+\sqrt {b}\right )^{3/4} \sqrt {b}}+\frac {a^{13/8} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}+\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{8\ 2^{7/8} \left (2 \sqrt {2} \sqrt {a}+\sqrt {b}\right )^{3/4} b}+\frac {a^{9/8} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}-\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{32\ 2^{3/8} \left (2 \sqrt {2} \sqrt {a}-\sqrt {b}\right )^{3/4} \sqrt {b}}-\frac {a^{13/8} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}-\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{8\ 2^{7/8} \left (2 \sqrt {2} \sqrt {a}-\sqrt {b}\right )^{3/4} b}-\frac {a^{9/8} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}+\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{32\ 2^{3/8} \left (2 \sqrt {2} \sqrt {a}+\sqrt {b}\right )^{3/4} \sqrt {b}}-\frac {a^{13/8} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}+\sqrt {b}} x}{2^{3/8} \sqrt [4]{a x^4-b}}\right )}{8\ 2^{7/8} \left (2 \sqrt {2} \sqrt {a}+\sqrt {b}\right )^{3/4} b}+\frac {a \sqrt [4]{a x^4-b}}{8 b x}+\frac {\left (a x^4-b\right )^{5/4}}{10 b x^5} \]
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Rule 209
Rule 212
Rule 270
Rule 283
Rule 304
Rule 338
Rule 508
Rule 524
Rule 525
Rule 1533
Rule 1543
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt [4]{-b+a x^4}}{2 x^6}-\frac {a \sqrt [4]{-b+a x^4}}{8 b x^2}-\frac {a x^2 \left (-4 b+a x^4\right ) \sqrt [4]{-b+a x^4}}{8 b \left (8 b-a x^8\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {\sqrt [4]{-b+a x^4}}{x^6} \, dx-\frac {a \int \frac {\sqrt [4]{-b+a x^4}}{x^2} \, dx}{8 b}-\frac {a \int \frac {x^2 \left (-4 b+a x^4\right ) \sqrt [4]{-b+a x^4}}{8 b-a x^8} \, dx}{8 b} \\ & = \frac {a \sqrt [4]{-b+a x^4}}{8 b x}+\frac {\left (-b+a x^4\right )^{5/4}}{10 b x^5}-\frac {a \int \left (-\frac {4 b x^2 \sqrt [4]{-b+a x^4}}{8 b-a x^8}-\frac {a x^6 \sqrt [4]{-b+a x^4}}{-8 b+a x^8}\right ) \, dx}{8 b}-\frac {a^2 \int \frac {x^2}{\left (-b+a x^4\right )^{3/4}} \, dx}{8 b} \\ & = \frac {a \sqrt [4]{-b+a x^4}}{8 b x}+\frac {\left (-b+a x^4\right )^{5/4}}{10 b x^5}+\frac {1}{2} a \int \frac {x^2 \sqrt [4]{-b+a x^4}}{8 b-a x^8} \, dx+\frac {a^2 \int \frac {x^6 \sqrt [4]{-b+a x^4}}{-8 b+a x^8} \, dx}{8 b}-\frac {a^2 \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{8 b} \\ & = \frac {a \sqrt [4]{-b+a x^4}}{8 b x}+\frac {\left (-b+a x^4\right )^{5/4}}{10 b x^5}+\frac {1}{2} a \int \left (\frac {\sqrt {a} x^2 \sqrt [4]{-b+a x^4}}{4 \sqrt {2} \sqrt {b} \left (2 \sqrt {2} \sqrt {a} \sqrt {b}-a x^4\right )}+\frac {\sqrt {a} x^2 \sqrt [4]{-b+a x^4}}{4 \sqrt {2} \sqrt {b} \left (2 \sqrt {2} \sqrt {a} \sqrt {b}+a x^4\right )}\right ) \, dx-\frac {a \int \frac {x^2 \left (-8 a b+a b x^4\right )}{\left (-b+a x^4\right )^{3/4} \left (-8 b+a x^8\right )} \, dx}{8 b}-\frac {a^{3/2} \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{16 b}+\frac {a^{3/2} \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{16 b}+\frac {a^2 \int \frac {x^2}{\left (-b+a x^4\right )^{3/4}} \, dx}{8 b} \\ & = \frac {a \sqrt [4]{-b+a x^4}}{8 b x}+\frac {\left (-b+a x^4\right )^{5/4}}{10 b x^5}+\frac {a^{5/4} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{16 b}-\frac {a^{5/4} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{16 b}-\frac {a \int \left (-\frac {8 a b x^2}{\left (-b+a x^4\right )^{3/4} \left (-8 b+a x^8\right )}+\frac {a b x^6}{\left (-b+a x^4\right )^{3/4} \left (-8 b+a x^8\right )}\right ) \, dx}{8 b}+\frac {a^2 \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{8 b}+\frac {a^{3/2} \int \frac {x^2 \sqrt [4]{-b+a x^4}}{2 \sqrt {2} \sqrt {a} \sqrt {b}-a x^4} \, dx}{8 \sqrt {2} \sqrt {b}}+\frac {a^{3/2} \int \frac {x^2 \sqrt [4]{-b+a x^4}}{2 \sqrt {2} \sqrt {a} \sqrt {b}+a x^4} \, dx}{8 \sqrt {2} \sqrt {b}} \\ & = \frac {a \sqrt [4]{-b+a x^4}}{8 b x}+\frac {\left (-b+a x^4\right )^{5/4}}{10 b x^5}+\frac {a^{5/4} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{16 b}-\frac {a^{5/4} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{16 b}-\frac {1}{8} a^2 \int \frac {x^6}{\left (-b+a x^4\right )^{3/4} \left (-8 b+a x^8\right )} \, dx+a^2 \int \frac {x^2}{\left (-b+a x^4\right )^{3/4} \left (-8 b+a x^8\right )} \, dx+\frac {a^{3/2} \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{16 b}-\frac {a^{3/2} \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{16 b}+\frac {\left (a^{3/2} \sqrt [4]{-b+a x^4}\right ) \int \frac {x^2 \sqrt [4]{1-\frac {a x^4}{b}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}-a x^4} \, dx}{8 \sqrt {2} \sqrt {b} \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {\left (a^{3/2} \sqrt [4]{-b+a x^4}\right ) \int \frac {x^2 \sqrt [4]{1-\frac {a x^4}{b}}}{2 \sqrt {2} \sqrt {a} \sqrt {b}+a x^4} \, dx}{8 \sqrt {2} \sqrt {b} \sqrt [4]{1-\frac {a x^4}{b}}} \\ & = \frac {a \sqrt [4]{-b+a x^4}}{8 b x}+\frac {\left (-b+a x^4\right )^{5/4}}{10 b x^5}+\frac {a x^3 \sqrt [4]{-b+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right )}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {a x^3 \sqrt [4]{-b+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right )}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {1}{8} a^2 \int \left (\frac {x^2}{2 \left (-2 \sqrt {2} \sqrt {a} \sqrt {b}+a x^4\right ) \left (-b+a x^4\right )^{3/4}}+\frac {x^2}{2 \left (2 \sqrt {2} \sqrt {a} \sqrt {b}+a x^4\right ) \left (-b+a x^4\right )^{3/4}}\right ) \, dx+a^2 \int \left (-\frac {\sqrt {a} x^2}{4 \sqrt {2} \sqrt {b} \left (2 \sqrt {2} \sqrt {a} \sqrt {b}-a x^4\right ) \left (-b+a x^4\right )^{3/4}}-\frac {\sqrt {a} x^2}{4 \sqrt {2} \sqrt {b} \left (2 \sqrt {2} \sqrt {a} \sqrt {b}+a x^4\right ) \left (-b+a x^4\right )^{3/4}}\right ) \, dx \\ & = \frac {a \sqrt [4]{-b+a x^4}}{8 b x}+\frac {\left (-b+a x^4\right )^{5/4}}{10 b x^5}+\frac {a x^3 \sqrt [4]{-b+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right )}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {a x^3 \sqrt [4]{-b+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right )}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {1}{16} a^2 \int \frac {x^2}{\left (-2 \sqrt {2} \sqrt {a} \sqrt {b}+a x^4\right ) \left (-b+a x^4\right )^{3/4}} \, dx-\frac {1}{16} a^2 \int \frac {x^2}{\left (2 \sqrt {2} \sqrt {a} \sqrt {b}+a x^4\right ) \left (-b+a x^4\right )^{3/4}} \, dx-\frac {a^{5/2} \int \frac {x^2}{\left (2 \sqrt {2} \sqrt {a} \sqrt {b}-a x^4\right ) \left (-b+a x^4\right )^{3/4}} \, dx}{4 \sqrt {2} \sqrt {b}}-\frac {a^{5/2} \int \frac {x^2}{\left (2 \sqrt {2} \sqrt {a} \sqrt {b}+a x^4\right ) \left (-b+a x^4\right )^{3/4}} \, dx}{4 \sqrt {2} \sqrt {b}} \\ & = \frac {a \sqrt [4]{-b+a x^4}}{8 b x}+\frac {\left (-b+a x^4\right )^{5/4}}{10 b x^5}+\frac {a x^3 \sqrt [4]{-b+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right )}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {a x^3 \sqrt [4]{-b+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right )}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {1}{16} a^2 \text {Subst}\left (\int \frac {x^2}{-2 \sqrt {2} \sqrt {a} \sqrt {b}-\left (-2 \sqrt {2} a^{3/2} \sqrt {b}+a b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )-\frac {1}{16} a^2 \text {Subst}\left (\int \frac {x^2}{2 \sqrt {2} \sqrt {a} \sqrt {b}-\left (2 \sqrt {2} a^{3/2} \sqrt {b}+a b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )-\frac {a^{5/2} \text {Subst}\left (\int \frac {x^2}{2 \sqrt {2} \sqrt {a} \sqrt {b}-\left (2 \sqrt {2} a^{3/2} \sqrt {b}-a b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt {2} \sqrt {b}}-\frac {a^{5/2} \text {Subst}\left (\int \frac {x^2}{2 \sqrt {2} \sqrt {a} \sqrt {b}-\left (2 \sqrt {2} a^{3/2} \sqrt {b}+a b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt {2} \sqrt {b}} \\ & = \frac {a \sqrt [4]{-b+a x^4}}{8 b x}+\frac {\left (-b+a x^4\right )^{5/4}}{10 b x^5}+\frac {a x^3 \sqrt [4]{-b+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right )}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {a x^3 \sqrt [4]{-b+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right )}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {a^{7/4} \text {Subst}\left (\int \frac {1}{2^{3/4}-\sqrt [4]{a} \sqrt {2 \sqrt {2} \sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{8 \sqrt {2} \sqrt {2 \sqrt {2} \sqrt {a}-\sqrt {b}} b}+\frac {a^{7/4} \text {Subst}\left (\int \frac {1}{2^{3/4}+\sqrt [4]{a} \sqrt {2 \sqrt {2} \sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{8 \sqrt {2} \sqrt {2 \sqrt {2} \sqrt {a}-\sqrt {b}} b}-\frac {a^{7/4} \text {Subst}\left (\int \frac {1}{2^{3/4}-\sqrt [4]{a} \sqrt {2 \sqrt {2} \sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{8 \sqrt {2} \sqrt {2 \sqrt {2} \sqrt {a}+\sqrt {b}} b}+\frac {a^{7/4} \text {Subst}\left (\int \frac {1}{2^{3/4}+\sqrt [4]{a} \sqrt {2 \sqrt {2} \sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{8 \sqrt {2} \sqrt {2 \sqrt {2} \sqrt {a}+\sqrt {b}} b}+\frac {a^{5/4} \text {Subst}\left (\int \frac {1}{2^{3/4}-\sqrt [4]{a} \sqrt {2 \sqrt {2} \sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{32 \sqrt {2 \sqrt {2} \sqrt {a}-\sqrt {b}} \sqrt {b}}-\frac {a^{5/4} \text {Subst}\left (\int \frac {1}{2^{3/4}+\sqrt [4]{a} \sqrt {2 \sqrt {2} \sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{32 \sqrt {2 \sqrt {2} \sqrt {a}-\sqrt {b}} \sqrt {b}}-\frac {a^{5/4} \text {Subst}\left (\int \frac {1}{2^{3/4}-\sqrt [4]{a} \sqrt {2 \sqrt {2} \sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{32 \sqrt {2 \sqrt {2} \sqrt {a}+\sqrt {b}} \sqrt {b}}+\frac {a^{5/4} \text {Subst}\left (\int \frac {1}{2^{3/4}+\sqrt [4]{a} \sqrt {2 \sqrt {2} \sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{32 \sqrt {2 \sqrt {2} \sqrt {a}+\sqrt {b}} \sqrt {b}} \\ & = \frac {a \sqrt [4]{-b+a x^4}}{8 b x}+\frac {\left (-b+a x^4\right )^{5/4}}{10 b x^5}+\frac {a x^3 \sqrt [4]{-b+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right )}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {a x^3 \sqrt [4]{-b+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},\frac {\sqrt {a} x^4}{2 \sqrt {2} \sqrt {b}},\frac {a x^4}{b}\right )}{96 b \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {a^{13/8} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}-\sqrt {b}} x}{2^{3/8} \sqrt [4]{-b+a x^4}}\right )}{8\ 2^{7/8} \left (2 \sqrt {2} \sqrt {a}-\sqrt {b}\right )^{3/4} b}-\frac {a^{9/8} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}-\sqrt {b}} x}{2^{3/8} \sqrt [4]{-b+a x^4}}\right )}{32\ 2^{3/8} \left (2 \sqrt {2} \sqrt {a}-\sqrt {b}\right )^{3/4} \sqrt {b}}+\frac {a^{13/8} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}+\sqrt {b}} x}{2^{3/8} \sqrt [4]{-b+a x^4}}\right )}{8\ 2^{7/8} \left (2 \sqrt {2} \sqrt {a}+\sqrt {b}\right )^{3/4} b}+\frac {a^{9/8} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}+\sqrt {b}} x}{2^{3/8} \sqrt [4]{-b+a x^4}}\right )}{32\ 2^{3/8} \left (2 \sqrt {2} \sqrt {a}+\sqrt {b}\right )^{3/4} \sqrt {b}}-\frac {a^{13/8} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}-\sqrt {b}} x}{2^{3/8} \sqrt [4]{-b+a x^4}}\right )}{8\ 2^{7/8} \left (2 \sqrt {2} \sqrt {a}-\sqrt {b}\right )^{3/4} b}+\frac {a^{9/8} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}-\sqrt {b}} x}{2^{3/8} \sqrt [4]{-b+a x^4}}\right )}{32\ 2^{3/8} \left (2 \sqrt {2} \sqrt {a}-\sqrt {b}\right )^{3/4} \sqrt {b}}-\frac {a^{13/8} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}+\sqrt {b}} x}{2^{3/8} \sqrt [4]{-b+a x^4}}\right )}{8\ 2^{7/8} \left (2 \sqrt {2} \sqrt {a}+\sqrt {b}\right )^{3/4} b}-\frac {a^{9/8} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{2 \sqrt {2} \sqrt {a}+\sqrt {b}} x}{2^{3/8} \sqrt [4]{-b+a x^4}}\right )}{32\ 2^{3/8} \left (2 \sqrt {2} \sqrt {a}+\sqrt {b}\right )^{3/4} \sqrt {b}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.98 \[ \int \frac {\left (-4 b+a x^4\right ) \sqrt [4]{-b+a x^4}}{x^6 \left (-8 b+a x^8\right )} \, dx=\frac {\frac {64 \sqrt [4]{-b+a x^4} \left (-4 b+9 a x^4\right )}{x^5}+5 a \text {RootSum}\left [8 a^2-a b-16 a \text {$\#$1}^4+8 \text {$\#$1}^8\&,\frac {8 a^2 \log (x)-a b \log (x)-8 a^2 \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+a b \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-8 a \log (x) \text {$\#$1}^4-4 b \log (x) \text {$\#$1}^4+8 a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+4 b \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]}{2560 b} \]
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Time = 0.00 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.58
method | result | size |
pseudoelliptic | \(\frac {5 a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{8}-16 a \,\textit {\_Z}^{4}+8 a^{2}-a b \right )}{\sum }\frac {\left (8 \textit {\_R}^{4} a +4 \textit {\_R}^{4} b -8 a^{2}+a b \right ) \ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (\textit {\_R}^{4}-a \right )}\right ) x^{5}+576 a \,x^{4} \left (a \,x^{4}-b \right )^{\frac {1}{4}}-256 b \left (a \,x^{4}-b \right )^{\frac {1}{4}}}{2560 b \,x^{5}}\) | \(127\) |
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Timed out. \[ \int \frac {\left (-4 b+a x^4\right ) \sqrt [4]{-b+a x^4}}{x^6 \left (-8 b+a x^8\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 75.90 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.14 \[ \int \frac {\left (-4 b+a x^4\right ) \sqrt [4]{-b+a x^4}}{x^6 \left (-8 b+a x^8\right )} \, dx=\int \frac {\left (a x^{4} - 4 b\right ) \sqrt [4]{a x^{4} - b}}{x^{6} \left (a x^{8} - 8 b\right )}\, dx \]
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Not integrable
Time = 0.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.17 \[ \int \frac {\left (-4 b+a x^4\right ) \sqrt [4]{-b+a x^4}}{x^6 \left (-8 b+a x^8\right )} \, dx=\int { \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}} {\left (a x^{4} - 4 \, b\right )}}{{\left (a x^{8} - 8 \, b\right )} x^{6}} \,d x } \]
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Exception generated. \[ \int \frac {\left (-4 b+a x^4\right ) \sqrt [4]{-b+a x^4}}{x^6 \left (-8 b+a x^8\right )} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.18 \[ \int \frac {\left (-4 b+a x^4\right ) \sqrt [4]{-b+a x^4}}{x^6 \left (-8 b+a x^8\right )} \, dx=\int \frac {{\left (a\,x^4-b\right )}^{1/4}\,\left (4\,b-a\,x^4\right )}{x^6\,\left (8\,b-a\,x^8\right )} \,d x \]
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