Integrand size = 35, antiderivative size = 155 \[ \int \frac {\sqrt [4]{-b+a x^4} \left (-8 b+a x^8\right )}{x^{10} \left (b+a x^4\right )} \, dx=\frac {\sqrt [4]{-b+a x^4} \left (8 b^2-16 a b x^4+80 a^2 x^8-9 a b x^8\right )}{9 b^2 x^9}+\frac {\left (8 a^{9/4}-a^{5/4} b\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2^{3/4} b^2}-\frac {\left (8 a^{9/4}-a^{5/4} b\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2^{3/4} b^2} \]
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Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.45 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.50, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6857, 277, 270, 283, 338, 304, 209, 212, 525, 524} \[ \int \frac {\sqrt [4]{-b+a x^4} \left (-8 b+a x^8\right )}{x^{10} \left (b+a x^4\right )} \, dx=\frac {a^{5/4} (8 a-b) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 b^2}-\frac {a^{5/4} (8 a-b) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 b^2}+\frac {a^2 x^3 (8 a-b) \sqrt [4]{a x^4-b} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {a x^4}{b},\frac {a x^4}{b}\right )}{3 b^3 \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {a (8 a-b) \sqrt [4]{a x^4-b}}{b^2 x}+\frac {8 a \left (a x^4-b\right )^{5/4}}{9 b^2 x^5}-\frac {8 \left (a x^4-b\right )^{5/4}}{9 b x^9} \]
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Rule 209
Rule 212
Rule 270
Rule 277
Rule 283
Rule 304
Rule 338
Rule 524
Rule 525
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {8 \sqrt [4]{-b+a x^4}}{x^{10}}+\frac {8 a \sqrt [4]{-b+a x^4}}{b x^6}-\frac {a (8 a-b) \sqrt [4]{-b+a x^4}}{b^2 x^2}+\frac {a^2 (8 a-b) x^2 \sqrt [4]{-b+a x^4}}{b^2 \left (b+a x^4\right )}\right ) \, dx \\ & = -\left (8 \int \frac {\sqrt [4]{-b+a x^4}}{x^{10}} \, dx\right )-\frac {(a (8 a-b)) \int \frac {\sqrt [4]{-b+a x^4}}{x^2} \, dx}{b^2}+\frac {\left (a^2 (8 a-b)\right ) \int \frac {x^2 \sqrt [4]{-b+a x^4}}{b+a x^4} \, dx}{b^2}+\frac {(8 a) \int \frac {\sqrt [4]{-b+a x^4}}{x^6} \, dx}{b} \\ & = \frac {a (8 a-b) \sqrt [4]{-b+a x^4}}{b^2 x}-\frac {8 \left (-b+a x^4\right )^{5/4}}{9 b x^9}+\frac {8 a \left (-b+a x^4\right )^{5/4}}{5 b^2 x^5}-\frac {\left (a^2 (8 a-b)\right ) \int \frac {x^2}{\left (-b+a x^4\right )^{3/4}} \, dx}{b^2}-\frac {(32 a) \int \frac {\sqrt [4]{-b+a x^4}}{x^6} \, dx}{9 b}+\frac {\left (a^2 (8 a-b) \sqrt [4]{-b+a x^4}\right ) \int \frac {x^2 \sqrt [4]{1-\frac {a x^4}{b}}}{b+a x^4} \, dx}{b^2 \sqrt [4]{1-\frac {a x^4}{b}}} \\ & = \frac {a (8 a-b) \sqrt [4]{-b+a x^4}}{b^2 x}-\frac {8 \left (-b+a x^4\right )^{5/4}}{9 b x^9}+\frac {8 a \left (-b+a x^4\right )^{5/4}}{9 b^2 x^5}+\frac {a^2 (8 a-b) x^3 \sqrt [4]{-b+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {a x^4}{b},\frac {a x^4}{b}\right )}{3 b^3 \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {\left (a^2 (8 a-b)\right ) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{b^2} \\ & = \frac {a (8 a-b) \sqrt [4]{-b+a x^4}}{b^2 x}-\frac {8 \left (-b+a x^4\right )^{5/4}}{9 b x^9}+\frac {8 a \left (-b+a x^4\right )^{5/4}}{9 b^2 x^5}+\frac {a^2 (8 a-b) x^3 \sqrt [4]{-b+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {a x^4}{b},\frac {a x^4}{b}\right )}{3 b^3 \sqrt [4]{1-\frac {a x^4}{b}}}-\frac {\left (a^{3/2} (8 a-b)\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 b^2}+\frac {\left (a^{3/2} (8 a-b)\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 b^2} \\ & = \frac {a (8 a-b) \sqrt [4]{-b+a x^4}}{b^2 x}-\frac {8 \left (-b+a x^4\right )^{5/4}}{9 b x^9}+\frac {8 a \left (-b+a x^4\right )^{5/4}}{9 b^2 x^5}+\frac {a^2 (8 a-b) x^3 \sqrt [4]{-b+a x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-\frac {a x^4}{b},\frac {a x^4}{b}\right )}{3 b^3 \sqrt [4]{1-\frac {a x^4}{b}}}+\frac {a^{5/4} (8 a-b) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 b^2}-\frac {a^{5/4} (8 a-b) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 b^2} \\ \end{align*}
Time = 0.71 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt [4]{-b+a x^4} \left (-8 b+a x^8\right )}{x^{10} \left (b+a x^4\right )} \, dx=\frac {\sqrt [4]{-b+a x^4} \left (8 b^2-16 a b x^4+80 a^2 x^8-9 a b x^8\right )}{9 b^2 x^9}+\frac {a^{5/4} (8 a-b) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2^{3/4} b^2}-\frac {a^{5/4} (8 a-b) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2^{3/4} b^2} \]
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Time = 0.53 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.01
method | result | size |
pseudoelliptic | \(\frac {9 x^{9} 2^{\frac {1}{4}} \left (a^{\frac {5}{4}} b -8 a^{\frac {9}{4}}\right ) \ln \left (\frac {-x 2^{\frac {1}{4}} a^{\frac {1}{4}}-\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{x 2^{\frac {1}{4}} a^{\frac {1}{4}}-\left (a \,x^{4}-b \right )^{\frac {1}{4}}}\right )+18 x^{9} 2^{\frac {1}{4}} \left (a^{\frac {5}{4}} b -8 a^{\frac {9}{4}}\right ) \arctan \left (\frac {2^{\frac {3}{4}} \left (a \,x^{4}-b \right )^{\frac {1}{4}}}{2 a^{\frac {1}{4}} x}\right )+320 \left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (a \left (a -\frac {9 b}{80}\right ) x^{8}-\frac {a b \,x^{4}}{5}+\frac {b^{2}}{10}\right )}{36 b^{2} x^{9}}\) | \(156\) |
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Timed out. \[ \int \frac {\sqrt [4]{-b+a x^4} \left (-8 b+a x^8\right )}{x^{10} \left (b+a x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt [4]{-b+a x^4} \left (-8 b+a x^8\right )}{x^{10} \left (b+a x^4\right )} \, dx=\int \frac {\sqrt [4]{a x^{4} - b} \left (a x^{8} - 8 b\right )}{x^{10} \left (a x^{4} + b\right )}\, dx \]
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\[ \int \frac {\sqrt [4]{-b+a x^4} \left (-8 b+a x^8\right )}{x^{10} \left (b+a x^4\right )} \, dx=\int { \frac {{\left (a x^{8} - 8 \, b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{{\left (a x^{4} + b\right )} x^{10}} \,d x } \]
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\[ \int \frac {\sqrt [4]{-b+a x^4} \left (-8 b+a x^8\right )}{x^{10} \left (b+a x^4\right )} \, dx=\int { \frac {{\left (a x^{8} - 8 \, b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{{\left (a x^{4} + b\right )} x^{10}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt [4]{-b+a x^4} \left (-8 b+a x^8\right )}{x^{10} \left (b+a x^4\right )} \, dx=\int -\frac {{\left (a\,x^4-b\right )}^{1/4}\,\left (8\,b-a\,x^8\right )}{x^{10}\,\left (a\,x^4+b\right )} \,d x \]
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