Integrand size = 16, antiderivative size = 78 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^5} \, dx=-\frac {\sqrt [4]{a-b x^4}}{4 x^4}+\frac {b \arctan \left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}+\frac {b \text {arctanh}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}} \]
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Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {272, 43, 65, 218, 212, 209} \[ \int \frac {\sqrt [4]{a-b x^4}}{x^5} \, dx=\frac {b \arctan \left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}+\frac {b \text {arctanh}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac {\sqrt [4]{a-b x^4}}{4 x^4} \]
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Rule 43
Rule 65
Rule 209
Rule 212
Rule 218
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {\sqrt [4]{a-b x}}{x^2} \, dx,x,x^4\right ) \\ & = -\frac {\sqrt [4]{a-b x^4}}{4 x^4}-\frac {1}{16} b \text {Subst}\left (\int \frac {1}{x (a-b x)^{3/4}} \, dx,x,x^4\right ) \\ & = -\frac {\sqrt [4]{a-b x^4}}{4 x^4}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{\frac {a}{b}-\frac {x^4}{b}} \, dx,x,\sqrt [4]{a-b x^4}\right ) \\ & = -\frac {\sqrt [4]{a-b x^4}}{4 x^4}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a}-x^2} \, dx,x,\sqrt [4]{a-b x^4}\right )}{8 \sqrt {a}}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a}+x^2} \, dx,x,\sqrt [4]{a-b x^4}\right )}{8 \sqrt {a}} \\ & = -\frac {\sqrt [4]{a-b x^4}}{4 x^4}+\frac {b \tan ^{-1}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}+\frac {b \tanh ^{-1}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^5} \, dx=\frac {1}{8} \left (-\frac {2 \sqrt [4]{a-b x^4}}{x^4}+\frac {b \arctan \left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac {b \text {arctanh}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{a^{3/4}}\right ) \]
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Time = 4.49 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.14
method | result | size |
pseudoelliptic | \(\frac {\ln \left (\frac {-\left (-b \,x^{4}+a \right )^{\frac {1}{4}}-a^{\frac {1}{4}}}{-\left (-b \,x^{4}+a \right )^{\frac {1}{4}}+a^{\frac {1}{4}}}\right ) b \,x^{4}+2 \arctan \left (\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right ) b \,x^{4}-4 \left (-b \,x^{4}+a \right )^{\frac {1}{4}} a^{\frac {3}{4}}}{16 x^{4} a^{\frac {3}{4}}}\) | \(89\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.26 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^5} \, dx=\frac {\left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} x^{4} \log \left ({\left (-b x^{4} + a\right )}^{\frac {1}{4}} b + a \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}}\right ) + i \, \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} x^{4} \log \left ({\left (-b x^{4} + a\right )}^{\frac {1}{4}} b + i \, a \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}}\right ) - i \, \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} x^{4} \log \left ({\left (-b x^{4} + a\right )}^{\frac {1}{4}} b - i \, a \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}}\right ) - \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} x^{4} \log \left ({\left (-b x^{4} + a\right )}^{\frac {1}{4}} b - a \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}}\right ) - 4 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{16 \, x^{4}} \]
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Result contains complex when optimal does not.
Time = 0.74 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.54 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^5} \, dx=\frac {\sqrt [4]{b} e^{- \frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {a}{b x^{4}}} \right )}}{4 x^{3} \Gamma \left (\frac {7}{4}\right )} \]
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none
Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^5} \, dx=\frac {b \arctan \left (\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{8 \, a^{\frac {3}{4}}} - \frac {b \log \left (\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}} - a^{\frac {1}{4}}}{{\left (-b x^{4} + a\right )}^{\frac {1}{4}} + a^{\frac {1}{4}}}\right )}{16 \, a^{\frac {3}{4}}} - \frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{4 \, x^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (58) = 116\).
Time = 0.28 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.82 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^5} \, dx=\frac {\frac {2 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a} + \frac {2 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a} + \frac {\sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \log \left (\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {-b x^{4} + a} + \sqrt {-a}\right )}{a} + \frac {\sqrt {2} b^{2} \log \left (-\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {-b x^{4} + a} + \sqrt {-a}\right )}{\left (-a\right )^{\frac {3}{4}}} - \frac {8 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b}{x^{4}}}{32 \, b} \]
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Time = 5.88 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^5} \, dx=\frac {b\,\mathrm {atan}\left (\frac {{\left (a-b\,x^4\right )}^{1/4}}{a^{1/4}}\right )}{8\,a^{3/4}}-\frac {{\left (a-b\,x^4\right )}^{1/4}}{4\,x^4}+\frac {b\,\mathrm {atanh}\left (\frac {{\left (a-b\,x^4\right )}^{1/4}}{a^{1/4}}\right )}{8\,a^{3/4}} \]
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