Integrand size = 15, antiderivative size = 55 \[ \int \frac {1}{x \left (b+a x^5\right )^{3/4}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{b}}\right )}{5 b^{3/4}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{b}}\right )}{5 b^{3/4}} \]
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Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {272, 65, 218, 212, 209} \[ \int \frac {1}{x \left (b+a x^5\right )^{3/4}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{a x^5+b}}{\sqrt [4]{b}}\right )}{5 b^{3/4}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{a x^5+b}}{\sqrt [4]{b}}\right )}{5 b^{3/4}} \]
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Rule 65
Rule 209
Rule 212
Rule 218
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \text {Subst}\left (\int \frac {1}{x (b+a x)^{3/4}} \, dx,x,x^5\right ) \\ & = \frac {4 \text {Subst}\left (\int \frac {1}{-\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{b+a x^5}\right )}{5 a} \\ & = -\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {b}-x^2} \, dx,x,\sqrt [4]{b+a x^5}\right )}{5 \sqrt {b}}-\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {b}+x^2} \, dx,x,\sqrt [4]{b+a x^5}\right )}{5 \sqrt {b}} \\ & = -\frac {2 \arctan \left (\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{b}}\right )}{5 b^{3/4}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{b}}\right )}{5 b^{3/4}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x \left (b+a x^5\right )^{3/4}} \, dx=-\frac {2 \left (\arctan \left (\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{b}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{b}}\right )\right )}{5 b^{3/4}} \]
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Time = 2.18 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00
method | result | size |
pseudoelliptic | \(-\frac {\ln \left (\frac {\left (a \,x^{5}+b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{\left (a \,x^{5}+b \right )^{\frac {1}{4}}-b^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (a \,x^{5}+b \right )^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )}{5 b^{\frac {3}{4}}}\) | \(55\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.89 \[ \int \frac {1}{x \left (b+a x^5\right )^{3/4}} \, dx=-\frac {1}{5} \, \frac {1}{b^{3}}^{\frac {1}{4}} \log \left (b \frac {1}{b^{3}}^{\frac {1}{4}} + {\left (a x^{5} + b\right )}^{\frac {1}{4}}\right ) - \frac {1}{5} i \, \frac {1}{b^{3}}^{\frac {1}{4}} \log \left (i \, b \frac {1}{b^{3}}^{\frac {1}{4}} + {\left (a x^{5} + b\right )}^{\frac {1}{4}}\right ) + \frac {1}{5} i \, \frac {1}{b^{3}}^{\frac {1}{4}} \log \left (-i \, b \frac {1}{b^{3}}^{\frac {1}{4}} + {\left (a x^{5} + b\right )}^{\frac {1}{4}}\right ) + \frac {1}{5} \, \frac {1}{b^{3}}^{\frac {1}{4}} \log \left (-b \frac {1}{b^{3}}^{\frac {1}{4}} + {\left (a x^{5} + b\right )}^{\frac {1}{4}}\right ) \]
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Result contains complex when optimal does not.
Time = 0.63 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x \left (b+a x^5\right )^{3/4}} \, dx=- \frac {\Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{5}}} \right )}}{5 a^{\frac {3}{4}} x^{\frac {15}{4}} \Gamma \left (\frac {7}{4}\right )} \]
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none
Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x \left (b+a x^5\right )^{3/4}} \, dx=-\frac {2 \, \arctan \left (\frac {{\left (a x^{5} + b\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )}{5 \, b^{\frac {3}{4}}} + \frac {\log \left (\frac {{\left (a x^{5} + b\right )}^{\frac {1}{4}} - b^{\frac {1}{4}}}{{\left (a x^{5} + b\right )}^{\frac {1}{4}} + b^{\frac {1}{4}}}\right )}{5 \, b^{\frac {3}{4}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (39) = 78\).
Time = 0.28 (sec) , antiderivative size = 186, normalized size of antiderivative = 3.38 \[ \int \frac {1}{x \left (b+a x^5\right )^{3/4}} \, dx=-\frac {\sqrt {2} \left (-b\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-b\right )^{\frac {1}{4}} + 2 \, {\left (a x^{5} + b\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-b\right )^{\frac {1}{4}}}\right )}{5 \, b} - \frac {\sqrt {2} \left (-b\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-b\right )^{\frac {1}{4}} - 2 \, {\left (a x^{5} + b\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-b\right )^{\frac {1}{4}}}\right )}{5 \, b} - \frac {\sqrt {2} \left (-b\right )^{\frac {1}{4}} \log \left (\sqrt {2} {\left (a x^{5} + b\right )}^{\frac {1}{4}} \left (-b\right )^{\frac {1}{4}} + \sqrt {a x^{5} + b} + \sqrt {-b}\right )}{10 \, b} + \frac {\sqrt {2} \left (-b\right )^{\frac {1}{4}} \log \left (-\sqrt {2} {\left (a x^{5} + b\right )}^{\frac {1}{4}} \left (-b\right )^{\frac {1}{4}} + \sqrt {a x^{5} + b} + \sqrt {-b}\right )}{10 \, b} \]
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Time = 5.99 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x \left (b+a x^5\right )^{3/4}} \, dx=-\frac {2\,\mathrm {atan}\left (\frac {{\left (a\,x^5+b\right )}^{1/4}}{b^{1/4}}\right )}{5\,b^{3/4}}-\frac {2\,\mathrm {atanh}\left (\frac {{\left (a\,x^5+b\right )}^{1/4}}{b^{1/4}}\right )}{5\,b^{3/4}} \]
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