Integrand size = 24, antiderivative size = 86 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{b+a x^4}}{x^2} \, dx=\frac {\sqrt [4]{b+a x^4} \left (4 b+a x^4\right )}{4 x}+\frac {3}{8} \sqrt [4]{a} b \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )-\frac {3}{8} \sqrt [4]{a} b \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {464, 285, 338, 304, 209, 212} \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{b+a x^4}}{x^2} \, dx=\frac {3}{8} \sqrt [4]{a} b \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )-\frac {3}{8} \sqrt [4]{a} b \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )+\frac {\left (a x^4+b\right )^{5/4}}{x}-\frac {3}{4} a x^3 \sqrt [4]{a x^4+b} \]
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Rule 209
Rule 212
Rule 285
Rule 304
Rule 338
Rule 464
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b+a x^4\right )^{5/4}}{x}-(3 a) \int x^2 \sqrt [4]{b+a x^4} \, dx \\ & = -\frac {3}{4} a x^3 \sqrt [4]{b+a x^4}+\frac {\left (b+a x^4\right )^{5/4}}{x}-\frac {1}{4} (3 a b) \int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx \\ & = -\frac {3}{4} a x^3 \sqrt [4]{b+a x^4}+\frac {\left (b+a x^4\right )^{5/4}}{x}-\frac {1}{4} (3 a b) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right ) \\ & = -\frac {3}{4} a x^3 \sqrt [4]{b+a x^4}+\frac {\left (b+a x^4\right )^{5/4}}{x}-\frac {1}{8} \left (3 \sqrt {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 )+\frac {1}{8} \left (3 \sqrt {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 ) \\ & = -\frac {3}{4} a x^3 \sqrt [4]{b+a x^4}+\frac {\left (b+a x^4\right )^{5/4}}{x}+\frac {3}{8} \sqrt [4]{a} b \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )-\frac {3}{8} \sqrt [4]{a} b \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right ) \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{b+a x^4}}{x^2} \, dx=\frac {\sqrt [4]{b+a x^4} \left (4 b+a x^4\right )}{4 x}+\frac {3}{8} \sqrt [4]{a} b \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )-\frac {3}{8} \sqrt [4]{a} b \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right ) \]
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Time = 1.22 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.08
method | result | size |
pseudoelliptic | \(\frac {-3 b x \left (2 \arctan \left (\frac {\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )+\ln \left (\frac {-a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right )\right ) a^{\frac {1}{4}}+4 \left (a \,x^{4}+b \right )^{\frac {1}{4}} \left (a \,x^{4}+4 b \right )}{16 x}\) | \(93\) |
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Timed out. \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{b+a x^4}}{x^2} \, dx=\text {Timed out} \]
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Result contains complex when optimal does not.
Time = 1.54 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{b+a x^4}}{x^2} \, dx=\frac {a \sqrt [4]{b} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {a x^{4} e^{i \pi }}{b}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} - \frac {b^{\frac {5}{4}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {a x^{4} e^{i \pi }}{b}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (66) = 132\).
Time = 0.29 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.21 \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{b+a x^4}}{x^2} \, dx=\frac {1}{16} \, {\left (\frac {2 \, b \arctan \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {3}{4}}} - \frac {b \log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {3}{4}}} - \frac {4 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}} b}{{\left (a - \frac {a x^{4} + b}{x^{4}}\right )} x}\right )} a - \frac {1}{4} \, {\left (2 \, a^{\frac {1}{4}} \arctan \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) - a^{\frac {1}{4}} \log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}}\right ) - \frac {4 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )} b \]
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\[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{b+a x^4}}{x^2} \, dx=\int { \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (-b+a x^4\right ) \sqrt [4]{b+a x^4}}{x^2} \, dx=-\int \frac {{\left (a\,x^4+b\right )}^{1/4}\,\left (b-a\,x^4\right )}{x^2} \,d x \]
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