Integrand size = 24, antiderivative size = 77 \[ \int \frac {x^2 \left (-b+a x^4\right )}{\left (b+a x^4\right )^{3/4}} \, dx=\frac {1}{4} x^3 \sqrt [4]{b+a x^4}+\frac {7 b \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{8 a^{3/4}}-\frac {7 b \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{8 a^{3/4}} \]
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Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {470, 338, 304, 209, 212} \[ \int \frac {x^2 \left (-b+a x^4\right )}{\left (b+a x^4\right )^{3/4}} \, dx=\frac {7 b \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{8 a^{3/4}}-\frac {7 b \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{8 a^{3/4}}+\frac {1}{4} x^3 \sqrt [4]{a x^4+b} \]
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Rule 209
Rule 212
Rule 304
Rule 338
Rule 470
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^3 \sqrt [4]{b+a x^4}-\frac {1}{4} (7 b) \int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx \\ & = \frac {1}{4} x^3 \sqrt [4]{b+a x^4}-\frac {1}{4} (7 b) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right ) \\ & = \frac {1}{4} x^3 \sqrt [4]{b+a x^4}-\frac {(7 b) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{8 \sqrt {a}}+\frac {(7 b) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{8 \sqrt {a}} \\ & = \frac {1}{4} x^3 \sqrt [4]{b+a x^4}+\frac {7 b \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{8 a^{3/4}}-\frac {7 b \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{8 a^{3/4}} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 \left (-b+a x^4\right )}{\left (b+a x^4\right )^{3/4}} \, dx=\frac {1}{4} x^3 \sqrt [4]{b+a x^4}+\frac {7 b \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{8 a^{3/4}}-\frac {7 b \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{8 a^{3/4}} \]
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Time = 1.32 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.10
method | result | size |
pseudoelliptic | \(-\frac {7 \left (-\frac {4 \left (a \,x^{4}+b \right )^{\frac {1}{4}} x^{3} a^{\frac {3}{4}}}{7}+\ln \left (\frac {-a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (a \,x^{4}+b \right )^{\frac {1}{4}}}\right ) b +2 \arctan \left (\frac {\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b \right )}{16 a^{\frac {3}{4}}}\) | \(85\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.39 \[ \int \frac {x^2 \left (-b+a x^4\right )}{\left (b+a x^4\right )^{3/4}} \, dx=\frac {1}{4} \, {\left (a x^{4} + b\right )}^{\frac {1}{4}} x^{3} - \frac {7}{16} \, \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} \log \left (\frac {7 \, {\left (a \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}} b\right )}}{x}\right ) + \frac {7}{16} \, \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} \log \left (-\frac {7 \, {\left (a \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} x - {\left (a x^{4} + b\right )}^{\frac {1}{4}} b\right )}}{x}\right ) + \frac {7}{16} i \, \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} \log \left (-\frac {7 \, {\left (i \, a \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} x - {\left (a x^{4} + b\right )}^{\frac {1}{4}} b\right )}}{x}\right ) - \frac {7}{16} i \, \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} \log \left (-\frac {7 \, {\left (-i \, a \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} x - {\left (a x^{4} + b\right )}^{\frac {1}{4}} b\right )}}{x}\right ) \]
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Result contains complex when optimal does not.
Time = 1.66 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.01 \[ \int \frac {x^2 \left (-b+a x^4\right )}{\left (b+a x^4\right )^{3/4}} \, dx=\frac {a x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {a x^{4} e^{i \pi }}{b}} \right )}}{4 b^{\frac {3}{4}} \Gamma \left (\frac {11}{4}\right )} - \frac {\sqrt [4]{b} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {a x^{4} e^{i \pi }}{b}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (57) = 114\).
Time = 0.27 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.40 \[ \int \frac {x^2 \left (-b+a x^4\right )}{\left (b+a x^4\right )^{3/4}} \, dx=-\frac {1}{4} \, b {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {3}{4}}} - \frac {\log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {3}{4}}}\right )} - \frac {1}{16} \, a {\left (\frac {3 \, {\left (\frac {2 \, b \arctan \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {3}{4}}} - \frac {b \log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {3}{4}}}\right )}}{a} + \frac {4 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}} b}{{\left (a^{2} - \frac {{\left (a x^{4} + b\right )} a}{x^{4}}\right )} x}\right )} \]
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\[ \int \frac {x^2 \left (-b+a x^4\right )}{\left (b+a x^4\right )^{3/4}} \, dx=\int { \frac {{\left (a x^{4} - b\right )} x^{2}}{{\left (a x^{4} + b\right )}^{\frac {3}{4}}} \,d x } \]
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Timed out. \[ \int \frac {x^2 \left (-b+a x^4\right )}{\left (b+a x^4\right )^{3/4}} \, dx=-\int \frac {x^2\,\left (b-a\,x^4\right )}{{\left (a\,x^4+b\right )}^{3/4}} \,d x \]
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