Integrand size = 24, antiderivative size = 121 \[ \int \frac {x^5}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {4}{27} \left (2-3 x^2\right )^{3/4}-\frac {2}{189} \left (2-3 x^2\right )^{7/4}+\frac {8}{27} \sqrt [4]{2} \arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\frac {8}{27} \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt {2}+\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {451, 267, 272, 45, 450} \[ \int \frac {x^5}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {8}{27} \sqrt [4]{2} \arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\frac {8}{27} \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt {2-3 x^2}+\sqrt {2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )-\frac {2}{189} \left (2-3 x^2\right )^{7/4}+\frac {4}{27} \left (2-3 x^2\right )^{3/4} \]
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Rule 45
Rule 267
Rule 272
Rule 450
Rule 451
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {4 x}{9 \sqrt [4]{2-3 x^2}}-\frac {x^3}{3 \sqrt [4]{2-3 x^2}}+\frac {16 x}{9 \sqrt [4]{2-3 x^2} \left (4-3 x^2\right )}\right ) \, dx \\ & = -\left (\frac {1}{3} \int \frac {x^3}{\sqrt [4]{2-3 x^2}} \, dx\right )-\frac {4}{9} \int \frac {x}{\sqrt [4]{2-3 x^2}} \, dx+\frac {16}{9} \int \frac {x}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx \\ & = \frac {8}{81} \left (2-3 x^2\right )^{3/4}+\frac {8}{27} \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\frac {8}{27} \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt {2}+\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )-\frac {1}{6} \text {Subst}\left (\int \frac {x}{\sqrt [4]{2-3 x}} \, dx,x,x^2\right ) \\ & = \frac {8}{81} \left (2-3 x^2\right )^{3/4}+\frac {8}{27} \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\frac {8}{27} \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt {2}+\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )-\frac {1}{6} \text {Subst}\left (\int \left (\frac {2}{3 \sqrt [4]{2-3 x}}-\frac {1}{3} (2-3 x)^{3/4}\right ) \, dx,x,x^2\right ) \\ & = \frac {4}{27} \left (2-3 x^2\right )^{3/4}-\frac {2}{189} \left (2-3 x^2\right )^{7/4}+\frac {8}{27} \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\frac {8}{27} \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt {2}+\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right ) \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.85 \[ \int \frac {x^5}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {1}{189} \left (6 \left (2-3 x^2\right )^{3/4} \left (4+x^2\right )+56 \sqrt [4]{2} \arctan \left (\frac {\sqrt {2}-\sqrt {2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+56 \sqrt [4]{2} \text {arctanh}\left (\frac {2 \sqrt [4]{4-6 x^2}}{2+\sqrt {4-6 x^2}}\right )\right ) \]
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Time = 4.94 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.08
method | result | size |
pseudoelliptic | \(\frac {2 x^{2} \left (-3 x^{2}+2\right )^{\frac {3}{4}}}{63}+\frac {8 \left (-3 x^{2}+2\right )^{\frac {3}{4}}}{63}-\frac {4 \ln \left (\frac {-2^{\frac {3}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\sqrt {2}+\sqrt {-3 x^{2}+2}}{2^{\frac {3}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+\sqrt {2}+\sqrt {-3 x^{2}+2}}\right ) 2^{\frac {1}{4}}}{27}-\frac {8 \arctan \left (2^{\frac {1}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}+1\right ) 2^{\frac {1}{4}}}{27}-\frac {8 \arctan \left (-1+2^{\frac {1}{4}} \left (-3 x^{2}+2\right )^{\frac {1}{4}}\right ) 2^{\frac {1}{4}}}{27}\) | \(131\) |
trager | \(\left (\frac {2 x^{2}}{63}+\frac {8}{63}\right ) \left (-3 x^{2}+2\right )^{\frac {3}{4}}+\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{3} \left (-3 x^{2}+2\right )^{\frac {3}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {-3 x^{2}+2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}+6 x^{2}}{3 x^{2}-4}\right )}{27}-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \left (-3 x^{2}+2\right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {-3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}+6 x^{2}}{3 x^{2}-4}\right )}{27}\) | \(206\) |
risch | \(-\frac {2 \left (3 x^{2}-2\right ) \left (x^{2}+4\right )}{63 \left (-3 x^{2}+2\right )^{\frac {1}{4}}}-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{3} \left (-3 x^{2}+2\right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {-3 x^{2}+2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}-6 x^{2}}{3 x^{2}-4}\right )}{27}+\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \left (-3 x^{2}+2\right )^{\frac {3}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2} \sqrt {-3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+8\right )^{2}\right ) \left (-3 x^{2}+2\right )^{\frac {1}{4}}-6 x^{2}}{3 x^{2}-4}\right )}{27}\) | \(210\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.88 \[ \int \frac {x^5}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {2}{63} \, {\left (x^{2} + 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} - \frac {4}{27} \, \left (-8\right )^{\frac {1}{4}} \log \left (\left (-8\right )^{\frac {3}{4}} + 4 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) + \frac {4}{27} i \, \left (-8\right )^{\frac {1}{4}} \log \left (i \, \left (-8\right )^{\frac {3}{4}} + 4 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) - \frac {4}{27} i \, \left (-8\right )^{\frac {1}{4}} \log \left (-i \, \left (-8\right )^{\frac {3}{4}} + 4 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) + \frac {4}{27} \, \left (-8\right )^{\frac {1}{4}} \log \left (-\left (-8\right )^{\frac {3}{4}} + 4 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right ) \]
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\[ \int \frac {x^5}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=- \int \frac {x^{5}}{3 x^{2} \sqrt [4]{2 - 3 x^{2}} - 4 \sqrt [4]{2 - 3 x^{2}}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.16 \[ \int \frac {x^5}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=-\frac {2}{189} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {7}{4}} - \frac {8}{27} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {8}{27} \cdot 2^{\frac {1}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) + \frac {4}{27} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) - \frac {4}{27} \cdot 2^{\frac {1}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {4}{27} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} \]
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Time = 0.29 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.16 \[ \int \frac {x^5}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=-\frac {2}{189} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {7}{4}} - \frac {2}{27} \cdot 8^{\frac {3}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} + 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) - \frac {2}{27} \cdot 8^{\frac {3}{4}} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} - 2 \, {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}\right )}\right ) + \frac {4}{27} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) - \frac {4}{27} \cdot 2^{\frac {1}{4}} \log \left (-2^{\frac {3}{4}} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} + \sqrt {2} + \sqrt {-3 \, x^{2} + 2}\right ) + \frac {4}{27} \, {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}} \]
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Time = 0.17 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.59 \[ \int \frac {x^5}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {4\,{\left (2-3\,x^2\right )}^{3/4}}{27}-\frac {2\,{\left (2-3\,x^2\right )}^{7/4}}{189}+2^{1/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {8}{27}+\frac {8}{27}{}\mathrm {i}\right )+2^{1/4}\,\mathrm {atan}\left (2^{1/4}\,{\left (2-3\,x^2\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {8}{27}-\frac {8}{27}{}\mathrm {i}\right ) \]
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