Integrand size = 24, antiderivative size = 63 \[ \int \frac {x^5}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\frac {2}{27} \left (-1+3 x^2\right )^{3/4}+\frac {2}{189} \left (-1+3 x^2\right )^{7/4}+\frac {4}{27} \arctan \left (\sqrt [4]{-1+3 x^2}\right )-\frac {4}{27} \text {arctanh}\left (\sqrt [4]{-1+3 x^2}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {457, 90, 65, 304, 209, 212} \[ \int \frac {x^5}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\frac {4}{27} \arctan \left (\sqrt [4]{3 x^2-1}\right )-\frac {4}{27} \text {arctanh}\left (\sqrt [4]{3 x^2-1}\right )+\frac {2}{189} \left (3 x^2-1\right )^{7/4}+\frac {2}{27} \left (3 x^2-1\right )^{3/4} \]
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Rule 65
Rule 90
Rule 209
Rule 212
Rule 304
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^2}{(-2+3 x) \sqrt [4]{-1+3 x}} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{3 \sqrt [4]{-1+3 x}}+\frac {4}{9 (-2+3 x) \sqrt [4]{-1+3 x}}+\frac {1}{9} (-1+3 x)^{3/4}\right ) \, dx,x,x^2\right ) \\ & = \frac {2}{27} \left (-1+3 x^2\right )^{3/4}+\frac {2}{189} \left (-1+3 x^2\right )^{7/4}+\frac {2}{9} \text {Subst}\left (\int \frac {1}{(-2+3 x) \sqrt [4]{-1+3 x}} \, dx,x,x^2\right ) \\ & = \frac {2}{27} \left (-1+3 x^2\right )^{3/4}+\frac {2}{189} \left (-1+3 x^2\right )^{7/4}+\frac {8}{27} \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt [4]{-1+3 x^2}\right ) \\ & = \frac {2}{27} \left (-1+3 x^2\right )^{3/4}+\frac {2}{189} \left (-1+3 x^2\right )^{7/4}-\frac {4}{27} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )+\frac {4}{27} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right ) \\ & = \frac {2}{27} \left (-1+3 x^2\right )^{3/4}+\frac {2}{189} \left (-1+3 x^2\right )^{7/4}+\frac {4}{27} \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac {4}{27} \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.84 \[ \int \frac {x^5}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\frac {2}{63} \left (2+x^2\right ) \left (-1+3 x^2\right )^{3/4}+\frac {4}{27} \arctan \left (\sqrt [4]{-1+3 x^2}\right )-\frac {4}{27} \text {arctanh}\left (\sqrt [4]{-1+3 x^2}\right ) \]
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Time = 3.90 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.06
method | result | size |
pseudoelliptic | \(\frac {2 x^{2} \left (3 x^{2}-1\right )^{\frac {3}{4}}}{63}+\frac {4 \left (3 x^{2}-1\right )^{\frac {3}{4}}}{63}+\frac {2 \ln \left (-1+\left (3 x^{2}-1\right )^{\frac {1}{4}}\right )}{27}-\frac {2 \ln \left (1+\left (3 x^{2}-1\right )^{\frac {1}{4}}\right )}{27}+\frac {4 \arctan \left (\left (3 x^{2}-1\right )^{\frac {1}{4}}\right )}{27}\) | \(67\) |
trager | \(\left (\frac {2 x^{2}}{63}+\frac {4}{63}\right ) \left (3 x^{2}-1\right )^{\frac {3}{4}}+\frac {2 \ln \left (\frac {2 \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \sqrt {3 x^{2}-1}-3 x^{2}+2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{27}-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (3 x^{2}-1\right )^{\frac {1}{4}}+2 \sqrt {3 x^{2}-1}-3 x^{2}}{3 x^{2}-2}\right )}{27}\) | \(141\) |
risch | \(\frac {2 \left (x^{2}+2\right ) \left (3 x^{2}-1\right )^{\frac {3}{4}}}{63}-\frac {2 \ln \left (-\frac {2 \left (3 x^{2}-1\right )^{\frac {3}{4}}+2 \sqrt {3 x^{2}-1}+3 x^{2}+2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{27}+\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (3 x^{2}-1\right )^{\frac {1}{4}}-2 \sqrt {3 x^{2}-1}+3 x^{2}}{3 x^{2}-2}\right )}{27}\) | \(142\) |
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Time = 0.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.90 \[ \int \frac {x^5}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\frac {2}{63} \, {\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}} {\left (x^{2} + 2\right )} + \frac {4}{27} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1\right ) \]
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Time = 4.99 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.19 \[ \int \frac {x^5}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\frac {2 \left (3 x^{2} - 1\right )^{\frac {7}{4}}}{189} + \frac {2 \left (3 x^{2} - 1\right )^{\frac {3}{4}}}{27} + \frac {2 \log {\left (\sqrt [4]{3 x^{2} - 1} - 1 \right )}}{27} - \frac {2 \log {\left (\sqrt [4]{3 x^{2} - 1} + 1 \right )}}{27} + \frac {4 \operatorname {atan}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{27} \]
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Time = 0.31 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\frac {2}{189} \, {\left (3 \, x^{2} - 1\right )}^{\frac {7}{4}} + \frac {2}{27} \, {\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}} + \frac {4}{27} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1\right ) \]
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Time = 0.36 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.02 \[ \int \frac {x^5}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\frac {2}{189} \, {\left (3 \, x^{2} - 1\right )}^{\frac {7}{4}} + \frac {2}{27} \, {\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}} + \frac {4}{27} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {2}{27} \, \log \left ({\left | {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
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Time = 5.42 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int \frac {x^5}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx=\frac {4\,\mathrm {atan}\left ({\left (3\,x^2-1\right )}^{1/4}\right )}{27}+\frac {2\,{\left (3\,x^2-1\right )}^{3/4}}{27}+\frac {2\,{\left (3\,x^2-1\right )}^{7/4}}{189}+\frac {\mathrm {atan}\left ({\left (3\,x^2-1\right )}^{1/4}\,1{}\mathrm {i}\right )\,4{}\mathrm {i}}{27} \]
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