Integrand size = 24, antiderivative size = 63 \[ \int \frac {x^5}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {2}{9} \sqrt [4]{-1+3 x^2}+\frac {2}{135} \left (-1+3 x^2\right )^{5/4}-\frac {4}{27} \arctan \left (\sqrt [4]{-1+3 x^2}\right )-\frac {4}{27} \text {arctanh}\left (\sqrt [4]{-1+3 x^2}\right ) \]
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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, 218, 212, 209} \[ \int \frac {x^5}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=-\frac {4}{27} \arctan \left (\sqrt [4]{3 x^2-1}\right )-\frac {4}{27} \text {arctanh}\left (\sqrt [4]{3 x^2-1}\right )+\frac {2}{135} \left (3 x^2-1\right )^{5/4}+\frac {2}{9} \sqrt [4]{3 x^2-1} \]
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Rule 65
Rule 90
Rule 209
Rule 212
Rule 218
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^2}{(-2+3 x) (-1+3 x)^{3/4}} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{3 (-1+3 x)^{3/4}}+\frac {4}{9 (-2+3 x) (-1+3 x)^{3/4}}+\frac {1}{9} \sqrt [4]{-1+3 x}\right ) \, dx,x,x^2\right ) \\ & = \frac {2}{9} \sqrt [4]{-1+3 x^2}+\frac {2}{135} \left (-1+3 x^2\right )^{5/4}+\frac {2}{9} \text {Subst}\left (\int \frac {1}{(-2+3 x) (-1+3 x)^{3/4}} \, dx,x,x^2\right ) \\ & = \frac {2}{9} \sqrt [4]{-1+3 x^2}+\frac {2}{135} \left (-1+3 x^2\right )^{5/4}+\frac {8}{27} \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt [4]{-1+3 x^2}\right ) \\ & = \frac {2}{9} \sqrt [4]{-1+3 x^2}+\frac {2}{135} \left (-1+3 x^2\right )^{5/4}-\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}{9} \sqrt [4]{-1+3 x^2}+\frac {2}{135} \left (-1+3 x^2\right )^{5/4}-\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 = 0.05 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int \frac {x^5}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {2}{135} \left (\sqrt [4]{-1+3 x^2} \left (14+3 x^2\right )-10 \arctan \left (\sqrt [4]{-1+3 x^2}\right )-10 \text {arctanh}\left (\sqrt [4]{-1+3 x^2}\right )\right ) \]
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Time = 4.00 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.06
method | result | size |
pseudoelliptic | \(\frac {2 x^{2} \left (3 x^{2}-1\right )^{\frac {1}{4}}}{45}+\frac {28 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{135}+\frac {2 \ln \left (-1+\left (3 x^{2}-1\right )^{\frac {1}{4}}\right )}{27}-\frac {2 \ln \left (1+\left (3 x^{2}-1\right )^{\frac {1}{4}}\right )}{27}-\frac {4 \arctan \left (\left (3 x^{2}-1\right )^{\frac {1}{4}}\right )}{27}\) | \(67\) |
trager | \(\left (\frac {2 x^{2}}{45}+\frac {28}{135}\right ) \left (3 x^{2}-1\right )^{\frac {1}{4}}-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (3 x^{2}-1\right )^{\frac {1}{4}}-2 \sqrt {3 x^{2}-1}+3 x^{2}}{3 x^{2}-2}\right )}{27}+\frac {2 \ln \left (\frac {2 \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \sqrt {3 x^{2}-1}-3 x^{2}+2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{27}\) | \(142\) |
risch | \(\frac {2 \left (3 x^{2}+14\right ) \left (3 x^{2}-1\right )^{\frac {1}{4}}}{135}+\frac {\left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-18 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}} x^{4}+27 x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {3}{4}}+12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}} x^{2}-6 \sqrt {27 x^{6}-27 x^{4}+9 x^{2}-1}\, x^{2}-18 x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}}+2 \sqrt {27 x^{6}-27 x^{4}+9 x^{2}-1}+3 x^{2}}{\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )^{2}}\right )}{27}+\frac {2 \ln \left (\frac {-27 x^{6}+18 \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}} x^{4}-6 \sqrt {27 x^{6}-27 x^{4}+9 x^{2}-1}\, x^{2}+18 x^{4}+2 \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {3}{4}}-12 \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}} x^{2}+2 \sqrt {27 x^{6}-27 x^{4}+9 x^{2}-1}-3 x^{2}+2 \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}}}{\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )^{2}}\right )}{27}\right ) {\left (\left (3 x^{2}-1\right )^{3}\right )}^{\frac {1}{4}}}{\left (3 x^{2}-1\right )^{\frac {3}{4}}}\) | \(419\) |
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Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.94 \[ \int \frac {x^5}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {2}{135} \, {\left (3 \, x^{2} + 14\right )} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - \frac {4}{27} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1\right ) \]
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Time = 5.14 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.19 \[ \int \frac {x^5}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {2 \left (3 x^{2} - 1\right )^{\frac {5}{4}}}{135} + \frac {2 \sqrt [4]{3 x^{2} - 1}}{9} + \frac {2 \log {\left (\sqrt [4]{3 x^{2} - 1} - 1 \right )}}{27} - \frac {2 \log {\left (\sqrt [4]{3 x^{2} - 1} + 1 \right )}}{27} - \frac {4 \operatorname {atan}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{27} \]
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Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {2}{135} \, {\left (3 \, x^{2} - 1\right )}^{\frac {5}{4}} + \frac {2}{9} \, {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - \frac {4}{27} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1\right ) \]
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Time = 0.38 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.02 \[ \int \frac {x^5}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {2}{135} \, {\left (3 \, x^{2} - 1\right )}^{\frac {5}{4}} + \frac {2}{9} \, {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - \frac {4}{27} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {2}{27} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {2}{27} \, \log \left ({\left | {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
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Time = 0.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int \frac {x^5}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {2\,{\left (3\,x^2-1\right )}^{1/4}}{9}-\frac {4\,\mathrm {atan}\left ({\left (3\,x^2-1\right )}^{1/4}\right )}{27}+\frac {2\,{\left (3\,x^2-1\right )}^{5/4}}{135}+\frac {\mathrm {atan}\left ({\left (3\,x^2-1\right )}^{1/4}\,1{}\mathrm {i}\right )\,4{}\mathrm {i}}{27} \]
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