Integrand size = 24, antiderivative size = 48 \[ \int \frac {x^3}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {2}{9} \sqrt [4]{-1+3 x^2}-\frac {2}{9} \arctan \left (\sqrt [4]{-1+3 x^2}\right )-\frac {2}{9} \text {arctanh}\left (\sqrt [4]{-1+3 x^2}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {457, 81, 65, 218, 212, 209} \[ \int \frac {x^3}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=-\frac {2}{9} \arctan \left (\sqrt [4]{3 x^2-1}\right )-\frac {2}{9} \text {arctanh}\left (\sqrt [4]{3 x^2-1}\right )+\frac {2}{9} \sqrt [4]{3 x^2-1} \]
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Rule 65
Rule 81
Rule 209
Rule 212
Rule 218
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x}{(-2+3 x) (-1+3 x)^{3/4}} \, dx,x,x^2\right ) \\ & = \frac {2}{9} \sqrt [4]{-1+3 x^2}+\frac {1}{3} \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 {4}{9} \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}{9} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )-\frac {2}{9} \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}{9} \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac {2}{9} \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.92 \[ \int \frac {x^3}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {2}{9} \left (\sqrt [4]{-1+3 x^2}-\arctan \left (\sqrt [4]{-1+3 x^2}\right )-\text {arctanh}\left (\sqrt [4]{-1+3 x^2}\right )\right ) \]
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Time = 4.22 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.10
method | result | size |
pseudoelliptic | \(\frac {2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{9}+\frac {\ln \left (-1+\left (3 x^{2}-1\right )^{\frac {1}{4}}\right )}{9}-\frac {\ln \left (1+\left (3 x^{2}-1\right )^{\frac {1}{4}}\right )}{9}-\frac {2 \arctan \left (\left (3 x^{2}-1\right )^{\frac {1}{4}}\right )}{9}\) | \(53\) |
trager | \(\frac {2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{9}-\frac {\ln \left (-\frac {2 \left (3 x^{2}-1\right )^{\frac {3}{4}}+2 \sqrt {3 x^{2}-1}+3 x^{2}+2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{9}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (3 x^{2}-1\right )^{\frac {1}{4}}-2 \sqrt {3 x^{2}-1}+3 x^{2}}{3 x^{2}-2}\right )}{9}\) | \(137\) |
risch | \(\frac {2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{9}+\frac {\left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-18 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}} x^{4}-27 x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {3}{4}}+12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}} x^{2}+6 \sqrt {27 x^{6}-27 x^{4}+9 x^{2}-1}\, x^{2}+18 x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}}-2 \sqrt {27 x^{6}-27 x^{4}+9 x^{2}-1}-3 x^{2}}{\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )^{2}}\right )}{9}+\frac {\ln \left (\frac {-27 x^{6}+18 \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}} x^{4}-6 \sqrt {27 x^{6}-27 x^{4}+9 x^{2}-1}\, x^{2}+18 x^{4}+2 \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {3}{4}}-12 \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}} x^{2}+2 \sqrt {27 x^{6}-27 x^{4}+9 x^{2}-1}-3 x^{2}+2 \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}}}{\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )^{2}}\right )}{9}\right ) {\left (\left (3 x^{2}-1\right )^{3}\right )}^{\frac {1}{4}}}{\left (3 x^{2}-1\right )^{\frac {3}{4}}}\) | \(411\) |
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Time = 0.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.08 \[ \int \frac {x^3}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {2}{9} \, {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - \frac {2}{9} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{9} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{9} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1\right ) \]
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Time = 4.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.21 \[ \int \frac {x^3}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {2 \sqrt [4]{3 x^{2} - 1}}{9} + \frac {\log {\left (\sqrt [4]{3 x^{2} - 1} - 1 \right )}}{9} - \frac {\log {\left (\sqrt [4]{3 x^{2} - 1} + 1 \right )}}{9} - \frac {2 \operatorname {atan}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{9} \]
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Time = 0.29 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.08 \[ \int \frac {x^3}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {2}{9} \, {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - \frac {2}{9} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{9} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{9} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1\right ) \]
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Time = 0.36 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.10 \[ \int \frac {x^3}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {2}{9} \, {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - \frac {2}{9} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{9} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{9} \, \log \left ({\left | {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
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Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.75 \[ \int \frac {x^3}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx=\frac {2\,{\left (3\,x^2-1\right )}^{1/4}}{9}-\frac {2\,\mathrm {atanh}\left ({\left (3\,x^2-1\right )}^{1/4}\right )}{9}-\frac {2\,\mathrm {atan}\left ({\left (3\,x^2-1\right )}^{1/4}\right )}{9} \]
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