Optimal. Leaf size=78 \[ \frac {2}{891} \left (3 x^2-1\right )^{11/4}+\frac {8}{567} \left (3 x^2-1\right )^{7/4}+\frac {14}{243} \left (3 x^2-1\right )^{3/4}+\frac {8}{81} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-\frac {8}{81} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {446, 88, 63, 298, 203, 206} \begin {gather*} \frac {2}{891} \left (3 x^2-1\right )^{11/4}+\frac {8}{567} \left (3 x^2-1\right )^{7/4}+\frac {14}{243} \left (3 x^2-1\right )^{3/4}+\frac {8}{81} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-\frac {8}{81} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 88
Rule 203
Rule 206
Rule 298
Rule 446
Rubi steps
\begin {align*} \int \frac {x^7}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3}{(-2+3 x) \sqrt [4]{-1+3 x}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {7}{27 \sqrt [4]{-1+3 x}}+\frac {8}{27 (-2+3 x) \sqrt [4]{-1+3 x}}+\frac {4}{27} (-1+3 x)^{3/4}+\frac {1}{27} (-1+3 x)^{7/4}\right ) \, dx,x,x^2\right )\\ &=\frac {14}{243} \left (-1+3 x^2\right )^{3/4}+\frac {8}{567} \left (-1+3 x^2\right )^{7/4}+\frac {2}{891} \left (-1+3 x^2\right )^{11/4}+\frac {4}{27} \operatorname {Subst}\left (\int \frac {1}{(-2+3 x) \sqrt [4]{-1+3 x}} \, dx,x,x^2\right )\\ &=\frac {14}{243} \left (-1+3 x^2\right )^{3/4}+\frac {8}{567} \left (-1+3 x^2\right )^{7/4}+\frac {2}{891} \left (-1+3 x^2\right )^{11/4}+\frac {16}{81} \operatorname {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt [4]{-1+3 x^2}\right )\\ &=\frac {14}{243} \left (-1+3 x^2\right )^{3/4}+\frac {8}{567} \left (-1+3 x^2\right )^{7/4}+\frac {2}{891} \left (-1+3 x^2\right )^{11/4}-\frac {8}{81} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )+\frac {8}{81} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )\\ &=\frac {14}{243} \left (-1+3 x^2\right )^{3/4}+\frac {8}{567} \left (-1+3 x^2\right )^{7/4}+\frac {2}{891} \left (-1+3 x^2\right )^{11/4}+\frac {8}{81} \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac {8}{81} \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 57, normalized size = 0.73 \begin {gather*} \frac {2 \left (924 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-924 \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right )+\left (3 x^2-1\right )^{3/4} \left (189 x^4+270 x^2+428\right )\right )}{18711} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 60, normalized size = 0.77 \begin {gather*} \frac {8}{81} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-\frac {8}{81} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right )+\frac {2 \left (3 x^2-1\right )^{3/4} \left (189 x^4+270 x^2+428\right )}{18711} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 64, normalized size = 0.82 \begin {gather*} \frac {2}{18711} \, {\left (189 \, x^{4} + 270 \, x^{2} + 428\right )} {\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}} + \frac {8}{81} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {4}{81} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {4}{81} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 75, normalized size = 0.96 \begin {gather*} \frac {2}{891} \, {\left (3 \, x^{2} - 1\right )}^{\frac {11}{4}} + \frac {8}{567} \, {\left (3 \, x^{2} - 1\right )}^{\frac {7}{4}} + \frac {14}{243} \, {\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}} + \frac {8}{81} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {4}{81} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {4}{81} \, \log \left ({\left | {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.08, size = 148, normalized size = 1.90 \begin {gather*} \frac {4 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {3 x^{2}+2 \left (3 x^{2}-1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right )-2 \left (3 x^{2}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right )-2 \sqrt {3 x^{2}-1}}{3 x^{2}-2}\right )}{81}+\frac {4 \ln \left (\frac {-3 x^{2}+2 \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \sqrt {3 x^{2}-1}+2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{81}+\frac {2 \left (189 x^{4}+270 x^{2}+428\right ) \left (3 x^{2}-1\right )^{\frac {3}{4}}}{18711} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.86, size = 74, normalized size = 0.95 \begin {gather*} \frac {2}{891} \, {\left (3 \, x^{2} - 1\right )}^{\frac {11}{4}} + \frac {8}{567} \, {\left (3 \, x^{2} - 1\right )}^{\frac {7}{4}} + \frac {14}{243} \, {\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}} + \frac {8}{81} \, \arctan \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}\right ) - \frac {4}{81} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {4}{81} \, \log \left ({\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 62, normalized size = 0.79 \begin {gather*} \frac {8\,\mathrm {atan}\left ({\left (3\,x^2-1\right )}^{1/4}\right )}{81}+\frac {14\,{\left (3\,x^2-1\right )}^{3/4}}{243}+\frac {8\,{\left (3\,x^2-1\right )}^{7/4}}{567}+\frac {2\,{\left (3\,x^2-1\right )}^{11/4}}{891}+\frac {\mathrm {atan}\left ({\left (3\,x^2-1\right )}^{1/4}\,1{}\mathrm {i}\right )\,8{}\mathrm {i}}{81} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 24.08, size = 88, normalized size = 1.13 \begin {gather*} \frac {2 \left (3 x^{2} - 1\right )^{\frac {11}{4}}}{891} + \frac {8 \left (3 x^{2} - 1\right )^{\frac {7}{4}}}{567} + \frac {14 \left (3 x^{2} - 1\right )^{\frac {3}{4}}}{243} + \frac {4 \log {\left (\sqrt [4]{3 x^{2} - 1} - 1 \right )}}{81} - \frac {4 \log {\left (\sqrt [4]{3 x^{2} - 1} + 1 \right )}}{81} + \frac {8 \operatorname {atan}{\left (\sqrt [4]{3 x^{2} - 1} \right )}}{81} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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