Optimal. Leaf size=78 \[ \frac {14}{81} \sqrt [4]{-1+3 x^2}+\frac {8}{405} \left (-1+3 x^2\right )^{5/4}+\frac {2}{729} \left (-1+3 x^2\right )^{9/4}-\frac {8}{81} \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-\frac {8}{81} \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right ) \]
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Rubi [A]
time = 0.04, 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 = {457, 90, 65,
218, 212, 209} \begin {gather*} -\frac {8}{81} \text {ArcTan}\left (\sqrt [4]{3 x^2-1}\right )+\frac {2}{729} \left (3 x^2-1\right )^{9/4}+\frac {8}{405} \left (3 x^2-1\right )^{5/4}+\frac {14}{81} \sqrt [4]{3 x^2-1}-\frac {8}{81} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 90
Rule 209
Rule 212
Rule 218
Rule 457
Rubi steps
\begin {align*} \int \frac {x^7}{\left (-2+3 x^2\right ) \left (-1+3 x^2\right )^{3/4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^3}{(-2+3 x) (-1+3 x)^{3/4}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {7}{27 (-1+3 x)^{3/4}}+\frac {8}{27 (-2+3 x) (-1+3 x)^{3/4}}+\frac {4}{27} \sqrt [4]{-1+3 x}+\frac {1}{27} (-1+3 x)^{5/4}\right ) \, dx,x,x^2\right )\\ &=\frac {14}{81} \sqrt [4]{-1+3 x^2}+\frac {8}{405} \left (-1+3 x^2\right )^{5/4}+\frac {2}{729} \left (-1+3 x^2\right )^{9/4}+\frac {4}{27} \text {Subst}\left (\int \frac {1}{(-2+3 x) (-1+3 x)^{3/4}} \, dx,x,x^2\right )\\ &=\frac {14}{81} \sqrt [4]{-1+3 x^2}+\frac {8}{405} \left (-1+3 x^2\right )^{5/4}+\frac {2}{729} \left (-1+3 x^2\right )^{9/4}+\frac {16}{81} \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt [4]{-1+3 x^2}\right )\\ &=\frac {14}{81} \sqrt [4]{-1+3 x^2}+\frac {8}{405} \left (-1+3 x^2\right )^{5/4}+\frac {2}{729} \left (-1+3 x^2\right )^{9/4}-\frac {8}{81} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )-\frac {8}{81} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{-1+3 x^2}\right )\\ &=\frac {14}{81} \sqrt [4]{-1+3 x^2}+\frac {8}{405} \left (-1+3 x^2\right )^{5/4}+\frac {2}{729} \left (-1+3 x^2\right )^{9/4}-\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 = 0.06, size = 57, normalized size = 0.73 \begin {gather*} \frac {2 \left (\sqrt [4]{-1+3 x^2} \left (284+78 x^2+45 x^4\right )-180 \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )-180 \tanh ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )\right )}{3645} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.65, size = 147, normalized size = 1.88
method | result | size |
trager | \(\left (\frac {2}{81} x^{4}+\frac {52}{1215} x^{2}+\frac {568}{3645}\right ) \left (3 x^{2}-1\right )^{\frac {1}{4}}-\frac {4 \ln \left (-\frac {2 \left (3 x^{2}-1\right )^{\frac {3}{4}}+2 \sqrt {3 x^{2}-1}+3 x^{2}+2 \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3 x^{2}-2}\right )}{81}+\frac {4 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (3 x^{2}-1\right )^{\frac {3}{4}}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (3 x^{2}-1\right )^{\frac {1}{4}}+2 \sqrt {3 x^{2}-1}-3 x^{2}}{3 x^{2}-2}\right )}{81}\) | \(147\) |
risch | \(\frac {2 \left (45 x^{4}+78 x^{2}+284\right ) \left (3 x^{2}-1\right )^{\frac {1}{4}}}{3645}+\frac {\left (\frac {4 \ln \left (\frac {-27 x^{6}+18 \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}} x^{4}-6 \sqrt {27 x^{6}-27 x^{4}+9 x^{2}-1}\, x^{2}+18 x^{4}+2 \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {3}{4}}-12 \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}} x^{2}+2 \sqrt {27 x^{6}-27 x^{4}+9 x^{2}-1}-3 x^{2}+2 \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}}}{\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )^{2}}\right )}{81}-\frac {4 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-18 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}} x^{4}-27 x^{6}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {3}{4}}+12 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}} x^{2}+6 \sqrt {27 x^{6}-27 x^{4}+9 x^{2}-1}\, x^{2}+18 x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (27 x^{6}-27 x^{4}+9 x^{2}-1\right )^{\frac {1}{4}}-2 \sqrt {27 x^{6}-27 x^{4}+9 x^{2}-1}-3 x^{2}}{\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )^{2}}\right )}{81}\right ) \left (\left (3 x^{2}-1\right )^{3}\right )^{\frac {1}{4}}}{\left (3 x^{2}-1\right )^{\frac {3}{4}}}\) | \(423\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 74, normalized size = 0.95 \begin {gather*} \frac {2}{729} \, {\left (3 \, x^{2} - 1\right )}^{\frac {9}{4}} + \frac {8}{405} \, {\left (3 \, x^{2} - 1\right )}^{\frac {5}{4}} + \frac {14}{81} \, {\left (3 \, x^{2} - 1\right )}^{\frac {1}{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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Fricas [A]
time = 0.50, size = 64, normalized size = 0.82 \begin {gather*} \frac {2}{3645} \, {\left (45 \, x^{4} + 78 \, x^{2} + 284\right )} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{7}}{\left (3 x^{2} - 2\right ) \left (3 x^{2} - 1\right )^{\frac {3}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.51, size = 75, normalized size = 0.96 \begin {gather*} \frac {2}{729} \, {\left (3 \, x^{2} - 1\right )}^{\frac {9}{4}} + \frac {8}{405} \, {\left (3 \, x^{2} - 1\right )}^{\frac {5}{4}} + \frac {14}{81} \, {\left (3 \, x^{2} - 1\right )}^{\frac {1}{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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Mupad [B]
time = 0.05, size = 62, normalized size = 0.79 \begin {gather*} \frac {14\,{\left (3\,x^2-1\right )}^{1/4}}{81}-\frac {8\,\mathrm {atan}\left ({\left (3\,x^2-1\right )}^{1/4}\right )}{81}+\frac {8\,{\left (3\,x^2-1\right )}^{5/4}}{405}+\frac {2\,{\left (3\,x^2-1\right )}^{9/4}}{729}+\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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