Optimal. Leaf size=78 \[ \frac{2}{729} \left (3 x^2-1\right )^{9/4}+\frac{8}{405} \left (3 x^2-1\right )^{5/4}+\frac{14}{81} \sqrt [4]{3 x^2-1}-\frac{8}{81} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-\frac{8}{81} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]
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Rubi [A] time = 0.0509349, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {446, 88, 63, 212, 206, 203} \[ \frac{2}{729} \left (3 x^2-1\right )^{9/4}+\frac{8}{405} \left (3 x^2-1\right )^{5/4}+\frac{14}{81} \sqrt [4]{3 x^2-1}-\frac{8}{81} \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-\frac{8}{81} \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right ) \]
Antiderivative was successfully verified.
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Rule 446
Rule 88
Rule 63
Rule 212
Rule 206
Rule 203
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} \operatorname{Subst}\left (\int \frac{x^3}{(-2+3 x) (-1+3 x)^{3/4}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{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} \operatorname{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} \operatorname{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} \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}{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*}
Mathematica [A] time = 0.0346687, size = 58, normalized size = 0.74 \[ \frac{2 \sqrt [4]{3 x^2-1} \left (45 x^4+78 x^2+284\right )-360 \tan ^{-1}\left (\sqrt [4]{3 x^2-1}\right )-360 \tanh ^{-1}\left (\sqrt [4]{3 x^2-1}\right )}{3645} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.075, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{7}}{3\,{x}^{2}-2} \left ( 3\,{x}^{2}-1 \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.5393, size = 100, normalized size = 1.28 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53744, size = 204, normalized size = 2.62 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{7}}{\left (3 x^{2} - 2\right ) \left (3 x^{2} - 1\right )^{\frac{3}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2302, size = 101, normalized size = 1.29 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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