Optimal. Leaf size=173 \[ -\frac{2}{135} \left (2-3 x^2\right )^{5/4}+\frac{4}{9} \sqrt [4]{2-3 x^2}+\frac{2}{27} 2^{3/4} \log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )-\frac{2}{27} 2^{3/4} \log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )-\frac{4}{27} 2^{3/4} \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )+\frac{4}{27} 2^{3/4} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right ) \]
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Rubi [A] time = 0.180285, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {443, 261, 266, 43, 444, 63, 211, 1165, 628, 1162, 617, 204} \[ -\frac{2}{135} \left (2-3 x^2\right )^{5/4}+\frac{4}{9} \sqrt [4]{2-3 x^2}+\frac{2}{27} 2^{3/4} \log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )-\frac{2}{27} 2^{3/4} \log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )-\frac{4}{27} 2^{3/4} \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )+\frac{4}{27} 2^{3/4} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 443
Rule 261
Rule 266
Rule 43
Rule 444
Rule 63
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{x^5}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx &=\int \left (-\frac{4 x}{9 \left (2-3 x^2\right )^{3/4}}-\frac{x^3}{3 \left (2-3 x^2\right )^{3/4}}+\frac{16 x}{9 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )}\right ) \, dx\\ &=-\left (\frac{1}{3} \int \frac{x^3}{\left (2-3 x^2\right )^{3/4}} \, dx\right )-\frac{4}{9} \int \frac{x}{\left (2-3 x^2\right )^{3/4}} \, dx+\frac{16}{9} \int \frac{x}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx\\ &=\frac{8}{27} \sqrt [4]{2-3 x^2}-\frac{1}{6} \operatorname{Subst}\left (\int \frac{x}{(2-3 x)^{3/4}} \, dx,x,x^2\right )+\frac{8}{9} \operatorname{Subst}\left (\int \frac{1}{(2-3 x)^{3/4} (4-3 x)} \, dx,x,x^2\right )\\ &=\frac{8}{27} \sqrt [4]{2-3 x^2}-\frac{1}{6} \operatorname{Subst}\left (\int \left (\frac{2}{3 (2-3 x)^{3/4}}-\frac{1}{3} \sqrt [4]{2-3 x}\right ) \, dx,x,x^2\right )-\frac{32}{27} \operatorname{Subst}\left (\int \frac{1}{2+x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )\\ &=\frac{4}{9} \sqrt [4]{2-3 x^2}-\frac{2}{135} \left (2-3 x^2\right )^{5/4}-\frac{1}{27} \left (8 \sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}-x^2}{2+x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )-\frac{1}{27} \left (8 \sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}+x^2}{2+x^4} \, dx,x,\sqrt [4]{2-3 x^2}\right )\\ &=\frac{4}{9} \sqrt [4]{2-3 x^2}-\frac{2}{135} \left (2-3 x^2\right )^{5/4}-\frac{1}{27} \left (4 \sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2}-2^{3/4} x+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )-\frac{1}{27} \left (4 \sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2}+2^{3/4} x+x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )+\frac{1}{27} \left (2\ 2^{3/4}\right ) \operatorname{Subst}\left (\int \frac{2^{3/4}+2 x}{-\sqrt{2}-2^{3/4} x-x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )+\frac{1}{27} \left (2\ 2^{3/4}\right ) \operatorname{Subst}\left (\int \frac{2^{3/4}-2 x}{-\sqrt{2}+2^{3/4} x-x^2} \, dx,x,\sqrt [4]{2-3 x^2}\right )\\ &=\frac{4}{9} \sqrt [4]{2-3 x^2}-\frac{2}{135} \left (2-3 x^2\right )^{5/4}+\frac{2}{27} 2^{3/4} \log \left (\sqrt{2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2-3 x^2}\right )-\frac{2}{27} 2^{3/4} \log \left (\sqrt{2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2-3 x^2}\right )-\frac{1}{27} \left (4\ 2^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt [4]{4-6 x^2}\right )+\frac{1}{27} \left (4\ 2^{3/4}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt [4]{4-6 x^2}\right )\\ &=\frac{4}{9} \sqrt [4]{2-3 x^2}-\frac{2}{135} \left (2-3 x^2\right )^{5/4}-\frac{4}{27} 2^{3/4} \tan ^{-1}\left (1+\sqrt [4]{4-6 x^2}\right )+\frac{4}{27} 2^{3/4} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )+\frac{2}{27} 2^{3/4} \log \left (\sqrt{2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2-3 x^2}\right )-\frac{2}{27} 2^{3/4} \log \left (\sqrt{2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2-3 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0639823, size = 163, normalized size = 0.94 \[ \frac{2}{135} \left (3 \sqrt [4]{2-3 x^2} x^2+28 \sqrt [4]{2-3 x^2}+5\ 2^{3/4} \log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )-5\ 2^{3/4} \log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )+10\ 2^{3/4} \tan ^{-1}\left (1-\sqrt [4]{4-6 x^2}\right )-10\ 2^{3/4} \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.048, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{5}}{-3\,{x}^{2}+4} \left ( -3\,{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52908, size = 189, normalized size = 1.09 \begin{align*} -\frac{4}{27} \cdot 2^{\frac{3}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{4}{27} \cdot 2^{\frac{3}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{2}{27} \cdot 2^{\frac{3}{4}} \log \left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{2}{27} \cdot 2^{\frac{3}{4}} \log \left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) - \frac{2}{135} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{5}{4}} + \frac{4}{9} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7383, size = 601, normalized size = 3.47 \begin{align*} \frac{8}{27} \cdot 2^{\frac{3}{4}} \arctan \left (2^{\frac{1}{4}} \sqrt{2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}} - 2^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} - 1\right ) + \frac{8}{27} \cdot 2^{\frac{3}{4}} \arctan \left (2^{\frac{1}{4}} \sqrt{-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}} - 2^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 1\right ) - \frac{2}{27} \cdot 2^{\frac{3}{4}} \log \left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{2}{27} \cdot 2^{\frac{3}{4}} \log \left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{2}{135} \,{\left (3 \, x^{2} + 28\right )}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} \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^{5}}{3 x^{2} \left (2 - 3 x^{2}\right )^{\frac{3}{4}} - 4 \left (2 - 3 x^{2}\right )^{\frac{3}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20761, size = 189, normalized size = 1.09 \begin{align*} -\frac{4}{27} \cdot 2^{\frac{3}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{4}{27} \cdot 2^{\frac{3}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{2}{27} \cdot 2^{\frac{3}{4}} \log \left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{2}{27} \cdot 2^{\frac{3}{4}} \log \left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) - \frac{2}{135} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{5}{4}} + \frac{4}{9} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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