Optimal. Leaf size=109 \[ \frac{3}{10} \left (1-x^2\right )^{5/3}+\frac{3}{2} \left (1-x^2\right )^{2/3}-\frac{9 \log \left (x^2+3\right )}{4\ 2^{2/3}}+\frac{27 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}}+\frac{9 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{2\ 2^{2/3}} \]
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Rubi [A] time = 0.220752, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{3}{10} \left (1-x^2\right )^{5/3}+\frac{3}{2} \left (1-x^2\right )^{2/3}-\frac{9 \log \left (x^2+3\right )}{4\ 2^{2/3}}+\frac{27 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{4\ 2^{2/3}}+\frac{9 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{2\ 2^{2/3}} \]
Antiderivative was successfully verified.
[In] Int[x^5/((1 - x^2)^(1/3)*(3 + x^2)),x]
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Rubi in Sympy [A] time = 13.586, size = 100, normalized size = 0.92 \[ \frac{3 \left (- x^{2} + 1\right )^{\frac{5}{3}}}{10} + \frac{3 \left (- x^{2} + 1\right )^{\frac{2}{3}}}{2} - \frac{9 \sqrt [3]{2} \log{\left (x^{2} + 3 \right )}}{8} + \frac{27 \sqrt [3]{2} \log{\left (- \sqrt [3]{- x^{2} + 1} + 2^{\frac{2}{3}} \right )}}{8} + \frac{9 \sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{\sqrt [3]{2} \sqrt [3]{- x^{2} + 1}}{3} + \frac{1}{3}\right ) \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5/(-x**2+1)**(1/3)/(x**2+3),x)
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Mathematica [C] time = 0.0595504, size = 63, normalized size = 0.58 \[ \frac{3 \left (-45 \sqrt [3]{\frac{x^2-1}{x^2+3}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{4}{x^2+3}\right )+x^4-7 x^2+6\right )}{10 \sqrt [3]{1-x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^5/((1 - x^2)^(1/3)*(3 + x^2)),x]
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Maple [F] time = 0.063, size = 0, normalized size = 0. \[ \int{\frac{{x}^{5}}{{x}^{2}+3}{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5/(-x^2+1)^(1/3)/(x^2+3),x)
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Maxima [A] time = 1.49327, size = 146, normalized size = 1.34 \[ \frac{9}{8} \cdot 4^{\frac{2}{3}} \sqrt{3} \arctan \left (\frac{1}{12} \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (4^{\frac{1}{3}} + 2 \,{\left (-x^{2} + 1\right )}^{\frac{1}{3}}\right )}\right ) + \frac{3}{10} \,{\left (-x^{2} + 1\right )}^{\frac{5}{3}} - \frac{9}{16} \cdot 4^{\frac{2}{3}} \log \left (4^{\frac{2}{3}} + 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} +{\left (-x^{2} + 1\right )}^{\frac{2}{3}}\right ) + \frac{9}{8} \cdot 4^{\frac{2}{3}} \log \left (-4^{\frac{1}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}}\right ) + \frac{3}{2} \,{\left (-x^{2} + 1\right )}^{\frac{2}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/((x^2 + 3)*(-x^2 + 1)^(1/3)),x, algorithm="maxima")
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Fricas [A] time = 0.234945, size = 135, normalized size = 1.24 \[ -\frac{3}{80} \cdot 4^{\frac{2}{3}}{\left (2 \cdot 4^{\frac{1}{3}}{\left (x^{2} - 6\right )}{\left (-x^{2} + 1\right )}^{\frac{2}{3}} - 30 \, \sqrt{3} \arctan \left (\frac{1}{6} \, \sqrt{3}{\left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 2\right )}\right ) + 15 \, \log \left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{2}{3}} + 4\right ) - 30 \, \log \left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 4\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/((x^2 + 3)*(-x^2 + 1)^(1/3)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5/(-x**2+1)**(1/3)/(x**2+3),x)
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/((x^2 + 3)*(-x^2 + 1)^(1/3)),x, algorithm="giac")
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