Optimal. Leaf size=121 \[ \frac{8}{63} \sqrt [4]{-3 x^2-2} x-\frac{8\ 2^{3/4} \sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{63 \sqrt{3} x}-\frac{2}{21} \sqrt [4]{-3 x^2-2} x^3 \]
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Rubi [A] time = 0.133272, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{8}{63} \sqrt [4]{-3 x^2-2} x-\frac{8\ 2^{3/4} \sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{63 \sqrt{3} x}-\frac{2}{21} \sqrt [4]{-3 x^2-2} x^3 \]
Antiderivative was successfully verified.
[In] Int[x^4/(-2 - 3*x^2)^(3/4),x]
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Rubi in Sympy [A] time = 5.7245, size = 80, normalized size = 0.66 \[ - \frac{2 x^{3} \sqrt [4]{- 3 x^{2} - 2}}{21} + \frac{8 x \sqrt [4]{- 3 x^{2} - 2}}{63} + \frac{32 \sqrt{6} \left (\frac{3 x^{2}}{2} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{6} x}{2} \right )}}{2}\middle | 2\right )}{189 \left (- 3 x^{2} - 2\right )^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(-3*x**2-2)**(3/4),x)
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Mathematica [C] time = 0.0350301, size = 63, normalized size = 0.52 \[ \frac{2 x \left (4 \sqrt [4]{2} \left (3 x^2+2\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{3 x^2}{2}\right )+9 x^4-6 x^2-8\right )}{63 \left (-3 x^2-2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^4/(-2 - 3*x^2)^(3/4),x]
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Maple [F] time = 0.021, size = 0, normalized size = 0. \[ \int{{x}^{4} \left ( -3\,{x}^{2}-2 \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(-3*x^2-2)^(3/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (-3 \, x^{2} - 2\right )}^{\frac{3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(-3*x^2 - 2)^(3/4),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[ -\frac{2}{63} \,{\left (3 \, x^{3} - 4 \, x\right )}{\left (-3 \, x^{2} - 2\right )}^{\frac{1}{4}} +{\rm integral}\left (-\frac{16 \,{\left (-3 \, x^{2} - 2\right )}^{\frac{1}{4}}}{63 \,{\left (3 \, x^{2} + 2\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(-3*x^2 - 2)^(3/4),x, algorithm="fricas")
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Sympy [A] time = 2.26923, size = 36, normalized size = 0.3 \[ \frac{\sqrt [4]{2} x^{5} e^{- \frac{3 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{i \pi }}{2}} \right )}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(-3*x**2-2)**(3/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (-3 \, x^{2} - 2\right )}^{\frac{3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(-3*x^2 - 2)^(3/4),x, algorithm="giac")
[Out]