Optimal. Leaf size=240 \[ \frac{8}{135} \left (3 x^2-2\right )^{3/4} x+\frac{32 \sqrt [4]{3 x^2-2} x}{135 \left (\sqrt{3 x^2-2}+\sqrt{2}\right )}+\frac{16 \sqrt [4]{2} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{135 \sqrt{3} x}-\frac{32 \sqrt [4]{2} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{135 \sqrt{3} x}+\frac{2}{27} \left (3 x^2-2\right )^{3/4} x^3 \]
[Out]
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Rubi [A] time = 0.282993, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{8}{135} \left (3 x^2-2\right )^{3/4} x+\frac{32 \sqrt [4]{3 x^2-2} x}{135 \left (\sqrt{3 x^2-2}+\sqrt{2}\right )}+\frac{16 \sqrt [4]{2} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{135 \sqrt{3} x}-\frac{32 \sqrt [4]{2} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{135 \sqrt{3} x}+\frac{2}{27} \left (3 x^2-2\right )^{3/4} x^3 \]
Antiderivative was successfully verified.
[In] Int[x^4/(-2 + 3*x^2)^(1/4),x]
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Rubi in Sympy [A] time = 5.85185, size = 75, normalized size = 0.31 \[ \frac{2 x^{3} \left (3 x^{2} - 2\right )^{\frac{3}{4}}}{27} + \frac{8 x \left (3 x^{2} - 2\right )^{\frac{3}{4}}}{135} + \frac{32 \sqrt{6} \sqrt [4]{- \frac{3 x^{2}}{2} + 1} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}}{2}\middle | 2\right )}{405 \sqrt [4]{3 x^{2} - 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(3*x**2-2)**(1/4),x)
[Out]
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Mathematica [C] time = 0.0357802, size = 63, normalized size = 0.26 \[ \frac{2 x \left (4\ 2^{3/4} \sqrt [4]{2-3 x^2} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{3 x^2}{2}\right )+15 x^4+2 x^2-8\right )}{135 \sqrt [4]{3 x^2-2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^4/(-2 + 3*x^2)^(1/4),x]
[Out]
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Maple [C] time = 0.051, size = 60, normalized size = 0.3 \[{\frac{2\,x \left ( 5\,{x}^{2}+4 \right ) }{135} \left ( 3\,{x}^{2}-2 \right ) ^{{\frac{3}{4}}}}+{\frac{8\,{2}^{3/4}x}{135}\sqrt [4]{-{\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) }{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,{\frac{3\,{x}^{2}}{2}})}{\frac{1}{\sqrt [4]{{\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(3*x^2-2)^(1/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (3 \, x^{2} - 2\right )}^{\frac{1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(3*x^2 - 2)^(1/4),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{4}}{{\left (3 \, x^{2} - 2\right )}^{\frac{1}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(3*x^2 - 2)^(1/4),x, algorithm="fricas")
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Sympy [A] time = 2.46017, size = 31, normalized size = 0.13 \[ \frac{2^{\frac{3}{4}} x^{5} e^{\frac{15 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{3 x^{2}}{2}} \right )}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(3*x**2-2)**(1/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (3 \, x^{2} - 2\right )}^{\frac{1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(3*x^2 - 2)^(1/4),x, algorithm="giac")
[Out]