Optimal. Leaf size=47 \[ \frac{2}{9} x \sqrt [4]{3 x^2+2}-\frac{4\ 2^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{9 \sqrt{3}} \]
[Out]
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Rubi [A] time = 0.0334056, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2}{9} x \sqrt [4]{3 x^2+2}-\frac{4\ 2^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{9 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[x^2/(2 + 3*x^2)^(3/4),x]
[Out]
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Rubi in Sympy [A] time = 3.58923, size = 41, normalized size = 0.87 \[ \frac{2 x \sqrt [4]{3 x^{2} + 2}}{9} - \frac{4 \cdot 2^{\frac{3}{4}} \sqrt{3} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{6} x}{2} \right )}}{2}\middle | 2\right )}{27} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(3*x**2+2)**(3/4),x)
[Out]
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Mathematica [C] time = 0.0207826, size = 41, normalized size = 0.87 \[ \frac{2}{9} x \left (\sqrt [4]{3 x^2+2}-\sqrt [4]{2} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{3 x^2}{2}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(2 + 3*x^2)^(3/4),x]
[Out]
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Maple [C] time = 0.029, size = 31, normalized size = 0.7 \[{\frac{2\,x}{9}\sqrt [4]{3\,{x}^{2}+2}}-{\frac{2\,\sqrt [4]{2}x}{9}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{3}{4}};\,{\frac{3}{2}};\,-{\frac{3\,{x}^{2}}{2}})}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(3*x^2+2)^(3/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(3*x^2 + 2)^(3/4),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(3*x^2 + 2)^(3/4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.12111, size = 27, normalized size = 0.57 \[ \frac{\sqrt [4]{2} x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{i \pi }}{2}} \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(3*x**2+2)**(3/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(3*x^2 + 2)^(3/4),x, algorithm="giac")
[Out]