Optimal. Leaf size=85 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{a} \sqrt [4]{3 x^2-a}}\right )}{3 \sqrt{6} \sqrt [4]{a}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{a} \sqrt [4]{3 x^2-a}}\right )}{3 \sqrt{6} \sqrt [4]{a}} \]
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Rubi [A] time = 0.101892, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{a} \sqrt [4]{3 x^2-a}}\right )}{3 \sqrt{6} \sqrt [4]{a}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{a} \sqrt [4]{3 x^2-a}}\right )}{3 \sqrt{6} \sqrt [4]{a}} \]
Antiderivative was successfully verified.
[In] Int[x^2/((-2*a + 3*x^2)*(-a + 3*x^2)^(3/4)),x]
[Out]
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Rubi in Sympy [A] time = 33.717, size = 49, normalized size = 0.58 \[ \frac{x^{3} \sqrt [4]{- a + 3 x^{2}} \operatorname{appellf_{1}}{\left (\frac{3}{2},\frac{3}{4},1,\frac{5}{2},\frac{3 x^{2}}{a},\frac{3 x^{2}}{2 a} \right )}}{6 a^{2} \sqrt [4]{1 - \frac{3 x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(3*x**2-2*a)/(3*x**2-a)**(3/4),x)
[Out]
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Mathematica [C] time = 0.30744, size = 164, normalized size = 1.93 \[ \frac{10 a x^3 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};\frac{3 x^2}{a},\frac{3 x^2}{2 a}\right )}{3 \left (3 x^2-2 a\right ) \left (3 x^2-a\right )^{3/4} \left (3 x^2 \left (2 F_1\left (\frac{5}{2};\frac{3}{4},2;\frac{7}{2};\frac{3 x^2}{a},\frac{3 x^2}{2 a}\right )+3 F_1\left (\frac{5}{2};\frac{7}{4},1;\frac{7}{2};\frac{3 x^2}{a},\frac{3 x^2}{2 a}\right )\right )+10 a F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};\frac{3 x^2}{a},\frac{3 x^2}{2 a}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[x^2/((-2*a + 3*x^2)*(-a + 3*x^2)^(3/4)),x]
[Out]
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Maple [F] time = 0.102, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2}}{3\,{x}^{2}-2\,a} \left ( 3\,{x}^{2}-a \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(3*x^2-2*a)/(3*x^2-a)^(3/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (3 \, x^{2} - a\right )}^{\frac{3}{4}}{\left (3 \, x^{2} - 2 \, a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((3*x^2 - a)^(3/4)*(3*x^2 - 2*a)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.23318, size = 185, normalized size = 2.18 \[ \frac{2 \, \left (\frac{1}{36}\right )^{\frac{1}{4}} \arctan \left (\frac{3 \, \left (\frac{1}{36}\right )^{\frac{1}{4}} x}{{\left (\sqrt{\frac{1}{2}} x \sqrt{\frac{\frac{3 \, x^{2}}{\sqrt{a}} + 2 \, \sqrt{3 \, x^{2} - a}}{x^{2}}} +{\left (3 \, x^{2} - a\right )}^{\frac{1}{4}}\right )} a^{\frac{1}{4}}}\right )}{3 \, a^{\frac{1}{4}}} - \frac{\left (\frac{1}{36}\right )^{\frac{1}{4}} \log \left (\frac{\frac{3 \, \left (\frac{1}{36}\right )^{\frac{1}{4}} x}{a^{\frac{1}{4}}} +{\left (3 \, x^{2} - a\right )}^{\frac{1}{4}}}{x}\right )}{6 \, a^{\frac{1}{4}}} + \frac{\left (\frac{1}{36}\right )^{\frac{1}{4}} \log \left (-\frac{\frac{3 \, \left (\frac{1}{36}\right )^{\frac{1}{4}} x}{a^{\frac{1}{4}}} -{\left (3 \, x^{2} - a\right )}^{\frac{1}{4}}}{x}\right )}{6 \, a^{\frac{1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((3*x^2 - a)^(3/4)*(3*x^2 - 2*a)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (- 2 a + 3 x^{2}\right ) \left (- a + 3 x^{2}\right )^{\frac{3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(3*x**2-2*a)/(3*x**2-a)**(3/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (3 \, x^{2} - a\right )}^{\frac{3}{4}}{\left (3 \, x^{2} - 2 \, a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((3*x^2 - a)^(3/4)*(3*x^2 - 2*a)),x, algorithm="giac")
[Out]