3.1053 \(\int \frac{x^2}{\left (2+3 x^2\right )^{3/4} \left (4+3 x^2\right )} \, dx\)

Optimal. Leaf size=129 \[ \frac{\tanh ^{-1}\left (\frac{2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt{3 x^2+2}}{2 \sqrt{3} x \sqrt [4]{3 x^2+2}}\right )}{3 \sqrt [4]{2} \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [4]{2} \sqrt{3 x^2+2}+2\ 2^{3/4}}{2 \sqrt{3} x \sqrt [4]{3 x^2+2}}\right )}{3 \sqrt [4]{2} \sqrt{3}} \]

[Out]

-ArcTan[(2*2^(3/4) + 2*2^(1/4)*Sqrt[2 + 3*x^2])/(2*Sqrt[3]*x*(2 + 3*x^2)^(1/4))]
/(3*2^(1/4)*Sqrt[3]) + ArcTanh[(2*2^(3/4) - 2*2^(1/4)*Sqrt[2 + 3*x^2])/(2*Sqrt[3
]*x*(2 + 3*x^2)^(1/4))]/(3*2^(1/4)*Sqrt[3])

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Rubi [A]  time = 0.0965152, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{\tanh ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{3 x^2+2}}{\sqrt{3} x \sqrt [4]{3 x^2+2}}\right )}{3 \sqrt [4]{2} \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt{3 x^2+2}+2^{3/4}}{\sqrt{3} x \sqrt [4]{3 x^2+2}}\right )}{3 \sqrt [4]{2} \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/((2 + 3*x^2)^(3/4)*(4 + 3*x^2)),x]

[Out]

-ArcTan[(2^(3/4) + 2^(1/4)*Sqrt[2 + 3*x^2])/(Sqrt[3]*x*(2 + 3*x^2)^(1/4))]/(3*2^
(1/4)*Sqrt[3]) + ArcTanh[(2^(3/4) - 2^(1/4)*Sqrt[2 + 3*x^2])/(Sqrt[3]*x*(2 + 3*x
^2)^(1/4))]/(3*2^(1/4)*Sqrt[3])

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Rubi in Sympy [A]  time = 9.65177, size = 31, normalized size = 0.24 \[ \frac{\sqrt [4]{2} x^{3} \operatorname{appellf_{1}}{\left (\frac{3}{2},\frac{3}{4},1,\frac{5}{2},- \frac{3 x^{2}}{2},- \frac{3 x^{2}}{4} \right )}}{24} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(3*x**2+2)**(3/4)/(3*x**2+4),x)

[Out]

2**(1/4)*x**3*appellf1(3/2, 3/4, 1, 5/2, -3*x**2/2, -3*x**2/4)/24

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Mathematica [C]  time = 0.233651, size = 142, normalized size = 1.1 \[ -\frac{20 x^3 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{3 x^2}{2},-\frac{3 x^2}{4}\right )}{3 \left (3 x^2+2\right )^{3/4} \left (3 x^2+4\right ) \left (3 x^2 \left (2 F_1\left (\frac{5}{2};\frac{3}{4},2;\frac{7}{2};-\frac{3 x^2}{2},-\frac{3 x^2}{4}\right )+3 F_1\left (\frac{5}{2};\frac{7}{4},1;\frac{7}{2};-\frac{3 x^2}{2},-\frac{3 x^2}{4}\right )\right )-20 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{3 x^2}{2},-\frac{3 x^2}{4}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^2/((2 + 3*x^2)^(3/4)*(4 + 3*x^2)),x]

[Out]

(-20*x^3*AppellF1[3/2, 3/4, 1, 5/2, (-3*x^2)/2, (-3*x^2)/4])/(3*(2 + 3*x^2)^(3/4
)*(4 + 3*x^2)*(-20*AppellF1[3/2, 3/4, 1, 5/2, (-3*x^2)/2, (-3*x^2)/4] + 3*x^2*(2
*AppellF1[5/2, 3/4, 2, 7/2, (-3*x^2)/2, (-3*x^2)/4] + 3*AppellF1[5/2, 7/4, 1, 7/
2, (-3*x^2)/2, (-3*x^2)/4])))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2}}{3\,{x}^{2}+4} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(3*x^2+2)^(3/4)/(3*x^2+4),x)

[Out]

int(x^2/(3*x^2+2)^(3/4)/(3*x^2+4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (3 \, x^{2} + 4\right )}{\left (3 \, x^{2} + 2\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((3*x^2 + 4)*(3*x^2 + 2)^(3/4)),x, algorithm="maxima")

[Out]

integrate(x^2/((3*x^2 + 4)*(3*x^2 + 2)^(3/4)), x)

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Fricas [A]  time = 0.240918, size = 351, normalized size = 2.72 \[ \frac{1}{864} \cdot 72^{\frac{3}{4}}{\left (4 \, \sqrt{2} \arctan \left (\frac{3 \, x}{\sqrt{6} x \sqrt{\frac{72^{\frac{1}{4}} \sqrt{2}{\left (3 \, x^{2} + 2\right )}^{\frac{1}{4}} x + 3 \, x^{2} + 2 \, \sqrt{2} \sqrt{3 \, x^{2} + 2}}{x^{2}}} + 72^{\frac{1}{4}} \sqrt{2}{\left (3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 3 \, x}\right ) + 4 \, \sqrt{2} \arctan \left (\frac{3 \, x}{\sqrt{6} x \sqrt{-\frac{72^{\frac{1}{4}} \sqrt{2}{\left (3 \, x^{2} + 2\right )}^{\frac{1}{4}} x - 3 \, x^{2} - 2 \, \sqrt{2} \sqrt{3 \, x^{2} + 2}}{x^{2}}} + 72^{\frac{1}{4}} \sqrt{2}{\left (3 \, x^{2} + 2\right )}^{\frac{1}{4}} - 3 \, x}\right ) - \sqrt{2} \log \left (\frac{6 \,{\left (72^{\frac{1}{4}} \sqrt{2}{\left (3 \, x^{2} + 2\right )}^{\frac{1}{4}} x + 3 \, x^{2} + 2 \, \sqrt{2} \sqrt{3 \, x^{2} + 2}\right )}}{x^{2}}\right ) + \sqrt{2} \log \left (-\frac{6 \,{\left (72^{\frac{1}{4}} \sqrt{2}{\left (3 \, x^{2} + 2\right )}^{\frac{1}{4}} x - 3 \, x^{2} - 2 \, \sqrt{2} \sqrt{3 \, x^{2} + 2}\right )}}{x^{2}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((3*x^2 + 4)*(3*x^2 + 2)^(3/4)),x, algorithm="fricas")

[Out]

1/864*72^(3/4)*(4*sqrt(2)*arctan(3*x/(sqrt(6)*x*sqrt((72^(1/4)*sqrt(2)*(3*x^2 +
2)^(1/4)*x + 3*x^2 + 2*sqrt(2)*sqrt(3*x^2 + 2))/x^2) + 72^(1/4)*sqrt(2)*(3*x^2 +
2)^(1/4) + 3*x)) + 4*sqrt(2)*arctan(3*x/(sqrt(6)*x*sqrt(-(72^(1/4)*sqrt(2)*(3*x
^2 + 2)^(1/4)*x - 3*x^2 - 2*sqrt(2)*sqrt(3*x^2 + 2))/x^2) + 72^(1/4)*sqrt(2)*(3*
x^2 + 2)^(1/4) - 3*x)) - sqrt(2)*log(6*(72^(1/4)*sqrt(2)*(3*x^2 + 2)^(1/4)*x + 3
*x^2 + 2*sqrt(2)*sqrt(3*x^2 + 2))/x^2) + sqrt(2)*log(-6*(72^(1/4)*sqrt(2)*(3*x^2
+ 2)^(1/4)*x - 3*x^2 - 2*sqrt(2)*sqrt(3*x^2 + 2))/x^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (3 x^{2} + 2\right )^{\frac{3}{4}} \left (3 x^{2} + 4\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(3*x**2+2)**(3/4)/(3*x**2+4),x)

[Out]

Integral(x**2/((3*x**2 + 2)**(3/4)*(3*x**2 + 4)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (3 \, x^{2} + 4\right )}{\left (3 \, x^{2} + 2\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((3*x^2 + 4)*(3*x^2 + 2)^(3/4)),x, algorithm="giac")

[Out]

integrate(x^2/((3*x^2 + 4)*(3*x^2 + 2)^(3/4)), x)