3.1031 \(\int \frac{x^7}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx\)

Optimal. Leaf size=136 \[ \frac{2}{891} \left (2-3 x^2\right )^{11/4}-\frac{16}{567} \left (2-3 x^2\right )^{7/4}+\frac{56}{243} \left (2-3 x^2\right )^{3/4}+\frac{32}{81} \sqrt [4]{2} \tan ^{-1}\left (\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\frac{32}{81} \sqrt [4]{2} \tanh ^{-1}\left (\frac{\sqrt{2-3 x^2}+\sqrt{2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right ) \]

[Out]

(56*(2 - 3*x^2)^(3/4))/243 - (16*(2 - 3*x^2)^(7/4))/567 + (2*(2 - 3*x^2)^(11/4))
/891 + (32*2^(1/4)*ArcTan[(Sqrt[2] - Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1/4)
)])/81 + (32*2^(1/4)*ArcTanh[(Sqrt[2] + Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1
/4))])/81

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Rubi [A]  time = 0.260037, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{2}{891} \left (2-3 x^2\right )^{11/4}-\frac{16}{567} \left (2-3 x^2\right )^{7/4}+\frac{56}{243} \left (2-3 x^2\right )^{3/4}+\frac{32}{81} \sqrt [4]{2} \tan ^{-1}\left (\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\frac{32}{81} \sqrt [4]{2} \tanh ^{-1}\left (\frac{\sqrt{2-3 x^2}+\sqrt{2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^7/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]

[Out]

(56*(2 - 3*x^2)^(3/4))/243 - (16*(2 - 3*x^2)^(7/4))/567 + (2*(2 - 3*x^2)^(11/4))
/891 + (32*2^(1/4)*ArcTan[(Sqrt[2] - Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1/4)
)])/81 + (32*2^(1/4)*ArcTanh[(Sqrt[2] + Sqrt[2 - 3*x^2])/(2^(3/4)*(2 - 3*x^2)^(1
/4))])/81

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Rubi in Sympy [A]  time = 35.4572, size = 175, normalized size = 1.29 \[ \frac{2 \left (- 3 x^{2} + 2\right )^{\frac{11}{4}}}{891} - \frac{16 \left (- 3 x^{2} + 2\right )^{\frac{7}{4}}}{567} + \frac{56 \left (- 3 x^{2} + 2\right )^{\frac{3}{4}}}{243} - \frac{16 \sqrt [4]{2} \log{\left (- 2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{81} + \frac{16 \sqrt [4]{2} \log{\left (2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{81} - \frac{32 \sqrt [4]{2} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} - 1 \right )}}{81} - \frac{32 \sqrt [4]{2} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} + 1 \right )}}{81} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(-3*x**2+2)**(1/4)/(-3*x**2+4),x)

[Out]

2*(-3*x**2 + 2)**(11/4)/891 - 16*(-3*x**2 + 2)**(7/4)/567 + 56*(-3*x**2 + 2)**(3
/4)/243 - 16*2**(1/4)*log(-2**(3/4)*(-3*x**2 + 2)**(1/4) + sqrt(-3*x**2 + 2) + s
qrt(2))/81 + 16*2**(1/4)*log(2**(3/4)*(-3*x**2 + 2)**(1/4) + sqrt(-3*x**2 + 2) +
sqrt(2))/81 - 32*2**(1/4)*atan(2**(1/4)*(-3*x**2 + 2)**(1/4) - 1)/81 - 32*2**(1
/4)*atan(2**(1/4)*(-3*x**2 + 2)**(1/4) + 1)/81

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Mathematica [C]  time = 0.103542, size = 76, normalized size = 0.56 \[ -\frac{2 \left (-14784 \sqrt [4]{\frac{2-3 x^2}{4-3 x^2}} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{2}{4-3 x^2}\right )+567 x^6+1242 x^4+4056 x^2-3424\right )}{18711 \sqrt [4]{2-3 x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/((2 - 3*x^2)^(1/4)*(4 - 3*x^2)),x]

[Out]

(-2*(-3424 + 4056*x^2 + 1242*x^4 + 567*x^6 - 14784*((2 - 3*x^2)/(4 - 3*x^2))^(1/
4)*Hypergeometric2F1[1/4, 1/4, 5/4, 2/(4 - 3*x^2)]))/(18711*(2 - 3*x^2)^(1/4))

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Maple [F]  time = 0.109, size = 0, normalized size = 0. \[ \int{\frac{{x}^{7}}{-3\,{x}^{2}+4}{\frac{1}{\sqrt [4]{-3\,{x}^{2}+2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(-3*x^2+2)^(1/4)/(-3*x^2+4),x)

[Out]

int(x^7/(-3*x^2+2)^(1/4)/(-3*x^2+4),x)

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Maxima [A]  time = 1.49637, size = 204, normalized size = 1.5 \[ \frac{2}{891} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{11}{4}} - \frac{16}{567} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{7}{4}} - \frac{32}{81} \cdot 2^{\frac{1}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{32}{81} \cdot 2^{\frac{1}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) + \frac{16}{81} \cdot 2^{\frac{1}{4}} \log \left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) - \frac{16}{81} \cdot 2^{\frac{1}{4}} \log \left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{56}{243} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{3}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^7/((3*x^2 - 4)*(-3*x^2 + 2)^(1/4)),x, algorithm="maxima")

[Out]

2/891*(-3*x^2 + 2)^(11/4) - 16/567*(-3*x^2 + 2)^(7/4) - 32/81*2^(1/4)*arctan(1/2
*2^(1/4)*(2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) - 32/81*2^(1/4)*arctan(-1/2*2^(1/4)*(
2^(3/4) - 2*(-3*x^2 + 2)^(1/4))) + 16/81*2^(1/4)*log(2^(3/4)*(-3*x^2 + 2)^(1/4)
+ sqrt(2) + sqrt(-3*x^2 + 2)) - 16/81*2^(1/4)*log(-2^(3/4)*(-3*x^2 + 2)^(1/4) +
sqrt(2) + sqrt(-3*x^2 + 2)) + 56/243*(-3*x^2 + 2)^(3/4)

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Fricas [A]  time = 0.25409, size = 351, normalized size = 2.58 \[ \frac{2}{18711} \,{\left (189 \, x^{4} + 540 \, x^{2} + 1712\right )}{\left (-3 \, x^{2} + 2\right )}^{\frac{3}{4}} + \frac{32}{81} \cdot 8^{\frac{1}{4}} \sqrt{2} \arctan \left (\frac{8^{\frac{3}{4}} \sqrt{2}}{8^{\frac{3}{4}} \sqrt{2} + 4 \, \sqrt{8^{\frac{3}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4 \, \sqrt{2} + 4 \, \sqrt{-3 \, x^{2} + 2}} + 8 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}}\right ) + \frac{32}{81} \cdot 8^{\frac{1}{4}} \sqrt{2} \arctan \left (-\frac{8^{\frac{3}{4}} \sqrt{2}}{8^{\frac{3}{4}} \sqrt{2} - 2 \, \sqrt{-4 \cdot 8^{\frac{3}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 16 \, \sqrt{2} + 16 \, \sqrt{-3 \, x^{2} + 2}} - 8 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}}\right ) + \frac{8}{81} \cdot 8^{\frac{1}{4}} \sqrt{2} \log \left (4 \cdot 8^{\frac{3}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 16 \, \sqrt{2} + 16 \, \sqrt{-3 \, x^{2} + 2}\right ) - \frac{8}{81} \cdot 8^{\frac{1}{4}} \sqrt{2} \log \left (-4 \cdot 8^{\frac{3}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 16 \, \sqrt{2} + 16 \, \sqrt{-3 \, x^{2} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^7/((3*x^2 - 4)*(-3*x^2 + 2)^(1/4)),x, algorithm="fricas")

[Out]

2/18711*(189*x^4 + 540*x^2 + 1712)*(-3*x^2 + 2)^(3/4) + 32/81*8^(1/4)*sqrt(2)*ar
ctan(8^(3/4)*sqrt(2)/(8^(3/4)*sqrt(2) + 4*sqrt(8^(3/4)*sqrt(2)*(-3*x^2 + 2)^(1/4
) + 4*sqrt(2) + 4*sqrt(-3*x^2 + 2)) + 8*(-3*x^2 + 2)^(1/4))) + 32/81*8^(1/4)*sqr
t(2)*arctan(-8^(3/4)*sqrt(2)/(8^(3/4)*sqrt(2) - 2*sqrt(-4*8^(3/4)*sqrt(2)*(-3*x^
2 + 2)^(1/4) + 16*sqrt(2) + 16*sqrt(-3*x^2 + 2)) - 8*(-3*x^2 + 2)^(1/4))) + 8/81
*8^(1/4)*sqrt(2)*log(4*8^(3/4)*sqrt(2)*(-3*x^2 + 2)^(1/4) + 16*sqrt(2) + 16*sqrt
(-3*x^2 + 2)) - 8/81*8^(1/4)*sqrt(2)*log(-4*8^(3/4)*sqrt(2)*(-3*x^2 + 2)^(1/4) +
16*sqrt(2) + 16*sqrt(-3*x^2 + 2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x^{7}}{3 x^{2} \sqrt [4]{- 3 x^{2} + 2} - 4 \sqrt [4]{- 3 x^{2} + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(-3*x**2+2)**(1/4)/(-3*x**2+4),x)

[Out]

-Integral(x**7/(3*x**2*(-3*x**2 + 2)**(1/4) - 4*(-3*x**2 + 2)**(1/4)), x)

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GIAC/XCAS [A]  time = 0.247951, size = 216, normalized size = 1.59 \[ \frac{2}{891} \,{\left (3 \, x^{2} - 2\right )}^{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{3}{4}} - \frac{16}{567} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{7}{4}} - \frac{8}{81} \cdot 8^{\frac{3}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{8}{81} \cdot 8^{\frac{3}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) + \frac{16}{81} \cdot 2^{\frac{1}{4}}{\rm ln}\left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) - \frac{16}{81} \cdot 2^{\frac{1}{4}}{\rm ln}\left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{56}{243} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{3}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^7/((3*x^2 - 4)*(-3*x^2 + 2)^(1/4)),x, algorithm="giac")

[Out]

2/891*(3*x^2 - 2)^2*(-3*x^2 + 2)^(3/4) - 16/567*(-3*x^2 + 2)^(7/4) - 8/81*8^(3/4
)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) - 8/81*8^(3/4)*arctan(-1/
2*2^(1/4)*(2^(3/4) - 2*(-3*x^2 + 2)^(1/4))) + 16/81*2^(1/4)*ln(2^(3/4)*(-3*x^2 +
2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 16/81*2^(1/4)*ln(-2^(3/4)*(-3*x^2 + 2)
^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 56/243*(-3*x^2 + 2)^(3/4)