3.1061 \(\int \frac{x^7}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx\)

Optimal. Leaf size=188 \[ \frac{2}{729} \left (2-3 x^2\right )^{9/4}-\frac{16}{405} \left (2-3 x^2\right )^{5/4}+\frac{56}{81} \sqrt [4]{2-3 x^2}+\frac{8}{81} 2^{3/4} \log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )-\frac{8}{81} 2^{3/4} \log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )-\frac{16}{81} 2^{3/4} \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )+\frac{16}{81} 2^{3/4} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right ) \]

[Out]

(56*(2 - 3*x^2)^(1/4))/81 - (16*(2 - 3*x^2)^(5/4))/405 + (2*(2 - 3*x^2)^(9/4))/7
29 - (16*2^(3/4)*ArcTan[1 + (4 - 6*x^2)^(1/4)])/81 + (16*2^(3/4)*ArcTan[1 - 2^(1
/4)*(2 - 3*x^2)^(1/4)])/81 + (8*2^(3/4)*Log[Sqrt[2] - 2^(3/4)*(2 - 3*x^2)^(1/4)
+ Sqrt[2 - 3*x^2]])/81 - (8*2^(3/4)*Log[Sqrt[2] + 2^(3/4)*(2 - 3*x^2)^(1/4) + Sq
rt[2 - 3*x^2]])/81

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Rubi [A]  time = 0.495926, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{2}{729} \left (2-3 x^2\right )^{9/4}-\frac{16}{405} \left (2-3 x^2\right )^{5/4}+\frac{56}{81} \sqrt [4]{2-3 x^2}+\frac{8}{81} 2^{3/4} \log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )-\frac{8}{81} 2^{3/4} \log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )-\frac{16}{81} 2^{3/4} \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )+\frac{16}{81} 2^{3/4} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(56*(2 - 3*x^2)^(1/4))/81 - (16*(2 - 3*x^2)^(5/4))/405 + (2*(2 - 3*x^2)^(9/4))/7
29 - (16*2^(3/4)*ArcTan[1 + (4 - 6*x^2)^(1/4)])/81 + (16*2^(3/4)*ArcTan[1 - 2^(1
/4)*(2 - 3*x^2)^(1/4)])/81 + (8*2^(3/4)*Log[Sqrt[2] - 2^(3/4)*(2 - 3*x^2)^(1/4)
+ Sqrt[2 - 3*x^2]])/81 - (8*2^(3/4)*Log[Sqrt[2] + 2^(3/4)*(2 - 3*x^2)^(1/4) + Sq
rt[2 - 3*x^2]])/81

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Rubi in Sympy [A]  time = 34.9502, size = 175, normalized size = 0.93 \[ \frac{2 \left (- 3 x^{2} + 2\right )^{\frac{9}{4}}}{729} - \frac{16 \left (- 3 x^{2} + 2\right )^{\frac{5}{4}}}{405} + \frac{56 \sqrt [4]{- 3 x^{2} + 2}}{81} + \frac{8 \cdot 2^{\frac{3}{4}} \log{\left (- 2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{81} - \frac{8 \cdot 2^{\frac{3}{4}} \log{\left (2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{81} - \frac{16 \cdot 2^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} - 1 \right )}}{81} - \frac{16 \cdot 2^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} + 1 \right )}}{81} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-3*x**2 + 2)**(9/4)/729 - 16*(-3*x**2 + 2)**(5/4)/405 + 56*(-3*x**2 + 2)**(1/
4)/81 + 8*2**(3/4)*log(-2**(3/4)*(-3*x**2 + 2)**(1/4) + sqrt(-3*x**2 + 2) + sqrt
(2))/81 - 8*2**(3/4)*log(2**(3/4)*(-3*x**2 + 2)**(1/4) + sqrt(-3*x**2 + 2) + sqr
t(2))/81 - 16*2**(3/4)*atan(2**(1/4)*(-3*x**2 + 2)**(1/4) - 1)/81 - 16*2**(3/4)*
atan(2**(1/4)*(-3*x**2 + 2)**(1/4) + 1)/81

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Mathematica [C]  time = 0.096009, size = 76, normalized size = 0.4 \[ -\frac{2 \left (-960 \left (\frac{2-3 x^2}{4-3 x^2}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{2}{4-3 x^2}\right )+135 x^6+378 x^4+3096 x^2-2272\right )}{3645 \left (2-3 x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(-2272 + 3096*x^2 + 378*x^4 + 135*x^6 - 960*((2 - 3*x^2)/(4 - 3*x^2))^(3/4)*
Hypergeometric2F1[3/4, 3/4, 7/4, 2/(4 - 3*x^2)]))/(3645*(2 - 3*x^2)^(3/4))

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{\frac{{x}^{7}}{-3\,{x}^{2}+4} \left ( -3\,{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.49938, size = 204, normalized size = 1.09 \[ \frac{2}{729} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{9}{4}} - \frac{16}{81} \cdot 2^{\frac{3}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{16}{81} \cdot 2^{\frac{3}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{8}{81} \cdot 2^{\frac{3}{4}} \log \left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{8}{81} \cdot 2^{\frac{3}{4}} \log \left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) - \frac{16}{405} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{5}{4}} + \frac{56}{81} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/729*(-3*x^2 + 2)^(9/4) - 16/81*2^(3/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(-3*x^2
+ 2)^(1/4))) - 16/81*2^(3/4)*arctan(-1/2*2^(1/4)*(2^(3/4) - 2*(-3*x^2 + 2)^(1/4
))) - 8/81*2^(3/4)*log(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2))
+ 8/81*2^(3/4)*log(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 1
6/405*(-3*x^2 + 2)^(5/4) + 56/81*(-3*x^2 + 2)^(1/4)

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Fricas [A]  time = 0.243048, size = 277, normalized size = 1.47 \[ \frac{32}{81} \cdot 2^{\frac{3}{4}} \arctan \left (\frac{2^{\frac{3}{4}}}{2^{\frac{3}{4}} + 2 \, \sqrt{2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}}\right ) + \frac{32}{81} \cdot 2^{\frac{3}{4}} \arctan \left (-\frac{2^{\frac{3}{4}}}{2^{\frac{3}{4}} - 2 \, \sqrt{-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}}\right ) - \frac{8}{81} \cdot 2^{\frac{3}{4}} \log \left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{8}{81} \cdot 2^{\frac{3}{4}} \log \left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{2}{3645} \,{\left (45 \, x^{4} + 156 \, x^{2} + 1136\right )}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32/81*2^(3/4)*arctan(2^(3/4)/(2^(3/4) + 2*sqrt(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt
(2) + sqrt(-3*x^2 + 2)) + 2*(-3*x^2 + 2)^(1/4))) + 32/81*2^(3/4)*arctan(-2^(3/4)
/(2^(3/4) - 2*sqrt(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 2
*(-3*x^2 + 2)^(1/4))) - 8/81*2^(3/4)*log(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) +
sqrt(-3*x^2 + 2)) + 8/81*2^(3/4)*log(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqr
t(-3*x^2 + 2)) + 2/3645*(45*x^4 + 156*x^2 + 1136)*(-3*x^2 + 2)^(1/4)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.242872, size = 216, normalized size = 1.15 \[ \frac{2}{729} \,{\left (3 \, x^{2} - 2\right )}^{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} - \frac{16}{81} \cdot 2^{\frac{3}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{16}{81} \cdot 2^{\frac{3}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{8}{81} \cdot 2^{\frac{3}{4}}{\rm ln}\left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{8}{81} \cdot 2^{\frac{3}{4}}{\rm ln}\left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) - \frac{16}{405} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{5}{4}} + \frac{56}{81} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/729*(3*x^2 - 2)^2*(-3*x^2 + 2)^(1/4) - 16/81*2^(3/4)*arctan(1/2*2^(1/4)*(2^(3/
4) + 2*(-3*x^2 + 2)^(1/4))) - 16/81*2^(3/4)*arctan(-1/2*2^(1/4)*(2^(3/4) - 2*(-3
*x^2 + 2)^(1/4))) - 8/81*2^(3/4)*ln(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(
-3*x^2 + 2)) + 8/81*2^(3/4)*ln(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x
^2 + 2)) - 16/405*(-3*x^2 + 2)^(5/4) + 56/81*(-3*x^2 + 2)^(1/4)