3.1062 \(\int \frac{x^5}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx\)

Optimal. Leaf size=173 \[ -\frac{2}{135} \left (2-3 x^2\right )^{5/4}+\frac{4}{9} \sqrt [4]{2-3 x^2}+\frac{2}{27} 2^{3/4} \log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )-\frac{2}{27} 2^{3/4} \log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )-\frac{4}{27} 2^{3/4} \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )+\frac{4}{27} 2^{3/4} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right ) \]

[Out]

(4*(2 - 3*x^2)^(1/4))/9 - (2*(2 - 3*x^2)^(5/4))/135 - (4*2^(3/4)*ArcTan[1 + (4 -
6*x^2)^(1/4)])/27 + (4*2^(3/4)*ArcTan[1 - 2^(1/4)*(2 - 3*x^2)^(1/4)])/27 + (2*2
^(3/4)*Log[Sqrt[2] - 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]])/27 - (2*2^(3/
4)*Log[Sqrt[2] + 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]])/27

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Rubi [A]  time = 0.424921, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{2}{135} \left (2-3 x^2\right )^{5/4}+\frac{4}{9} \sqrt [4]{2-3 x^2}+\frac{2}{27} 2^{3/4} \log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )-\frac{2}{27} 2^{3/4} \log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )-\frac{4}{27} 2^{3/4} \tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )+\frac{4}{27} 2^{3/4} \tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^5/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]

[Out]

(4*(2 - 3*x^2)^(1/4))/9 - (2*(2 - 3*x^2)^(5/4))/135 - (4*2^(3/4)*ArcTan[1 + (4 -
6*x^2)^(1/4)])/27 + (4*2^(3/4)*ArcTan[1 - 2^(1/4)*(2 - 3*x^2)^(1/4)])/27 + (2*2
^(3/4)*Log[Sqrt[2] - 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]])/27 - (2*2^(3/
4)*Log[Sqrt[2] + 2^(3/4)*(2 - 3*x^2)^(1/4) + Sqrt[2 - 3*x^2]])/27

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Rubi in Sympy [A]  time = 33.0568, size = 162, normalized size = 0.94 \[ - \frac{2 \left (- 3 x^{2} + 2\right )^{\frac{5}{4}}}{135} + \frac{4 \sqrt [4]{- 3 x^{2} + 2}}{9} + \frac{2 \cdot 2^{\frac{3}{4}} \log{\left (- 2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{27} - \frac{2 \cdot 2^{\frac{3}{4}} \log{\left (2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{27} - \frac{4 \cdot 2^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} - 1 \right )}}{27} - \frac{4 \cdot 2^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} + 1 \right )}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(-3*x**2+2)**(3/4)/(-3*x**2+4),x)

[Out]

-2*(-3*x**2 + 2)**(5/4)/135 + 4*(-3*x**2 + 2)**(1/4)/9 + 2*2**(3/4)*log(-2**(3/4
)*(-3*x**2 + 2)**(1/4) + sqrt(-3*x**2 + 2) + sqrt(2))/27 - 2*2**(3/4)*log(2**(3/
4)*(-3*x**2 + 2)**(1/4) + sqrt(-3*x**2 + 2) + sqrt(2))/27 - 4*2**(3/4)*atan(2**(
1/4)*(-3*x**2 + 2)**(1/4) - 1)/27 - 4*2**(3/4)*atan(2**(1/4)*(-3*x**2 + 2)**(1/4
) + 1)/27

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Mathematica [C]  time = 0.0795577, size = 74, normalized size = 0.43 \[ -\frac{2 \left (3 \left (9 x^4+78 x^2-56\right )-80 \left (\frac{2-3 x^2}{4-3 x^2}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{2}{4-3 x^2}\right )\right )}{405 \left (2-3 x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/((2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]

[Out]

(-2*(3*(-56 + 78*x^2 + 9*x^4) - 80*((2 - 3*x^2)/(4 - 3*x^2))^(3/4)*Hypergeometri
c2F1[3/4, 3/4, 7/4, 2/(4 - 3*x^2)]))/(405*(2 - 3*x^2)^(3/4))

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int{\frac{{x}^{5}}{-3\,{x}^{2}+4} \left ( -3\,{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(-3*x^2+2)^(3/4)/(-3*x^2+4),x)

[Out]

int(x^5/(-3*x^2+2)^(3/4)/(-3*x^2+4),x)

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Maxima [A]  time = 1.51104, size = 189, normalized size = 1.09 \[ -\frac{4}{27} \cdot 2^{\frac{3}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{4}{27} \cdot 2^{\frac{3}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{2}{27} \cdot 2^{\frac{3}{4}} \log \left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{2}{27} \cdot 2^{\frac{3}{4}} \log \left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) - \frac{2}{135} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{5}{4}} + \frac{4}{9} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^5/((3*x^2 - 4)*(-3*x^2 + 2)^(3/4)),x, algorithm="maxima")

[Out]

-4/27*2^(3/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) - 4/27*2^(3/4
)*arctan(-1/2*2^(1/4)*(2^(3/4) - 2*(-3*x^2 + 2)^(1/4))) - 2/27*2^(3/4)*log(2^(3/
4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 2/27*2^(3/4)*log(-2^(3/4)*
(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 2/135*(-3*x^2 + 2)^(5/4) + 4/
9*(-3*x^2 + 2)^(1/4)

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Fricas [A]  time = 0.243554, size = 270, normalized size = 1.56 \[ \frac{8}{27} \cdot 2^{\frac{3}{4}} \arctan \left (\frac{2^{\frac{3}{4}}}{2^{\frac{3}{4}} + 2 \, \sqrt{2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}}\right ) + \frac{8}{27} \cdot 2^{\frac{3}{4}} \arctan \left (-\frac{2^{\frac{3}{4}}}{2^{\frac{3}{4}} - 2 \, \sqrt{-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}}\right ) - \frac{2}{27} \cdot 2^{\frac{3}{4}} \log \left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{2}{27} \cdot 2^{\frac{3}{4}} \log \left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{2}{135} \,{\left (3 \, x^{2} + 28\right )}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^5/((3*x^2 - 4)*(-3*x^2 + 2)^(3/4)),x, algorithm="fricas")

[Out]

8/27*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))) + 8/27*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))) - 2/27*2^(3/4)*log(2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sq
rt(-3*x^2 + 2)) + 2/27*2^(3/4)*log(-2^(3/4)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(
-3*x^2 + 2)) + 2/135*(3*x^2 + 28)*(-3*x^2 + 2)^(1/4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x^{5}}{3 x^{2} \left (- 3 x^{2} + 2\right )^{\frac{3}{4}} - 4 \left (- 3 x^{2} + 2\right )^{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(-3*x**2+2)**(3/4)/(-3*x**2+4),x)

[Out]

-Integral(x**5/(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 = 0.249008, size = 189, normalized size = 1.09 \[ -\frac{4}{27} \cdot 2^{\frac{3}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{4}{27} \cdot 2^{\frac{3}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{2}{27} \cdot 2^{\frac{3}{4}}{\rm ln}\left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{2}{27} \cdot 2^{\frac{3}{4}}{\rm ln}\left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) - \frac{2}{135} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{5}{4}} + \frac{4}{9} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^5/((3*x^2 - 4)*(-3*x^2 + 2)^(3/4)),x, algorithm="giac")

[Out]

-4/27*2^(3/4)*arctan(1/2*2^(1/4)*(2^(3/4) + 2*(-3*x^2 + 2)^(1/4))) - 4/27*2^(3/4
)*arctan(-1/2*2^(1/4)*(2^(3/4) - 2*(-3*x^2 + 2)^(1/4))) - 2/27*2^(3/4)*ln(2^(3/4
)*(-3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) + 2/27*2^(3/4)*ln(-2^(3/4)*(-
3*x^2 + 2)^(1/4) + sqrt(2) + sqrt(-3*x^2 + 2)) - 2/135*(-3*x^2 + 2)^(5/4) + 4/9*
(-3*x^2 + 2)^(1/4)