3.1014 \(\int \frac{x^4}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx\)

Optimal. Leaf size=536 \[ -\frac{3}{7} \left (1-x^2\right )^{2/3} x+\frac{54 x}{7 \left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )}+\frac{3 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3}}+\frac{9 \tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}-\frac{18 \sqrt{2} 3^{3/4} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{7 \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}+\frac{27 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{7 \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}+\frac{3 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3}}-\frac{3 \tanh ^{-1}(x)}{2\ 2^{2/3}} \]

[Out]

(-3*x*(1 - x^2)^(2/3))/7 + (54*x)/(7*(1 - Sqrt[3] - (1 - x^2)^(1/3))) + (3*Sqrt[
3]*ArcTan[Sqrt[3]/x])/(2*2^(2/3)) + (3*Sqrt[3]*ArcTan[(Sqrt[3]*(1 - 2^(1/3)*(1 -
x^2)^(1/3)))/x])/(2*2^(2/3)) - (3*ArcTanh[x])/(2*2^(2/3)) + (9*ArcTanh[x/(1 + 2
^(1/3)*(1 - x^2)^(1/3))])/(2*2^(2/3)) + (27*3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - (1 -
x^2)^(1/3))*Sqrt[(1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - Sqrt[3] - (1 - x^2
)^(1/3))^2]*EllipticE[ArcSin[(1 + Sqrt[3] - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 -
x^2)^(1/3))], -7 + 4*Sqrt[3]])/(7*x*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] -
(1 - x^2)^(1/3))^2)]) - (18*Sqrt[2]*3^(3/4)*(1 - (1 - x^2)^(1/3))*Sqrt[(1 + (1
- x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2]*EllipticF[Arc
Sin[(1 + Sqrt[3] - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))], -7 + 4*Sqr
t[3]])/(7*x*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2)])

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Rubi [A]  time = 0.621052, antiderivative size = 536, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ -\frac{3}{7} \left (1-x^2\right )^{2/3} x+\frac{54 x}{7 \left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )}+\frac{3 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3}}+\frac{9 \tanh ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}-\frac{18 \sqrt{2} 3^{3/4} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{7 \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}+\frac{27 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt{\frac{\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{1-x^2}+\sqrt{3}+1}{-\sqrt [3]{1-x^2}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{7 \sqrt{-\frac{1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt{3}+1\right )^2}} x}+\frac{3 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3}}-\frac{3 \tanh ^{-1}(x)}{2\ 2^{2/3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-3*x*(1 - x^2)^(2/3))/7 + (54*x)/(7*(1 - Sqrt[3] - (1 - x^2)^(1/3))) + (3*Sqrt[
3]*ArcTan[Sqrt[3]/x])/(2*2^(2/3)) + (3*Sqrt[3]*ArcTan[(Sqrt[3]*(1 - 2^(1/3)*(1 -
x^2)^(1/3)))/x])/(2*2^(2/3)) - (3*ArcTanh[x])/(2*2^(2/3)) + (9*ArcTanh[x/(1 + 2
^(1/3)*(1 - x^2)^(1/3))])/(2*2^(2/3)) + (27*3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - (1 -
x^2)^(1/3))*Sqrt[(1 + (1 - x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - Sqrt[3] - (1 - x^2
)^(1/3))^2]*EllipticE[ArcSin[(1 + Sqrt[3] - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 -
x^2)^(1/3))], -7 + 4*Sqrt[3]])/(7*x*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] -
(1 - x^2)^(1/3))^2)]) - (18*Sqrt[2]*3^(3/4)*(1 - (1 - x^2)^(1/3))*Sqrt[(1 + (1
- x^2)^(1/3) + (1 - x^2)^(2/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2]*EllipticF[Arc
Sin[(1 + Sqrt[3] - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))], -7 + 4*Sqr
t[3]])/(7*x*Sqrt[-((1 - (1 - x^2)^(1/3))/(1 - Sqrt[3] - (1 - x^2)^(1/3))^2)])

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Rubi in Sympy [A]  time = 7.5991, size = 19, normalized size = 0.04 \[ \frac{x^{5} \operatorname{appellf_{1}}{\left (\frac{5}{2},\frac{1}{3},1,\frac{7}{2},x^{2},- \frac{x^{2}}{3} \right )}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**5*appellf1(5/2, 1/3, 1, 7/2, x**2, -x**2/3)/15

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Mathematica [C]  time = 0.334782, size = 236, normalized size = 0.44 \[ \frac{3 x \left (\frac{30 x^2 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )}{\left (x^2+3\right ) \left (2 x^2 \left (F_1\left (\frac{5}{2};\frac{1}{3},2;\frac{7}{2};x^2,-\frac{x^2}{3}\right )-F_1\left (\frac{5}{2};\frac{4}{3},1;\frac{7}{2};x^2,-\frac{x^2}{3}\right )\right )-15 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )\right )}-\frac{27 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )}{\left (x^2+3\right ) \left (2 x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right )-F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )\right )-9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )\right )}+x^2-1\right )}{7 \sqrt [3]{1-x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(3*x*(-1 + x^2 - (27*AppellF1[1/2, 1/3, 1, 3/2, x^2, -x^2/3])/((3 + x^2)*(-9*App
ellF1[1/2, 1/3, 1, 3/2, x^2, -x^2/3] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, x^2, -x
^2/3] - AppellF1[3/2, 4/3, 1, 5/2, x^2, -x^2/3]))) + (30*x^2*AppellF1[3/2, 1/3,
1, 5/2, x^2, -x^2/3])/((3 + x^2)*(-15*AppellF1[3/2, 1/3, 1, 5/2, x^2, -x^2/3] +
2*x^2*(AppellF1[5/2, 1/3, 2, 7/2, x^2, -x^2/3] - AppellF1[5/2, 4/3, 1, 7/2, x^2,
-x^2/3])))))/(7*(1 - x^2)^(1/3))

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \[ \int{\frac{{x}^{4}}{{x}^{2}+3}{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (x^{2} + 3\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (x^{2} + 3\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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