3.1019 \(\int \frac{x^7}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx\)

Optimal. Leaf size=133 \[ \frac{9 \left (1-x^2\right )^{2/3} \left (14 x^2+69\right )}{40 \left (x^2+3\right )}-\frac{99 \log \left (x^2+3\right )}{16\ 2^{2/3}}+\frac{297 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}+\frac{99 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{8\ 2^{2/3}}-\frac{3 \left (1-x^2\right )^{2/3} x^4}{10 \left (x^2+3\right )} \]

[Out]

(-3*x^4*(1 - x^2)^(2/3))/(10*(3 + x^2)) + (9*(1 - x^2)^(2/3)*(69 + 14*x^2))/(40*
(3 + x^2)) + (99*Sqrt[3]*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]])/(8*2^(2/3)) -
(99*Log[3 + x^2])/(16*2^(2/3)) + (297*Log[2^(2/3) - (1 - x^2)^(1/3)])/(16*2^(2/3
))

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Rubi [A]  time = 0.279958, antiderivative size = 133, 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 \[ \frac{9 \left (1-x^2\right )^{2/3} \left (14 x^2+69\right )}{40 \left (x^2+3\right )}-\frac{99 \log \left (x^2+3\right )}{16\ 2^{2/3}}+\frac{297 \log \left (2^{2/3}-\sqrt [3]{1-x^2}\right )}{16\ 2^{2/3}}+\frac{99 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2-2 x^2}+1}{\sqrt{3}}\right )}{8\ 2^{2/3}}-\frac{3 \left (1-x^2\right )^{2/3} x^4}{10 \left (x^2+3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-3*x^4*(1 - x^2)^(2/3))/(10*(3 + x^2)) + (9*(1 - x^2)^(2/3)*(69 + 14*x^2))/(40*
(3 + x^2)) + (99*Sqrt[3]*ArcTan[(1 + (2 - 2*x^2)^(1/3))/Sqrt[3]])/(8*2^(2/3)) -
(99*Log[3 + x^2])/(16*2^(2/3)) + (297*Log[2^(2/3) - (1 - x^2)^(1/3)])/(16*2^(2/3
))

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Rubi in Sympy [A]  time = 15.4049, size = 121, normalized size = 0.91 \[ - \frac{3 x^{4} \left (- x^{2} + 1\right )^{\frac{2}{3}}}{10 \left (x^{2} + 3\right )} + \frac{9 \left (- x^{2} + 1\right )^{\frac{2}{3}} \left (28 x^{2} + 138\right )}{80 \left (x^{2} + 3\right )} - \frac{99 \sqrt [3]{2} \log{\left (x^{2} + 3 \right )}}{32} + \frac{297 \sqrt [3]{2} \log{\left (- \sqrt [3]{- x^{2} + 1} + 2^{\frac{2}{3}} \right )}}{32} + \frac{99 \sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{\sqrt [3]{2} \sqrt [3]{- x^{2} + 1}}{3} + \frac{1}{3}\right ) \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*x**4*(-x**2 + 1)**(2/3)/(10*(x**2 + 3)) + 9*(-x**2 + 1)**(2/3)*(28*x**2 + 138
)/(80*(x**2 + 3)) - 99*2**(1/3)*log(x**2 + 3)/32 + 297*2**(1/3)*log(-(-x**2 + 1)
**(1/3) + 2**(2/3))/32 + 99*2**(1/3)*sqrt(3)*atan(sqrt(3)*(2**(1/3)*(-x**2 + 1)*
*(1/3)/3 + 1/3))/16

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Mathematica [C]  time = 0.0793001, size = 82, normalized size = 0.62 \[ \frac{3 \left (-495 \sqrt [3]{\frac{x^2-1}{x^2+3}} \left (x^2+3\right ) \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{4}{x^2+3}\right )+4 x^6-46 x^4-165 x^2+207\right )}{40 \sqrt [3]{1-x^2} \left (x^2+3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(3*(207 - 165*x^2 - 46*x^4 + 4*x^6 - 495*((-1 + x^2)/(3 + x^2))^(1/3)*(3 + x^2)*
Hypergeometric2F1[1/3, 1/3, 4/3, 4/(3 + x^2)]))/(40*(1 - x^2)^(1/3)*(3 + x^2))

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \[ \int{\frac{{x}^{7}}{ \left ({x}^{2}+3 \right ) ^{2}}{\frac{1}{\sqrt [3]{-{x}^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.50764, size = 170, normalized size = 1.28 \[ \frac{99}{32} \cdot 4^{\frac{2}{3}} \sqrt{3} \arctan \left (\frac{1}{12} \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (4^{\frac{1}{3}} + 2 \,{\left (-x^{2} + 1\right )}^{\frac{1}{3}}\right )}\right ) + \frac{3}{10} \,{\left (-x^{2} + 1\right )}^{\frac{5}{3}} - \frac{99}{64} \cdot 4^{\frac{2}{3}} \log \left (4^{\frac{2}{3}} + 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} +{\left (-x^{2} + 1\right )}^{\frac{2}{3}}\right ) + \frac{99}{32} \cdot 4^{\frac{2}{3}} \log \left (-4^{\frac{1}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}}\right ) + \frac{15}{4} \,{\left (-x^{2} + 1\right )}^{\frac{2}{3}} + \frac{27 \,{\left (-x^{2} + 1\right )}^{\frac{2}{3}}}{8 \,{\left (x^{2} + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

99/32*4^(2/3)*sqrt(3)*arctan(1/12*4^(2/3)*sqrt(3)*(4^(1/3) + 2*(-x^2 + 1)^(1/3))
) + 3/10*(-x^2 + 1)^(5/3) - 99/64*4^(2/3)*log(4^(2/3) + 4^(1/3)*(-x^2 + 1)^(1/3)
+ (-x^2 + 1)^(2/3)) + 99/32*4^(2/3)*log(-4^(1/3) + (-x^2 + 1)^(1/3)) + 15/4*(-x
^2 + 1)^(2/3) + 27/8*(-x^2 + 1)^(2/3)/(x^2 + 3)

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Fricas [A]  time = 0.235657, size = 174, normalized size = 1.31 \[ \frac{3 \cdot 4^{\frac{2}{3}}{\left (330 \, \sqrt{3}{\left (x^{2} + 3\right )} \arctan \left (\frac{1}{6} \, \sqrt{3}{\left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 2\right )}\right ) - 2 \cdot 4^{\frac{1}{3}}{\left (4 \, x^{4} - 42 \, x^{2} - 207\right )}{\left (-x^{2} + 1\right )}^{\frac{2}{3}} - 165 \,{\left (x^{2} + 3\right )} \log \left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{2}{3}} + 4\right ) + 330 \,{\left (x^{2} + 3\right )} \log \left (4^{\frac{2}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 4\right )\right )}}{320 \,{\left (x^{2} + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/320*4^(2/3)*(330*sqrt(3)*(x^2 + 3)*arctan(1/6*sqrt(3)*(4^(2/3)*(-x^2 + 1)^(1/3
) + 2)) - 2*4^(1/3)*(4*x^4 - 42*x^2 - 207)*(-x^2 + 1)^(2/3) - 165*(x^2 + 3)*log(
4^(2/3)*(-x^2 + 1)^(1/3) + 4^(1/3)*(-x^2 + 1)^(2/3) + 4) + 330*(x^2 + 3)*log(4^(
2/3)*(-x^2 + 1)^(1/3) - 4))/(x^2 + 3)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError