3.697 \(\int \frac{2+3 i x}{\sqrt [3]{4-27 x^2}} \, dx\)

Optimal. Leaf size=531 \[ -\frac{6 x}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}-\frac{1}{12} i \left (4-27 x^2\right )^{2/3}+\frac{2\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{9 \sqrt [4]{3} \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} x}-\frac{\sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{3\ 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} x} \]

[Out]

(-I/12)*(4 - 27*x^2)^(2/3) - (6*x)/(2^(2/3)*(1 - Sqrt[3]) - (4 - 27*x^2)^(1/3))
- (2^(1/3)*Sqrt[2 + Sqrt[3]]*(2^(2/3) - (4 - 27*x^2)^(1/3))*Sqrt[(2*2^(1/3) + 2^
(2/3)*(4 - 27*x^2)^(1/3) + (4 - 27*x^2)^(2/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 - 27*
x^2)^(1/3))^2]*EllipticE[ArcSin[(2^(2/3)*(1 + Sqrt[3]) - (4 - 27*x^2)^(1/3))/(2^
(2/3)*(1 - Sqrt[3]) - (4 - 27*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(3*3^(3/4)*x*Sqrt[-
((2^(2/3) - (4 - 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 - 27*x^2)^(1/3))^2)]
) + (2*2^(5/6)*(2^(2/3) - (4 - 27*x^2)^(1/3))*Sqrt[(2*2^(1/3) + 2^(2/3)*(4 - 27*
x^2)^(1/3) + (4 - 27*x^2)^(2/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 - 27*x^2)^(1/3))^2]
*EllipticF[ArcSin[(2^(2/3)*(1 + Sqrt[3]) - (4 - 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqr
t[3]) - (4 - 27*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(9*3^(1/4)*x*Sqrt[-((2^(2/3) - (4
- 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 - 27*x^2)^(1/3))^2)])

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Rubi [A]  time = 0.651782, antiderivative size = 531, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{6 x}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}-\frac{1}{12} i \left (4-27 x^2\right )^{2/3}+\frac{2\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{9 \sqrt [4]{3} \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} x}-\frac{\sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{3\ 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} x} \]

Warning: Unable to verify antiderivative.

[In]  Int[(2 + (3*I)*x)/(4 - 27*x^2)^(1/3),x]

[Out]

(-I/12)*(4 - 27*x^2)^(2/3) - (6*x)/(2^(2/3)*(1 - Sqrt[3]) - (4 - 27*x^2)^(1/3))
- (2^(1/3)*Sqrt[2 + Sqrt[3]]*(2^(2/3) - (4 - 27*x^2)^(1/3))*Sqrt[(2*2^(1/3) + 2^
(2/3)*(4 - 27*x^2)^(1/3) + (4 - 27*x^2)^(2/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 - 27*
x^2)^(1/3))^2]*EllipticE[ArcSin[(2^(2/3)*(1 + Sqrt[3]) - (4 - 27*x^2)^(1/3))/(2^
(2/3)*(1 - Sqrt[3]) - (4 - 27*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(3*3^(3/4)*x*Sqrt[-
((2^(2/3) - (4 - 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 - 27*x^2)^(1/3))^2)]
) + (2*2^(5/6)*(2^(2/3) - (4 - 27*x^2)^(1/3))*Sqrt[(2*2^(1/3) + 2^(2/3)*(4 - 27*
x^2)^(1/3) + (4 - 27*x^2)^(2/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 - 27*x^2)^(1/3))^2]
*EllipticF[ArcSin[(2^(2/3)*(1 + Sqrt[3]) - (4 - 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqr
t[3]) - (4 - 27*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(9*3^(1/4)*x*Sqrt[-((2^(2/3) - (4
- 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 - 27*x^2)^(1/3))^2)])

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Rubi in Sympy [A]  time = 20.5693, size = 457, normalized size = 0.86 \[ - \frac{6 \sqrt [3]{2} x}{- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2} - \frac{i \left (- 27 x^{2} + 4\right )^{\frac{2}{3}}}{12} - \frac{2^{\frac{2}{3}} \sqrt [4]{3} \sqrt{\frac{2^{\frac{2}{3}} \left (- 27 x^{2} + 4\right )^{\frac{2}{3}} + 2 \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} + 4}{\left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- 2 \sqrt [3]{- 27 x^{2} + 4} + 2 \cdot 2^{\frac{2}{3}}\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} + 2 + 2 \sqrt{3}}{- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2} \right )}\middle | -7 + 4 \sqrt{3}\right )}{18 x \sqrt{\frac{2 \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 4}{\left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}}} + \frac{2 \sqrt [6]{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{2^{\frac{2}{3}} \left (- 27 x^{2} + 4\right )^{\frac{2}{3}} + 2 \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} + 4}{\left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}} \left (- 2 \sqrt [3]{- 27 x^{2} + 4} + 2 \cdot 2^{\frac{2}{3}}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} + 2 + 2 \sqrt{3}}{- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2} \right )}\middle | -7 + 4 \sqrt{3}\right )}{27 x \sqrt{\frac{2 \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 4}{\left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*I*x)/(-27*x**2+4)**(1/3),x)

[Out]

-6*2**(1/3)*x/(-2**(1/3)*(-27*x**2 + 4)**(1/3) - 2*sqrt(3) + 2) - I*(-27*x**2 +
4)**(2/3)/12 - 2**(2/3)*3**(1/4)*sqrt((2**(2/3)*(-27*x**2 + 4)**(2/3) + 2*2**(1/
3)*(-27*x**2 + 4)**(1/3) + 4)/(-2**(1/3)*(-27*x**2 + 4)**(1/3) - 2*sqrt(3) + 2)*
*2)*sqrt(sqrt(3) + 2)*(-2*(-27*x**2 + 4)**(1/3) + 2*2**(2/3))*elliptic_e(asin((-
2**(1/3)*(-27*x**2 + 4)**(1/3) + 2 + 2*sqrt(3))/(-2**(1/3)*(-27*x**2 + 4)**(1/3)
- 2*sqrt(3) + 2)), -7 + 4*sqrt(3))/(18*x*sqrt((2*2**(1/3)*(-27*x**2 + 4)**(1/3)
- 4)/(-2**(1/3)*(-27*x**2 + 4)**(1/3) - 2*sqrt(3) + 2)**2)) + 2*2**(1/6)*3**(3/
4)*sqrt((2**(2/3)*(-27*x**2 + 4)**(2/3) + 2*2**(1/3)*(-27*x**2 + 4)**(1/3) + 4)/
(-2**(1/3)*(-27*x**2 + 4)**(1/3) - 2*sqrt(3) + 2)**2)*(-2*(-27*x**2 + 4)**(1/3)
+ 2*2**(2/3))*elliptic_f(asin((-2**(1/3)*(-27*x**2 + 4)**(1/3) + 2 + 2*sqrt(3))/
(-2**(1/3)*(-27*x**2 + 4)**(1/3) - 2*sqrt(3) + 2)), -7 + 4*sqrt(3))/(27*x*sqrt((
2*2**(1/3)*(-27*x**2 + 4)**(1/3) - 4)/(-2**(1/3)*(-27*x**2 + 4)**(1/3) - 2*sqrt(
3) + 2)**2))

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Mathematica [C]  time = 0.0257542, size = 42, normalized size = 0.08 \[ \sqrt [3]{2} x \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{27 x^2}{4}\right )-\frac{1}{12} i \left (4-27 x^2\right )^{2/3} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + (3*I)*x)/(4 - 27*x^2)^(1/3),x]

[Out]

(-I/12)*(4 - 27*x^2)^(2/3) + 2^(1/3)*x*Hypergeometric2F1[1/3, 1/2, 3/2, (27*x^2)
/4]

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Maple [C]  time = 0.06, size = 37, normalized size = 0.1 \[{{\frac{i}{12}} \left ( 27\,{x}^{2}-4 \right ){\frac{1}{\sqrt [3]{-27\,{x}^{2}+4}}}}+\sqrt [3]{2}x{\mbox{$_2$F$_1$}({\frac{1}{3}},{\frac{1}{2}};\,{\frac{3}{2}};\,{\frac{27\,{x}^{2}}{4}})} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*I*x)/(-27*x^2+4)^(1/3),x)

[Out]

1/12*I*(27*x^2-4)/(-27*x^2+4)^(1/3)+2^(1/3)*x*hypergeom([1/3,1/2],[3/2],27/4*x^2
)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{3 i \, x + 2}{{\left (-27 \, x^{2} + 4\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*I*x + 2)/(-27*x^2 + 4)^(1/3),x, algorithm="maxima")

[Out]

integrate((3*I*x + 2)/(-27*x^2 + 4)^(1/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[ \frac{36 \, x{\rm integral}\left (\frac{8 \,{\left (-27 \, x^{2} + 4\right )}^{\frac{2}{3}}}{9 \,{\left (27 \, x^{4} - 4 \, x^{2}\right )}}, x\right ) +{\left (-27 \, x^{2} + 4\right )}^{\frac{2}{3}}{\left (-3 i \, x - 8\right )}}{36 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*I*x + 2)/(-27*x^2 + 4)^(1/3),x, algorithm="fricas")

[Out]

1/36*(36*x*integral(8/9*(-27*x^2 + 4)^(2/3)/(27*x^4 - 4*x^2), x) + (-27*x^2 + 4)
^(2/3)*(-3*I*x - 8))/x

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Sympy [A]  time = 3.88009, size = 39, normalized size = 0.07 \[ \sqrt [3]{2} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{27 x^{2} e^{2 i \pi }}{4}} \right )} - \frac{i \left (- 27 x^{2} + 4\right )^{\frac{2}{3}}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+3*I*x)/(-27*x**2+4)**(1/3),x)

[Out]

2**(1/3)*x*hyper((1/3, 1/2), (3/2,), 27*x**2*exp_polar(2*I*pi)/4) - I*(-27*x**2
+ 4)**(2/3)/12

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{3 i \, x + 2}{{\left (-27 \, x^{2} + 4\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*I*x + 2)/(-27*x^2 + 4)^(1/3),x, algorithm="giac")

[Out]

integrate((3*I*x + 2)/(-27*x^2 + 4)^(1/3), x)