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

Optimal. Leaf size=557 \[ -\frac{1}{21} i \left (4-27 x^2\right )^{2/3} (2+3 i x)-\frac{5}{21} i \left (4-27 x^2\right )^{2/3}-\frac{72 x}{7 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )}+\frac{8\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{21 \sqrt [4]{3} x \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}-\frac{4 \sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{7\ 3^{3/4} x \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}} \]

[Out]

((-5*I)/21)*(4 - 27*x^2)^(2/3) - (I/21)*(2 + (3*I)*x)*(4 - 27*x^2)^(2/3) - (72*x
)/(7*(2^(2/3)*(1 - Sqrt[3]) - (4 - 27*x^2)^(1/3))) - (4*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[Ar
cSin[(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]])/(7*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)]) + (8*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 - Sqrt[3]) - (4 - 27*x^2)^(1/3))
], -7 + 4*Sqrt[3]])/(21*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)])

_______________________________________________________________________________________

Rubi [A]  time = 0.719622, antiderivative size = 557, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{1}{21} i \left (4-27 x^2\right )^{2/3} (2+3 i x)-\frac{5}{21} i \left (4-27 x^2\right )^{2/3}-\frac{72 x}{7 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )}+\frac{8\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{21 \sqrt [4]{3} x \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}-\frac{4 \sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{7\ 3^{3/4} x \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-5*I)/21)*(4 - 27*x^2)^(2/3) - (I/21)*(2 + (3*I)*x)*(4 - 27*x^2)^(2/3) - (72*x
)/(7*(2^(2/3)*(1 - Sqrt[3]) - (4 - 27*x^2)^(1/3))) - (4*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[Ar
cSin[(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]])/(7*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)]) + (8*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 - Sqrt[3]) - (4 - 27*x^2)^(1/3))
], -7 + 4*Sqrt[3]])/(21*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)])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 25.8046, size = 483, normalized size = 0.87 \[ - \frac{72 \sqrt [3]{2} x}{7 \left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )} - \frac{i \left (- 27 x^{2} + 4\right )^{\frac{2}{3}} \left (3 i x + 2\right )}{21} - \frac{5 i \left (- 27 x^{2} + 4\right )^{\frac{2}{3}}}{21} - \frac{2 \cdot 2^{\frac{2}{3}} \sqrt [4]{3} \sqrt{\frac{2^{\frac{2}{3}} \left (- 27 x^{2} + 4\right )^{\frac{2}{3}} + 2 \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} + 4}{\left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- 2 \sqrt [3]{- 27 x^{2} + 4} + 2 \cdot 2^{\frac{2}{3}}\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} + 2 + 2 \sqrt{3}}{- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2} \right )}\middle | -7 + 4 \sqrt{3}\right )}{21 x \sqrt{\frac{2 \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 4}{\left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}}} + \frac{8 \sqrt [6]{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{2^{\frac{2}{3}} \left (- 27 x^{2} + 4\right )^{\frac{2}{3}} + 2 \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} + 4}{\left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}} \left (- 2 \sqrt [3]{- 27 x^{2} + 4} + 2 \cdot 2^{\frac{2}{3}}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} + 2 + 2 \sqrt{3}}{- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2} \right )}\middle | -7 + 4 \sqrt{3}\right )}{63 x \sqrt{\frac{2 \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 4}{\left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-72*2**(1/3)*x/(7*(-2**(1/3)*(-27*x**2 + 4)**(1/3) - 2*sqrt(3) + 2)) - I*(-27*x*
*2 + 4)**(2/3)*(3*I*x + 2)/21 - 5*I*(-27*x**2 + 4)**(2/3)/21 - 2*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*sqr
t(3))/(21*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)) + 8*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))/(63*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))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0521691, size = 51, normalized size = 0.09 \[ \frac{12}{7} \sqrt [3]{2} x \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{27 x^2}{4}\right )+\left (4-27 x^2\right )^{2/3} \left (\frac{x}{7}-\frac{i}{3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-I/3 + x/7)*(4 - 27*x^2)^(2/3) + (12*2^(1/3)*x*Hypergeometric2F1[1/3, 1/2, 3/2,
(27*x^2)/4])/7

_______________________________________________________________________________________

Maple [C]  time = 0.068, size = 43, normalized size = 0.1 \[ -{\frac{ \left ( -7\,i+3\,x \right ) \left ( 27\,{x}^{2}-4 \right ) }{21}{\frac{1}{\sqrt [3]{-27\,{x}^{2}+4}}}}+{\frac{12\,\sqrt [3]{2}x}{7}{\mbox{$_2$F$_1$}({\frac{1}{3}},{\frac{1}{2}};\,{\frac{3}{2}};\,{\frac{27\,{x}^{2}}{4}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/21*(-7*I+3*x)*(27*x^2-4)/(-27*x^2+4)^(1/3)+12/7*2^(1/3)*x*hypergeom([1/3,1/2]
,[3/2],27/4*x^2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[ \frac{21 \, x{\rm integral}\left (\frac{32 \,{\left (-27 \, x^{2} + 4\right )}^{\frac{2}{3}}}{21 \,{\left (27 \, x^{4} - 4 \, x^{2}\right )}}, x\right ) +{\left (3 \, x^{2} - 7 i \, x - 8\right )}{\left (-27 \, x^{2} + 4\right )}^{\frac{2}{3}}}{21 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21*(21*x*integral(32/21*(-27*x^2 + 4)^(2/3)/(27*x^4 - 4*x^2), x) + (3*x^2 - 7*
I*x - 8)*(-27*x^2 + 4)^(2/3))/x

_______________________________________________________________________________________

Sympy [A]  time = 6.81824, size = 73, normalized size = 0.13 \[ - \frac{3 \sqrt [3]{2} x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{27 x^{2} e^{2 i \pi }}{4}} \right )}}{2} + 2 \sqrt [3]{2} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{27 x^{2} e^{2 i \pi }}{4}} \right )} - \frac{i \left (- 27 x^{2} + 4\right )^{\frac{2}{3}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*2**(1/3)*x**3*hyper((1/3, 3/2), (5/2,), 27*x**2*exp_polar(2*I*pi)/4)/2 + 2*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)/3

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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