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

Optimal. Leaf size=564 \[ -\frac{1}{30} i \left (4-27 x^2\right )^{2/3} (2+3 i x)^2-\frac{4}{35} (-4 x+7 i) \left (4-27 x^2\right )^{2/3}-\frac{96 x}{7 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )}+\frac{32\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{63 \sqrt [4]{3} x \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}-\frac{16 \sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{21\ 3^{3/4} x \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}} \]

[Out]

(-4*(7*I - 4*x)*(4 - 27*x^2)^(2/3))/35 - (I/30)*(2 + (3*I)*x)^2*(4 - 27*x^2)^(2/
3) - (96*x)/(7*(2^(2/3)*(1 - Sqrt[3]) - (4 - 27*x^2)^(1/3))) - (16*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]*E
llipticE[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^(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)]) + (32*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[Arc
Sin[(2^(2/3)*(1 + Sqrt[3]) - (4 - 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 - 2
7*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(63*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 = 0.793092, antiderivative size = 564, 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 \[ -\frac{1}{30} i \left (4-27 x^2\right )^{2/3} (2+3 i x)^2-\frac{4}{35} (-4 x+7 i) \left (4-27 x^2\right )^{2/3}-\frac{96 x}{7 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )}+\frac{32\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{63 \sqrt [4]{3} x \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}}-\frac{16 \sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{21\ 3^{3/4} x \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-4*(7*I - 4*x)*(4 - 27*x^2)^(2/3))/35 - (I/30)*(2 + (3*I)*x)^2*(4 - 27*x^2)^(2/
3) - (96*x)/(7*(2^(2/3)*(1 - Sqrt[3]) - (4 - 27*x^2)^(1/3))) - (16*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]*E
llipticE[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^(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)]) + (32*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[Arc
Sin[(2^(2/3)*(1 + Sqrt[3]) - (4 - 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 - 2
7*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(63*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 = 29.1245, size = 488, normalized size = 0.87 \[ - \frac{96 \sqrt [3]{2} x}{7 \left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )} - \frac{\left (- 3456 x + 6048 i\right ) \left (- 27 x^{2} + 4\right )^{\frac{2}{3}}}{7560} - \frac{i \left (- 27 x^{2} + 4\right )^{\frac{2}{3}} \left (3 i x + 2\right )^{2}}{30} - \frac{8 \cdot 2^{\frac{2}{3}} \sqrt [4]{3} \sqrt{\frac{2^{\frac{2}{3}} \left (- 27 x^{2} + 4\right )^{\frac{2}{3}} + 2 \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} + 4}{\left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- 2 \sqrt [3]{- 27 x^{2} + 4} + 2 \cdot 2^{\frac{2}{3}}\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} + 2 + 2 \sqrt{3}}{- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2} \right )}\middle | -7 + 4 \sqrt{3}\right )}{63 x \sqrt{\frac{2 \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 4}{\left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}}} + \frac{32 \sqrt [6]{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{2^{\frac{2}{3}} \left (- 27 x^{2} + 4\right )^{\frac{2}{3}} + 2 \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} + 4}{\left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}} \left (- 2 \sqrt [3]{- 27 x^{2} + 4} + 2 \cdot 2^{\frac{2}{3}}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} + 2 + 2 \sqrt{3}}{- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2} \right )}\middle | -7 + 4 \sqrt{3}\right )}{189 x \sqrt{\frac{2 \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 4}{\left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-96*2**(1/3)*x/(7*(-2**(1/3)*(-27*x**2 + 4)**(1/3) - 2*sqrt(3) + 2)) - (-3456*x
+ 6048*I)*(-27*x**2 + 4)**(2/3)/7560 - I*(-27*x**2 + 4)**(2/3)*(3*I*x + 2)**2/30
- 8*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(s
qrt(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))/(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)) + 32*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))/(189*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 = 0.0447698, size = 60, normalized size = 0.11 \[ \frac{16}{7} \sqrt [3]{2} x \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{27 x^2}{4}\right )+\left (4-27 x^2\right )^{2/3} \left (\frac{3 i x^2}{10}+\frac{6 x}{7}-\frac{14 i}{15}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(4 - 27*x^2)^(2/3)*((-14*I)/15 + (6*x)/7 + ((3*I)/10)*x^2) + (16*2^(1/3)*x*Hyper
geometric2F1[1/3, 1/2, 3/2, (27*x^2)/4])/7

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Maple [C]  time = 0.097, size = 49, normalized size = 0.1 \[{-{\frac{i}{210}} \left ( -180\,ix+63\,{x}^{2}-196 \right ) \left ( 27\,{x}^{2}-4 \right ){\frac{1}{\sqrt [3]{-27\,{x}^{2}+4}}}}+{\frac{16\,\sqrt [3]{2}x}{7}{\mbox{$_2$F$_1$}({\frac{1}{3}},{\frac{1}{2}};\,{\frac{3}{2}};\,{\frac{27\,{x}^{2}}{4}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/210*I*(-180*I*x+63*x^2-196)*(27*x^2-4)/(-27*x^2+4)^(1/3)+16/7*2^(1/3)*x*hyper
geom([1/3,1/2],[3/2],27/4*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[ \frac{630 \, x{\rm integral}\left (\frac{128 \,{\left (-27 \, x^{2} + 4\right )}^{\frac{2}{3}}}{63 \,{\left (27 \, x^{4} - 4 \, x^{2}\right )}}, x\right ) +{\left (189 i \, x^{3} + 540 \, x^{2} - 588 i \, x - 320\right )}{\left (-27 \, x^{2} + 4\right )}^{\frac{2}{3}}}{630 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/630*(630*x*integral(128/63*(-27*x^2 + 4)^(2/3)/(27*x^4 - 4*x^2), x) + (189*I*x
^3 + 540*x^2 - 588*I*x - 320)*(-27*x^2 + 4)^(2/3))/x

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Sympy [A]  time = 10.7967, size = 153, normalized size = 0.27 \[ - 9 \sqrt [3]{2} x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{27 x^{2} e^{2 i \pi }}{4}} \right )} + 4 \sqrt [3]{2} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{27 x^{2} e^{2 i \pi }}{4}} \right )} - i \left (- 27 x^{2} + 4\right )^{\frac{2}{3}} - 27 i \left (\begin{cases} \frac{x^{2} \left (27 x^{2} - 4\right )^{\frac{2}{3}} e^{\frac{5 i \pi }{3}}}{90} + \frac{\left (27 x^{2} - 4\right )^{\frac{2}{3}} e^{\frac{5 i \pi }{3}}}{405} & \text{for}\: \frac{27 \left |{x^{2}}\right |}{4} > 1 \\- \frac{x^{2} \left (- 27 x^{2} + 4\right )^{\frac{2}{3}}}{90} - \frac{\left (- 27 x^{2} + 4\right )^{\frac{2}{3}}}{405} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9*2**(1/3)*x**3*hyper((1/3, 3/2), (5/2,), 27*x**2*exp_polar(2*I*pi)/4) + 4*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) - 27*I*Piecewise((x**2*(27*x**2 - 4)**(2/3)*exp(5*I*pi/3)/90 + (27*x**2
- 4)**(2/3)*exp(5*I*pi/3)/405, 27*Abs(x**2)/4 > 1), (-x**2*(-27*x**2 + 4)**(2/3)
/90 - (-27*x**2 + 4)**(2/3)/405, True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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