3.699 \(\int \frac{1}{(2+3 i x)^2 \sqrt [3]{4-27 x^2}} \, dx\)

Optimal. Leaf size=650 \[ -\frac{3 x}{16 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )}+\frac{i \left (4-27 x^2\right )^{2/3}}{48 (2+3 i x)}-\frac{i \log \left (27\ 2^{2/3} \sqrt [3]{4-27 x^2}+81 i x-54\right )}{48 \sqrt [3]{2}}+\frac{i \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 i x)}{\sqrt{3} \sqrt [3]{4-27 x^2}}\right )}{24 \sqrt [3]{2} \sqrt{3}}+\frac{\left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{72 \sqrt [6]{2} \sqrt [4]{3} \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} x}-\frac{\sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{48\ 2^{2/3} 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} x}+\frac{i \log (2+3 i x)}{48 \sqrt [3]{2}} \]

[Out]

((I/48)*(4 - 27*x^2)^(2/3))/(2 + (3*I)*x) - (3*x)/(16*(2^(2/3)*(1 - Sqrt[3]) - (
4 - 27*x^2)^(1/3))) + ((I/24)*ArcTan[1/Sqrt[3] + (2^(1/3)*(2 - (3*I)*x))/(Sqrt[3
]*(4 - 27*x^2)^(1/3))])/(2^(1/3)*Sqrt[3]) - (Sqrt[2 + Sqrt[3]]*(2^(2/3) - (4 - 2
7*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]])/(48*2^(2/3)*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^(2/3) - (4 - 27*x^2)^(1/3))*S
qrt[(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]])/
(72*2^(1/6)*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)]) + ((I/48)*Log[2 + (3*I)*x])/2^(1/3) - ((I/48)*Log[
-54 + (81*I)*x + 27*2^(2/3)*(4 - 27*x^2)^(1/3)])/2^(1/3)

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Rubi [A]  time = 0.828996, antiderivative size = 650, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{3 x}{16 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )}+\frac{i \left (4-27 x^2\right )^{2/3}}{48 (2+3 i x)}-\frac{i \log \left (27\ 2^{2/3} \sqrt [3]{4-27 x^2}+81 i x-54\right )}{48 \sqrt [3]{2}}+\frac{i \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 i x)}{\sqrt{3} \sqrt [3]{4-27 x^2}}\right )}{24 \sqrt [3]{2} \sqrt{3}}+\frac{\left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{72 \sqrt [6]{2} \sqrt [4]{3} \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} x}-\frac{\sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{48\ 2^{2/3} 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} x}+\frac{i \log (2+3 i x)}{48 \sqrt [3]{2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((I/48)*(4 - 27*x^2)^(2/3))/(2 + (3*I)*x) - (3*x)/(16*(2^(2/3)*(1 - Sqrt[3]) - (
4 - 27*x^2)^(1/3))) + ((I/24)*ArcTan[1/Sqrt[3] + (2^(1/3)*(2 - (3*I)*x))/(Sqrt[3
]*(4 - 27*x^2)^(1/3))])/(2^(1/3)*Sqrt[3]) - (Sqrt[2 + Sqrt[3]]*(2^(2/3) - (4 - 2
7*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]])/(48*2^(2/3)*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^(2/3) - (4 - 27*x^2)^(1/3))*S
qrt[(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]])/
(72*2^(1/6)*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)]) + ((I/48)*Log[2 + (3*I)*x])/2^(1/3) - ((I/48)*Log[
-54 + (81*I)*x + 27*2^(2/3)*(4 - 27*x^2)^(1/3)])/2^(1/3)

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Rubi in Sympy [A]  time = 33.5139, size = 564, normalized size = 0.87 \[ - \frac{3 \sqrt [3]{2} x}{16 \left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )} + \frac{i \left (- 27 x^{2} + 4\right )^{\frac{2}{3}}}{48 \left (3 i x + 2\right )} + \frac{2^{\frac{2}{3}} i \log{\left (3 i x + 2 \right )}}{96} - \frac{2^{\frac{2}{3}} i \log{\left (81 i x + 27 \cdot 2^{\frac{2}{3}} \sqrt [3]{- 27 x^{2} + 4} - 54 \right )}}{96} + \frac{2^{\frac{2}{3}} \sqrt{3} i \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{\sqrt [3]{2} \sqrt{3} \left (- 3 i x + 2\right )}{3 \sqrt [3]{- 27 x^{2} + 4}} \right )}}{144} - \frac{2^{\frac{2}{3}} \sqrt [4]{3} \sqrt{\frac{2^{\frac{2}{3}} \left (- 27 x^{2} + 4\right )^{\frac{2}{3}} + 2 \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} + 4}{\left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- 2 \sqrt [3]{- 27 x^{2} + 4} + 2 \cdot 2^{\frac{2}{3}}\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} + 2 + 2 \sqrt{3}}{- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2} \right )}\middle | -7 + 4 \sqrt{3}\right )}{576 x \sqrt{\frac{2 \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 4}{\left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}}} + \frac{\sqrt [6]{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{2^{\frac{2}{3}} \left (- 27 x^{2} + 4\right )^{\frac{2}{3}} + 2 \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} + 4}{\left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}} \left (- 2 \sqrt [3]{- 27 x^{2} + 4} + 2 \cdot 2^{\frac{2}{3}}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} + 2 + 2 \sqrt{3}}{- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2} \right )}\middle | -7 + 4 \sqrt{3}\right )}{432 x \sqrt{\frac{2 \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 4}{\left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*2**(1/3)*x/(16*(-2**(1/3)*(-27*x**2 + 4)**(1/3) - 2*sqrt(3) + 2)) + I*(-27*x*
*2 + 4)**(2/3)/(48*(3*I*x + 2)) + 2**(2/3)*I*log(3*I*x + 2)/96 - 2**(2/3)*I*log(
81*I*x + 27*2**(2/3)*(-27*x**2 + 4)**(1/3) - 54)/96 + 2**(2/3)*sqrt(3)*I*atan(sq
rt(3)/3 + 2**(1/3)*sqrt(3)*(-3*I*x + 2)/(3*(-27*x**2 + 4)**(1/3)))/144 - 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))/(576*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**(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))/(432*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 = 0.194052, size = 132, normalized size = 0.2 \[ \frac{\sqrt [3]{\frac{2 \sqrt{3}-9 x}{-3 x+2 i}} \sqrt [3]{\frac{9 x+2 \sqrt{3}}{3 x-2 i}} F_1\left (\frac{5}{3};\frac{1}{3},\frac{1}{3};\frac{8}{3};\frac{2 \left (3 i+\sqrt{3}\right )}{6 i-9 x},\frac{2 \left (-3 i+\sqrt{3}\right )}{9 x-6 i}\right )}{5\ 3^{2/3} (3 x-2 i) \sqrt [3]{4-27 x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(((2*Sqrt[3] - 9*x)/(2*I - 3*x))^(1/3)*((2*Sqrt[3] + 9*x)/(-2*I + 3*x))^(1/3)*Ap
pellF1[5/3, 1/3, 1/3, 8/3, (2*(3*I + Sqrt[3]))/(6*I - 9*x), (2*(-3*I + Sqrt[3]))
/(-6*I + 9*x)])/(5*3^(2/3)*(-2*I + 3*x)*(4 - 27*x^2)^(1/3))

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Maple [F]  time = 0.123, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( 2+3\,ix \right ) ^{2}}{\frac{1}{\sqrt [3]{-27\,{x}^{2}+4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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