Optimal. Leaf size=676 \[ -\frac{3 x}{32 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )}+\frac{i \left (4-27 x^2\right )^{2/3}}{96 (2+3 i x)}+\frac{i \left (4-27 x^2\right )^{2/3}}{96 (2+3 i x)^2}-\frac{i \log \left (27\ 2^{2/3} \sqrt [3]{4-27 x^2}+81 i x-54\right )}{192 \sqrt [3]{2}}+\frac{i \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 i x)}{\sqrt{3} \sqrt [3]{4-27 x^2}}\right )}{96 \sqrt [3]{2} \sqrt{3}}+\frac{\left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{144 \sqrt [6]{2} \sqrt [4]{3} \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} x}-\frac{\sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{96\ 2^{2/3} 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} x}+\frac{i \log (2+3 i x)}{192 \sqrt [3]{2}} \]
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Rubi [A] time = 0.997436, antiderivative size = 676, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ -\frac{3 x}{32 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )}+\frac{i \left (4-27 x^2\right )^{2/3}}{96 (2+3 i x)}+\frac{i \left (4-27 x^2\right )^{2/3}}{96 (2+3 i x)^2}-\frac{i \log \left (27\ 2^{2/3} \sqrt [3]{4-27 x^2}+81 i x-54\right )}{192 \sqrt [3]{2}}+\frac{i \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 i x)}{\sqrt{3} \sqrt [3]{4-27 x^2}}\right )}{96 \sqrt [3]{2} \sqrt{3}}+\frac{\left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{144 \sqrt [6]{2} \sqrt [4]{3} \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} x}-\frac{\sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{4-27 x^2}\right ) \sqrt{\frac{\left (4-27 x^2\right )^{2/3}+2^{2/3} \sqrt [3]{4-27 x^2}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}}\right )|-7+4 \sqrt{3}\right )}{96\ 2^{2/3} 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt [3]{4-27 x^2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{4-27 x^2}\right )^2}} x}+\frac{i \log (2+3 i x)}{192 \sqrt [3]{2}} \]
Warning: Unable to verify antiderivative.
[In] Int[1/((2 + (3*I)*x)^3*(4 - 27*x^2)^(1/3)),x]
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Rubi in Sympy [A] time = 41.4353, size = 586, normalized size = 0.87 \[ - \frac{3 \sqrt [3]{2} x}{32 \left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )} + \frac{i \left (- 27 x^{2} + 4\right )^{\frac{2}{3}}}{96 \left (3 i x + 2\right )} + \frac{i \left (- 27 x^{2} + 4\right )^{\frac{2}{3}}}{96 \left (3 i x + 2\right )^{2}} + \frac{2^{\frac{2}{3}} i \log{\left (3 i x + 2 \right )}}{384} - \frac{2^{\frac{2}{3}} i \log{\left (81 i x + 27 \cdot 2^{\frac{2}{3}} \sqrt [3]{- 27 x^{2} + 4} - 54 \right )}}{384} + \frac{2^{\frac{2}{3}} \sqrt{3} i \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{\sqrt [3]{2} \sqrt{3} \left (- 3 i x + 2\right )}{3 \sqrt [3]{- 27 x^{2} + 4}} \right )}}{576} - \frac{2^{\frac{2}{3}} \sqrt [4]{3} \sqrt{\frac{2^{\frac{2}{3}} \left (- 27 x^{2} + 4\right )^{\frac{2}{3}} + 2 \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} + 4}{\left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- 2 \sqrt [3]{- 27 x^{2} + 4} + 2 \cdot 2^{\frac{2}{3}}\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} + 2 + 2 \sqrt{3}}{- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2} \right )}\middle | -7 + 4 \sqrt{3}\right )}{1152 x \sqrt{\frac{2 \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 4}{\left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}}} + \frac{\sqrt [6]{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{2^{\frac{2}{3}} \left (- 27 x^{2} + 4\right )^{\frac{2}{3}} + 2 \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} + 4}{\left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}} \left (- 2 \sqrt [3]{- 27 x^{2} + 4} + 2 \cdot 2^{\frac{2}{3}}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} + 2 + 2 \sqrt{3}}{- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2} \right )}\middle | -7 + 4 \sqrt{3}\right )}{864 x \sqrt{\frac{2 \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 4}{\left (- \sqrt [3]{2} \sqrt [3]{- 27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(2+3*I*x)**3/(-27*x**2+4)**(1/3),x)
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Mathematica [C] time = 0.162708, size = 134, normalized size = 0.2 \[ -\frac{i \sqrt [3]{\frac{2 \sqrt{3}-9 x}{-3 x+2 i}} \sqrt [3]{\frac{9 x+2 \sqrt{3}}{3 x-2 i}} F_1\left (\frac{8}{3};\frac{1}{3},\frac{1}{3};\frac{11}{3};\frac{2 \left (3 i+\sqrt{3}\right )}{6 i-9 x},\frac{2 \left (-3 i+\sqrt{3}\right )}{9 x-6 i}\right )}{8\ 3^{2/3} (-3 x+2 i)^2 \sqrt [3]{4-27 x^2}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/((2 + (3*I)*x)^3*(4 - 27*x^2)^(1/3)),x]
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Maple [F] time = 0.128, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( 2+3\,ix \right ) ^{3}}{\frac{1}{\sqrt [3]{-27\,{x}^{2}+4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(2+3*I*x)^3/(-27*x^2+4)^(1/3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-27 \, x^{2} + 4\right )}^{\frac{1}{3}}{\left (3 i \, x + 2\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-27*x^2 + 4)^(1/3)*(3*I*x + 2)^3),x, algorithm="maxima")
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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)^3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [3]{- 27 x^{2} + 4} \left (3 i x + 2\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(2+3*I*x)**3/(-27*x**2+4)**(1/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-27 \, x^{2} + 4\right )}^{\frac{1}{3}}{\left (3 i \, x + 2\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-27*x^2 + 4)^(1/3)*(3*I*x + 2)^3),x, algorithm="giac")
[Out]