Optimal. Leaf size=656 \[ -\frac{3 x}{32 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )}-\frac{\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)}-\frac{\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)^2}+\frac{\log \left (-27\ 2^{2/3} \sqrt [3]{27 x^2+4}-81 x+54\right )}{192 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2} (2-3 x)}{\sqrt{3} \sqrt [3]{27 x^2+4}}+\frac{1}{\sqrt{3}}\right )}{96 \sqrt [3]{2} \sqrt{3}}-\frac{\left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{144 \sqrt [6]{2} \sqrt [4]{3} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}+\frac{\sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{96\ 2^{2/3} 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}-\frac{\log (3 x+2)}{192 \sqrt [3]{2}} \]
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Rubi [A] time = 0.945986, antiderivative size = 656, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{3 x}{32 \left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )}-\frac{\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)}-\frac{\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)^2}+\frac{\log \left (-27\ 2^{2/3} \sqrt [3]{27 x^2+4}-81 x+54\right )}{192 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2} (2-3 x)}{\sqrt{3} \sqrt [3]{27 x^2+4}}+\frac{1}{\sqrt{3}}\right )}{96 \sqrt [3]{2} \sqrt{3}}-\frac{\left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} F\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{144 \sqrt [6]{2} \sqrt [4]{3} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}+\frac{\sqrt{2+\sqrt{3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt{\frac{\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\sin ^{-1}\left (\frac{2^{2/3} \left (1+\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt{3}\right )}{96\ 2^{2/3} 3^{3/4} \sqrt{-\frac{2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt{3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}-\frac{\log (3 x+2)}{192 \sqrt [3]{2}} \]
Warning: Unable to verify antiderivative.
[In] Int[1/((2 + 3*x)^3*(4 + 27*x^2)^(1/3)),x]
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Rubi in Sympy [A] time = 36.6448, size = 570, normalized size = 0.87 \[ - \frac{3 \sqrt [3]{2} x}{32 \left (- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2\right )} - \frac{2^{\frac{2}{3}} \log{\left (3 x + 2 \right )}}{384} + \frac{2^{\frac{2}{3}} \log{\left (- 81 x - 27 \cdot 2^{\frac{2}{3}} \sqrt [3]{27 x^{2} + 4} + 54 \right )}}{384} - \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt [3]{2} \sqrt{3} \left (- 3 x + 2\right )}{3 \sqrt [3]{27 x^{2} + 4}} + \frac{\sqrt{3}}{3} \right )}}{576} - \frac{\left (27 x^{2} + 4\right )^{\frac{2}{3}}}{96 \left (3 x + 2\right )} - \frac{\left (27 x^{2} + 4\right )^{\frac{2}{3}}}{96 \left (3 x + 2\right )^{2}} + \frac{2^{\frac{2}{3}} \sqrt [4]{3} \sqrt{\frac{2^{\frac{2}{3}} \left (27 x^{2} + 4\right )^{\frac{2}{3}} + 2 \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} + 4}{\left (- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- 2 \sqrt [3]{27 x^{2} + 4} + 2 \cdot 2^{\frac{2}{3}}\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} + 2 + 2 \sqrt{3}}{- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2} \right )}\middle | -7 + 4 \sqrt{3}\right )}{1152 x \sqrt{\frac{2 \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 4}{\left (- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}}} - \frac{\sqrt [6]{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{2^{\frac{2}{3}} \left (27 x^{2} + 4\right )^{\frac{2}{3}} + 2 \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} + 4}{\left (- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}} \left (- 2 \sqrt [3]{27 x^{2} + 4} + 2 \cdot 2^{\frac{2}{3}}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} + 2 + 2 \sqrt{3}}{- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2} \right )}\middle | -7 + 4 \sqrt{3}\right )}{864 x \sqrt{\frac{2 \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 4}{\left (- \sqrt [3]{2} \sqrt [3]{27 x^{2} + 4} - 2 \sqrt{3} + 2\right )^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(2+3*x)**3/(27*x**2+4)**(1/3),x)
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Mathematica [C] time = 0.554723, size = 379, normalized size = 0.58 \[ \frac{-\frac{540 (3 x+2) \left (9 x-2 i \sqrt{3}\right ) \left (9 x+2 i \sqrt{3}\right ) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{6-2 i \sqrt{3}}{9 x+6},\frac{6+2 i \sqrt{3}}{9 x+6}\right )}{15 (3 x+2) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{6-2 i \sqrt{3}}{9 x+6},\frac{6+2 i \sqrt{3}}{9 x+6}\right )+\left (6+2 i \sqrt{3}\right ) F_1\left (\frac{5}{3};\frac{1}{3},\frac{4}{3};\frac{8}{3};\frac{6-2 i \sqrt{3}}{9 x+6},\frac{6+2 i \sqrt{3}}{9 x+6}\right )+2 \left (3-i \sqrt{3}\right ) F_1\left (\frac{5}{3};\frac{4}{3},\frac{1}{3};\frac{8}{3};\frac{6-2 i \sqrt{3}}{9 x+6},\frac{6+2 i \sqrt{3}}{9 x+6}\right )}+3\ 3^{5/6} \sqrt [3]{4 \sqrt{3}-18 i x} \left (9 x-2 i \sqrt{3}\right ) \left (27 x^2+4\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{3}{4} i \sqrt{3} x+\frac{1}{2}\right )-\frac{108 (x+1) \left (27 x^2+4\right )^2}{(3 x+2)^2}}{3456 \left (27 x^2+4\right )^{4/3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/((2 + 3*x)^3*(4 + 27*x^2)^(1/3)),x]
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Maple [F] time = 0.058, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( 2+3\,x \right ) ^{3}}{\frac{1}{\sqrt [3]{27\,{x}^{2}+4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(2+3*x)^3/(27*x^2+4)^(1/3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (27 \, x^{2} + 4\right )}^{\frac{1}{3}}{\left (3 \, x + 2\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((27*x^2 + 4)^(1/3)*(3*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*x + 2)^3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (3 x + 2\right )^{3} \sqrt [3]{27 x^{2} + 4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(2+3*x)**3/(27*x**2+4)**(1/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (27 \, x^{2} + 4\right )}^{\frac{1}{3}}{\left (3 \, x + 2\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((27*x^2 + 4)^(1/3)*(3*x + 2)^3),x, algorithm="giac")
[Out]