Optimal. Leaf size=108 \[ \frac{\log \left (-27 \sqrt [3]{10} \sqrt [3]{27 x^2-54 x+52}-81 x+216\right )}{6\ 10^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (8-3 x)}{\sqrt{3} \sqrt [3]{5} \sqrt [3]{27 x^2-54 x+52}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} 10^{2/3}}-\frac{\log (3 x+2)}{6\ 10^{2/3}} \]
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Rubi [A] time = 0.0662618, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{\log \left (-27 \sqrt [3]{10} \sqrt [3]{27 x^2-54 x+52}-81 x+216\right )}{6\ 10^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (8-3 x)}{\sqrt{3} \sqrt [3]{5} \sqrt [3]{27 x^2-54 x+52}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} 10^{2/3}}-\frac{\log (3 x+2)}{6\ 10^{2/3}} \]
Antiderivative was successfully verified.
[In] Int[1/((2 + 3*x)*(52 - 54*x + 27*x^2)^(1/3)),x]
[Out]
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Rubi in Sympy [A] time = 7.52415, size = 97, normalized size = 0.9 \[ - \frac{\sqrt [3]{10} \log{\left (3 x + 2 \right )}}{60} + \frac{\sqrt [3]{10} \log{\left (- 81 x - 27 \sqrt [3]{10} \sqrt [3]{27 x^{2} - 54 x + 52} + 216 \right )}}{60} - \frac{\sqrt [3]{10} \sqrt{3} \operatorname{atan}{\left (\frac{10^{\frac{2}{3}} \sqrt{3} \left (- 81 x + 216\right )}{405 \sqrt [3]{27 x^{2} - 54 x + 52}} + \frac{\sqrt{3}}{3} \right )}}{90} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(2+3*x)/(27*x**2-54*x+52)**(1/3),x)
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Mathematica [C] time = 0.972841, size = 288, normalized size = 2.67 \[ -\frac{(3 x+2) \left (9 x-5 i \sqrt{3}-9\right ) \left (9 x+5 i \sqrt{3}-9\right ) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{15-5 i \sqrt{3}}{9 x+6},\frac{15+5 i \sqrt{3}}{9 x+6}\right )}{2 \left (27 x^2-54 x+52\right )^{4/3} \left ((9 x+6) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{15-5 i \sqrt{3}}{9 x+6},\frac{15+5 i \sqrt{3}}{9 x+6}\right )+\left (3+i \sqrt{3}\right ) F_1\left (\frac{5}{3};\frac{1}{3},\frac{4}{3};\frac{8}{3};\frac{15-5 i \sqrt{3}}{9 x+6},\frac{15+5 i \sqrt{3}}{9 x+6}\right )+\left (3-i \sqrt{3}\right ) F_1\left (\frac{5}{3};\frac{4}{3},\frac{1}{3};\frac{8}{3};\frac{15-5 i \sqrt{3}}{9 x+6},\frac{15+5 i \sqrt{3}}{9 x+6}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/((2 + 3*x)*(52 - 54*x + 27*x^2)^(1/3)),x]
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Maple [F] time = 0.101, size = 0, normalized size = 0. \[ \int{\frac{1}{2+3\,x}{\frac{1}{\sqrt [3]{27\,{x}^{2}-54\,x+52}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(2+3*x)/(27*x^2-54*x+52)^(1/3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac{1}{3}}{\left (3 \, x + 2\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((27*x^2 - 54*x + 52)^(1/3)*(3*x + 2)),x, algorithm="maxima")
[Out]
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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 - 54*x + 52)^(1/3)*(3*x + 2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (3 x + 2\right ) \sqrt [3]{27 x^{2} - 54 x + 52}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(2+3*x)/(27*x**2-54*x+52)**(1/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac{1}{3}}{\left (3 \, x + 2\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((27*x^2 - 54*x + 52)^(1/3)*(3*x + 2)),x, algorithm="giac")
[Out]