3.2495 \(\int \frac{1}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx\)

Optimal. Leaf size=103 \[ \frac{\log \left (27 \sqrt [3]{2} \sqrt [3]{27 x^2+54 x+28}-81 x-108\right )}{6\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (3 x+4)}{\sqrt{3} \sqrt [3]{27 x^2+54 x+28}}+\frac{1}{\sqrt{3}}\right )}{3\ 2^{2/3} \sqrt{3}}-\frac{\log (3 x+2)}{6\ 2^{2/3}} \]

[Out]

-ArcTan[1/Sqrt[3] + (2^(2/3)*(4 + 3*x))/(Sqrt[3]*(28 + 54*x + 27*x^2)^(1/3))]/(3
*2^(2/3)*Sqrt[3]) - Log[2 + 3*x]/(6*2^(2/3)) + Log[-108 - 81*x + 27*2^(1/3)*(28
+ 54*x + 27*x^2)^(1/3)]/(6*2^(2/3))

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Rubi [A]  time = 0.0561205, antiderivative size = 103, 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 \[ \frac{\log \left (27 \sqrt [3]{2} \sqrt [3]{27 x^2+54 x+28}-81 x-108\right )}{6\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (3 x+4)}{\sqrt{3} \sqrt [3]{27 x^2+54 x+28}}+\frac{1}{\sqrt{3}}\right )}{3\ 2^{2/3} \sqrt{3}}-\frac{\log (3 x+2)}{6\ 2^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[1/((2 + 3*x)*(28 + 54*x + 27*x^2)^(1/3)),x]

[Out]

-ArcTan[1/Sqrt[3] + (2^(2/3)*(4 + 3*x))/(Sqrt[3]*(28 + 54*x + 27*x^2)^(1/3))]/(3
*2^(2/3)*Sqrt[3]) - Log[2 + 3*x]/(6*2^(2/3)) + Log[-108 - 81*x + 27*2^(1/3)*(28
+ 54*x + 27*x^2)^(1/3)]/(6*2^(2/3))

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Rubi in Sympy [A]  time = 7.58725, size = 99, normalized size = 0.96 \[ - \frac{\sqrt [3]{2} \log{\left (3 x + 2 \right )}}{12} + \frac{\sqrt [3]{2} \log{\left (- 81 x + 27 \sqrt [3]{2} \sqrt [3]{27 x^{2} + 54 x + 28} - 108 \right )}}{12} - \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (- \frac{2^{\frac{2}{3}} \sqrt{3} \left (- 81 x - 108\right )}{81 \sqrt [3]{27 x^{2} + 54 x + 28}} + \frac{\sqrt{3}}{3} \right )}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2+3*x)/(27*x**2+54*x+28)**(1/3),x)

[Out]

-2**(1/3)*log(3*x + 2)/12 + 2**(1/3)*log(-81*x + 27*2**(1/3)*(27*x**2 + 54*x + 2
8)**(1/3) - 108)/12 - 2**(1/3)*sqrt(3)*atan(-2**(2/3)*sqrt(3)*(-81*x - 108)/(81*
(27*x**2 + 54*x + 28)**(1/3)) + sqrt(3)/3)/18

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Mathematica [C]  time = 1.04245, size = 294, normalized size = 2.85 \[ -\frac{5 (3 x+2) \left (9 x-i \sqrt{3}+9\right ) \left (9 x+i \sqrt{3}+9\right ) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{3+i \sqrt{3}}{9 x+6},\frac{-3+i \sqrt{3}}{9 x+6}\right )}{2 \left (27 x^2+54 x+28\right )^{4/3} \left (15 (3 x+2) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{3+i \sqrt{3}}{9 x+6},\frac{-3+i \sqrt{3}}{9 x+6}\right )+i \left (\sqrt{3}+3 i\right ) F_1\left (\frac{5}{3};\frac{1}{3},\frac{4}{3};\frac{8}{3};-\frac{3+i \sqrt{3}}{9 x+6},\frac{-3+i \sqrt{3}}{9 x+6}\right )+\left (-3-i \sqrt{3}\right ) F_1\left (\frac{5}{3};\frac{4}{3},\frac{1}{3};\frac{8}{3};-\frac{3+i \sqrt{3}}{9 x+6},\frac{-3+i \sqrt{3}}{9 x+6}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((2 + 3*x)*(28 + 54*x + 27*x^2)^(1/3)),x]

[Out]

(-5*(2 + 3*x)*(9 - I*Sqrt[3] + 9*x)*(9 + I*Sqrt[3] + 9*x)*AppellF1[2/3, 1/3, 1/3
, 5/3, -((3 + I*Sqrt[3])/(6 + 9*x)), (-3 + I*Sqrt[3])/(6 + 9*x)])/(2*(28 + 54*x
+ 27*x^2)^(4/3)*(15*(2 + 3*x)*AppellF1[2/3, 1/3, 1/3, 5/3, -((3 + I*Sqrt[3])/(6
+ 9*x)), (-3 + I*Sqrt[3])/(6 + 9*x)] + I*(3*I + Sqrt[3])*AppellF1[5/3, 1/3, 4/3,
8/3, -((3 + I*Sqrt[3])/(6 + 9*x)), (-3 + I*Sqrt[3])/(6 + 9*x)] + (-3 - I*Sqrt[3
])*AppellF1[5/3, 4/3, 1/3, 8/3, -((3 + I*Sqrt[3])/(6 + 9*x)), (-3 + I*Sqrt[3])/(
6 + 9*x)]))

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Maple [F]  time = 0.101, size = 0, normalized size = 0. \[ \int{\frac{1}{2+3\,x}{\frac{1}{\sqrt [3]{27\,{x}^{2}+54\,x+28}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2+3*x)/(27*x^2+54*x+28)^(1/3),x)

[Out]

int(1/(2+3*x)/(27*x^2+54*x+28)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac{1}{3}}{\left (3 \, x + 2\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((27*x^2 + 54*x + 28)^(1/3)*(3*x + 2)),x, algorithm="maxima")

[Out]

integrate(1/((27*x^2 + 54*x + 28)^(1/3)*(3*x + 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 + 54*x + 28)^(1/3)*(3*x + 2)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (3 x + 2\right ) \sqrt [3]{27 x^{2} + 54 x + 28}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2+3*x)/(27*x**2+54*x+28)**(1/3),x)

[Out]

Integral(1/((3*x + 2)*(27*x**2 + 54*x + 28)**(1/3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac{1}{3}}{\left (3 \, x + 2\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((27*x^2 + 54*x + 28)^(1/3)*(3*x + 2)),x, algorithm="giac")

[Out]

integrate(1/((27*x^2 + 54*x + 28)^(1/3)*(3*x + 2)), x)