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

Optimal. Leaf size=97 \[ \frac{\log \left (-27\ 2^{2/3} \sqrt [3]{27 x^2+4}-81 x+54\right )}{12 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2} (2-3 x)}{\sqrt{3} \sqrt [3]{27 x^2+4}}+\frac{1}{\sqrt{3}}\right )}{6 \sqrt [3]{2} \sqrt{3}}-\frac{\log (3 x+2)}{12 \sqrt [3]{2}} \]

[Out]

-ArcTan[1/Sqrt[3] + (2^(1/3)*(2 - 3*x))/(Sqrt[3]*(4 + 27*x^2)^(1/3))]/(6*2^(1/3)
*Sqrt[3]) - Log[2 + 3*x]/(12*2^(1/3)) + Log[54 - 81*x - 27*2^(2/3)*(4 + 27*x^2)^
(1/3)]/(12*2^(1/3))

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Rubi [A]  time = 0.0635902, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{\log \left (-27\ 2^{2/3} \sqrt [3]{27 x^2+4}-81 x+54\right )}{12 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2} (2-3 x)}{\sqrt{3} \sqrt [3]{27 x^2+4}}+\frac{1}{\sqrt{3}}\right )}{6 \sqrt [3]{2} \sqrt{3}}-\frac{\log (3 x+2)}{12 \sqrt [3]{2}} \]

Antiderivative was successfully verified.

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

[Out]

-ArcTan[1/Sqrt[3] + (2^(1/3)*(2 - 3*x))/(Sqrt[3]*(4 + 27*x^2)^(1/3))]/(6*2^(1/3)
*Sqrt[3]) - Log[2 + 3*x]/(12*2^(1/3)) + Log[54 - 81*x - 27*2^(2/3)*(4 + 27*x^2)^
(1/3)]/(12*2^(1/3))

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Rubi in Sympy [A]  time = 4.28696, size = 90, normalized size = 0.93 \[ - \frac{2^{\frac{2}{3}} \log{\left (3 x + 2 \right )}}{24} + \frac{2^{\frac{2}{3}} \log{\left (- 81 x - 27 \cdot 2^{\frac{2}{3}} \sqrt [3]{27 x^{2} + 4} + 54 \right )}}{24} - \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt [3]{2} \sqrt{3} \left (- 3 x + 2\right )}{3 \sqrt [3]{27 x^{2} + 4}} + \frac{\sqrt{3}}{3} \right )}}{36} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2**(2/3)*log(3*x + 2)/24 + 2**(2/3)*log(-81*x - 27*2**(2/3)*(27*x**2 + 4)**(1/3
) + 54)/24 - 2**(2/3)*sqrt(3)*atan(2**(1/3)*sqrt(3)*(-3*x + 2)/(3*(27*x**2 + 4)*
*(1/3)) + sqrt(3)/3)/36

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Mathematica [C]  time = 1.01335, size = 285, normalized size = 2.94 \[ -\frac{5 (3 x+2) \left (9 x-2 i \sqrt{3}\right ) \left (9 x+2 i \sqrt{3}\right ) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{6-2 i \sqrt{3}}{9 x+6},\frac{6+2 i \sqrt{3}}{9 x+6}\right )}{2 \left (27 x^2+4\right )^{4/3} \left (15 (3 x+2) F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{6-2 i \sqrt{3}}{9 x+6},\frac{6+2 i \sqrt{3}}{9 x+6}\right )+\left (6+2 i \sqrt{3}\right ) F_1\left (\frac{5}{3};\frac{1}{3},\frac{4}{3};\frac{8}{3};\frac{6-2 i \sqrt{3}}{9 x+6},\frac{6+2 i \sqrt{3}}{9 x+6}\right )+2 \left (3-i \sqrt{3}\right ) F_1\left (\frac{5}{3};\frac{4}{3},\frac{1}{3};\frac{8}{3};\frac{6-2 i \sqrt{3}}{9 x+6},\frac{6+2 i \sqrt{3}}{9 x+6}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int{\frac{1}{2+3\,x}{\frac{1}{\sqrt [3]{27\,{x}^{2}+4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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