3.2489 \(\int \frac{1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx\)

Optimal. Leaf size=108 \[ \frac{\log \left (-27 \sqrt [3]{10} \sqrt [3]{27 x^2-54 x+52}-81 x+216\right )}{6\ 10^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (8-3 x)}{\sqrt{3} \sqrt [3]{5} \sqrt [3]{27 x^2-54 x+52}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} 10^{2/3}}-\frac{\log (3 x+2)}{6\ 10^{2/3}} \]

[Out]

-ArcTan[1/Sqrt[3] + (2^(2/3)*(8 - 3*x))/(Sqrt[3]*5^(1/3)*(52 - 54*x + 27*x^2)^(1
/3))]/(3*Sqrt[3]*10^(2/3)) - Log[2 + 3*x]/(6*10^(2/3)) + Log[216 - 81*x - 27*10^
(1/3)*(52 - 54*x + 27*x^2)^(1/3)]/(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]

-ArcTan[1/Sqrt[3] + (2^(2/3)*(8 - 3*x))/(Sqrt[3]*5^(1/3)*(52 - 54*x + 27*x^2)^(1
/3))]/(3*Sqrt[3]*10^(2/3)) - Log[2 + 3*x]/(6*10^(2/3)) + Log[216 - 81*x - 27*10^
(1/3)*(52 - 54*x + 27*x^2)^(1/3)]/(6*10^(2/3))

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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)

[Out]

-10**(1/3)*log(3*x + 2)/60 + 10**(1/3)*log(-81*x - 27*10**(1/3)*(27*x**2 - 54*x
+ 52)**(1/3) + 216)/60 - 10**(1/3)*sqrt(3)*atan(10**(2/3)*sqrt(3)*(-81*x + 216)/
(405*(27*x**2 - 54*x + 52)**(1/3)) + sqrt(3)/3)/90

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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]

[Out]

-((2 + 3*x)*(-9 - (5*I)*Sqrt[3] + 9*x)*(-9 + (5*I)*Sqrt[3] + 9*x)*AppellF1[2/3,
1/3, 1/3, 5/3, (15 - (5*I)*Sqrt[3])/(6 + 9*x), (15 + (5*I)*Sqrt[3])/(6 + 9*x)])/
(2*(52 - 54*x + 27*x^2)^(4/3)*((6 + 9*x)*AppellF1[2/3, 1/3, 1/3, 5/3, (15 - (5*I
)*Sqrt[3])/(6 + 9*x), (15 + (5*I)*Sqrt[3])/(6 + 9*x)] + (3 + I*Sqrt[3])*AppellF1
[5/3, 1/3, 4/3, 8/3, (15 - (5*I)*Sqrt[3])/(6 + 9*x), (15 + (5*I)*Sqrt[3])/(6 + 9
*x)] + (3 - I*Sqrt[3])*AppellF1[5/3, 4/3, 1/3, 8/3, (15 - (5*I)*Sqrt[3])/(6 + 9*
x), (15 + (5*I)*Sqrt[3])/(6 + 9*x)]))

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Maple [F]  time = 0.101, size = 0, normalized size = 0. \[ \int{\frac{1}{2+3\,x}{\frac{1}{\sqrt [3]{27\,{x}^{2}-54\,x+52}}}}\, dx \]

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]

int(1/(2+3*x)/(27*x^2-54*x+52)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac{1}{3}}{\left (3 \, x + 2\right )}}\,{d x} \]

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]

integrate(1/((27*x^2 - 54*x + 52)^(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 + 52)^(1/3)*(3*x + 2)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (3 x + 2\right ) \sqrt [3]{27 x^{2} - 54 x + 52}}\, dx \]

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]

Integral(1/((3*x + 2)*(27*x**2 - 54*x + 52)**(1/3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac{1}{3}}{\left (3 \, x + 2\right )}}\,{d x} \]

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]

integrate(1/((27*x^2 - 54*x + 52)^(1/3)*(3*x + 2)), x)