3.548 \(\int \frac{3+2 x}{729-64 x^6} \, dx\)

Optimal. Leaf size=50 \[ \frac{1}{972} \log \left (4 x^2+6 x+9\right )-\frac{1}{486} \log (3-2 x)-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{162 \sqrt{3}} \]

[Out]

-ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(162*Sqrt[3]) - Log[3 - 2*x]/486 + Log[9 + 6*x +
4*x^2]/972

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Rubi [A]  time = 0.0847005, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{1}{972} \log \left (4 x^2+6 x+9\right )-\frac{1}{486} \log (3-2 x)-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{162 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(3 + 2*x)/(729 - 64*x^6),x]

[Out]

-ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(162*Sqrt[3]) - Log[3 - 2*x]/486 + Log[9 + 6*x +
4*x^2]/972

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{- 32 x^{5} + 48 x^{4} - 72 x^{3} + 108 x^{2} - 162 x + 243}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+2*x)/(-64*x**6+729),x)

[Out]

Integral(1/(-32*x**5 + 48*x**4 - 72*x**3 + 108*x**2 - 162*x + 243), x)

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Mathematica [A]  time = 0.0277928, size = 46, normalized size = 0.92 \[ \frac{1}{972} \left (\log \left (4 x^2+6 x+9\right )-2 \log (3-2 x)+2 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(3 + 2*x)/(729 - 64*x^6),x]

[Out]

(2*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] - 2*Log[3 - 2*x] + Log[9 + 6*x + 4*x^2
])/972

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Maple [A]  time = 0.01, size = 39, normalized size = 0.8 \[ -{\frac{\ln \left ( -3+2\,x \right ) }{486}}+{\frac{\ln \left ( 4\,{x}^{2}+6\,x+9 \right ) }{972}}+{\frac{\sqrt{3}}{486}\arctan \left ({\frac{ \left ( 8\,x-6 \right ) \sqrt{3}}{18}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x+3)/(-64*x^6+729),x)

[Out]

-1/486*ln(-3+2*x)+1/972*ln(4*x^2+6*x+9)+1/486*3^(1/2)*arctan(1/18*(8*x-6)*3^(1/2
))

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Maxima [A]  time = 1.55074, size = 51, normalized size = 1.02 \[ \frac{1}{486} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{1}{972} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{1}{486} \, \log \left (2 \, x - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)/(64*x^6 - 729),x, algorithm="maxima")

[Out]

1/486*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) + 1/972*log(4*x^2 + 6*x + 9) - 1/486
*log(2*x - 3)

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Fricas [A]  time = 0.210587, size = 61, normalized size = 1.22 \[ \frac{1}{2916} \, \sqrt{3}{\left (\sqrt{3} \log \left (4 \, x^{2} + 6 \, x + 9\right ) - 2 \, \sqrt{3} \log \left (2 \, x - 3\right ) + 6 \, \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)/(64*x^6 - 729),x, algorithm="fricas")

[Out]

1/2916*sqrt(3)*(sqrt(3)*log(4*x^2 + 6*x + 9) - 2*sqrt(3)*log(2*x - 3) + 6*arctan
(1/9*sqrt(3)*(4*x - 3)))

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Sympy [A]  time = 0.264688, size = 46, normalized size = 0.92 \[ - \frac{\log{\left (x - \frac{3}{2} \right )}}{486} + \frac{\log{\left (4 x^{2} + 6 x + 9 \right )}}{972} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} - \frac{\sqrt{3}}{3} \right )}}{486} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3+2*x)/(-64*x**6+729),x)

[Out]

-log(x - 3/2)/486 + log(4*x**2 + 6*x + 9)/972 + sqrt(3)*atan(4*sqrt(3)*x/9 - sqr
t(3)/3)/486

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GIAC/XCAS [A]  time = 0.217976, size = 53, normalized size = 1.06 \[ \frac{1}{486} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{1}{972} \,{\rm ln}\left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{1}{486} \,{\rm ln}\left ({\left | 2 \, x - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)/(64*x^6 - 729),x, algorithm="giac")

[Out]

1/486*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) + 1/972*ln(4*x^2 + 6*x + 9) - 1/486*
ln(abs(2*x - 3))