3.554 \(\int \frac{243+162 x+108 x^2+72 x^3+48 x^4+32 x^5}{\left (729-64 x^6\right )^2} \, dx\)

Optimal. Leaf size=110 \[ \frac{\log \left (4 x^2-6 x+9\right )}{17496}+\frac{\log \left (4 x^2+6 x+9\right )}{17496}+\frac{1}{2916 (3-2 x)}-\frac{5 \log (3-2 x)}{17496}+\frac{\log (2 x+3)}{17496}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{2916 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{8748 \sqrt{3}} \]

[Out]

1/(2916*(3 - 2*x)) - ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(2916*Sqrt[3]) + ArcTan[(3 +
4*x)/(3*Sqrt[3])]/(8748*Sqrt[3]) - (5*Log[3 - 2*x])/17496 + Log[3 + 2*x]/17496 +
Log[9 - 6*x + 4*x^2]/17496 + Log[9 + 6*x + 4*x^2]/17496

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Rubi [A]  time = 0.206841, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ \frac{\log \left (4 x^2-6 x+9\right )}{17496}+\frac{\log \left (4 x^2+6 x+9\right )}{17496}+\frac{1}{2916 (3-2 x)}-\frac{5 \log (3-2 x)}{17496}+\frac{\log (2 x+3)}{17496}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{2916 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{8748 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(243 + 162*x + 108*x^2 + 72*x^3 + 48*x^4 + 32*x^5)/(729 - 64*x^6)^2,x]

[Out]

1/(2916*(3 - 2*x)) - ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(2916*Sqrt[3]) + ArcTan[(3 +
4*x)/(3*Sqrt[3])]/(8748*Sqrt[3]) - (5*Log[3 - 2*x])/17496 + Log[3 + 2*x]/17496 +
Log[9 - 6*x + 4*x^2]/17496 + Log[9 + 6*x + 4*x^2]/17496

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((32*x**5+48*x**4+72*x**3+108*x**2+162*x+243)/(-64*x**6+729)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.135762, size = 97, normalized size = 0.88 \[ \frac{3 \left (\log \left (4 x^2-6 x+9\right )+\log \left (4 x^2+6 x+9\right )+\frac{6}{3-2 x}-5 \log (3-2 x)+\log (2 x+3)\right )+6 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{52488} \]

Antiderivative was successfully verified.

[In]  Integrate[(243 + 162*x + 108*x^2 + 72*x^3 + 48*x^4 + 32*x^5)/(729 - 64*x^6)^2,x]

[Out]

(6*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] + 2*Sqrt[3]*ArcTan[(3 + 4*x)/(3*Sqrt[3
])] + 3*(6/(3 - 2*x) - 5*Log[3 - 2*x] + Log[3 + 2*x] + Log[9 - 6*x + 4*x^2] + Lo
g[9 + 6*x + 4*x^2]))/52488

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Maple [A]  time = 0.015, size = 85, normalized size = 0.8 \[{\frac{\ln \left ( 2\,x+3 \right ) }{17496}}-{\frac{1}{-8748+5832\,x}}-{\frac{5\,\ln \left ( -3+2\,x \right ) }{17496}}+{\frac{\ln \left ( 4\,{x}^{2}+6\,x+9 \right ) }{17496}}+{\frac{\sqrt{3}}{26244}\arctan \left ({\frac{ \left ( 8\,x+6 \right ) \sqrt{3}}{18}} \right ) }+{\frac{\ln \left ( 4\,{x}^{2}-6\,x+9 \right ) }{17496}}+{\frac{\sqrt{3}}{8748}\arctan \left ({\frac{ \left ( 8\,x-6 \right ) \sqrt{3}}{18}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((32*x^5+48*x^4+72*x^3+108*x^2+162*x+243)/(-64*x^6+729)^2,x)

[Out]

1/17496*ln(2*x+3)-1/2916/(-3+2*x)-5/17496*ln(-3+2*x)+1/17496*ln(4*x^2+6*x+9)+1/2
6244*3^(1/2)*arctan(1/18*(8*x+6)*3^(1/2))+1/17496*ln(4*x^2-6*x+9)+1/8748*3^(1/2)
*arctan(1/18*(8*x-6)*3^(1/2))

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Maxima [A]  time = 1.50101, size = 113, normalized size = 1.03 \[ \frac{1}{26244} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{1}{8748} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - \frac{1}{2916 \,{\left (2 \, x - 3\right )}} + \frac{1}{17496} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac{1}{17496} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{1}{17496} \, \log \left (2 \, x + 3\right ) - \frac{5}{17496} \, \log \left (2 \, x - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((32*x^5 + 48*x^4 + 72*x^3 + 108*x^2 + 162*x + 243)/(64*x^6 - 729)^2,x, algorithm="maxima")

[Out]

1/26244*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 1/8748*sqrt(3)*arctan(1/9*sqrt(3
)*(4*x - 3)) - 1/2916/(2*x - 3) + 1/17496*log(4*x^2 + 6*x + 9) + 1/17496*log(4*x
^2 - 6*x + 9) + 1/17496*log(2*x + 3) - 5/17496*log(2*x - 3)

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Fricas [A]  time = 0.214526, size = 169, normalized size = 1.54 \[ \frac{\sqrt{3}{\left (\sqrt{3}{\left (2 \, x - 3\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \sqrt{3}{\left (2 \, x - 3\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \sqrt{3}{\left (2 \, x - 3\right )} \log \left (2 \, x + 3\right ) - 5 \, \sqrt{3}{\left (2 \, x - 3\right )} \log \left (2 \, x - 3\right ) + 2 \,{\left (2 \, x - 3\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + 6 \,{\left (2 \, x - 3\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - 6 \, \sqrt{3}\right )}}{52488 \,{\left (2 \, x - 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((32*x^5 + 48*x^4 + 72*x^3 + 108*x^2 + 162*x + 243)/(64*x^6 - 729)^2,x, algorithm="fricas")

[Out]

1/52488*sqrt(3)*(sqrt(3)*(2*x - 3)*log(4*x^2 + 6*x + 9) + sqrt(3)*(2*x - 3)*log(
4*x^2 - 6*x + 9) + sqrt(3)*(2*x - 3)*log(2*x + 3) - 5*sqrt(3)*(2*x - 3)*log(2*x
- 3) + 2*(2*x - 3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 6*(2*x - 3)*arctan(1/9*sqrt(3
)*(4*x - 3)) - 6*sqrt(3))/(2*x - 3)

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Sympy [A]  time = 0.593222, size = 105, normalized size = 0.95 \[ - \frac{5 \log{\left (x - \frac{3}{2} \right )}}{17496} + \frac{\log{\left (x + \frac{3}{2} \right )}}{17496} + \frac{\log{\left (x^{2} - \frac{3 x}{2} + \frac{9}{4} \right )}}{17496} + \frac{\log{\left (x^{2} + \frac{3 x}{2} + \frac{9}{4} \right )}}{17496} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} - \frac{\sqrt{3}}{3} \right )}}{8748} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} + \frac{\sqrt{3}}{3} \right )}}{26244} - \frac{1}{5832 x - 8748} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((32*x**5+48*x**4+72*x**3+108*x**2+162*x+243)/(-64*x**6+729)**2,x)

[Out]

-5*log(x - 3/2)/17496 + log(x + 3/2)/17496 + log(x**2 - 3*x/2 + 9/4)/17496 + log
(x**2 + 3*x/2 + 9/4)/17496 + sqrt(3)*atan(4*sqrt(3)*x/9 - sqrt(3)/3)/8748 + sqrt
(3)*atan(4*sqrt(3)*x/9 + sqrt(3)/3)/26244 - 1/(5832*x - 8748)

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GIAC/XCAS [A]  time = 0.218703, size = 116, normalized size = 1.05 \[ \frac{1}{26244} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{1}{8748} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - \frac{1}{2916 \,{\left (2 \, x - 3\right )}} + \frac{1}{17496} \,{\rm ln}\left (4 \, x^{2} + 6 \, x + 9\right ) + \frac{1}{17496} \,{\rm ln}\left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{1}{17496} \,{\rm ln}\left ({\left | 2 \, x + 3 \right |}\right ) - \frac{5}{17496} \,{\rm ln}\left ({\left | 2 \, x - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((32*x^5 + 48*x^4 + 72*x^3 + 108*x^2 + 162*x + 243)/(64*x^6 - 729)^2,x, algorithm="giac")

[Out]

1/26244*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 1/8748*sqrt(3)*arctan(1/9*sqrt(3
)*(4*x - 3)) - 1/2916/(2*x - 3) + 1/17496*ln(4*x^2 + 6*x + 9) + 1/17496*ln(4*x^2
- 6*x + 9) + 1/17496*ln(abs(2*x + 3)) - 5/17496*ln(abs(2*x - 3))