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

Optimal. Leaf size=146 \[ \frac{x}{236196 \left (4 x^2-6 x+9\right )}-\frac{x+3}{708588 \left (4 x^2+6 x+9\right )}-\frac{\log \left (4 x^2-6 x+9\right )}{8503056}+\frac{\log \left (4 x^2+6 x+9\right )}{944784}+\frac{1}{708588 (3-2 x)}-\frac{\log (3-2 x)}{472392}+\frac{\log (2 x+3)}{4251528}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{157464 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{1417176 \sqrt{3}} \]

[Out]

1/(708588*(3 - 2*x)) + x/(236196*(9 - 6*x + 4*x^2)) - (3 + x)/(708588*(9 + 6*x +
4*x^2)) - ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(157464*Sqrt[3]) + ArcTan[(3 + 4*x)/(3*
Sqrt[3])]/(1417176*Sqrt[3]) - Log[3 - 2*x]/472392 + Log[3 + 2*x]/4251528 - Log[9
- 6*x + 4*x^2]/8503056 + Log[9 + 6*x + 4*x^2]/944784

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Rubi [A]  time = 0.314212, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467 \[ \frac{x}{236196 \left (4 x^2-6 x+9\right )}-\frac{x+3}{708588 \left (4 x^2+6 x+9\right )}-\frac{\log \left (4 x^2-6 x+9\right )}{8503056}+\frac{\log \left (4 x^2+6 x+9\right )}{944784}+\frac{1}{708588 (3-2 x)}-\frac{\log (3-2 x)}{472392}+\frac{\log (2 x+3)}{4251528}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{157464 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{1417176 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

1/(708588*(3 - 2*x)) + x/(236196*(9 - 6*x + 4*x^2)) - (3 + x)/(708588*(9 + 6*x +
4*x^2)) - ArcTan[(3 - 4*x)/(3*Sqrt[3])]/(157464*Sqrt[3]) + ArcTan[(3 + 4*x)/(3*
Sqrt[3])]/(1417176*Sqrt[3]) - Log[3 - 2*x]/472392 + Log[3 + 2*x]/4251528 - Log[9
- 6*x + 4*x^2]/8503056 + Log[9 + 6*x + 4*x^2]/944784

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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((3+2*x)/(-64*x**6+729)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.136821, size = 121, normalized size = 0.83 \[ \frac{-\log \left (4 x^2-6 x+9\right )+9 \log \left (4 x^2+6 x+9\right )+\frac{1944 x}{-32 x^5+48 x^4-72 x^3+108 x^2-162 x+243}-18 \log (3-2 x)+2 \log (2 x+3)+18 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{8503056} \]

Antiderivative was successfully verified.

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

[Out]

((1944*x)/(243 - 162*x + 108*x^2 - 72*x^3 + 48*x^4 - 32*x^5) + 18*Sqrt[3]*ArcTan
[(-3 + 4*x)/(3*Sqrt[3])] + 2*Sqrt[3]*ArcTan[(3 + 4*x)/(3*Sqrt[3])] - 18*Log[3 -
2*x] + 2*Log[3 + 2*x] - Log[9 - 6*x + 4*x^2] + 9*Log[9 + 6*x + 4*x^2])/8503056

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Maple [A]  time = 0.023, size = 115, normalized size = 0.8 \[{\frac{\ln \left ( 2\,x+3 \right ) }{4251528}}-{\frac{1}{-2125764+1417176\,x}}-{\frac{\ln \left ( -3+2\,x \right ) }{472392}}+{\frac{1}{708588} \left ( -{\frac{x}{4}}-{\frac{3}{4}} \right ) \left ({x}^{2}+{\frac{3\,x}{2}}+{\frac{9}{4}} \right ) ^{-1}}+{\frac{\ln \left ( 16\,{x}^{2}+24\,x+36 \right ) }{944784}}+{\frac{\sqrt{3}}{4251528}\arctan \left ({\frac{ \left ( 32\,x+24 \right ) \sqrt{3}}{72}} \right ) }+{\frac{x}{944784} \left ({x}^{2}-{\frac{3\,x}{2}}+{\frac{9}{4}} \right ) ^{-1}}-{\frac{\ln \left ( 16\,{x}^{2}-24\,x+36 \right ) }{8503056}}+{\frac{\sqrt{3}}{472392}\arctan \left ({\frac{ \left ( 32\,x-24 \right ) \sqrt{3}}{72}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x+3)/(-64*x^6+729)^2,x)

[Out]

1/4251528*ln(2*x+3)-1/708588/(-3+2*x)-1/472392*ln(-3+2*x)+1/708588*(-1/4*x-3/4)/
(x^2+3/2*x+9/4)+1/944784*ln(16*x^2+24*x+36)+1/4251528*3^(1/2)*arctan(1/72*(32*x+
24)*3^(1/2))+1/944784*x/(x^2-3/2*x+9/4)-1/8503056*ln(16*x^2-24*x+36)+1/472392*3^
(1/2)*arctan(1/72*(32*x-24)*3^(1/2))

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Maxima [A]  time = 1.72804, size = 142, normalized size = 0.97 \[ \frac{1}{4251528} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{1}{472392} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - \frac{x}{4374 \,{\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )}} + \frac{1}{944784} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{1}{8503056} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{1}{4251528} \, \log \left (2 \, x + 3\right ) - \frac{1}{472392} \, \log \left (2 \, x - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4251528*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 1/472392*sqrt(3)*arctan(1/9*sq
rt(3)*(4*x - 3)) - 1/4374*x/(32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243) +
1/944784*log(4*x^2 + 6*x + 9) - 1/8503056*log(4*x^2 - 6*x + 9) + 1/4251528*log(
2*x + 3) - 1/472392*log(2*x - 3)

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Fricas [A]  time = 0.213924, size = 363, normalized size = 2.49 \[ \frac{\sqrt{3}{\left (9 \, \sqrt{3}{\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \sqrt{3}{\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) + 2 \, \sqrt{3}{\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \log \left (2 \, x + 3\right ) - 18 \, \sqrt{3}{\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \log \left (2 \, x - 3\right ) + 6 \,{\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + 54 \,{\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - 1944 \, \sqrt{3} x\right )}}{25509168 \,{\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/25509168*sqrt(3)*(9*sqrt(3)*(32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243)
*log(4*x^2 + 6*x + 9) - sqrt(3)*(32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 24
3)*log(4*x^2 - 6*x + 9) + 2*sqrt(3)*(32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x
- 243)*log(2*x + 3) - 18*sqrt(3)*(32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 2
43)*log(2*x - 3) + 6*(32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243)*arctan(1
/9*sqrt(3)*(4*x + 3)) + 54*(32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243)*ar
ctan(1/9*sqrt(3)*(4*x - 3)) - 1944*sqrt(3)*x)/(32*x^5 - 48*x^4 + 72*x^3 - 108*x^
2 + 162*x - 243)

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Sympy [A]  time = 0.864815, size = 124, normalized size = 0.85 \[ - \frac{x}{139968 x^{5} - 209952 x^{4} + 314928 x^{3} - 472392 x^{2} + 708588 x - 1062882} - \frac{\log{\left (x - \frac{3}{2} \right )}}{472392} + \frac{\log{\left (x + \frac{3}{2} \right )}}{4251528} - \frac{\log{\left (x^{2} - \frac{3 x}{2} + \frac{9}{4} \right )}}{8503056} + \frac{\log{\left (x^{2} + \frac{3 x}{2} + \frac{9}{4} \right )}}{944784} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} - \frac{\sqrt{3}}{3} \right )}}{472392} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} + \frac{\sqrt{3}}{3} \right )}}{4251528} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/(139968*x**5 - 209952*x**4 + 314928*x**3 - 472392*x**2 + 708588*x - 1062882)
- log(x - 3/2)/472392 + log(x + 3/2)/4251528 - log(x**2 - 3*x/2 + 9/4)/8503056 +
log(x**2 + 3*x/2 + 9/4)/944784 + sqrt(3)*atan(4*sqrt(3)*x/9 - sqrt(3)/3)/472392
+ sqrt(3)*atan(4*sqrt(3)*x/9 + sqrt(3)/3)/4251528

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GIAC/XCAS [A]  time = 0.222503, size = 150, normalized size = 1.03 \[ \frac{1}{4251528} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{1}{472392} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) - \frac{x}{4374 \,{\left (4 \, x^{2} + 6 \, x + 9\right )}{\left (4 \, x^{2} - 6 \, x + 9\right )}{\left (2 \, x - 3\right )}} + \frac{1}{944784} \,{\rm ln}\left (4 \, x^{2} + 6 \, x + 9\right ) - \frac{1}{8503056} \,{\rm ln}\left (4 \, x^{2} - 6 \, x + 9\right ) + \frac{1}{4251528} \,{\rm ln}\left ({\left | 2 \, x + 3 \right |}\right ) - \frac{1}{472392} \,{\rm ln}\left ({\left | 2 \, x - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4251528*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 1/472392*sqrt(3)*arctan(1/9*sq
rt(3)*(4*x - 3)) - 1/4374*x/((4*x^2 + 6*x + 9)*(4*x^2 - 6*x + 9)*(2*x - 3)) + 1/
944784*ln(4*x^2 + 6*x + 9) - 1/8503056*ln(4*x^2 - 6*x + 9) + 1/4251528*ln(abs(2*
x + 3)) - 1/472392*ln(abs(2*x - 3))