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

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

________________________________________________________________________________________

Rubi [A]  time = 0.172511, 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, Rules used = {1586, 2074, 638, 618, 204, 634, 628} \[ \frac{x}{236196 \left (4 x^2-6 x+9\right )}-\frac{x+3}{708588 \left (4 x^2+6 x+9\right )}-\frac{\log \left (4 x^2-6 x+9\right )}{8503056}+\frac{\log \left (4 x^2+6 x+9\right )}{944784}+\frac{1}{708588 (3-2 x)}-\frac{\log (3-2 x)}{472392}+\frac{\log (2 x+3)}{4251528}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{157464 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{1417176 \sqrt{3}} \]

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

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{3+2 x}{\left (729-64 x^6\right )^2} \, dx &=\int \frac{1}{(3+2 x) \left (243-162 x+108 x^2-72 x^3+48 x^4-32 x^5\right )^2} \, dx\\ &=\int \left (\frac{1}{354294 (-3+2 x)^2}-\frac{1}{236196 (-3+2 x)}+\frac{1}{2125764 (3+2 x)}+\frac{3-x}{39366 \left (9-6 x+4 x^2\right )^2}+\frac{33-2 x}{2125764 \left (9-6 x+4 x^2\right )}+\frac{x}{39366 \left (9+6 x+4 x^2\right )^2}+\frac{7+6 x}{708588 \left (9+6 x+4 x^2\right )}\right ) \, dx\\ &=\frac{1}{708588 (3-2 x)}-\frac{\log (3-2 x)}{472392}+\frac{\log (3+2 x)}{4251528}+\frac{\int \frac{33-2 x}{9-6 x+4 x^2} \, dx}{2125764}+\frac{\int \frac{7+6 x}{9+6 x+4 x^2} \, dx}{708588}+\frac{\int \frac{3-x}{\left (9-6 x+4 x^2\right )^2} \, dx}{39366}+\frac{\int \frac{x}{\left (9+6 x+4 x^2\right )^2} \, dx}{39366}\\ &=\frac{1}{708588 (3-2 x)}+\frac{x}{236196 \left (9-6 x+4 x^2\right )}-\frac{3+x}{708588 \left (9+6 x+4 x^2\right )}-\frac{\log (3-2 x)}{472392}+\frac{\log (3+2 x)}{4251528}-\frac{\int \frac{-6+8 x}{9-6 x+4 x^2} \, dx}{8503056}+\frac{\int \frac{6+8 x}{9+6 x+4 x^2} \, dx}{944784}-\frac{\int \frac{1}{9+6 x+4 x^2} \, dx}{708588}+\frac{5 \int \frac{1}{9+6 x+4 x^2} \, dx}{1417176}+\frac{\int \frac{1}{9-6 x+4 x^2} \, dx}{236196}+\frac{7 \int \frac{1}{9-6 x+4 x^2} \, dx}{472392}\\ &=\frac{1}{708588 (3-2 x)}+\frac{x}{236196 \left (9-6 x+4 x^2\right )}-\frac{3+x}{708588 \left (9+6 x+4 x^2\right )}-\frac{\log (3-2 x)}{472392}+\frac{\log (3+2 x)}{4251528}-\frac{\log \left (9-6 x+4 x^2\right )}{8503056}+\frac{\log \left (9+6 x+4 x^2\right )}{944784}+\frac{\operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,6+8 x\right )}{354294}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,6+8 x\right )}{708588}-\frac{\operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,-6+8 x\right )}{118098}-\frac{7 \operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,-6+8 x\right )}{236196}\\ &=\frac{1}{708588 (3-2 x)}+\frac{x}{236196 \left (9-6 x+4 x^2\right )}-\frac{3+x}{708588 \left (9+6 x+4 x^2\right )}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{157464 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{3+4 x}{3 \sqrt{3}}\right )}{1417176 \sqrt{3}}-\frac{\log (3-2 x)}{472392}+\frac{\log (3+2 x)}{4251528}-\frac{\log \left (9-6 x+4 x^2\right )}{8503056}+\frac{\log \left (9+6 x+4 x^2\right )}{944784}\\ \end{align*}

Mathematica [A]  time = 0.0817094, size = 121, normalized size = 0.83 \[ \frac{\frac{1944 x}{-32 x^5+48 x^4-72 x^3+108 x^2-162 x+243}-\log \left (4 x^2-6 x+9\right )+9 \log \left (4 x^2+6 x+9\right )-18 \log (3-2 x)+2 \log (2 x+3)+18 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{8503056} \]

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*S
qrt[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 = 0.013, size = 115, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( 3+2\,x \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 ( 4\,{x}^{2}+6\,x+9 \right ) }{944784}}+{\frac{\sqrt{3}}{4251528}\arctan \left ({\frac{ \left ( 8\,x+6 \right ) \sqrt{3}}{18}} \right ) }+{\frac{x}{944784} \left ({x}^{2}-{\frac{3\,x}{2}}+{\frac{9}{4}} \right ) ^{-1}}-{\frac{\ln \left ( 4\,{x}^{2}-6\,x+9 \right ) }{8503056}}+{\frac{\sqrt{3}}{472392}\arctan \left ({\frac{ \left ( 8\,x-6 \right ) \sqrt{3}}{18}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4251528*ln(3+2*x)-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(4*
x^2+6*x+9)+1/4251528*3^(1/2)*arctan(1/18*(8*x+6)*3^(1/2))+1/944784*x/(x^2-3/2*x+9/4)-1/8503056*ln(4*x^2-6*x+9)
+1/472392*3^(1/2)*arctan(1/18*(8*x-6)*3^(1/2))

________________________________________________________________________________________

Maxima [A]  time = 1.38129, size = 142, normalized size = 0.97 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+2*x)/(-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*sqrt(3)*(4*x - 3)) - 1/4374*x/(3
2*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 [B]  time = 1.50641, size = 736, normalized size = 5.04 \begin{align*} \frac{2 \, \sqrt{3}{\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 ) + 18 \, \sqrt{3}{\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 ) + 9 \,{\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 ) -{\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 \,{\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \log \left (2 \, x + 3\right ) - 18 \,{\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )} \log \left (2 \, x - 3\right ) - 1944 \, x}{8503056 \,{\left (32 \, x^{5} - 48 \, x^{4} + 72 \, x^{3} - 108 \, x^{2} + 162 \, x - 243\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8503056*(2*sqrt(3)*(32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243)*arctan(1/9*sqrt(3)*(4*x + 3)) + 18*sqr
t(3)*(32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243)*arctan(1/9*sqrt(3)*(4*x - 3)) + 9*(32*x^5 - 48*x^4 + 7
2*x^3 - 108*x^2 + 162*x - 243)*log(4*x^2 + 6*x + 9) - (32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243)*log(4
*x^2 - 6*x + 9) + 2*(32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243)*log(2*x + 3) - 18*(32*x^5 - 48*x^4 + 72
*x^3 - 108*x^2 + 162*x - 243)*log(2*x - 3) - 1944*x)/(32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243)

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Sympy [A]  time = 0.501011, size = 124, normalized size = 0.85 \begin{align*} - \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} \end{align*}

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 [A]  time = 1.06199, size = 150, normalized size = 1.03 \begin{align*} \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} \, \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 ({\left | 2 \, x + 3 \right |}\right ) - \frac{1}{472392} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+2*x)/(-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*sqrt(3)*(4*x - 3)) - 1/4374*x/((
4*x^2 + 6*x + 9)*(4*x^2 - 6*x + 9)*(2*x - 3)) + 1/944784*log(4*x^2 + 6*x + 9) - 1/8503056*log(4*x^2 - 6*x + 9)
+ 1/4251528*log(abs(2*x + 3)) - 1/472392*log(abs(2*x - 3))