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

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 = 0.117263, 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, Rules used = {1586, 2074, 634, 618, 204, 628} \[ \frac{\log \left (4 x^2-6 x+9\right )}{17496}+\frac{\log \left (4 x^2+6 x+9\right )}{17496}+\frac{1}{2916 (3-2 x)}-\frac{5 \log (3-2 x)}{17496}+\frac{\log (2 x+3)}{17496}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{2916 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{8748 \sqrt{3}} \]

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

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 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 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 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{243+162 x+108 x^2+72 x^3+48 x^4+32 x^5}{\left (729-64 x^6\right )^2} \, dx &=\int \frac{1}{(3-2 x)^2 \left (243+162 x+108 x^2+72 x^3+48 x^4+32 x^5\right )} \, dx\\ &=\int \left (\frac{1}{1458 (-3+2 x)^2}-\frac{5}{8748 (-3+2 x)}+\frac{1}{8748 (3+2 x)}+\frac{3+2 x}{4374 \left (9-6 x+4 x^2\right )}+\frac{3+2 x}{4374 \left (9+6 x+4 x^2\right )}\right ) \, dx\\ &=\frac{1}{2916 (3-2 x)}-\frac{5 \log (3-2 x)}{17496}+\frac{\log (3+2 x)}{17496}+\frac{\int \frac{3+2 x}{9-6 x+4 x^2} \, dx}{4374}+\frac{\int \frac{3+2 x}{9+6 x+4 x^2} \, dx}{4374}\\ &=\frac{1}{2916 (3-2 x)}-\frac{5 \log (3-2 x)}{17496}+\frac{\log (3+2 x)}{17496}+\frac{\int \frac{-6+8 x}{9-6 x+4 x^2} \, dx}{17496}+\frac{\int \frac{6+8 x}{9+6 x+4 x^2} \, dx}{17496}+\frac{\int \frac{1}{9+6 x+4 x^2} \, dx}{2916}+\frac{1}{972} \int \frac{1}{9-6 x+4 x^2} \, dx\\ &=\frac{1}{2916 (3-2 x)}-\frac{5 \log (3-2 x)}{17496}+\frac{\log (3+2 x)}{17496}+\frac{\log \left (9-6 x+4 x^2\right )}{17496}+\frac{\log \left (9+6 x+4 x^2\right )}{17496}-\frac{\operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,6+8 x\right )}{1458}-\frac{1}{486} \operatorname{Subst}\left (\int \frac{1}{-108-x^2} \, dx,x,-6+8 x\right )\\ &=\frac{1}{2916 (3-2 x)}-\frac{\tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{2916 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{3+4 x}{3 \sqrt{3}}\right )}{8748 \sqrt{3}}-\frac{5 \log (3-2 x)}{17496}+\frac{\log (3+2 x)}{17496}+\frac{\log \left (9-6 x+4 x^2\right )}{17496}+\frac{\log \left (9+6 x+4 x^2\right )}{17496}\\ \end{align*}

Mathematica [A]  time = 0.076873, 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] + Log[9 + 6*x + 4*x^2]))/52488

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Maple [A]  time = 0.009, size = 85, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( 3+2\,x \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 ) } \end{align*}

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(3+2*x)-1/2916/(-3+2*x)-5/17496*ln(-3+2*x)+1/17496*ln(4*x^2+6*x+9)+1/26244*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 = 1.38071, size = 113, normalized size = 1.03 \begin{align*} \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 ) \end{align*}

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 = 1.43763, size = 342, normalized size = 3.11 \begin{align*} \frac{2 \, \sqrt{3}{\left (2 \, x - 3\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + 6 \, \sqrt{3}{\left (2 \, x - 3\right )} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + 3 \,{\left (2 \, x - 3\right )} \log \left (4 \, x^{2} + 6 \, x + 9\right ) + 3 \,{\left (2 \, x - 3\right )} \log \left (4 \, x^{2} - 6 \, x + 9\right ) + 3 \,{\left (2 \, x - 3\right )} \log \left (2 \, x + 3\right ) - 15 \,{\left (2 \, x - 3\right )} \log \left (2 \, x - 3\right ) - 18}{52488 \,{\left (2 \, x - 3\right )}} \end{align*}

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*(2*sqrt(3)*(2*x - 3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 6*sqrt(3)*(2*x - 3)*arctan(1/9*sqrt(3)*(4*x - 3))
+ 3*(2*x - 3)*log(4*x^2 + 6*x + 9) + 3*(2*x - 3)*log(4*x^2 - 6*x + 9) + 3*(2*x - 3)*log(2*x + 3) - 15*(2*x -
3)*log(2*x - 3) - 18)/(2*x - 3)

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Sympy [A]  time = 0.351426, size = 105, normalized size = 0.95 \begin{align*} - \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} \end{align*}

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 + s
qrt(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 [A]  time = 1.07353, size = 116, normalized size = 1.05 \begin{align*} \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 ({\left | 2 \, x + 3 \right |}\right ) - \frac{5}{17496} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \end{align*}

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*log(4*x^2 + 6*x + 9) + 1/17496*log(4*x^2 - 6*x + 9) + 1/17496*log(abs(2*x + 3)) - 5/17496*log(abs(
2*x - 3))