3.575 \(\int \frac{x (27-2 x^3)}{729-64 x^6} \, dx\)

Optimal. Leaf size=99 \[ \frac{5}{576} \log \left (4 x^2-6 x+9\right )+\frac{1}{192} \log \left (4 x^2+6 x+9\right )-\frac{1}{96} \log (3-2 x)-\frac{5}{288} \log (2 x+3)-\frac{5 \tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{96 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{32 \sqrt{3}} \]

[Out]

(-5*ArcTan[(3 - 4*x)/(3*Sqrt[3])])/(96*Sqrt[3]) - ArcTan[(3 + 4*x)/(3*Sqrt[3])]/(32*Sqrt[3]) - Log[3 - 2*x]/96
- (5*Log[3 + 2*x])/288 + (5*Log[9 - 6*x + 4*x^2])/576 + Log[9 + 6*x + 4*x^2]/192

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Rubi [A]  time = 0.0627053, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {1511, 292, 31, 634, 618, 204, 628} \[ \frac{5}{576} \log \left (4 x^2-6 x+9\right )+\frac{1}{192} \log \left (4 x^2+6 x+9\right )-\frac{1}{96} \log (3-2 x)-\frac{5}{288} \log (2 x+3)-\frac{5 \tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{96 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )}{32 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(x*(27 - 2*x^3))/(729 - 64*x^6),x]

[Out]

(-5*ArcTan[(3 - 4*x)/(3*Sqrt[3])])/(96*Sqrt[3]) - ArcTan[(3 + 4*x)/(3*Sqrt[3])]/(32*Sqrt[3]) - Log[3 - 2*x]/96
- (5*Log[3 + 2*x])/288 + (5*Log[9 - 6*x + 4*x^2])/576 + Log[9 + 6*x + 4*x^2]/192

Rule 1511

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_)))/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[-(a*c),
2]}, -Dist[e/2 + (c*d)/(2*q), Int[(f*x)^m/(q - c*x^n), x], x] + Dist[e/2 - (c*d)/(2*q), Int[(f*x)^m/(q + c*x^
n), x], x]] /; FreeQ[{a, c, d, e, f, m}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, 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{x \left (27-2 x^3\right )}{729-64 x^6} \, dx &=3 \int \frac{x}{216-64 x^3} \, dx+5 \int \frac{x}{216+64 x^3} \, dx\\ &=\frac{1}{24} \int \frac{1}{6-4 x} \, dx-\frac{1}{24} \int \frac{6-4 x}{36+24 x+16 x^2} \, dx-\frac{5}{72} \int \frac{1}{6+4 x} \, dx+\frac{5}{72} \int \frac{6+4 x}{36-24 x+16 x^2} \, dx\\ &=-\frac{1}{96} \log (3-2 x)-\frac{5}{288} \log (3+2 x)+\frac{1}{192} \int \frac{24+32 x}{36+24 x+16 x^2} \, dx+\frac{5}{576} \int \frac{-24+32 x}{36-24 x+16 x^2} \, dx-\frac{3}{8} \int \frac{1}{36+24 x+16 x^2} \, dx+\frac{5}{8} \int \frac{1}{36-24 x+16 x^2} \, dx\\ &=-\frac{1}{96} \log (3-2 x)-\frac{5}{288} \log (3+2 x)+\frac{5}{576} \log \left (9-6 x+4 x^2\right )+\frac{1}{192} \log \left (9+6 x+4 x^2\right )+\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{-1728-x^2} \, dx,x,24+32 x\right )-\frac{5}{4} \operatorname{Subst}\left (\int \frac{1}{-1728-x^2} \, dx,x,-24+32 x\right )\\ &=-\frac{5 \tan ^{-1}\left (\frac{3-4 x}{3 \sqrt{3}}\right )}{96 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{3+4 x}{3 \sqrt{3}}\right )}{32 \sqrt{3}}-\frac{1}{96} \log (3-2 x)-\frac{5}{288} \log (3+2 x)+\frac{5}{576} \log \left (9-6 x+4 x^2\right )+\frac{1}{192} \log \left (9+6 x+4 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0139777, size = 91, normalized size = 0.92 \[ \frac{1}{576} \left (5 \log \left (4 x^2-6 x+9\right )+3 \log \left (4 x^2+6 x+9\right )-6 \log (3-2 x)-10 \log (2 x+3)+10 \sqrt{3} \tan ^{-1}\left (\frac{4 x-3}{3 \sqrt{3}}\right )-6 \sqrt{3} \tan ^{-1}\left (\frac{4 x+3}{3 \sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(27 - 2*x^3))/(729 - 64*x^6),x]

[Out]

(10*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] - 6*Sqrt[3]*ArcTan[(3 + 4*x)/(3*Sqrt[3])] - 6*Log[3 - 2*x] - 10*Log
[3 + 2*x] + 5*Log[9 - 6*x + 4*x^2] + 3*Log[9 + 6*x + 4*x^2])/576

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Maple [A]  time = 0.007, size = 76, normalized size = 0.8 \begin{align*} -{\frac{5\,\ln \left ( 3+2\,x \right ) }{288}}-{\frac{\ln \left ( -3+2\,x \right ) }{96}}+{\frac{\ln \left ( 4\,{x}^{2}+6\,x+9 \right ) }{192}}-{\frac{\sqrt{3}}{96}\arctan \left ({\frac{ \left ( 8\,x+6 \right ) \sqrt{3}}{18}} \right ) }+{\frac{5\,\ln \left ( 4\,{x}^{2}-6\,x+9 \right ) }{576}}+{\frac{5\,\sqrt{3}}{288}\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(x*(-2*x^3+27)/(-64*x^6+729),x)

[Out]

-5/288*ln(3+2*x)-1/96*ln(-3+2*x)+1/192*ln(4*x^2+6*x+9)-1/96*3^(1/2)*arctan(1/18*(8*x+6)*3^(1/2))+5/576*ln(4*x^
2-6*x+9)+5/288*3^(1/2)*arctan(1/18*(8*x-6)*3^(1/2))

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Maxima [A]  time = 1.37515, size = 101, normalized size = 1.02 \begin{align*} -\frac{1}{96} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{5}{288} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{1}{192} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac{5}{576} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) - \frac{5}{288} \, \log \left (2 \, x + 3\right ) - \frac{1}{96} \, \log \left (2 \, x - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-2*x^3+27)/(-64*x^6+729),x, algorithm="maxima")

[Out]

-1/96*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 5/288*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) + 1/192*log(4*x^2 +
6*x + 9) + 5/576*log(4*x^2 - 6*x + 9) - 5/288*log(2*x + 3) - 1/96*log(2*x - 3)

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Fricas [A]  time = 1.44422, size = 257, normalized size = 2.6 \begin{align*} -\frac{1}{96} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{5}{288} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{1}{192} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) + \frac{5}{576} \, \log \left (4 \, x^{2} - 6 \, x + 9\right ) - \frac{5}{288} \, \log \left (2 \, x + 3\right ) - \frac{1}{96} \, \log \left (2 \, x - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-2*x^3+27)/(-64*x^6+729),x, algorithm="fricas")

[Out]

-1/96*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 5/288*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) + 1/192*log(4*x^2 +
6*x + 9) + 5/576*log(4*x^2 - 6*x + 9) - 5/288*log(2*x + 3) - 1/96*log(2*x - 3)

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Sympy [A]  time = 0.327847, size = 102, normalized size = 1.03 \begin{align*} - \frac{\log{\left (x - \frac{3}{2} \right )}}{96} - \frac{5 \log{\left (x + \frac{3}{2} \right )}}{288} + \frac{5 \log{\left (x^{2} - \frac{3 x}{2} + \frac{9}{4} \right )}}{576} + \frac{\log{\left (x^{2} + \frac{3 x}{2} + \frac{9}{4} \right )}}{192} + \frac{5 \sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} - \frac{\sqrt{3}}{3} \right )}}{288} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{4 \sqrt{3} x}{9} + \frac{\sqrt{3}}{3} \right )}}{96} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-2*x**3+27)/(-64*x**6+729),x)

[Out]

-log(x - 3/2)/96 - 5*log(x + 3/2)/288 + 5*log(x**2 - 3*x/2 + 9/4)/576 + log(x**2 + 3*x/2 + 9/4)/192 + 5*sqrt(3
)*atan(4*sqrt(3)*x/9 - sqrt(3)/3)/288 - sqrt(3)*atan(4*sqrt(3)*x/9 + sqrt(3)/3)/96

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Giac [A]  time = 1.06298, size = 93, normalized size = 0.94 \begin{align*} -\frac{1}{96} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x + 3\right )}\right ) + \frac{5}{288} \, \sqrt{3} \arctan \left (\frac{1}{9} \, \sqrt{3}{\left (4 \, x - 3\right )}\right ) + \frac{1}{192} \, \log \left (x^{2} + \frac{3}{2} \, x + \frac{9}{4}\right ) + \frac{5}{576} \, \log \left (x^{2} - \frac{3}{2} \, x + \frac{9}{4}\right ) - \frac{5}{288} \, \log \left ({\left | x + \frac{3}{2} \right |}\right ) - \frac{1}{96} \, \log \left ({\left | x - \frac{3}{2} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-2*x^3+27)/(-64*x^6+729),x, algorithm="giac")

[Out]

-1/96*sqrt(3)*arctan(1/9*sqrt(3)*(4*x + 3)) + 5/288*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) + 1/192*log(x^2 + 3/
2*x + 9/4) + 5/576*log(x^2 - 3/2*x + 9/4) - 5/288*log(abs(x + 3/2)) - 1/96*log(abs(x - 3/2))