3.6.75 \(\int \frac {x (27-2 x^3)}{729-64 x^6} \, dx\) [575]

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

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 99, normalized size of antiderivative = 1.00, 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 = {1525, 298, 31, 648, 632, 210, 642} \begin {gather*} -\frac {5 \text {ArcTan}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{96 \sqrt {3}}-\frac {\text {ArcTan}\left (\frac {4 x+3}{3 \sqrt {3}}\right )}{32 \sqrt {3}}+\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) \end {gather*}

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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

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

Rule 298

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 632

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 642

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]

Rule 648

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 1525

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]

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} \text {Subst}\left (\int \frac {1}{-1728-x^2} \, dx,x,24+32 x\right )-\frac {5}{4} \text {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*}

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Mathematica [A]
time = 0.01, size = 91, normalized size = 0.92 \begin {gather*} \frac {1}{576} \left (10 \sqrt {3} \tan ^{-1}\left (\frac {-3+4 x}{3 \sqrt {3}}\right )-6 \sqrt {3} \tan ^{-1}\left (\frac {3+4 x}{3 \sqrt {3}}\right )-6 \log (3-2 x)-10 \log (3+2 x)+5 \log \left (9-6 x+4 x^2\right )+3 \log \left (9+6 x+4 x^2\right )\right ) \end {gather*}

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 = 0.39, size = 76, normalized size = 0.77

method	result	size
default	\(\frac {5 \ln \left (4 x^{2}-6 x +9\right )}{576}+\frac {5 \sqrt {3}\, \arctan \left (\frac {\left (8 x -6\right ) \sqrt {3}}{18}\right )}{288}+\frac {\ln \left (4 x^{2}+6 x +9\right )}{192}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x +6\right ) \sqrt {3}}{18}\right )}{96}-\frac {5 \ln \left (2 x +3\right )}{288}-\frac {\ln \left (-3+2 x \right )}{96}\)	\(76\)
risch	\(\frac {5 \ln \left (16 x^{2}-24 x +36\right )}{576}+\frac {5 \sqrt {3}\, \arctan \left (\frac {\left (-3+4 x \right ) \sqrt {3}}{9}\right )}{288}-\frac {\ln \left (-3+2 x \right )}{96}+\frac {\ln \left (4 x^{2}+6 x +9\right )}{192}-\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (2 x +\frac {3}{2}\right ) \sqrt {3}}{9}\right )}{96}-\frac {5 \ln \left (2 x +3\right )}{288}\)	\(76\)
meijerg	\(\frac {x^{5} \left (\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )-\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )+\frac {\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{288 \left (x^{6}\right )^{\frac {5}{6}}}-\frac {x^{2} \left (\ln \left (1-\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )-\frac {\ln \left (1+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}+\frac {16 \left (x^{6}\right )^{\frac {2}{3}}}{81}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (x^{6}\right )^{\frac {1}{3}}}{9 \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}\right )\right )}{72 \left (x^{6}\right )^{\frac {1}{3}}}\)	\(192\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(-2*x^3+27)/(-64*x^6+729),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.51, size = 75, normalized size = 0.76 \begin {gather*} -\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 {gather*}

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 = 0.37, size = 75, normalized size = 0.76 \begin {gather*} -\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 {gather*}

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 = 0.19, size = 102, normalized size = 1.03 \begin {gather*} - \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 {gather*}

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 = 0.88, size = 69, normalized size = 0.70 \begin {gather*} -\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 {gather*}

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))

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Mupad [B]
time = 5.10, size = 91, normalized size = 0.92 \begin {gather*} -\frac {\ln \left (x-\frac {3}{2}\right )}{96}-\frac {5\,\ln \left (x+\frac {3}{2}\right )}{288}+\ln \left (x+\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {1}{192}+\frac {\sqrt {3}\,1{}\mathrm {i}}{192}\right )-\ln \left (x+\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{192}+\frac {\sqrt {3}\,1{}\mathrm {i}}{192}\right )-\ln \left (x-\frac {3}{4}-\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (-\frac {5}{576}+\frac {\sqrt {3}\,5{}\mathrm {i}}{576}\right )+\ln \left (x-\frac {3}{4}+\frac {\sqrt {3}\,3{}\mathrm {i}}{4}\right )\,\left (\frac {5}{576}+\frac {\sqrt {3}\,5{}\mathrm {i}}{576}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(2*x^3 - 27))/(64*x^6 - 729),x)

[Out]

log(x - (3^(1/2)*3i)/4 + 3/4)*((3^(1/2)*1i)/192 + 1/192) - (5*log(x + 3/2))/288 - log(x - 3/2)/96 - log(x + (3
^(1/2)*3i)/4 + 3/4)*((3^(1/2)*1i)/192 - 1/192) - log(x - (3^(1/2)*3i)/4 - 3/4)*((3^(1/2)*5i)/576 - 5/576) + lo
g(x + (3^(1/2)*3i)/4 - 3/4)*((3^(1/2)*5i)/576 + 5/576)

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