3.6.60 \(\int \frac {3+2 x}{729-64 x^6} \, dx\) [560]

Optimal. Leaf size=50 \[ -\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{162 \sqrt {3}}-\frac {1}{486} \log (3-2 x)+\frac {1}{972} \log \left (9+6 x+4 x^2\right ) \]

[Out]

-1/486*ln(3-2*x)+1/972*ln(4*x^2+6*x+9)-1/486*arctan(1/9*(3-4*x)*3^(1/2))*3^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1600, 2083, 632, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{162 \sqrt {3}}+\frac {1}{972} \log \left (4 x^2+6 x+9\right )-\frac {1}{486} \log (3-2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 2*x)/(729 - 64*x^6),x]

[Out]

-1/162*ArcTan[(3 - 4*x)/(3*Sqrt[3])]/Sqrt[3] - Log[3 - 2*x]/486 + Log[9 + 6*x + 4*x^2]/972

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 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 1600

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 2083

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P]}, Int[ExpandIntegrand[u^p, x], x] /;  !SumQ[NonfreeFactors[u,
x]]] /; PolyQ[P, x] && ILtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {3+2 x}{729-64 x^6} \, dx &=\int \frac {1}{243-162 x+108 x^2-72 x^3+48 x^4-32 x^5} \, dx\\ &=\int \left (-\frac {1}{243 (-3+2 x)}+\frac {1}{54 \left (9-6 x+4 x^2\right )}+\frac {3+4 x}{486 \left (9+6 x+4 x^2\right )}\right ) \, dx\\ &=-\frac {1}{486} \log (3-2 x)+\frac {1}{486} \int \frac {3+4 x}{9+6 x+4 x^2} \, dx+\frac {1}{54} \int \frac {1}{9-6 x+4 x^2} \, dx\\ &=-\frac {1}{486} \log (3-2 x)+\frac {1}{972} \log \left (9+6 x+4 x^2\right )-\frac {1}{27} \text {Subst}\left (\int \frac {1}{-108-x^2} \, dx,x,-6+8 x\right )\\ &=-\frac {\tan ^{-1}\left (\frac {3-4 x}{3 \sqrt {3}}\right )}{162 \sqrt {3}}-\frac {1}{486} \log (3-2 x)+\frac {1}{972} \log \left (9+6 x+4 x^2\right )\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 46, normalized size = 0.92 \begin {gather*} \frac {1}{972} \left (2 \sqrt {3} \tan ^{-1}\left (\frac {-3+4 x}{3 \sqrt {3}}\right )-2 \log (3-2 x)+\log \left (9+6 x+4 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 2*x)/(729 - 64*x^6),x]

[Out]

(2*Sqrt[3]*ArcTan[(-3 + 4*x)/(3*Sqrt[3])] - 2*Log[3 - 2*x] + Log[9 + 6*x + 4*x^2])/972

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Maple [A]
time = 0.37, size = 39, normalized size = 0.78

method	result	size
default	\(\frac {\sqrt {3}\, \arctan \left (\frac {\left (8 x -6\right ) \sqrt {3}}{18}\right )}{486}+\frac {\ln \left (4 x^{2}+6 x +9\right )}{972}-\frac {\ln \left (-3+2 x \right )}{486}\)	\(39\)
risch	\(-\frac {\ln \left (-3+2 x \right )}{486}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (-3+4 x \right ) \sqrt {3}}{9}\right )}{486}+\frac {\ln \left (4 x^{2}+6 x +9\right )}{972}\)	\(39\)
meijerg	\(-\frac {x \left (\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )-\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}\right )+\frac {\ln \left (1-\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{6}}}{3}+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{3+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{972 \left (x^{6}\right )^{\frac {1}{6}}}-\frac {x^{2} \left (\ln \left (1-\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )-\frac {\ln \left (1+\frac {4 \left (x^{6}\right )^{\frac {1}{3}}}{9}+\frac {16 \left (x^{6}\right )^{\frac {2}{3}}}{81}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (x^{6}\right )^{\frac {1}{3}}}{9 \left (1+\frac {2 \left (x^{6}\right )^{\frac {1}{3}}}{9}\right )}\right )\right )}{972 \left (x^{6}\right )^{\frac {1}{3}}}\)	\(192\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/486*3^(1/2)*arctan(1/18*(8*x-6)*3^(1/2))+1/972*ln(4*x^2+6*x+9)-1/486*ln(-3+2*x)

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Maxima [A]
time = 0.50, size = 38, normalized size = 0.76 \begin {gather*} \frac {1}{486} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) + \frac {1}{972} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac {1}{486} \, \log \left (2 \, x - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/486*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) + 1/972*log(4*x^2 + 6*x + 9) - 1/486*log(2*x - 3)

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Fricas [A]
time = 0.39, size = 38, normalized size = 0.76 \begin {gather*} \frac {1}{486} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) + \frac {1}{972} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac {1}{486} \, \log \left (2 \, x - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/486*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) + 1/972*log(4*x^2 + 6*x + 9) - 1/486*log(2*x - 3)

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Sympy [A]
time = 0.07, size = 46, normalized size = 0.92 \begin {gather*} - \frac {\log {\left (x - \frac {3}{2} \right )}}{486} + \frac {\log {\left (4 x^{2} + 6 x + 9 \right )}}{972} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {4 \sqrt {3} x}{9} - \frac {\sqrt {3}}{3} \right )}}{486} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x - 3/2)/486 + log(4*x**2 + 6*x + 9)/972 + sqrt(3)*atan(4*sqrt(3)*x/9 - sqrt(3)/3)/486

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Giac [A]
time = 0.80, size = 39, normalized size = 0.78 \begin {gather*} \frac {1}{486} \, \sqrt {3} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, x - 3\right )}\right ) + \frac {1}{972} \, \log \left (4 \, x^{2} + 6 \, x + 9\right ) - \frac {1}{486} \, \log \left ({\left | 2 \, x - 3 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/486*sqrt(3)*arctan(1/9*sqrt(3)*(4*x - 3)) + 1/972*log(4*x^2 + 6*x + 9) - 1/486*log(abs(2*x - 3))

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Mupad [B]
time = 4.99, size = 48, normalized size = 0.96 \begin {gather*} \frac {\ln \left (x^2+\frac {3\,x}{2}+\frac {9}{4}\right )}{972}-\frac {\ln \left (x-\frac {3}{2}\right )}{486}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}}{1327104\,\left (\frac {x}{884736}-\frac {1}{884736}\right )}+\frac {\sqrt {3}\,x}{7962624\,\left (\frac {x}{884736}-\frac {1}{884736}\right )}\right )}{486} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x + 3)/(64*x^6 - 729),x)

[Out]

log((3*x)/2 + x^2 + 9/4)/972 - log(x - 3/2)/486 - (3^(1/2)*atan(3^(1/2)/(1327104*(x/884736 - 1/884736)) + (3^(
1/2)*x)/(7962624*(x/884736 - 1/884736))))/486

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