3.139 \(\int \frac{x}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx\)

Optimal. Leaf size=469 \[ -\frac{\log \left (3 a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{162 a^{7/3} c^{2/3}}+\frac{(-1)^{2/3} \log \left (-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{7/3} c^{2/3}}-\frac{(-1)^{2/3} \log \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{162 a^{7/3} c^{2/3}}-\frac{\tan ^{-1}\left (\frac{3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt{3} \left (1+\sqrt [3]{-1}\right )^2 a^{13/6} \sqrt [3]{c} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}-\frac{\tan ^{-1}\left (\frac{3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt{3} a^{13/6} \sqrt [3]{c} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac{\sqrt [3]{-1} \tan ^{-1}\left (\frac{3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{9 \sqrt{3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{13/6} \sqrt [3]{c} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}} \]

[Out]

-ArcTan[(3*(-1)^(1/3)*a^(2/3)*c^(1/3) - 2*b*x)/(Sqrt[3]*Sqrt[a]*Sqrt[4*b - 3*(-1)^(2/3)*a^(1/3)*c^(2/3)])]/(9*
Sqrt[3]*(1 + (-1)^(1/3))^2*a^(13/6)*Sqrt[4*b - 3*(-1)^(2/3)*a^(1/3)*c^(2/3)]*c^(1/3)) - ArcTan[(3*a^(2/3)*c^(1
/3) + 2*b*x)/(Sqrt[3]*Sqrt[a]*Sqrt[4*b - 3*a^(1/3)*c^(2/3)])]/(27*Sqrt[3]*a^(13/6)*Sqrt[4*b - 3*a^(1/3)*c^(2/3
)]*c^(1/3)) + ((-1)^(1/3)*ArcTan[(3*(-1)^(2/3)*a^(2/3)*c^(1/3) + 2*b*x)/(Sqrt[3]*Sqrt[a]*Sqrt[4*b + 3*(-1)^(1/
3)*a^(1/3)*c^(2/3)])])/(9*Sqrt[3]*(1 - (-1)^(1/3))*(1 + (-1)^(1/3))^2*a^(13/6)*Sqrt[4*b + 3*(-1)^(1/3)*a^(1/3)
*c^(2/3)]*c^(1/3)) - Log[3*a + 3*a^(2/3)*c^(1/3)*x + b*x^2]/(162*a^(7/3)*c^(2/3)) + ((-1)^(2/3)*Log[3*a - 3*(-
1)^(1/3)*a^(2/3)*c^(1/3)*x + b*x^2])/(54*(1 + (-1)^(1/3))^2*a^(7/3)*c^(2/3)) - ((-1)^(2/3)*Log[3*a + 3*(-1)^(2
/3)*a^(2/3)*c^(1/3)*x + b*x^2])/(162*a^(7/3)*c^(2/3))

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Rubi [A]  time = 0.684432, antiderivative size = 469, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {2097, 634, 618, 204, 628} \[ -\frac{\log \left (3 a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{162 a^{7/3} c^{2/3}}+\frac{(-1)^{2/3} \log \left (-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{7/3} c^{2/3}}-\frac{(-1)^{2/3} \log \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{162 a^{7/3} c^{2/3}}-\frac{\tan ^{-1}\left (\frac{3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt{3} \left (1+\sqrt [3]{-1}\right )^2 a^{13/6} \sqrt [3]{c} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}-\frac{\tan ^{-1}\left (\frac{3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt{3} a^{13/6} \sqrt [3]{c} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac{\sqrt [3]{-1} \tan ^{-1}\left (\frac{3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{9 \sqrt{3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{13/6} \sqrt [3]{c} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}} \]

Antiderivative was successfully verified.

[In]

Int[x/(27*a^3 + 27*a^2*b*x^2 + 27*a^2*c*x^3 + 9*a*b^2*x^4 + b^3*x^6),x]

[Out]

-ArcTan[(3*(-1)^(1/3)*a^(2/3)*c^(1/3) - 2*b*x)/(Sqrt[3]*Sqrt[a]*Sqrt[4*b - 3*(-1)^(2/3)*a^(1/3)*c^(2/3)])]/(9*
Sqrt[3]*(1 + (-1)^(1/3))^2*a^(13/6)*Sqrt[4*b - 3*(-1)^(2/3)*a^(1/3)*c^(2/3)]*c^(1/3)) - ArcTan[(3*a^(2/3)*c^(1
/3) + 2*b*x)/(Sqrt[3]*Sqrt[a]*Sqrt[4*b - 3*a^(1/3)*c^(2/3)])]/(27*Sqrt[3]*a^(13/6)*Sqrt[4*b - 3*a^(1/3)*c^(2/3
)]*c^(1/3)) + ((-1)^(1/3)*ArcTan[(3*(-1)^(2/3)*a^(2/3)*c^(1/3) + 2*b*x)/(Sqrt[3]*Sqrt[a]*Sqrt[4*b + 3*(-1)^(1/
3)*a^(1/3)*c^(2/3)])])/(9*Sqrt[3]*(1 - (-1)^(1/3))*(1 + (-1)^(1/3))^2*a^(13/6)*Sqrt[4*b + 3*(-1)^(1/3)*a^(1/3)
*c^(2/3)]*c^(1/3)) - Log[3*a + 3*a^(2/3)*c^(1/3)*x + b*x^2]/(162*a^(7/3)*c^(2/3)) + ((-1)^(2/3)*Log[3*a - 3*(-
1)^(1/3)*a^(2/3)*c^(1/3)*x + b*x^2])/(54*(1 + (-1)^(1/3))^2*a^(7/3)*c^(2/3)) - ((-1)^(2/3)*Log[3*a + 3*(-1)^(2
/3)*a^(2/3)*c^(1/3)*x + b*x^2])/(162*a^(7/3)*c^(2/3))

Rule 2097

Int[(Q6_)^(p_)*(u_), x_Symbol] :> With[{a = Coeff[Q6, x, 0], b = Coeff[Q6, x, 2], c = Coeff[Q6, x, 3], d = Coe
ff[Q6, x, 4], e = Coeff[Q6, x, 6]}, Dist[1/(3^(3*p)*a^(2*p)), Int[ExpandIntegrand[u*(3*a + 3*Rt[a, 3]^2*Rt[c,
3]*x + b*x^2)^p*(3*a - 3*(-1)^(1/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p*(3*a + 3*(-1)^(2/3)*Rt[a, 3]^2*Rt[c, 3]*x
+ b*x^2)^p, x], x], x] /; EqQ[b^2 - 3*a*d, 0] && EqQ[b^3 - 27*a^2*e, 0]] /; ILtQ[p, 0] && PolyQ[Q6, x, 6] &&
EqQ[Coeff[Q6, x, 1], 0] && EqQ[Coeff[Q6, x, 5], 0] && RationalFunctionQ[u, 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}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx &=\left (19683 a^6\right ) \int \left (\frac{-3 a^{2/3} \sqrt [3]{c}-(-1)^{2/3} b x}{531441 \left (1+\sqrt [3]{-1}\right )^2 a^{25/3} c^{2/3} \left (-3 a+3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x-b x^2\right )}+\frac{-3 a^{2/3} \sqrt [3]{c}-b x}{1594323 a^{25/3} c^{2/3} \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}+\frac{(-1)^{2/3} \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+b x\right )}{531441 \left (-1+\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{25/3} c^{2/3} \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{-3 a^{2/3} \sqrt [3]{c}-b x}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{81 a^{7/3} c^{2/3}}-\frac{(-1)^{2/3} \int \frac{3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+b x}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{81 a^{7/3} c^{2/3}}+\frac{\int \frac{-3 a^{2/3} \sqrt [3]{c}-(-1)^{2/3} b x}{-3 a+3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x-b x^2} \, dx}{27 \left (1+\sqrt [3]{-1}\right )^2 a^{7/3} c^{2/3}}\\ &=-\frac{\int \frac{3 a^{2/3} \sqrt [3]{c}+2 b x}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{162 a^{7/3} c^{2/3}}-\frac{(-1)^{2/3} \int \frac{3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{162 a^{7/3} c^{2/3}}+\frac{(-1)^{2/3} \int \frac{3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{-3 a+3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x-b x^2} \, dx}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{7/3} c^{2/3}}-\frac{\int \frac{1}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{54 a^{5/3} \sqrt [3]{c}}+\frac{\sqrt [3]{-1} \int \frac{1}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{54 a^{5/3} \sqrt [3]{c}}-\frac{\int \frac{1}{-3 a+3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x-b x^2} \, dx}{18 \left (1+\sqrt [3]{-1}\right )^2 a^{5/3} \sqrt [3]{c}}\\ &=-\frac{\log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{7/3} c^{2/3}}+\frac{(-1)^{2/3} \log \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{7/3} c^{2/3}}-\frac{(-1)^{2/3} \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{7/3} c^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-3 a \left (4 b-3 \sqrt [3]{a} c^{2/3}\right )-x^2} \, dx,x,3 a^{2/3} \sqrt [3]{c}+2 b x\right )}{27 a^{5/3} \sqrt [3]{c}}-\frac{\sqrt [3]{-1} \operatorname{Subst}\left (\int \frac{1}{-3 a \left (4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}\right )-x^2} \, dx,x,3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x\right )}{27 a^{5/3} \sqrt [3]{c}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-3 a \left (4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}\right )-x^2} \, dx,x,3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x\right )}{9 \left (1+\sqrt [3]{-1}\right )^2 a^{5/3} \sqrt [3]{c}}\\ &=-\frac{\tan ^{-1}\left (\frac{3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt{3} \left (1+\sqrt [3]{-1}\right )^2 a^{13/6} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}-\frac{\tan ^{-1}\left (\frac{3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt{3} a^{13/6} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}+\frac{\sqrt [3]{-1} \tan ^{-1}\left (\frac{3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt{3} a^{13/6} \sqrt{4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}-\frac{\log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{7/3} c^{2/3}}+\frac{(-1)^{2/3} \log \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{7/3} c^{2/3}}-\frac{(-1)^{2/3} \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{7/3} c^{2/3}}\\ \end{align*}

Mathematica [C]  time = 0.0450699, size = 95, normalized size = 0.2 \[ \frac{1}{3} \text{RootSum}\left [27 \text{$\#$1}^2 a^2 b+27 \text{$\#$1}^3 a^2 c+9 \text{$\#$1}^4 a b^2+\text{$\#$1}^6 b^3+27 a^3\& ,\frac{\log (x-\text{$\#$1})}{12 \text{$\#$1}^2 a b^2+2 \text{$\#$1}^4 b^3+27 \text{$\#$1} a^2 c+18 a^2 b}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[x/(27*a^3 + 27*a^2*b*x^2 + 27*a^2*c*x^3 + 9*a*b^2*x^4 + b^3*x^6),x]

[Out]

RootSum[27*a^3 + 27*a^2*b*#1^2 + 27*a^2*c*#1^3 + 9*a*b^2*#1^4 + b^3*#1^6 & , Log[x - #1]/(18*a^2*b + 27*a^2*c*
#1 + 12*a*b^2*#1^2 + 2*b^3*#1^4) & ]/3

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Maple [C]  time = 0.003, size = 91, normalized size = 0.2 \begin{align*}{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({b}^{3}{{\it \_Z}}^{6}+9\,a{b}^{2}{{\it \_Z}}^{4}+27\,{a}^{2}c{{\it \_Z}}^{3}+27\,{a}^{2}b{{\it \_Z}}^{2}+27\,{a}^{3} \right ) }{\frac{{\it \_R}\,\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}{b}^{3}+12\,{{\it \_R}}^{3}a{b}^{2}+27\,{{\it \_R}}^{2}{a}^{2}c+18\,{\it \_R}\,{a}^{2}b}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(b^3*x^6+9*a*b^2*x^4+27*a^2*c*x^3+27*a^2*b*x^2+27*a^3),x)

[Out]

1/3*sum(_R/(2*_R^5*b^3+12*_R^3*a*b^2+27*_R^2*a^2*c+18*_R*a^2*b)*ln(x-_R),_R=RootOf(_Z^6*b^3+9*_Z^4*a*b^2+27*_Z
^3*a^2*c+27*_Z^2*a^2*b+27*a^3))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b^3*x^6+9*a*b^2*x^4+27*a^2*c*x^3+27*a^2*b*x^2+27*a^3),x, algorithm="maxima")

[Out]

integrate(x/(b^3*x^6 + 9*a*b^2*x^4 + 27*a^2*c*x^3 + 27*a^2*b*x^2 + 27*a^3), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b^3*x^6+9*a*b^2*x^4+27*a^2*c*x^3+27*a^2*b*x^2+27*a^3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b**3*x**6+9*a*b**2*x**4+27*a**2*c*x**3+27*a**2*b*x**2+27*a**3),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b^3*x^6+9*a*b^2*x^4+27*a^2*c*x^3+27*a^2*b*x^2+27*a^3),x, algorithm="giac")

[Out]

integrate(x/(b^3*x^6 + 9*a*b^2*x^4 + 27*a^2*c*x^3 + 27*a^2*b*x^2 + 27*a^3), x)