3.142 \(\int \frac{1}{x^2 (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6)} \, dx\)

Optimal. Leaf size=645 \[ \frac{\left (9 a^{2/3} c^{4/3}+12 \sqrt [3]{-1} \sqrt [3]{a} b c^{2/3}+2 (-1)^{2/3} b^2\right ) \tan ^{-1}\left (\frac{3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{81 \sqrt{3} \left (1+\sqrt [3]{-1}\right )^2 a^{23/6} c^{2/3} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}+\frac{\left (9 a^{2/3} c^{4/3}-12 \sqrt [3]{a} b c^{2/3}+2 b^2\right ) \tan ^{-1}\left (\frac{3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{243 \sqrt{3} a^{23/6} c^{2/3} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac{(-1)^{2/3} \left (9 (-1)^{2/3} a^{2/3} c^{4/3}+12 \sqrt [3]{-1} \sqrt [3]{a} b c^{2/3}+2 b^2\right ) \tan ^{-1}\left (\frac{3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{81 \sqrt{3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{23/6} c^{2/3} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}-\frac{\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \log \left (3 a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{486 a^{11/3} \sqrt [3]{c}}+\frac{\left (2 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \log \left (-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{162 \left (1+\sqrt [3]{-1}\right )^2 a^{11/3} \sqrt [3]{c}}+\frac{\sqrt [3]{-1} \left (3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+2 b\right ) \log \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{486 a^{11/3} \sqrt [3]{c}}-\frac{1}{27 a^3 x} \]

[Out]

-1/(27*a^3*x) + ((2*(-1)^(2/3)*b^2 + 12*(-1)^(1/3)*a^(1/3)*b*c^(2/3) + 9*a^(2/3)*c^(4/3))*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)])])/(81*Sqrt[3]*(1 + (-1)^(
1/3))^2*a^(23/6)*Sqrt[4*b - 3*(-1)^(2/3)*a^(1/3)*c^(2/3)]*c^(2/3)) + ((2*b^2 - 12*a^(1/3)*b*c^(2/3) + 9*a^(2/3
)*c^(4/3))*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)])])/(243*Sqrt[3]*a
^(23/6)*Sqrt[4*b - 3*a^(1/3)*c^(2/3)]*c^(2/3)) + ((-1)^(2/3)*(2*b^2 + 12*(-1)^(1/3)*a^(1/3)*b*c^(2/3) + 9*(-1)
^(2/3)*a^(2/3)*c^(4/3))*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)])])/(81*Sqrt[3]*(1 - (-1)^(1/3))*(1 + (-1)^(1/3))^2*a^(23/6)*Sqrt[4*b + 3*(-1)^(1/3)*a^(1/3)*
c^(2/3)]*c^(2/3)) - ((2*b - 3*a^(1/3)*c^(2/3))*Log[3*a + 3*a^(2/3)*c^(1/3)*x + b*x^2])/(486*a^(11/3)*c^(1/3))
+ ((2*b - 3*(-1)^(2/3)*a^(1/3)*c^(2/3))*Log[3*a - 3*(-1)^(1/3)*a^(2/3)*c^(1/3)*x + b*x^2])/(162*(1 + (-1)^(1/3
))^2*a^(11/3)*c^(1/3)) + ((-1)^(1/3)*(2*b + 3*(-1)^(1/3)*a^(1/3)*c^(2/3))*Log[3*a + 3*(-1)^(2/3)*a^(2/3)*c^(1/
3)*x + b*x^2])/(486*a^(11/3)*c^(1/3))

________________________________________________________________________________________

Rubi [A]  time = 1.37957, antiderivative size = 640, normalized size of antiderivative = 0.99, number of steps used = 14, number of rules used = 5, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.109, Rules used = {2097, 634, 618, 204, 628} \[ \frac{\left (9 a^{2/3} c^{4/3}+12 \sqrt [3]{-1} \sqrt [3]{a} b c^{2/3}+2 (-1)^{2/3} b^2\right ) \tan ^{-1}\left (\frac{3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{81 \sqrt{3} \left (1+\sqrt [3]{-1}\right )^2 a^{23/6} c^{2/3} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}+\frac{\left (9 a^{2/3} c^{4/3}-12 \sqrt [3]{a} b c^{2/3}+2 b^2\right ) \tan ^{-1}\left (\frac{3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{243 \sqrt{3} a^{23/6} c^{2/3} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac{\left (-9 \sqrt [3]{-1} a^{2/3} c^{4/3}-12 \sqrt [3]{a} b c^{2/3}+2 (-1)^{2/3} b^2\right ) \tan ^{-1}\left (\frac{3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{81 \sqrt{3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{23/6} c^{2/3} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}-\frac{\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \log \left (3 a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{486 a^{11/3} \sqrt [3]{c}}+\frac{\left (2 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \log \left (-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{162 \left (1+\sqrt [3]{-1}\right )^2 a^{11/3} \sqrt [3]{c}}+\frac{\sqrt [3]{-1} \left (3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+2 b\right ) \log \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{486 a^{11/3} \sqrt [3]{c}}-\frac{1}{27 a^3 x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(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]

-1/(27*a^3*x) + ((2*(-1)^(2/3)*b^2 + 12*(-1)^(1/3)*a^(1/3)*b*c^(2/3) + 9*a^(2/3)*c^(4/3))*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)])])/(81*Sqrt[3]*(1 + (-1)^(
1/3))^2*a^(23/6)*Sqrt[4*b - 3*(-1)^(2/3)*a^(1/3)*c^(2/3)]*c^(2/3)) + ((2*b^2 - 12*a^(1/3)*b*c^(2/3) + 9*a^(2/3
)*c^(4/3))*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)])])/(243*Sqrt[3]*a
^(23/6)*Sqrt[4*b - 3*a^(1/3)*c^(2/3)]*c^(2/3)) + ((2*(-1)^(2/3)*b^2 - 12*a^(1/3)*b*c^(2/3) - 9*(-1)^(1/3)*a^(2
/3)*c^(4/3))*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)])])/(81*Sqrt[3]*(1 - (-1)^(1/3))*(1 + (-1)^(1/3))^2*a^(23/6)*Sqrt[4*b + 3*(-1)^(1/3)*a^(1/3)*c^(2/3)]*c^
(2/3)) - ((2*b - 3*a^(1/3)*c^(2/3))*Log[3*a + 3*a^(2/3)*c^(1/3)*x + b*x^2])/(486*a^(11/3)*c^(1/3)) + ((2*b - 3
*(-1)^(2/3)*a^(1/3)*c^(2/3))*Log[3*a - 3*(-1)^(1/3)*a^(2/3)*c^(1/3)*x + b*x^2])/(162*(1 + (-1)^(1/3))^2*a^(11/
3)*c^(1/3)) + ((-1)^(1/3)*(2*b + 3*(-1)^(1/3)*a^(1/3)*c^(2/3))*Log[3*a + 3*(-1)^(2/3)*a^(2/3)*c^(1/3)*x + b*x^
2])/(486*a^(11/3)*c^(1/3))

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{1}{x^2 \left (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\right )} \, dx &=\left (19683 a^6\right ) \int \left (\frac{1}{531441 a^9 x^2}+\frac{\sqrt [3]{a} \left (b^2-9 \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\right )-b \left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \sqrt [3]{c} x}{1594323 \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{29/3} c^{2/3} \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}+\frac{-\sqrt [3]{a} \left ((-1)^{2/3} b^2+9 \sqrt [3]{-1} \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\right )+b \left (2 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \sqrt [3]{c} x}{1594323 \left (1+\sqrt [3]{-1}\right )^2 a^{29/3} c^{2/3} \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}+\frac{\sqrt [3]{a} \left ((-1)^{2/3} b^2-9 \sqrt [3]{a} b c^{2/3}-9 \sqrt [3]{-1} a^{2/3} c^{4/3}\right )+\sqrt [3]{-1} b \left (2 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}\right ) \sqrt [3]{c} x}{1594323 \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{29/3} c^{2/3} \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}\right ) \, dx\\ &=-\frac{1}{27 a^3 x}+\frac{\int \frac{\sqrt [3]{a} \left (b^2-9 \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\right )-b \left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \sqrt [3]{c} x}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{243 a^{11/3} c^{2/3}}+\frac{\int \frac{\sqrt [3]{a} \left ((-1)^{2/3} b^2-9 \sqrt [3]{a} b c^{2/3}-9 \sqrt [3]{-1} a^{2/3} c^{4/3}\right )+\sqrt [3]{-1} b \left (2 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}\right ) \sqrt [3]{c} x}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{243 a^{11/3} c^{2/3}}+\frac{\int \frac{-\sqrt [3]{a} \left ((-1)^{2/3} b^2+9 \sqrt [3]{-1} \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\right )+b \left (2 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \sqrt [3]{c} x}{3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{81 \left (1+\sqrt [3]{-1}\right )^2 a^{11/3} c^{2/3}}\\ &=-\frac{1}{27 a^3 x}-\frac{\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \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}{486 a^{11/3} \sqrt [3]{c}}+\frac{\left (\sqrt [3]{-1} \left (2 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}\right )\right ) \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}{486 a^{11/3} \sqrt [3]{c}}+\frac{\left (2 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \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}{162 \left (1+\sqrt [3]{-1}\right )^2 a^{11/3} \sqrt [3]{c}}+\frac{\left (2 b^2-12 \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\right ) \int \frac{1}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{486 a^{10/3} c^{2/3}}-\frac{\left (2 (-1)^{2/3} b^2+12 \sqrt [3]{-1} \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\right ) \int \frac{1}{3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{162 \left (1+\sqrt [3]{-1}\right )^2 a^{10/3} c^{2/3}}+\frac{\left (2 (-1)^{2/3} b^2-12 \sqrt [3]{a} b c^{2/3}-9 \sqrt [3]{-1} a^{2/3} c^{4/3}\right ) \int \frac{1}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{486 a^{10/3} c^{2/3}}\\ &=-\frac{1}{27 a^3 x}-\frac{\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{486 a^{11/3} \sqrt [3]{c}}+\frac{\left (2 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \log \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 \left (1+\sqrt [3]{-1}\right )^2 a^{11/3} \sqrt [3]{c}}+\frac{\sqrt [3]{-1} \left (2 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}\right ) \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{486 a^{11/3} \sqrt [3]{c}}-\frac{\left (2 b^2-12 \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\right ) \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 )}{243 a^{10/3} c^{2/3}}+\frac{\left (2 (-1)^{2/3} b^2+12 \sqrt [3]{-1} \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\right ) \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 )}{81 \left (1+\sqrt [3]{-1}\right )^2 a^{10/3} c^{2/3}}-\frac{\left (2 (-1)^{2/3} b^2-12 \sqrt [3]{a} b c^{2/3}-9 \sqrt [3]{-1} a^{2/3} c^{4/3}\right ) \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 )}{243 a^{10/3} c^{2/3}}\\ &=-\frac{1}{27 a^3 x}+\frac{\left (2 (-1)^{2/3} b^2+12 \sqrt [3]{-1} \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\right ) \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 )}{81 \sqrt{3} \left (1+\sqrt [3]{-1}\right )^2 a^{23/6} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}} c^{2/3}}+\frac{\left (2 b^2-12 \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\right ) \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 )}{243 \sqrt{3} a^{23/6} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}} c^{2/3}}+\frac{\left (2 (-1)^{2/3} b^2-12 \sqrt [3]{a} b c^{2/3}-9 \sqrt [3]{-1} a^{2/3} c^{4/3}\right ) \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 )}{243 \sqrt{3} a^{23/6} \sqrt{4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}} c^{2/3}}-\frac{\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{486 a^{11/3} \sqrt [3]{c}}+\frac{\left (2 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \log \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 \left (1+\sqrt [3]{-1}\right )^2 a^{11/3} \sqrt [3]{c}}+\frac{\sqrt [3]{-1} \left (2 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}\right ) \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{486 a^{11/3} \sqrt [3]{c}}\\ \end{align*}

Mathematica [C]  time = 0.119681, size = 163, normalized size = 0.25 \[ -\frac{x \text{RootSum}\left [27 \text{$\#$1}^2 a^2 b+27 \text{$\#$1}^3 a^2 c+9 \text{$\#$1}^4 a b^2+\text{$\#$1}^6 b^3+27 a^3\& ,\frac{9 \text{$\#$1}^2 a b^2 \log (x-\text{$\#$1})+\text{$\#$1}^4 b^3 \log (x-\text{$\#$1})+27 a^2 b \log (x-\text{$\#$1})+27 \text{$\#$1} a^2 c \log (x-\text{$\#$1})}{27 \text{$\#$1}^2 a^2 c+12 \text{$\#$1}^3 a b^2+2 \text{$\#$1}^5 b^3+18 \text{$\#$1} a^2 b}\& \right ]+3}{81 a^3 x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(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]

-(3 + x*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 & , (27*a^2*b*Log[x - #1] + 2
7*a^2*c*Log[x - #1]*#1 + 9*a*b^2*Log[x - #1]*#1^2 + b^3*Log[x - #1]*#1^4)/(18*a^2*b*#1 + 27*a^2*c*#1^2 + 12*a*
b^2*#1^3 + 2*b^3*#1^5) & ])/(81*a^3*x)

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Maple [C]  time = 0.007, size = 133, normalized size = 0.2 \begin{align*}{\frac{1}{81\,{a}^{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{ \left ( -{{\it \_R}}^{4}{b}^{3}-9\,{{\it \_R}}^{2}a{b}^{2}-27\,{\it \_R}\,{a}^{2}c-27\,{a}^{2}b \right ) \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}}}-{\frac{1}{27\,{a}^{3}x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(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]

1/81/a^3*sum((-_R^4*b^3-9*_R^2*a*b^2-27*_R*a^2*c-27*a^2*b)/(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))-1/27/a^3/x

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Maxima [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(1/x^2/(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]

Timed out

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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(1/x^2/(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(1/x**2/(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{1}{{\left (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}\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(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(1/((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^2), x)