\(\int \frac {1}{x^2 (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6)} \, dx\) [142]

Optimal result

Integrand size = 46, antiderivative size = 645 \[ \int \frac {1}{x^2 \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx=-\frac {1}{27 a^3 x}+\frac {\left (2 (-1)^{2/3} b^2+12 \sqrt [3]{-1} \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\right ) \arctan \left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{81 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{23/6} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}} c^{2/3}}+\frac {\left (2 b^2-12 \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\right ) \arctan \left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{243 \sqrt {3} a^{23/6} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}} c^{2/3}}+\frac {(-1)^{2/3} \left (2 b^2+12 \sqrt [3]{-1} \sqrt [3]{a} b c^{2/3}+9 (-1)^{2/3} a^{2/3} c^{4/3}\right ) \arctan \left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}}}\right )}{81 \sqrt {3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{23/6} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}} c^{2/3}}-\frac {\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{486 a^{11/3} \sqrt [3]{c}}+\frac {\left (2 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \log \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 \left (1+\sqrt [3]{-1}\right )^2 a^{11/3} \sqrt [3]{c}}+\frac {\sqrt [3]{-1} \left (2 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}\right ) \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{486 a^{11/3} \sqrt [3]{c}} \]

[Out]

-1/27/a^3/x-1/486*(2*b-3*a^(1/3)*c^(2/3))*ln(3*a+3*a^(2/3)*c^(1/3)*x+b*x^2)/a^(11/3)/c^(1/3)+1/162*(2*b-3*(-1)
^(2/3)*a^(1/3)*c^(2/3))*ln(3*a-3*(-1)^(1/3)*a^(2/3)*c^(1/3)*x+b*x^2)/(1+(-1)^(1/3))^2/a^(11/3)/c^(1/3)+1/486*(
-1)^(1/3)*(2*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3))*ln(3*a+3*(-1)^(2/3)*a^(2/3)*c^(1/3)*x+b*x^2)/a^(11/3)/c^(1/3)+1/7
29*(2*b^2-12*a^(1/3)*b*c^(2/3)+9*a^(2/3)*c^(4/3))*arctan(1/3*(3*a^(2/3)*c^(1/3)+2*b*x)*3^(1/2)/a^(1/2)/(4*b-3*
a^(1/3)*c^(2/3))^(1/2))/a^(23/6)/c^(2/3)*3^(1/2)/(4*b-3*a^(1/3)*c^(2/3))^(1/2)+1/243*(-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(1/3*(3*(-1)^(2/3)*a^(2/3)*c^(1/3)+2*b*x)*3^(1/2)
/a^(1/2)/(4*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3))^(1/2))/(1-(-1)^(1/3))/(1+(-1)^(1/3))^2/a^(23/6)/c^(2/3)*3^(1/2)/(4
*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3))^(1/2)+1/243*(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(1/3*(3*(-1)^(1/3)*a^(2/3)*c^(1/3)-2*b*x)*3^(1/2)/a^(1/2)/(4*b-3*(-1)^(2/3)*a^(1/3)*c^(2/3))^(1/2))/
(1+(-1)^(1/3))^2/a^(23/6)/c^(2/3)*3^(1/2)/(4*b-3*(-1)^(2/3)*a^(1/3)*c^(2/3))^(1/2)

Rubi [A] (verified)

Time = 1.39 (sec) , antiderivative size = 640, normalized size of antiderivative = 0.99, number of steps used = 14, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {2122, 648, 632, 210, 642} \[ \int \frac {1}{x^2 \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx=\frac {\left (9 a^{2/3} c^{4/3}+12 \sqrt [3]{-1} \sqrt [3]{a} b c^{2/3}+2 (-1)^{2/3} b^2\right ) \arctan \left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{81 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{23/6} c^{2/3} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}+\frac {\left (9 a^{2/3} c^{4/3}-12 \sqrt [3]{a} b c^{2/3}+2 b^2\right ) \arctan \left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{243 \sqrt {3} a^{23/6} c^{2/3} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac {\left (-9 \sqrt [3]{-1} a^{2/3} c^{4/3}-12 \sqrt [3]{a} b c^{2/3}+2 (-1)^{2/3} b^2\right ) \arctan \left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{81 \sqrt {3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{23/6} c^{2/3} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}-\frac {\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \log \left (3 a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{486 a^{11/3} \sqrt [3]{c}}+\frac {\left (2 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \log \left (-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{162 \left (1+\sqrt [3]{-1}\right )^2 a^{11/3} \sqrt [3]{c}}+\frac {\sqrt [3]{-1} \left (3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+2 b\right ) \log \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{486 a^{11/3} \sqrt [3]{c}}-\frac {1}{27 a^3 x} \]

[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*1/(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^(1
1/3)*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 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 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 2122

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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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 ) \text {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] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.08 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.25 \[ \int \frac {1}{x^2 \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx=-\frac {3+x \text {RootSum}\left [27 a^3+27 a^2 b \text {$\#$1}^2+27 a^2 c \text {$\#$1}^3+9 a b^2 \text {$\#$1}^4+b^3 \text {$\#$1}^6\&,\frac {27 a^2 b \log (x-\text {$\#$1})+27 a^2 c \log (x-\text {$\#$1}) \text {$\#$1}+9 a b^2 \log (x-\text {$\#$1}) \text {$\#$1}^2+b^3 \log (x-\text {$\#$1}) \text {$\#$1}^4}{18 a^2 b \text {$\#$1}+27 a^2 c \text {$\#$1}^2+12 a b^2 \text {$\#$1}^3+2 b^3 \text {$\#$1}^5}\&\right ]}{81 a^3 x} \]

[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]

-1/81*(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
] + 27*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) & ])/(a^3*x)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.11 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.21

method	result	size
default	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 c \,a^{2} \textit {\_Z}^{3}+27 a^{2} b \,\textit {\_Z}^{2}+27 a^{3}\right )}{\sum }\frac {\left (-\textit {\_R}^{4} b^{3}-9 \textit {\_R}^{2} a \,b^{2}-27 \textit {\_R} \,a^{2} c -27 a^{2} b \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} b^{3}+12 \textit {\_R}^{3} a \,b^{2}+27 \textit {\_R}^{2} a^{2} c +18 a^{2} b \textit {\_R}}}{81 a^{3}}-\frac {1}{27 a^{3} x}\)	\(133\)
risch	\(-\frac {1}{27 a^{3} x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (729 a^{24} c^{6}-1728 a^{23} b^{3} c^{4}\right ) \textit {\_Z}^{6}+\left (13122 a^{17} b \,c^{6}-31347 a^{16} b^{4} c^{4}\right ) \textit {\_Z}^{4}+\left (-19683 c^{7} a^{14}+52488 c^{5} b^{3} a^{13}-14472 c^{3} b^{6} a^{12}\right ) \textit {\_Z}^{3}+\left (-4374 a^{9} b^{5} c^{4}-1701 a^{8} b^{8} c^{2}\right ) \textit {\_Z}^{2}-72 a^{4} b^{10} c \textit {\_Z} -b^{12}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-8748 a^{24} c^{6}+20574 a^{23} b^{3} c^{4}\right ) \textit {\_R}^{6}+\left (-3645 a^{20} b^{2} c^{5}+8100 a^{19} b^{5} c^{3}\right ) \textit {\_R}^{5}+\left (-118098 a^{17} b \,c^{6}+278478 a^{16} b^{4} c^{4}+1728 a^{15} b^{7} c^{2}\right ) \textit {\_R}^{4}+\left (177147 c^{7} a^{14}-472392 c^{5} b^{3} a^{13}+130329 c^{3} b^{6} a^{12}+108 a^{11} b^{9} c \right ) \textit {\_R}^{3}+\left (39366 a^{9} b^{5} c^{4}+15309 a^{8} b^{8} c^{2}+2 a^{7} b^{11}\right ) \textit {\_R}^{2}+648 \textit {\_R} \,a^{4} b^{10} c +9 b^{12}\right ) x +\left (729 a^{24} b \,c^{5}-2160 a^{23} b^{4} c^{3}\right ) \textit {\_R}^{6}+\left (-6561 a^{21} c^{6}+15066 a^{20} b^{3} c^{4}-144 a^{19} b^{6} c^{2}\right ) \textit {\_R}^{5}+\left (-6561 a^{17} b^{2} c^{5}+5832 a^{16} b^{5} c^{3}\right ) \textit {\_R}^{4}+54 a^{12} b^{7} c^{2} \textit {\_R}^{3}-9 a^{8} b^{9} c \,\textit {\_R}^{2}\right )\right )}{243}\)	\(444\)

[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,method=_RETURNVERBOSE)

[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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx=\text {Timed out} \]

[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

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx=\text {Timed out} \]

[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

Maxima [F]

\[ \int \frac {1}{x^2 \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx=\int { \frac {1}{{\left (b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}\right )} x^{2}} \,d x } \]

[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]

-1/27*integrate((b^3*x^4 + 9*a*b^2*x^2 + 27*a^2*c*x + 27*a^2*b)/(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)/a^3 - 1/27/(a^3*x)

Giac [F]

\[ \int \frac {1}{x^2 \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx=\int { \frac {1}{{\left (b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}\right )} x^{2}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 9.68 (sec) , antiderivative size = 2663, normalized size of antiderivative = 4.13 \[ \int \frac {1}{x^2 \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx=\text {Too large to display} \]

[In]

int(1/(x^2*(27*a^3 + b^3*x^6 + 27*a^2*b*x^2 + 9*a*b^2*x^4 + 27*a^2*c*x^3)),x)

[Out]

symsum(log(-282429536481*root(355779876259553472*a^23*b^3*c^4*z^6 - 150094635296999121*a^24*c^6*z^6 - 45753584
909922*a^17*b*c^6*z^4 + 109300230618147*a^16*b^4*c^4*z^4 - 753145430616*a^13*b^3*c^5*z^3 + 207657382104*a^12*b
^6*c^3*z^3 + 282429536481*a^14*c^7*z^3 + 258280326*a^9*b^5*c^4*z^2 + 100442349*a^8*b^8*c^2*z^2 + 17496*a^4*b^1
0*c*z + b^12, z, k)*a^23*b^9*(2*b^10*x + 2541865828329*root(355779876259553472*a^23*b^3*c^4*z^6 - 150094635296
999121*a^24*c^6*z^6 - 45753584909922*a^17*b*c^6*z^4 + 109300230618147*a^16*b^4*c^4*z^4 - 753145430616*a^13*b^3
*c^5*z^3 + 207657382104*a^12*b^6*c^3*z^3 + 282429536481*a^14*c^7*z^3 + 258280326*a^9*b^5*c^4*z^2 + 100442349*a
^8*b^8*c^2*z^2 + 17496*a^4*b^10*c*z + b^12, z, k)^4*a^17*c^5 - 45*a*b^8*c + 387420489*root(355779876259553472*
a^23*b^3*c^4*z^6 - 150094635296999121*a^24*c^6*z^6 - 45753584909922*a^17*b*c^6*z^4 + 109300230618147*a^16*b^4*
c^4*z^4 - 753145430616*a^13*b^3*c^5*z^3 + 207657382104*a^12*b^6*c^3*z^3 + 282429536481*a^14*c^7*z^3 + 25828032
6*a^9*b^5*c^4*z^2 + 100442349*a^8*b^8*c^2*z^2 + 17496*a^4*b^10*c*z + b^12, z, k)^2*a^10*c^6*x - 401769396*root
(355779876259553472*a^23*b^3*c^4*z^6 - 150094635296999121*a^24*c^6*z^6 - 45753584909922*a^17*b*c^6*z^4 + 10930
0230618147*a^16*b^4*c^4*z^4 - 753145430616*a^13*b^3*c^5*z^3 + 207657382104*a^12*b^6*c^3*z^3 + 282429536481*a^1
4*c^7*z^3 + 258280326*a^9*b^5*c^4*z^2 + 100442349*a^8*b^8*c^2*z^2 + 17496*a^4*b^10*c*z + b^12, z, k)^2*a^9*b^4
*c^3 - 2066242608*root(355779876259553472*a^23*b^3*c^4*z^6 - 150094635296999121*a^24*c^6*z^6 - 45753584909922*
a^17*b*c^6*z^4 + 109300230618147*a^16*b^4*c^4*z^4 - 753145430616*a^13*b^3*c^5*z^3 + 207657382104*a^12*b^6*c^3*
z^3 + 282429536481*a^14*c^7*z^3 + 258280326*a^9*b^5*c^4*z^2 + 100442349*a^8*b^8*c^2*z^2 + 17496*a^4*b^10*c*z +
b^12, z, k)^3*a^12*b^5*c^2 + 6973568802*root(355779876259553472*a^23*b^3*c^4*z^6 - 150094635296999121*a^24*c^
6*z^6 - 45753584909922*a^17*b*c^6*z^4 + 109300230618147*a^16*b^4*c^4*z^4 - 753145430616*a^13*b^3*c^5*z^3 + 207
657382104*a^12*b^6*c^3*z^3 + 282429536481*a^14*c^7*z^3 + 258280326*a^9*b^5*c^4*z^2 + 100442349*a^8*b^8*c^2*z^2
+ 17496*a^4*b^10*c*z + b^12, z, k)^3*a^13*b^2*c^4 - 4518872583696*root(355779876259553472*a^23*b^3*c^4*z^6 -
150094635296999121*a^24*c^6*z^6 - 45753584909922*a^17*b*c^6*z^4 + 109300230618147*a^16*b^4*c^4*z^4 - 753145430
616*a^13*b^3*c^5*z^3 + 207657382104*a^12*b^6*c^3*z^3 + 282429536481*a^14*c^7*z^3 + 258280326*a^9*b^5*c^4*z^2 +
100442349*a^8*b^8*c^2*z^2 + 17496*a^4*b^10*c*z + b^12, z, k)^4*a^16*b^3*c^3 - 328050*root(355779876259553472*
a^23*b^3*c^4*z^6 - 150094635296999121*a^24*c^6*z^6 - 45753584909922*a^17*b*c^6*z^4 + 109300230618147*a^16*b^4*
c^4*z^4 - 753145430616*a^13*b^3*c^5*z^3 + 207657382104*a^12*b^6*c^3*z^3 + 282429536481*a^14*c^7*z^3 + 25828032
6*a^9*b^5*c^4*z^2 + 100442349*a^8*b^8*c^2*z^2 + 17496*a^4*b^10*c*z + b^12, z, k)*a^5*b^6*c^2 - 177147*root(355
779876259553472*a^23*b^3*c^4*z^6 - 150094635296999121*a^24*c^6*z^6 - 45753584909922*a^17*b*c^6*z^4 + 109300230
618147*a^16*b^4*c^4*z^4 - 753145430616*a^13*b^3*c^5*z^3 + 207657382104*a^12*b^6*c^3*z^3 + 282429536481*a^14*c^
7*z^3 + 258280326*a^9*b^5*c^4*z^2 + 100442349*a^8*b^8*c^2*z^2 + 17496*a^4*b^10*c*z + b^12, z, k)*a^6*b^3*c^4 +
387420489*root(355779876259553472*a^23*b^3*c^4*z^6 - 150094635296999121*a^24*c^6*z^6 - 45753584909922*a^17*b*
c^6*z^4 + 109300230618147*a^16*b^4*c^4*z^4 - 753145430616*a^13*b^3*c^5*z^3 + 207657382104*a^12*b^6*c^3*z^3 + 2
82429536481*a^14*c^7*z^3 + 258280326*a^9*b^5*c^4*z^2 + 100442349*a^8*b^8*c^2*z^2 + 17496*a^4*b^10*c*z + b^12,
z, k)^2*a^10*b*c^5 + 23328*root(355779876259553472*a^23*b^3*c^4*z^6 - 150094635296999121*a^24*c^6*z^6 - 457535
84909922*a^17*b*c^6*z^4 + 109300230618147*a^16*b^4*c^4*z^4 - 753145430616*a^13*b^3*c^5*z^3 + 207657382104*a^12
*b^6*c^3*z^3 + 282429536481*a^14*c^7*z^3 + 258280326*a^9*b^5*c^4*z^2 + 100442349*a^8*b^8*c^2*z^2 + 17496*a^4*b
^10*c*z + b^12, z, k)*a^4*b^8*c*x + 196830*root(355779876259553472*a^23*b^3*c^4*z^6 - 150094635296999121*a^24*
c^6*z^6 - 45753584909922*a^17*b*c^6*z^4 + 109300230618147*a^16*b^4*c^4*z^4 - 753145430616*a^13*b^3*c^5*z^3 + 2
07657382104*a^12*b^6*c^3*z^3 + 282429536481*a^14*c^7*z^3 + 258280326*a^9*b^5*c^4*z^2 + 100442349*a^8*b^8*c^2*z
^2 + 17496*a^4*b^10*c*z + b^12, z, k)*a^5*b^5*c^3*x - 20920706406*root(355779876259553472*a^23*b^3*c^4*z^6 - 1
50094635296999121*a^24*c^6*z^6 - 45753584909922*a^17*b*c^6*z^4 + 109300230618147*a^16*b^4*c^4*z^4 - 7531454306
16*a^13*b^3*c^5*z^3 + 207657382104*a^12*b^6*c^3*z^3 + 282429536481*a^14*c^7*z^3 + 258280326*a^9*b^5*c^4*z^2 +
100442349*a^8*b^8*c^2*z^2 + 17496*a^4*b^10*c*z + b^12, z, k)^3*a^13*b*c^5*x + 74401740*root(355779876259553472
*a^23*b^3*c^4*z^6 - 150094635296999121*a^24*c^6*z^6 - 45753584909922*a^17*b*c^6*z^4 + 109300230618147*a^16*b^4
*c^4*z^4 - 753145430616*a^13*b^3*c^5*z^3 + 207657382104*a^12*b^6*c^3*z^3 + 282429536481*a^14*c^7*z^3 + 2582803
26*a^9*b^5*c^4*z^2 + 100442349*a^8*b^8*c^2*z^2 + 17496*a^4*b^10*c*z + b^12, z, k)^2*a^8*b^6*c^2*x - 746143164*
root(355779876259553472*a^23*b^3*c^4*z^6 - 150094635296999121*a^24*c^6*z^6 - 45753584909922*a^17*b*c^6*z^4 + 1
09300230618147*a^16*b^4*c^4*z^4 - 753145430616*a^13*b^3*c^5*z^3 + 207657382104*a^12*b^6*c^3*z^3 + 282429536481
*a^14*c^7*z^3 + 258280326*a^9*b^5*c^4*z^2 + 100442349*a^8*b^8*c^2*z^2 + 17496*a^4*b^10*c*z + b^12, z, k)^2*a^9
*b^3*c^4*x + 55788550416*root(355779876259553472*a^23*b^3*c^4*z^6 - 150094635296999121*a^24*c^6*z^6 - 45753584
909922*a^17*b*c^6*z^4 + 109300230618147*a^16*b^4*c^4*z^4 - 753145430616*a^13*b^3*c^5*z^3 + 207657382104*a^12*b
^6*c^3*z^3 + 282429536481*a^14*c^7*z^3 + 258280326*a^9*b^5*c^4*z^2 + 100442349*a^8*b^8*c^2*z^2 + 17496*a^4*b^1
0*c*z + b^12, z, k)^3*a^12*b^4*c^3*x + 564859072962*root(355779876259553472*a^23*b^3*c^4*z^6 - 150094635296999
121*a^24*c^6*z^6 - 45753584909922*a^17*b*c^6*z^4 + 109300230618147*a^16*b^4*c^4*z^4 - 753145430616*a^13*b^3*c^
5*z^3 + 207657382104*a^12*b^6*c^3*z^3 + 282429536481*a^14*c^7*z^3 + 258280326*a^9*b^5*c^4*z^2 + 100442349*a^8*
b^8*c^2*z^2 + 17496*a^4*b^10*c*z + b^12, z, k)^4*a^16*b^2*c^4*x))*root(355779876259553472*a^23*b^3*c^4*z^6 - 1
50094635296999121*a^24*c^6*z^6 - 45753584909922*a^17*b*c^6*z^4 + 109300230618147*a^16*b^4*c^4*z^4 - 7531454306
16*a^13*b^3*c^5*z^3 + 207657382104*a^12*b^6*c^3*z^3 + 282429536481*a^14*c^7*z^3 + 258280326*a^9*b^5*c^4*z^2 +
100442349*a^8*b^8*c^2*z^2 + 17496*a^4*b^10*c*z + b^12, z, k), k, 1, 6) - 1/(27*a^3*x)