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-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)

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Rubi [A]  time = 1.38, 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 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 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 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]

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 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]

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*}

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Mathematica [C]  time = 0.14, size = 163, normalized size = 0.25 \[ -\frac {x \text {RootSum}\left [\text {$\#$1}^6 b^3+9 \text {$\#$1}^4 a b^2+27 \text {$\#$1}^3 a^2 c+27 \text {$\#$1}^2 a^2 b+27 a^3\& ,\frac {\text {$\#$1}^4 b^3 \log (x-\text {$\#$1})+9 \text {$\#$1}^2 a b^2 \log (x-\text {$\#$1})+27 a^2 b \log (x-\text {$\#$1})+27 \text {$\#$1} a^2 c \log (x-\text {$\#$1})}{2 \text {$\#$1}^5 b^3+12 \text {$\#$1}^3 a b^2+27 \text {$\#$1}^2 a^2 c+18 \text {$\#$1} a^2 b}\& \right ]+3}{81 a^3 x} \]

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]

-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)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}\right )} x^{2}}\,{d x} \]

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)

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maple [C]  time = 0.01, size = 133, normalized size = 0.21 \[ \frac {\left (-\RootOf \left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right )^{4} b^{3}-9 \RootOf \left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right )^{2} a \,b^{2}-27 \RootOf \left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right ) a^{2} c -27 b \,a^{2}\right ) \ln \left (-\RootOf \left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right )+x \right )}{81 a^{3} \left (2 \RootOf \left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right )^{5} b^{3}+12 \RootOf \left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right )^{3} a \,b^{2}+27 \RootOf \left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right )^{2} a^{2} c +18 \RootOf \left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right ) a^{2} b \right )}-\frac {1}{27 a^{3} x} \]

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]

-1/27/a^3/x+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(-_R+x),_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(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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mupad [B]  time = 2.72, size = 2663, normalized size = 4.13 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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