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

Optimal. Leaf size=334 \[ \frac {2 (-1)^{2/3} \tan ^{-1}\left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{11/6} c^{2/3} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}+\frac {2 \tan ^{-1}\left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt {3} a^{11/6} c^{2/3} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac {2 (-1)^{2/3} \tan ^{-1}\left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{9 \sqrt {3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{11/6} c^{2/3} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}} \]

[Out]

2/81*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^(11/6)/c^(2/3)*3^(1
/2)/(4*b-3*a^(1/3)*c^(2/3))^(1/2)+2/27*(-1)^(2/3)*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^(11/6)/c^(2/3)*3^(1/2)/(4*b+3*
(-1)^(1/3)*a^(1/3)*c^(2/3))^(1/2)+2/27*(-1)^(2/3)*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^(11/6)/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 = 0.47, antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2097, 618, 204} \[ \frac {2 (-1)^{2/3} \tan ^{-1}\left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{11/6} c^{2/3} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}+\frac {2 \tan ^{-1}\left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt {3} a^{11/6} c^{2/3} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac {2 (-1)^{2/3} \tan ^{-1}\left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{9 \sqrt {3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{11/6} c^{2/3} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}} \]

Antiderivative was successfully verified.

[In]

Int[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]

(2*(-1)^(2/3)*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)])])/(9*Sqrt[3]*(1 + (-1)^(1/3))^2*a^(11/6)*Sqrt[4*b - 3*(-1)^(2/3)*a^(1/3)*c^(2/3)]*c^(2/3)) + (2*ArcTa
n[(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^(11/6)*Sqrt[4*b
- 3*a^(1/3)*c^(2/3)]*c^(2/3)) + (2*(-1)^(2/3)*ArcTan[(3*(-1)^(2/3)*a^(2/3)*c^(1/3) + 2*b*x)/(Sqrt[3]*Sqrt[a]*S
qrt[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^(11/6)*Sqrt[4*b +
3*(-1)^(1/3)*a^(1/3)*c^(2/3)]*c^(2/3))

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 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 {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 &=\left (19683 a^6\right ) \int \left (\frac {(-1)^{2/3}}{177147 \left (1+\sqrt [3]{-1}\right )^2 a^{22/3} c^{2/3} \left (-3 a+3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x-b x^2\right )}+\frac {1}{531441 a^{22/3} c^{2/3} \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}-\frac {(-1)^{2/3}}{177147 \left (-1+\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{22/3} 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 {1}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{27 a^{4/3} c^{2/3}}+\frac {(-1)^{2/3} \int \frac {1}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{27 a^{4/3} c^{2/3}}+\frac {(-1)^{2/3} \int \frac {1}{-3 a+3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x-b x^2} \, dx}{9 \left (1+\sqrt [3]{-1}\right )^2 a^{4/3} c^{2/3}}\\ &=-\frac {2 \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^{4/3} c^{2/3}}-\frac {\left (2 (-1)^{2/3}\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 )}{27 a^{4/3} c^{2/3}}-\frac {\left (2 (-1)^{2/3}\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 )}{9 \left (1+\sqrt [3]{-1}\right )^2 a^{4/3} c^{2/3}}\\ &=\frac {2 (-1)^{2/3} \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^{11/6} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}} c^{2/3}}+\frac {2 \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^{11/6} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}} c^{2/3}}+\frac {2 (-1)^{2/3} \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^{11/6} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}} c^{2/3}}\\ \end {align*}

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Mathematica [C]  time = 0.06, size = 97, normalized size = 0.29 \[ \frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6 b^3+9 \text {$\#$1}^4 a b^2+27 \text {$\#$1}^3 a^2 c+27 \text {$\#$1}^2 a^2 b+27 a^3\& ,\frac {\text {$\#$1} \log (x-\text {$\#$1})}{2 \text {$\#$1}^4 b^3+12 \text {$\#$1}^2 a b^2+27 \text {$\#$1} a^2 c+18 a^2 b}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[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]

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]*#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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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(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 \[ \int \frac {x^{2}}{b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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(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)

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maple [C]  time = 0.00, size = 93, normalized size = 0.28 \[ \frac {\RootOf \left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right )^{2} \ln \left (-\RootOf \left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right )+x \right )}{6 \RootOf \left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right )^{5} b^{3}+36 \RootOf \left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right )^{3} a \,b^{2}+81 \RootOf \left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right )^{2} a^{2} c +54 \RootOf \left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right ) a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(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]

1/3*sum(_R^2/(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]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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]

integrate(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)

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mupad [B]  time = 3.34, size = 825, normalized size = 2.47 \[ \sum _{k=1}^6\ln \left (-a^3\,b^9\,\left (-{\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )}^2\,a^4\,c^2\,13122-{\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )}^3\,a^6\,c^3\,1062882+{\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )}^4\,a^8\,c^4\,43046721+{\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )}^5\,a^{10}\,c^5\,3486784401+\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )\,a^2\,c\,81+\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )\,a\,b^2\,x\,18-{\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )}^4\,a^7\,b^3\,c^2\,25509168-{\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )}^5\,a^9\,b^3\,c^3\,6198727824+{\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )}^2\,a^3\,b^2\,c\,x\,5832+{\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )}^3\,a^5\,b^2\,c^2\,x\,708588+{\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )}^4\,a^7\,b^2\,c^3\,x\,38263752+{\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right )}^5\,a^9\,b^2\,c^4\,x\,774840978+1\right )\,27\right )\,\mathrm {root}\left (669462604992\,a^{11}\,b^3\,c^4\,z^6-282429536481\,a^{12}\,c^6\,z^6+129140163\,a^8\,c^4\,z^4-19683\,a^4\,c^2\,z^2+1,z,k\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(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(-27*a^3*b^9*(43046721*root(669462604992*a^11*b^3*c^4*z^6 - 282429536481*a^12*c^6*z^6 + 129140163*a^
8*c^4*z^4 - 19683*a^4*c^2*z^2 + 1, z, k)^4*a^8*c^4 - 1062882*root(669462604992*a^11*b^3*c^4*z^6 - 282429536481
*a^12*c^6*z^6 + 129140163*a^8*c^4*z^4 - 19683*a^4*c^2*z^2 + 1, z, k)^3*a^6*c^3 - 13122*root(669462604992*a^11*
b^3*c^4*z^6 - 282429536481*a^12*c^6*z^6 + 129140163*a^8*c^4*z^4 - 19683*a^4*c^2*z^2 + 1, z, k)^2*a^4*c^2 + 348
6784401*root(669462604992*a^11*b^3*c^4*z^6 - 282429536481*a^12*c^6*z^6 + 129140163*a^8*c^4*z^4 - 19683*a^4*c^2
*z^2 + 1, z, k)^5*a^10*c^5 + 81*root(669462604992*a^11*b^3*c^4*z^6 - 282429536481*a^12*c^6*z^6 + 129140163*a^8
*c^4*z^4 - 19683*a^4*c^2*z^2 + 1, z, k)*a^2*c + 18*root(669462604992*a^11*b^3*c^4*z^6 - 282429536481*a^12*c^6*
z^6 + 129140163*a^8*c^4*z^4 - 19683*a^4*c^2*z^2 + 1, z, k)*a*b^2*x - 25509168*root(669462604992*a^11*b^3*c^4*z
^6 - 282429536481*a^12*c^6*z^6 + 129140163*a^8*c^4*z^4 - 19683*a^4*c^2*z^2 + 1, z, k)^4*a^7*b^3*c^2 - 61987278
24*root(669462604992*a^11*b^3*c^4*z^6 - 282429536481*a^12*c^6*z^6 + 129140163*a^8*c^4*z^4 - 19683*a^4*c^2*z^2
+ 1, z, k)^5*a^9*b^3*c^3 + 5832*root(669462604992*a^11*b^3*c^4*z^6 - 282429536481*a^12*c^6*z^6 + 129140163*a^8
*c^4*z^4 - 19683*a^4*c^2*z^2 + 1, z, k)^2*a^3*b^2*c*x + 708588*root(669462604992*a^11*b^3*c^4*z^6 - 2824295364
81*a^12*c^6*z^6 + 129140163*a^8*c^4*z^4 - 19683*a^4*c^2*z^2 + 1, z, k)^3*a^5*b^2*c^2*x + 38263752*root(6694626
04992*a^11*b^3*c^4*z^6 - 282429536481*a^12*c^6*z^6 + 129140163*a^8*c^4*z^4 - 19683*a^4*c^2*z^2 + 1, z, k)^4*a^
7*b^2*c^3*x + 774840978*root(669462604992*a^11*b^3*c^4*z^6 - 282429536481*a^12*c^6*z^6 + 129140163*a^8*c^4*z^4
- 19683*a^4*c^2*z^2 + 1, z, k)^5*a^9*b^2*c^4*x + 1))*root(669462604992*a^11*b^3*c^4*z^6 - 282429536481*a^12*c
^6*z^6 + 129140163*a^8*c^4*z^4 - 19683*a^4*c^2*z^2 + 1, z, k), k, 1, 6)

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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(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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