3.148 \(\int \frac {1}{a+b x^3+c x^6} \, dx\)

Optimal. Leaf size=558 \[ -\frac {c^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt {b^2-4 a c}}+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{3 \sqrt [3]{2} \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {c^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt {b^2-4 a c}+b}+\left (\sqrt {b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{3 \sqrt [3]{2} \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}-\frac {2^{2/3} c^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} c^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}} \]

[Out]

1/3*2^(2/3)*c^(2/3)*ln(2^(1/3)*c^(1/3)*x+(b-(-4*a*c+b^2)^(1/2))^(1/3))/(b-(-4*a*c+b^2)^(1/2))^(2/3)/(-4*a*c+b^
2)^(1/2)-1/6*c^(2/3)*ln(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b-(-4*a*c+b^2)^(1/2))^(1/3)+(b-(-4*a*c+b^2)^(1/
2))^(2/3))*2^(2/3)/(b-(-4*a*c+b^2)^(1/2))^(2/3)/(-4*a*c+b^2)^(1/2)-1/3*2^(2/3)*c^(2/3)*arctan(1/3*(1-2*2^(1/3)
*c^(1/3)*x/(b-(-4*a*c+b^2)^(1/2))^(1/3))*3^(1/2))*3^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(2/3)/(-4*a*c+b^2)^(1/2)-1/3*
2^(2/3)*c^(2/3)*ln(2^(1/3)*c^(1/3)*x+(b+(-4*a*c+b^2)^(1/2))^(1/3))/(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(
2/3)+1/6*c^(2/3)*ln(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b+(-4*a*c+b^2)^(1/2))^(1/3)+(b+(-4*a*c+b^2)^(1/2))^
(2/3))*2^(2/3)/(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(2/3)+1/3*2^(2/3)*c^(2/3)*arctan(1/3*(1-2*2^(1/3)*c^(
1/3)*x/(b+(-4*a*c+b^2)^(1/2))^(1/3))*3^(1/2))*3^(1/2)/(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(2/3)

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Rubi [A]  time = 0.60, antiderivative size = 558, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1347, 200, 31, 634, 617, 204, 628} \[ -\frac {c^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt {b^2-4 a c}}+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{3 \sqrt [3]{2} \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {c^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt {b^2-4 a c}+b}+\left (\sqrt {b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{3 \sqrt [3]{2} \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}-\frac {2^{2/3} c^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} c^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3 + c*x^6)^(-1),x]

[Out]

-((2^(2/3)*c^(2/3)*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b - Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(Sqrt[3]*Sqrt[b^
2 - 4*a*c]*(b - Sqrt[b^2 - 4*a*c])^(2/3))) + (2^(2/3)*c^(2/3)*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b + Sqrt[b^2
- 4*a*c])^(1/3))/Sqrt[3]])/(Sqrt[3]*Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c])^(2/3)) + (2^(2/3)*c^(2/3)*Log[(b
- Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*Sqrt[b^2 - 4*a*c]*(b - Sqrt[b^2 - 4*a*c])^(2/3)) - (2^(2/
3)*c^(2/3)*Log[(b + Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c]
)^(2/3)) - (c^(2/3)*Log[(b - Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3)*x + 2^(2
/3)*c^(2/3)*x^2])/(3*2^(1/3)*Sqrt[b^2 - 4*a*c]*(b - Sqrt[b^2 - 4*a*c])^(2/3)) + (c^(2/3)*Log[(b + Sqrt[b^2 - 4
*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x^2])/(3*2^(1/3)*Sqrt[b^2 - 4
*a*c]*(b + Sqrt[b^2 - 4*a*c])^(2/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

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 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
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 1347

Int[((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, In
t[1/(b/2 - q/2 + c*x^n), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c}, x] && EqQ[n
2, 2*n] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {1}{a+b x^3+c x^6} \, dx &=\frac {c \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^3} \, dx}{\sqrt {b^2-4 a c}}-\frac {c \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^3} \, dx}{\sqrt {b^2-4 a c}}\\ &=\frac {\left (2^{2/3} c\right ) \int \frac {1}{\frac {\sqrt [3]{b-\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx}{3 \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {\left (2^{2/3} c\right ) \int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-\sqrt [3]{c} x}{\frac {\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{3 \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {\left (2^{2/3} c\right ) \int \frac {1}{\frac {\sqrt [3]{b+\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx}{3 \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {\left (2^{2/3} c\right ) \int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-\sqrt [3]{c} x}{\frac {\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{3 \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}\\ &=\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {c^{2/3} \int \frac {-\frac {\sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}+2 c^{2/3} x}{\frac {\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{3 \sqrt [3]{2} \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {c \int \frac {1}{\frac {\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{2^{2/3} \sqrt {b^2-4 a c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}+\frac {c^{2/3} \int \frac {-\frac {\sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}+2 c^{2/3} x}{\frac {\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{3 \sqrt [3]{2} \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {c \int \frac {1}{\frac {\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{2^{2/3} \sqrt {b^2-4 a c} \sqrt [3]{b+\sqrt {b^2-4 a c}}}\\ &=\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {c^{2/3} \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{3 \sqrt [3]{2} \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {c^{2/3} \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{3 \sqrt [3]{2} \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {\left (2^{2/3} c^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {\left (2^{2/3} c^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}\\ &=-\frac {2^{2/3} c^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} c^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {c^{2/3} \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{3 \sqrt [3]{2} \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {c^{2/3} \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{3 \sqrt [3]{2} \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 45, normalized size = 0.08 \[ \frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6 c+\text {$\#$1}^3 b+a\& ,\frac {\log (x-\text {$\#$1})}{2 \text {$\#$1}^5 c+\text {$\#$1}^2 b}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3 + c*x^6)^(-1),x]

[Out]

RootSum[a + b*#1^3 + c*#1^6 & , Log[x - #1]/(b*#1^2 + 2*c*#1^5) & ]/3

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fricas [B]  time = 1.38, size = 3978, normalized size = 7.13 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^6+b*x^3+a),x, algorithm="fricas")

[Out]

2/3*sqrt(3)*(1/2)^(1/3)*(((a^2*b^2 - 4*a^3*c)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*
a^6*b^2*c^2 - 64*a^7*c^3)) + b)/(a^2*b^2 - 4*a^3*c))^(1/3)*arctan(-1/6*(2*(1/2)^(2/3)*(sqrt(3)*(a^2*b^6 - 12*a
^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*x*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*
b^2*c^2 - 64*a^7*c^3)) - sqrt(3)*(b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*x)*(((a^2*b^2 - 4*a^3*c)*sqrt((b^4 - 4*a*b^2*
c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)) + b)/(a^2*b^2 - 4*a^3*c))^(2/3) - (1/2)
^(1/6)*(sqrt(3)*(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4
*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)) - sqrt(3)*(b^5 - 6*a*b^3*c + 8*a^2*b*c^2))*sqrt((2*(b^2*c^
2 - 2*a*c^3)*x^2 + (1/2)^(2/3)*(b^6 - 8*a*b^4*c + 20*a^2*b^2*c^2 - 16*a^3*c^3 - (a^2*b^7 - 12*a^3*b^5*c + 48*a
^4*b^3*c^2 - 64*a^5*b*c^3)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^
7*c^3)))*(((a^2*b^2 - 4*a^3*c)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 6
4*a^7*c^3)) + b)/(a^2*b^2 - 4*a^3*c))^(2/3) - (1/2)^(1/3)*((a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*x*sqrt((
b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)) - (b^4*c - 6*a*b^2*c^2 +
8*a^2*c^3)*x)*(((a^2*b^2 - 4*a^3*c)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^
2 - 64*a^7*c^3)) + b)/(a^2*b^2 - 4*a^3*c))^(1/3))/(b^2*c^2 - 2*a*c^3))*(((a^2*b^2 - 4*a^3*c)*sqrt((b^4 - 4*a*b
^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)) + b)/(a^2*b^2 - 4*a^3*c))^(2/3) + 2*
sqrt(3)*(b^2*c - 2*a*c^2))/(b^2*c - 2*a*c^2)) - 2/3*sqrt(3)*(1/2)^(1/3)*(-((a^2*b^2 - 4*a^3*c)*sqrt((b^4 - 4*a
*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)) - b)/(a^2*b^2 - 4*a^3*c))^(1/3)*ar
ctan(-1/6*(2*(1/2)^(2/3)*(sqrt(3)*(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*x*sqrt((b^4 - 4*a*b^2
*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)) + sqrt(3)*(b^5 - 6*a*b^3*c + 8*a^2*b*c
^2)*x)*(-((a^2*b^2 - 4*a^3*c)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64
*a^7*c^3)) - b)/(a^2*b^2 - 4*a^3*c))^(2/3) - (1/2)^(1/6)*(sqrt(3)*(a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 6
4*a^5*c^3)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)) + sqrt(3
)*(b^5 - 6*a*b^3*c + 8*a^2*b*c^2))*sqrt((2*(b^2*c^2 - 2*a*c^3)*x^2 + (1/2)^(2/3)*(b^6 - 8*a*b^4*c + 20*a^2*b^2
*c^2 - 16*a^3*c^3 + (a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2
)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)))*(-((a^2*b^2 - 4*a^3*c)*sqrt((b^4 - 4*a*b^2*c + 4*a^
2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)) - b)/(a^2*b^2 - 4*a^3*c))^(2/3) + (1/2)^(1/3)*(
(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*x*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a
^6*b^2*c^2 - 64*a^7*c^3)) + (b^4*c - 6*a*b^2*c^2 + 8*a^2*c^3)*x)*(-((a^2*b^2 - 4*a^3*c)*sqrt((b^4 - 4*a*b^2*c
+ 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)) - b)/(a^2*b^2 - 4*a^3*c))^(1/3))/(b^2*c^2
- 2*a*c^3))*(-((a^2*b^2 - 4*a^3*c)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^
2 - 64*a^7*c^3)) - b)/(a^2*b^2 - 4*a^3*c))^(2/3) - 2*sqrt(3)*(b^2*c - 2*a*c^2))/(b^2*c - 2*a*c^2)) - 1/6*(1/2)
^(1/3)*(((a^2*b^2 - 4*a^3*c)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*
a^7*c^3)) + b)/(a^2*b^2 - 4*a^3*c))^(1/3)*log(-2*(b^2*c^2 - 2*a*c^3)*x^2 - (1/2)^(2/3)*(b^6 - 8*a*b^4*c + 20*a
^2*b^2*c^2 - 16*a^3*c^3 - (a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*sqrt((b^4 - 4*a*b^2*c + 4*a
^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)))*(((a^2*b^2 - 4*a^3*c)*sqrt((b^4 - 4*a*b^2*c +
4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)) + b)/(a^2*b^2 - 4*a^3*c))^(2/3) + (1/2)^(1
/3)*((a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*x*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c +
48*a^6*b^2*c^2 - 64*a^7*c^3)) - (b^4*c - 6*a*b^2*c^2 + 8*a^2*c^3)*x)*(((a^2*b^2 - 4*a^3*c)*sqrt((b^4 - 4*a*b^
2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)) + b)/(a^2*b^2 - 4*a^3*c))^(1/3)) - 1/
6*(1/2)^(1/3)*(-((a^2*b^2 - 4*a^3*c)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c
^2 - 64*a^7*c^3)) - b)/(a^2*b^2 - 4*a^3*c))^(1/3)*log(-2*(b^2*c^2 - 2*a*c^3)*x^2 - (1/2)^(2/3)*(b^6 - 8*a*b^4*
c + 20*a^2*b^2*c^2 - 16*a^3*c^3 + (a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*sqrt((b^4 - 4*a*b^2
*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)))*(-((a^2*b^2 - 4*a^3*c)*sqrt((b^4 - 4*
a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)) - b)/(a^2*b^2 - 4*a^3*c))^(2/3) -
(1/2)^(1/3)*((a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*x*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^
5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)) + (b^4*c - 6*a*b^2*c^2 + 8*a^2*c^3)*x)*(-((a^2*b^2 - 4*a^3*c)*sqrt((b^
4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)) - b)/(a^2*b^2 - 4*a^3*c))^(
1/3)) + 1/3*(1/2)^(1/3)*(((a^2*b^2 - 4*a^3*c)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*
a^6*b^2*c^2 - 64*a^7*c^3)) + b)/(a^2*b^2 - 4*a^3*c))^(1/3)*log(-2*(b^2*c - 2*a*c^2)*x + (1/2)^(1/3)*(b^4 - 6*a
*b^2*c + 8*a^2*c^2 - (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a
^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)))*(((a^2*b^2 - 4*a^3*c)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 -
12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)) + b)/(a^2*b^2 - 4*a^3*c))^(1/3)) + 1/3*(1/2)^(1/3)*(-((a^2*b^2 -
4*a^3*c)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)) - b)/(a^2*
b^2 - 4*a^3*c))^(1/3)*log(-2*(b^2*c - 2*a*c^2)*x + (1/2)^(1/3)*(b^4 - 6*a*b^2*c + 8*a^2*c^2 + (a^2*b^5 - 8*a^3
*b^3*c + 16*a^4*b*c^2)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^
3)))*(-((a^2*b^2 - 4*a^3*c)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a
^7*c^3)) - b)/(a^2*b^2 - 4*a^3*c))^(1/3))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{c x^{6} + b x^{3} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^6+b*x^3+a),x, algorithm="giac")

[Out]

integrate(1/(c*x^6 + b*x^3 + a), x)

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maple [C]  time = 0.00, size = 40, normalized size = 0.07 \[ \frac {\ln \left (-\RootOf \left (c \,\textit {\_Z}^{6}+b \,\textit {\_Z}^{3}+a \right )+x \right )}{6 \RootOf \left (c \,\textit {\_Z}^{6}+b \,\textit {\_Z}^{3}+a \right )^{5} c +3 \RootOf \left (c \,\textit {\_Z}^{6}+b \,\textit {\_Z}^{3}+a \right )^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x^6+b*x^3+a),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{c x^{6} + b x^{3} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^6+b*x^3+a),x, algorithm="maxima")

[Out]

integrate(1/(c*x^6 + b*x^3 + a), x)

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mupad [B]  time = 8.49, size = 2597, normalized size = 4.65 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x^3 + c*x^6),x)

[Out]

log(6*c^5*x + (2^(2/3)*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a*b^3*c - 2*a*c*(-(4*a*c - b^2
)^3)^(1/2))/(a^2*(4*a*c - b^2)^3))^(1/3)*(36*a*c^5 - 9*b^2*c^4 + (9*2^(1/3)*b*c^3*(x + (2^(2/3)*a*(-(b^5 + b^2
*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a*b^3*c - 2*a*c*(-(4*a*c - b^2)^3)^(1/2))/(a^2*(4*a*c - b^2)^3))^
(1/3))/2)*(4*a*c - b^2)^2*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a*b^3*c - 2*a*c*(-(4*a*c -
b^2)^3)^(1/2))/(a^2*(4*a*c - b^2)^3))^(2/3))/2))/6)*((b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a*
b^3*c - 2*a*c*(-(4*a*c - b^2)^3)^(1/2))/(54*(a^2*b^6 - 64*a^5*c^3 - 12*a^3*b^4*c + 48*a^4*b^2*c^2)))^(1/3) + l
og(6*c^5*x + (2^(2/3)*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a*b^3*c + 2*a*c*(-(4*a*c - b^2)
^3)^(1/2))/(a^2*(4*a*c - b^2)^3))^(1/3)*(36*a*c^5 - 9*b^2*c^4 + (9*2^(1/3)*b*c^3*(x + (2^(2/3)*a*(-(b^5 - b^2*
(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a*b^3*c + 2*a*c*(-(4*a*c - b^2)^3)^(1/2))/(a^2*(4*a*c - b^2)^3))^(
1/3))/2)*(4*a*c - b^2)^2*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a*b^3*c + 2*a*c*(-(4*a*c - b
^2)^3)^(1/2))/(a^2*(4*a*c - b^2)^3))^(2/3))/2))/6)*((b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a*b
^3*c + 2*a*c*(-(4*a*c - b^2)^3)^(1/2))/(54*(a^2*b^6 - 64*a^5*c^3 - 12*a^3*b^4*c + 48*a^4*b^2*c^2)))^(1/3) + lo
g(6*c^5*x - (2^(2/3)*(3^(1/2)*1i - 1)*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a*b^3*c - 2*a*c
*(-(4*a*c - b^2)^3)^(1/2))/(a^2*(4*a*c - b^2)^3))^(1/3)*(9*b^2*c^4 - 36*a*c^5 + (2^(1/3)*(3^(1/2)*1i + 1)*(81*
b*c^3*x*(4*a*c - b^2)^2 + (81*2^(2/3)*a*b*c^3*(3^(1/2)*1i - 1)*(4*a*c - b^2)^2*(-(b^5 + b^2*(-(4*a*c - b^2)^3)
^(1/2) + 16*a^2*b*c^2 - 8*a*b^3*c - 2*a*c*(-(4*a*c - b^2)^3)^(1/2))/(a^2*(4*a*c - b^2)^3))^(1/3))/4)*(-(b^5 +
b^2*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a*b^3*c - 2*a*c*(-(4*a*c - b^2)^3)^(1/2))/(a^2*(4*a*c - b^2)^3
))^(2/3))/36))/12)*((3^(1/2)*1i)/2 - 1/2)*((b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a*b^3*c - 2*
a*c*(-(4*a*c - b^2)^3)^(1/2))/(54*(a^2*b^6 - 64*a^5*c^3 - 12*a^3*b^4*c + 48*a^4*b^2*c^2)))^(1/3) - log(6*c^5*x
- (2^(2/3)*(3^(1/2)*1i + 1)*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a*b^3*c - 2*a*c*(-(4*a*c
- b^2)^3)^(1/2))/(a^2*(4*a*c - b^2)^3))^(1/3)*(36*a*c^5 - 9*b^2*c^4 + (2^(1/3)*(3^(1/2)*1i - 1)*(81*b*c^3*x*(
4*a*c - b^2)^2 - (81*2^(2/3)*a*b*c^3*(3^(1/2)*1i + 1)*(4*a*c - b^2)^2*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) +
16*a^2*b*c^2 - 8*a*b^3*c - 2*a*c*(-(4*a*c - b^2)^3)^(1/2))/(a^2*(4*a*c - b^2)^3))^(1/3))/4)*(-(b^5 + b^2*(-(4*
a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a*b^3*c - 2*a*c*(-(4*a*c - b^2)^3)^(1/2))/(a^2*(4*a*c - b^2)^3))^(2/3))
/36))/12)*((3^(1/2)*1i)/2 + 1/2)*((b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a*b^3*c - 2*a*c*(-(4*
a*c - b^2)^3)^(1/2))/(54*(a^2*b^6 - 64*a^5*c^3 - 12*a^3*b^4*c + 48*a^4*b^2*c^2)))^(1/3) + log(6*c^5*x - (2^(2/
3)*(3^(1/2)*1i - 1)*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a*b^3*c + 2*a*c*(-(4*a*c - b^2)^3
)^(1/2))/(a^2*(4*a*c - b^2)^3))^(1/3)*(9*b^2*c^4 - 36*a*c^5 + (2^(1/3)*(3^(1/2)*1i + 1)*(81*b*c^3*x*(4*a*c - b
^2)^2 + (81*2^(2/3)*a*b*c^3*(3^(1/2)*1i - 1)*(4*a*c - b^2)^2*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*
c^2 - 8*a*b^3*c + 2*a*c*(-(4*a*c - b^2)^3)^(1/2))/(a^2*(4*a*c - b^2)^3))^(1/3))/4)*(-(b^5 - b^2*(-(4*a*c - b^2
)^3)^(1/2) + 16*a^2*b*c^2 - 8*a*b^3*c + 2*a*c*(-(4*a*c - b^2)^3)^(1/2))/(a^2*(4*a*c - b^2)^3))^(2/3))/36))/12)
*((3^(1/2)*1i)/2 - 1/2)*((b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a*b^3*c + 2*a*c*(-(4*a*c - b^2
)^3)^(1/2))/(54*(a^2*b^6 - 64*a^5*c^3 - 12*a^3*b^4*c + 48*a^4*b^2*c^2)))^(1/3) - log(6*c^5*x - (2^(2/3)*(3^(1/
2)*1i + 1)*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a*b^3*c + 2*a*c*(-(4*a*c - b^2)^3)^(1/2))/
(a^2*(4*a*c - b^2)^3))^(1/3)*(36*a*c^5 - 9*b^2*c^4 + (2^(1/3)*(3^(1/2)*1i - 1)*(81*b*c^3*x*(4*a*c - b^2)^2 - (
81*2^(2/3)*a*b*c^3*(3^(1/2)*1i + 1)*(4*a*c - b^2)^2*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a
*b^3*c + 2*a*c*(-(4*a*c - b^2)^3)^(1/2))/(a^2*(4*a*c - b^2)^3))^(1/3))/4)*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2
) + 16*a^2*b*c^2 - 8*a*b^3*c + 2*a*c*(-(4*a*c - b^2)^3)^(1/2))/(a^2*(4*a*c - b^2)^3))^(2/3))/36))/12)*((3^(1/2
)*1i)/2 + 1/2)*((b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 16*a^2*b*c^2 - 8*a*b^3*c + 2*a*c*(-(4*a*c - b^2)^3)^(1/2
))/(54*(a^2*b^6 - 64*a^5*c^3 - 12*a^3*b^4*c + 48*a^4*b^2*c^2)))^(1/3)

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sympy [A]  time = 4.35, size = 155, normalized size = 0.28 \[ \operatorname {RootSum} {\left (t^{6} \left (46656 a^{5} c^{3} - 34992 a^{4} b^{2} c^{2} + 8748 a^{3} b^{4} c - 729 a^{2} b^{6}\right ) + t^{3} \left (432 a^{2} b c^{2} - 216 a b^{3} c + 27 b^{5}\right ) + c^{2}, \left (t \mapsto t \log {\left (x + \frac {- 1296 t^{4} a^{4} b c^{2} + 648 t^{4} a^{3} b^{3} c - 81 t^{4} a^{2} b^{5} + 12 t a^{2} c^{2} - 15 t a b^{2} c + 3 t b^{4}}{2 a c^{2} - b^{2} c} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x**6+b*x**3+a),x)

[Out]

RootSum(_t**6*(46656*a**5*c**3 - 34992*a**4*b**2*c**2 + 8748*a**3*b**4*c - 729*a**2*b**6) + _t**3*(432*a**2*b*
c**2 - 216*a*b**3*c + 27*b**5) + c**2, Lambda(_t, _t*log(x + (-1296*_t**4*a**4*b*c**2 + 648*_t**4*a**3*b**3*c
- 81*_t**4*a**2*b**5 + 12*_t*a**2*c**2 - 15*_t*a*b**2*c + 3*_t*b**4)/(2*a*c**2 - b**2*c))))

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