\(\int \frac {1}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx\) [140]

Optimal result

Integrand size = 42, antiderivative size = 522 \[ \int \frac {1}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=-\frac {\sqrt [3]{-1} \left (2 \sqrt [3]{-1} b+3 \sqrt [3]{a} c^{2/3}\right ) \arctan \left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{17/6} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}} c^{2/3}}-\frac {\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \arctan \left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{81 \sqrt {3} a^{17/6} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}} c^{2/3}}-\frac {\left (2 (-1)^{2/3} b-3 \sqrt [3]{a} c^{2/3}\right ) \arctan \left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt {3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{17/6} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}} c^{2/3}}+\frac {\log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}}-\frac {\log \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} \sqrt [3]{c}}-\frac {\sqrt [3]{-1} \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}} \]

[Out]

1/162*ln(3*a+3*a^(2/3)*c^(1/3)*x+b*x^2)/a^(8/3)/c^(1/3)-1/54*ln(3*a-3*(-1)^(1/3)*a^(2/3)*c^(1/3)*x+b*x^2)/(1+(
-1)^(1/3))^2/a^(8/3)/c^(1/3)-1/162*(-1)^(1/3)*ln(3*a+3*(-1)^(2/3)*a^(2/3)*c^(1/3)*x+b*x^2)/a^(8/3)/c^(1/3)-1/2
43*(2*b-3*a^(1/3)*c^(2/3))*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^(17/6)/c^(2/3)*3^(1/2)/(4*b-3*a^(1/3)*c^(2/3))^(1/2)-1/81*(2*(-1)^(2/3)*b-3*a^(1/3)*c^(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^(17/6)/c^(2/3)*3^(1/2)/(4*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3))^(1/2)-1/81*(-1)^(1/3)*(2*(-1)^(1/3)*
b+3*a^(1/3)*c^(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^(17/6)/c^(2/3)*3^(1/2)/(4*b-3*(-1)^(2/3)*a^(1/3)*c^(2/3))^(1/2)

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {2095, 648, 632, 210, 642} \[ \int \frac {1}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=-\frac {\sqrt [3]{-1} \left (3 \sqrt [3]{a} c^{2/3}+2 \sqrt [3]{-1} b\right ) \arctan \left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{17/6} c^{2/3} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}-\frac {\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \arctan \left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{81 \sqrt {3} a^{17/6} c^{2/3} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}-\frac {\left (2 (-1)^{2/3} b-3 \sqrt [3]{a} c^{2/3}\right ) \arctan \left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{27 \sqrt {3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{17/6} c^{2/3} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}+\frac {\log \left (3 a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}}-\frac {\log \left (-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} \sqrt [3]{c}}-\frac {\sqrt [3]{-1} \log \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}} \]

[In]

Int[(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)^(-1),x]

[Out]

-1/27*((-1)^(1/3)*(2*(-1)^(1/3)*b + 3*a^(1/3)*c^(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)])])/(Sqrt[3]*(1 + (-1)^(1/3))^2*a^(17/6)*Sqrt[4*b - 3*(-1)^(2/
3)*a^(1/3)*c^(2/3)]*c^(2/3)) - ((2*b - 3*a^(1/3)*c^(2/3))*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)])])/(81*Sqrt[3]*a^(17/6)*Sqrt[4*b - 3*a^(1/3)*c^(2/3)]*c^(2/3)) - ((2*(-1)^(2/3)*
b - 3*a^(1/3)*c^(2/3))*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)])])/(27*Sqrt[3]*(1 - (-1)^(1/3))*(1 + (-1)^(1/3))^2*a^(17/6)*Sqrt[4*b + 3*(-1)^(1/3)*a^(1/3)*c
^(2/3)]*c^(2/3)) + Log[3*a + 3*a^(2/3)*c^(1/3)*x + b*x^2]/(162*a^(8/3)*c^(1/3)) - Log[3*a - 3*(-1)^(1/3)*a^(2/
3)*c^(1/3)*x + b*x^2]/(54*(1 + (-1)^(1/3))^2*a^(8/3)*c^(1/3)) - ((-1)^(1/3)*Log[3*a + 3*(-1)^(2/3)*a^(2/3)*c^(
1/3)*x + b*x^2])/(162*a^(8/3)*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 2095

Int[(Q6_)^(p_), x_Symbol] :> With[{a = Coeff[Q6, x, 0], b = Coeff[Q6, x, 2], c = Coeff[Q6, x, 3], d = Coeff[Q6
, x, 4], e = Coeff[Q6, x, 6]}, Dist[1/(3^(3*p)*a^(2*p)), Int[ExpandIntegrand[(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[Coe
ff[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)^{2/3} \sqrt [3]{a} b-3 \sqrt [3]{-1} a^{2/3} c^{2/3}+b \sqrt [3]{c} x}{531441 \left (1+\sqrt [3]{-1}\right )^2 a^{26/3} 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} b+3 a^{2/3} c^{2/3}+b \sqrt [3]{c} x}{1594323 a^{26/3} c^{2/3} \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}+\frac {(-1)^{2/3} \sqrt [3]{a} b-3 a^{2/3} c^{2/3}+\sqrt [3]{-1} b \sqrt [3]{c} x}{531441 \left (-1+\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{26/3} 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 {-\sqrt [3]{a} b+3 a^{2/3} c^{2/3}+b \sqrt [3]{c} x}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{81 a^{8/3} c^{2/3}}-\frac {\int \frac {(-1)^{2/3} \sqrt [3]{a} b-3 a^{2/3} c^{2/3}+\sqrt [3]{-1} b \sqrt [3]{c} x}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{81 a^{8/3} c^{2/3}}+\frac {\int \frac {-(-1)^{2/3} \sqrt [3]{a} b-3 \sqrt [3]{-1} a^{2/3} c^{2/3}+b \sqrt [3]{c} x}{-3 a+3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x-b x^2} \, dx}{27 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} c^{2/3}} \\ & = -\frac {\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \int \frac {1}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{162 a^{7/3} c^{2/3}}-\frac {\left (2 (-1)^{2/3} b-3 \sqrt [3]{a} c^{2/3}\right ) \int \frac {1}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{162 a^{7/3} c^{2/3}}-\frac {\left (-2 b \left (-(-1)^{2/3} \sqrt [3]{a} b-3 \sqrt [3]{-1} a^{2/3} c^{2/3}\right )-3 \sqrt [3]{-1} a^{2/3} b c^{2/3}\right ) \int \frac {1}{-3 a+3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x-b x^2} \, dx}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} b c^{2/3}}+\frac {\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}{162 a^{8/3} \sqrt [3]{c}}-\frac {\sqrt [3]{-1} \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}{162 a^{8/3} \sqrt [3]{c}}-\frac {\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}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} \sqrt [3]{c}} \\ & = \frac {\log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}}-\frac {\log \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} \sqrt [3]{c}}-\frac {\sqrt [3]{-1} \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}}+\frac {\left (2 b-3 \sqrt [3]{a} c^{2/3}\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 )}{81 a^{7/3} c^{2/3}}+\frac {\left (2 (-1)^{2/3} b-3 \sqrt [3]{a} c^{2/3}\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 )}{81 a^{7/3} c^{2/3}}+\frac {\left (-2 b \left (-(-1)^{2/3} \sqrt [3]{a} b-3 \sqrt [3]{-1} a^{2/3} c^{2/3}\right )-3 \sqrt [3]{-1} a^{2/3} b c^{2/3}\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 )}{27 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} b c^{2/3}} \\ & = -\frac {\left (2 (-1)^{2/3} b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}\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 )}{27 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{17/6} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}} c^{2/3}}-\frac {\left (2 b-3 \sqrt [3]{a} c^{2/3}\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 )}{81 \sqrt {3} a^{17/6} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}} c^{2/3}}-\frac {\left (2 (-1)^{2/3} b-3 \sqrt [3]{a} c^{2/3}\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 )}{81 \sqrt {3} a^{17/6} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}} c^{2/3}}+\frac {\log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}}-\frac {\log \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} \sqrt [3]{c}}-\frac {\sqrt [3]{-1} \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.19 \[ \int \frac {1}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\frac {1}{3} \text {RootSum}\left [27 a^3+27 a^2 b \text {$\#$1}^2+27 a^2 c \text {$\#$1}^3+9 a b^2 \text {$\#$1}^4+b^3 \text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1})}{18 a^2 b \text {$\#$1}+27 a^2 c \text {$\#$1}^2+12 a b^2 \text {$\#$1}^3+2 b^3 \text {$\#$1}^5}\&\right ] \]

[In]

Integrate[(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)^(-1),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]/(18*a^2*b*#1 + 27*a^2
*c*#1^2 + 12*a*b^2*#1^3 + 2*b^3*#1^5) & ]/3

Maple [C] (verified)

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

Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.17

method	result	size
default	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 c \,a^{2} \textit {\_Z}^{3}+27 a^{2} b \,\textit {\_Z}^{2}+27 a^{3}\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} b^{3}+12 \textit {\_R}^{3} a \,b^{2}+27 \textit {\_R}^{2} a^{2} c +18 a^{2} b \textit {\_R}}\right )}{3}\)	\(90\)
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 c \,a^{2} \textit {\_Z}^{3}+27 a^{2} b \,\textit {\_Z}^{2}+27 a^{3}\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} b^{3}+12 \textit {\_R}^{3} a \,b^{2}+27 \textit {\_R}^{2} a^{2} c +18 a^{2} b \textit {\_R}}\right )}{3}\)	\(90\)

[In]

int(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,method=_RETURNVERBOSE)

[Out]

1/3*sum(1/(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))

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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, algorithm="fricas")

[Out]

Exception raised: RuntimeError >> no explicit roots found

Sympy [F(-1)]

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

[In]

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)

[Out]

Timed out

Maxima [F]

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

[In]

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, algorithm="maxima")

[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)

Giac [F]

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

[In]

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, 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)

Mupad [B] (verification not implemented)

Time = 9.50 (sec) , antiderivative size = 1394, normalized size of antiderivative = 2.67 \[ \int \frac {1}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\text {Too large to display} \]

[In]

int(1/(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(6561*root(488038239039168*a^17*b^3*c^4*z^6 - 205891132094649*a^18*c^6*z^6 + 10460353203*a^12*b^2*c^
4*z^4 - 746143164*a^9*b^3*c^3*z^3 + 387420489*a^10*c^5*z^3 + 2657205*a^6*b^4*c^2*z^2 - 2916*a^3*b^5*c*z + b^6,
z, k)^2*a^4*b^12*c^2 - 6*root(488038239039168*a^17*b^3*c^4*z^6 - 205891132094649*a^18*c^6*z^6 + 10460353203*a
^12*b^2*c^4*z^4 - 746143164*a^9*b^3*c^3*z^3 + 387420489*a^10*c^5*z^3 + 2657205*a^6*b^4*c^2*z^2 - 2916*a^3*b^5*
c*z + b^6, z, k)*b^15*x - 4782969*root(488038239039168*a^17*b^3*c^4*z^6 - 205891132094649*a^18*c^6*z^6 + 10460
353203*a^12*b^2*c^4*z^4 - 746143164*a^9*b^3*c^3*z^3 + 387420489*a^10*c^5*z^3 + 2657205*a^6*b^4*c^2*z^2 - 2916*
a^3*b^5*c*z + b^6, z, k)^3*a^7*b^11*c^3 - 229582512*root(488038239039168*a^17*b^3*c^4*z^6 - 205891132094649*a^
18*c^6*z^6 + 10460353203*a^12*b^2*c^4*z^4 - 746143164*a^9*b^3*c^3*z^3 + 387420489*a^10*c^5*z^3 + 2657205*a^6*b
^4*c^2*z^2 - 2916*a^3*b^5*c*z + b^6, z, k)^4*a^9*b^13*c^2 - 387420489*root(488038239039168*a^17*b^3*c^4*z^6 -
205891132094649*a^18*c^6*z^6 + 10460353203*a^12*b^2*c^4*z^4 - 746143164*a^9*b^3*c^3*z^3 + 387420489*a^10*c^5*z
^3 + 2657205*a^6*b^4*c^2*z^2 - 2916*a^3*b^5*c*z + b^6, z, k)^4*a^10*b^10*c^4 + 167365651248*root(4880382390391
68*a^17*b^3*c^4*z^6 - 205891132094649*a^18*c^6*z^6 + 10460353203*a^12*b^2*c^4*z^4 - 746143164*a^9*b^3*c^3*z^3
+ 387420489*a^10*c^5*z^3 + 2657205*a^6*b^4*c^2*z^2 - 2916*a^3*b^5*c*z + b^6, z, k)^5*a^12*b^12*c^3 - 941431788
27*root(488038239039168*a^17*b^3*c^4*z^6 - 205891132094649*a^18*c^6*z^6 + 10460353203*a^12*b^2*c^4*z^4 - 74614
3164*a^9*b^3*c^3*z^3 + 387420489*a^10*c^5*z^3 + 2657205*a^6*b^4*c^2*z^2 - 2916*a^3*b^5*c*z + b^6, z, k)^5*a^13
*b^9*c^5 + 14580*root(488038239039168*a^17*b^3*c^4*z^6 - 205891132094649*a^18*c^6*z^6 + 10460353203*a^12*b^2*c
^4*z^4 - 746143164*a^9*b^3*c^3*z^3 + 387420489*a^10*c^5*z^3 + 2657205*a^6*b^4*c^2*z^2 - 2916*a^3*b^5*c*z + b^6
, z, k)^2*a^3*b^14*c*x - 10628820*root(488038239039168*a^17*b^3*c^4*z^6 - 205891132094649*a^18*c^6*z^6 + 10460
353203*a^12*b^2*c^4*z^4 - 746143164*a^9*b^3*c^3*z^3 + 387420489*a^10*c^5*z^3 + 2657205*a^6*b^4*c^2*z^2 - 2916*
a^3*b^5*c*z + b^6, z, k)^3*a^6*b^13*c^2*x + 2238429492*root(488038239039168*a^17*b^3*c^4*z^6 - 205891132094649
*a^18*c^6*z^6 + 10460353203*a^12*b^2*c^4*z^4 - 746143164*a^9*b^3*c^3*z^3 + 387420489*a^10*c^5*z^3 + 2657205*a^
6*b^4*c^2*z^2 - 2916*a^3*b^5*c*z + b^6, z, k)^4*a^9*b^12*c^3*x - 1162261467*root(488038239039168*a^17*b^3*c^4*
z^6 - 205891132094649*a^18*c^6*z^6 + 10460353203*a^12*b^2*c^4*z^4 - 746143164*a^9*b^3*c^3*z^3 + 387420489*a^10
*c^5*z^3 + 2657205*a^6*b^4*c^2*z^2 - 2916*a^3*b^5*c*z + b^6, z, k)^4*a^10*b^9*c^5*x - 20920706406*root(4880382
39039168*a^17*b^3*c^4*z^6 - 205891132094649*a^18*c^6*z^6 + 10460353203*a^12*b^2*c^4*z^4 - 746143164*a^9*b^3*c^
3*z^3 + 387420489*a^10*c^5*z^3 + 2657205*a^6*b^4*c^2*z^2 - 2916*a^3*b^5*c*z + b^6, z, k)^5*a^12*b^11*c^4*x)*ro
ot(488038239039168*a^17*b^3*c^4*z^6 - 205891132094649*a^18*c^6*z^6 + 10460353203*a^12*b^2*c^4*z^4 - 746143164*
a^9*b^3*c^3*z^3 + 387420489*a^10*c^5*z^3 + 2657205*a^6*b^4*c^2*z^2 - 2916*a^3*b^5*c*z + b^6, z, k), k, 1, 6)