\(\int \frac {x}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx\) [139]

Optimal result

Integrand size = 44, antiderivative size = 469 \[ \int \frac {x}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=-\frac {\arctan \left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{13/6} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}-\frac {\arctan \left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt {3} a^{13/6} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}+\frac {\sqrt [3]{-1} \arctan \left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt {3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{13/6} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}-\frac {\log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{7/3} c^{2/3}}+\frac {(-1)^{2/3} \log \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{7/3} c^{2/3}}-\frac {(-1)^{2/3} \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{7/3} c^{2/3}} \]

[Out]

-1/162*ln(3*a+3*a^(2/3)*c^(1/3)*x+b*x^2)/a^(7/3)/c^(2/3)+1/54*(-1)^(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^(7/3)/c^(2/3)-1/162*(-1)^(2/3)*ln(3*a+3*(-1)^(2/3)*a^(2/3)*c^(1/3)*x+b*x^2)/a^(7/3)
/c^(2/3)-1/81*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^(13/6)/c^(
1/3)*3^(1/2)/(4*b-3*a^(1/3)*c^(2/3))^(1/2)+1/27*(-1)^(1/3)*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^(13/6)/c^(1/3)*3^(1/2
)/(4*b+3*(-1)^(1/3)*a^(1/3)*c^(2/3))^(1/2)-1/27*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^(13/6)/c^(1/3)*3^(1/2)/(4*b-3*(-1)^(2/3)*a^(1/3
)*c^(2/3))^(1/2)

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2122, 648, 632, 210, 642} \[ \int \frac {x}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=-\frac {\arctan \left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{13/6} \sqrt [3]{c} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}-\frac {\arctan \left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt {3} a^{13/6} \sqrt [3]{c} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac {\sqrt [3]{-1} \arctan \left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{9 \sqrt {3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{13/6} \sqrt [3]{c} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}-\frac {\log \left (3 a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{162 a^{7/3} c^{2/3}}+\frac {(-1)^{2/3} \log \left (-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{7/3} c^{2/3}}-\frac {(-1)^{2/3} \log \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{162 a^{7/3} c^{2/3}} \]

[In]

Int[x/(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]

-1/9*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^(13/6)*Sqrt[4*b - 3*(-1)^(2/3)*a^(1/3)*c^(2/3)]*c^(1/3)) - 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)])]/(27*Sqrt[3]*a^(13/6)*Sqrt[4*b - 3*a^(1/3)*c^(2
/3)]*c^(1/3)) + ((-1)^(1/3)*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)])])/(9*Sqrt[3]*(1 - (-1)^(1/3))*(1 + (-1)^(1/3))^2*a^(13/6)*Sqrt[4*b + 3*(-1)^(1/3)*a^(1/
3)*c^(2/3)]*c^(1/3)) - Log[3*a + 3*a^(2/3)*c^(1/3)*x + b*x^2]/(162*a^(7/3)*c^(2/3)) + ((-1)^(2/3)*Log[3*a - 3*
(-1)^(1/3)*a^(2/3)*c^(1/3)*x + b*x^2])/(54*(1 + (-1)^(1/3))^2*a^(7/3)*c^(2/3)) - ((-1)^(2/3)*Log[3*a + 3*(-1)^
(2/3)*a^(2/3)*c^(1/3)*x + b*x^2])/(162*a^(7/3)*c^(2/3))

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 {-3 a^{2/3} \sqrt [3]{c}-(-1)^{2/3} b x}{531441 \left (1+\sqrt [3]{-1}\right )^2 a^{25/3} c^{2/3} \left (-3 a+3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x-b x^2\right )}+\frac {-3 a^{2/3} \sqrt [3]{c}-b x}{1594323 a^{25/3} c^{2/3} \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}+\frac {(-1)^{2/3} \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+b x\right )}{531441 \left (-1+\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{25/3} 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 {-3 a^{2/3} \sqrt [3]{c}-b x}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{81 a^{7/3} c^{2/3}}-\frac {(-1)^{2/3} \int \frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+b x}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{81 a^{7/3} c^{2/3}}+\frac {\int \frac {-3 a^{2/3} \sqrt [3]{c}-(-1)^{2/3} b 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^{7/3} 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^{7/3} c^{2/3}}-\frac {(-1)^{2/3} \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^{7/3} c^{2/3}}+\frac {(-1)^{2/3} \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^{7/3} c^{2/3}}-\frac {\int \frac {1}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{54 a^{5/3} \sqrt [3]{c}}+\frac {\sqrt [3]{-1} \int \frac {1}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{54 a^{5/3} \sqrt [3]{c}}-\frac {\int \frac {1}{-3 a+3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x-b x^2} \, dx}{18 \left (1+\sqrt [3]{-1}\right )^2 a^{5/3} \sqrt [3]{c}} \\ & = -\frac {\log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{7/3} c^{2/3}}+\frac {(-1)^{2/3} \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^{7/3} c^{2/3}}-\frac {(-1)^{2/3} \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{7/3} c^{2/3}}+\frac {\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 )}{27 a^{5/3} \sqrt [3]{c}}-\frac {\sqrt [3]{-1} \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 )}{27 a^{5/3} \sqrt [3]{c}}+\frac {\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 )}{9 \left (1+\sqrt [3]{-1}\right )^2 a^{5/3} \sqrt [3]{c}} \\ & = -\frac {\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^{13/6} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}-\frac {\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^{13/6} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}+\frac {\sqrt [3]{-1} \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^{13/6} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}-\frac {\log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{7/3} c^{2/3}}+\frac {(-1)^{2/3} \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^{7/3} c^{2/3}}-\frac {(-1)^{2/3} \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{7/3} c^{2/3}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.20 \[ \int \frac {x}{27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6} \, dx=\frac {1}{3} \text {RootSum}\left [27 a^3+27 a^2 b \text {$\#$1}^2+27 a^2 c \text {$\#$1}^3+9 a b^2 \text {$\#$1}^4+b^3 \text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1})}{18 a^2 b+27 a^2 c \text {$\#$1}+12 a b^2 \text {$\#$1}^2+2 b^3 \text {$\#$1}^4}\&\right ] \]

[In]

Integrate[x/(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]/(18*a^2*b + 27*a^2*c*
#1 + 12*a*b^2*#1^2 + 2*b^3*#1^4) & ]/3

Maple [C] (verified)

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

Time = 0.08 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.19

method	result	size
default	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 c \,a^{2} \textit {\_Z}^{3}+27 a^{2} b \,\textit {\_Z}^{2}+27 a^{3}\right )}{\sum }\frac {\textit {\_R} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} b^{3}+12 \textit {\_R}^{3} a \,b^{2}+27 \textit {\_R}^{2} a^{2} c +18 a^{2} b \textit {\_R}}\right )}{3}\)	\(91\)
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 c \,a^{2} \textit {\_Z}^{3}+27 a^{2} b \,\textit {\_Z}^{2}+27 a^{3}\right )}{\sum }\frac {\textit {\_R} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} b^{3}+12 \textit {\_R}^{3} a \,b^{2}+27 \textit {\_R}^{2} a^{2} c +18 a^{2} b \textit {\_R}}\right )}{3}\)	\(91\)

[In]

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

[In]

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

[In]

integrate(x/(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]

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

[In]

integrate(x/(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/(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]

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

[In]

integrate(x/(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/(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 = 9.61 (sec) , antiderivative size = 1057, normalized size of antiderivative = 2.25 \[ \int \frac {x}{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=\sum _{k=1}^6\ln \left (b^{12}\,x+{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^4\,a^{10}\,b^{11}\,c^3\,1033121304+{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^5\,a^{12}\,b^{12}\,c^3\,167365651248-{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^5\,a^{13}\,b^9\,c^5\,94143178827+\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )\,a^2\,b^{13}\,x\,54+{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^2\,a^5\,b^{11}\,c^2\,x\,177147+{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^3\,a^7\,b^{12}\,c^2\,x\,17006112-{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^3\,a^8\,b^9\,c^4\,x\,14348907+{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^4\,a^9\,b^{13}\,c^2\,x\,229582512+{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^4\,a^{10}\,b^{10}\,c^4\,x\,387420489-{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^5\,a^{12}\,b^{11}\,c^4\,x\,20920706406\right )\,\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right ) \]

[In]

int(x/(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(b^12*x + 1033121304*root(18075490334784*a^14*b^3*c^4*z^6 - 7625597484987*a^15*c^6*z^6 + 1162261467*
a^10*b*c^4*z^4 + 8503056*a^7*b^3*c^2*z^3 - 14348907*a^8*c^4*z^3 + 177147*a^5*b^2*c^2*z^2 + b^3, z, k)^4*a^10*b
^11*c^3 + 167365651248*root(18075490334784*a^14*b^3*c^4*z^6 - 7625597484987*a^15*c^6*z^6 + 1162261467*a^10*b*c
^4*z^4 + 8503056*a^7*b^3*c^2*z^3 - 14348907*a^8*c^4*z^3 + 177147*a^5*b^2*c^2*z^2 + b^3, z, k)^5*a^12*b^12*c^3
- 94143178827*root(18075490334784*a^14*b^3*c^4*z^6 - 7625597484987*a^15*c^6*z^6 + 1162261467*a^10*b*c^4*z^4 +
8503056*a^7*b^3*c^2*z^3 - 14348907*a^8*c^4*z^3 + 177147*a^5*b^2*c^2*z^2 + b^3, z, k)^5*a^13*b^9*c^5 + 54*root(
18075490334784*a^14*b^3*c^4*z^6 - 7625597484987*a^15*c^6*z^6 + 1162261467*a^10*b*c^4*z^4 + 8503056*a^7*b^3*c^2
*z^3 - 14348907*a^8*c^4*z^3 + 177147*a^5*b^2*c^2*z^2 + b^3, z, k)*a^2*b^13*x + 177147*root(18075490334784*a^14
*b^3*c^4*z^6 - 7625597484987*a^15*c^6*z^6 + 1162261467*a^10*b*c^4*z^4 + 8503056*a^7*b^3*c^2*z^3 - 14348907*a^8
*c^4*z^3 + 177147*a^5*b^2*c^2*z^2 + b^3, z, k)^2*a^5*b^11*c^2*x + 17006112*root(18075490334784*a^14*b^3*c^4*z^
6 - 7625597484987*a^15*c^6*z^6 + 1162261467*a^10*b*c^4*z^4 + 8503056*a^7*b^3*c^2*z^3 - 14348907*a^8*c^4*z^3 +
177147*a^5*b^2*c^2*z^2 + b^3, z, k)^3*a^7*b^12*c^2*x - 14348907*root(18075490334784*a^14*b^3*c^4*z^6 - 7625597
484987*a^15*c^6*z^6 + 1162261467*a^10*b*c^4*z^4 + 8503056*a^7*b^3*c^2*z^3 - 14348907*a^8*c^4*z^3 + 177147*a^5*
b^2*c^2*z^2 + b^3, z, k)^3*a^8*b^9*c^4*x + 229582512*root(18075490334784*a^14*b^3*c^4*z^6 - 7625597484987*a^15
*c^6*z^6 + 1162261467*a^10*b*c^4*z^4 + 8503056*a^7*b^3*c^2*z^3 - 14348907*a^8*c^4*z^3 + 177147*a^5*b^2*c^2*z^2
+ b^3, z, k)^4*a^9*b^13*c^2*x + 387420489*root(18075490334784*a^14*b^3*c^4*z^6 - 7625597484987*a^15*c^6*z^6 +
1162261467*a^10*b*c^4*z^4 + 8503056*a^7*b^3*c^2*z^3 - 14348907*a^8*c^4*z^3 + 177147*a^5*b^2*c^2*z^2 + b^3, z,
k)^4*a^10*b^10*c^4*x - 20920706406*root(18075490334784*a^14*b^3*c^4*z^6 - 7625597484987*a^15*c^6*z^6 + 116226
1467*a^10*b*c^4*z^4 + 8503056*a^7*b^3*c^2*z^3 - 14348907*a^8*c^4*z^3 + 177147*a^5*b^2*c^2*z^2 + b^3, z, k)^5*a
^12*b^11*c^4*x)*root(18075490334784*a^14*b^3*c^4*z^6 - 7625597484987*a^15*c^6*z^6 + 1162261467*a^10*b*c^4*z^4
+ 8503056*a^7*b^3*c^2*z^3 - 14348907*a^8*c^4*z^3 + 177147*a^5*b^2*c^2*z^2 + b^3, z, k), k, 1, 6)