3.141 \(\int \frac{1}{x \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx\)

Optimal. Leaf size=563 \[ -\frac{\left (3 \sqrt [3]{a}-\frac{b}{c^{2/3}}\right ) \log \left (3 a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{486 a^{10/3}}-\frac{\left (6 \sqrt [3]{a} c^{2/3}+i \sqrt{3} b+b\right ) \log \left (-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{972 a^{10/3} c^{2/3}}-\frac{\left (3 \sqrt [3]{a}-\frac{(-1)^{2/3} b}{c^{2/3}}\right ) \log \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{486 a^{10/3}}+\frac{\left (b-(-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \tan ^{-1}\left (\frac{3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt{3} \left (1+\sqrt [3]{-1}\right )^2 a^{19/6} \sqrt [3]{c} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}+\frac{\left (b-\sqrt [3]{a} c^{2/3}\right ) \tan ^{-1}\left (\frac{3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt{3} a^{19/6} \sqrt [3]{c} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac{(-1)^{2/3} \left ((-1)^{2/3} b-\sqrt [3]{a} c^{2/3}\right ) \tan ^{-1}\left (\frac{3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{9 \sqrt{3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{19/6} \sqrt [3]{c} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}+\frac{\log (x)}{27 a^3} \]

[Out]

((b - (-1)^(2/3)*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)])])/(9*Sqrt[3]*(1 + (-1
)^(1/3))^2*a^(19/6)*Sqrt[4*b - 3*(-1)^(2/3)*a^(1/3)*c^(2/3)]*c^(1/3)) + ((b - 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)])])/(27*Sqrt[3]*a^(19/6)*Sqrt[4*b - 3*a^(1/3)*c^(2/3)]*c^(1/3))
+ ((-1)^(2/3)*((-1)^(2/3)*b - 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)])])/(9*Sqr
t[3]*(1 - (-1)^(1/3))*(1 + (-1)^(1/3))^2*a^(19/6)*Sqrt[4*b + 3*(-1)^(1/3)*a^(1/3
)*c^(2/3)]*c^(1/3)) + Log[x]/(27*a^3) - ((3*a^(1/3) - b/c^(2/3))*Log[3*a + 3*a^(
2/3)*c^(1/3)*x + b*x^2])/(486*a^(10/3)) - ((b + I*Sqrt[3]*b + 6*a^(1/3)*c^(2/3))
*Log[3*a - 3*(-1)^(1/3)*a^(2/3)*c^(1/3)*x + b*x^2])/(972*a^(10/3)*c^(2/3)) - ((3
*a^(1/3) - ((-1)^(2/3)*b)/c^(2/3))*Log[3*a + 3*(-1)^(2/3)*a^(2/3)*c^(1/3)*x + b*
x^2])/(486*a^(10/3))

_______________________________________________________________________________________

Rubi [A]  time = 3.91037, antiderivative size = 563, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.109 \[ -\frac{\left (3 \sqrt [3]{a}-\frac{b}{c^{2/3}}\right ) \log \left (3 a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{486 a^{10/3}}-\frac{\left (6 \sqrt [3]{a} c^{2/3}+i \sqrt{3} b+b\right ) \log \left (-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{972 a^{10/3} c^{2/3}}-\frac{\left (3 \sqrt [3]{a}-\frac{(-1)^{2/3} b}{c^{2/3}}\right ) \log \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{486 a^{10/3}}+\frac{\left (b-(-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \tan ^{-1}\left (\frac{3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt{3} \left (1+\sqrt [3]{-1}\right )^2 a^{19/6} \sqrt [3]{c} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}+\frac{\left (b-\sqrt [3]{a} c^{2/3}\right ) \tan ^{-1}\left (\frac{3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt{3} a^{19/6} \sqrt [3]{c} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac{(-1)^{2/3} \left ((-1)^{2/3} b-\sqrt [3]{a} c^{2/3}\right ) \tan ^{-1}\left (\frac{3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{9 \sqrt{3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{19/6} \sqrt [3]{c} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}+\frac{\log (x)}{27 a^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(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]

((b - (-1)^(2/3)*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)])])/(9*Sqrt[3]*(1 + (-1
)^(1/3))^2*a^(19/6)*Sqrt[4*b - 3*(-1)^(2/3)*a^(1/3)*c^(2/3)]*c^(1/3)) + ((b - 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)])])/(27*Sqrt[3]*a^(19/6)*Sqrt[4*b - 3*a^(1/3)*c^(2/3)]*c^(1/3))
+ ((-1)^(2/3)*((-1)^(2/3)*b - 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)])])/(9*Sqr
t[3]*(1 - (-1)^(1/3))*(1 + (-1)^(1/3))^2*a^(19/6)*Sqrt[4*b + 3*(-1)^(1/3)*a^(1/3
)*c^(2/3)]*c^(1/3)) + Log[x]/(27*a^3) - ((3*a^(1/3) - b/c^(2/3))*Log[3*a + 3*a^(
2/3)*c^(1/3)*x + b*x^2])/(486*a^(10/3)) - ((b + I*Sqrt[3]*b + 6*a^(1/3)*c^(2/3))
*Log[3*a - 3*(-1)^(1/3)*a^(2/3)*c^(1/3)*x + b*x^2])/(972*a^(10/3)*c^(2/3)) - ((3
*a^(1/3) - ((-1)^(2/3)*b)/c^(2/3))*Log[3*a + 3*(-1)^(2/3)*a^(2/3)*c^(1/3)*x + b*
x^2])/(486*a^(10/3))

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/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

_______________________________________________________________________________________

Mathematica [C]  time = 0.147325, size = 157, normalized size = 0.28 \[ -\frac{\text{RootSum}\left [\text{$\#$1}^6 b^3+9 \text{$\#$1}^4 a b^2+27 \text{$\#$1}^3 a^2 c+27 \text{$\#$1}^2 a^2 b+27 a^3\&,\frac{\text{$\#$1}^4 b^3 \log (x-\text{$\#$1})+9 \text{$\#$1}^2 a b^2 \log (x-\text{$\#$1})+27 a^2 b \log (x-\text{$\#$1})+27 \text{$\#$1} a^2 c \log (x-\text{$\#$1})}{2 \text{$\#$1}^4 b^3+12 \text{$\#$1}^2 a b^2+27 \text{$\#$1} a^2 c+18 a^2 b}\&\right ]-3 \log (x)}{81 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(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]

-(-3*Log[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 + 27*a^2*c*#1 + 12*a*b^2*#1^2 + 2*b^3*#1
^4) & ])/(81*a^3)

_______________________________________________________________________________________

Maple [C]  time = 0.013, size = 134, normalized size = 0.2 \[ -{\frac{1}{81\,{a}^{3}}\sum _{{\it \_R}={\it RootOf} \left ({b}^{3}{{\it \_Z}}^{6}+9\,a{b}^{2}{{\it \_Z}}^{4}+27\,{a}^{2}c{{\it \_Z}}^{3}+27\,{a}^{2}b{{\it \_Z}}^{2}+27\,{a}^{3} \right ) }{\frac{ \left ({{\it \_R}}^{5}{b}^{3}+9\,{{\it \_R}}^{3}a{b}^{2}+27\,{{\it \_R}}^{2}{a}^{2}c+27\,{\it \_R}\,{a}^{2}b \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}{b}^{3}+12\,{{\it \_R}}^{3}a{b}^{2}+27\,{{\it \_R}}^{2}{a}^{2}c+18\,{\it \_R}\,{a}^{2}b}}}+{\frac{\ln \left ( x \right ) }{27\,{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/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]

-1/81/a^3*sum((_R^5*b^3+9*_R^3*a*b^2+27*_R^2*a^2*c+27*_R*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(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))+1/27*ln(x)/a^3

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{\int \frac{b^{3} x^{5} + 9 \, a b^{2} x^{3} + 27 \, a^{2} c x^{2} + 27 \, a^{2} b x}{b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}}\,{d x}}{27 \, a^{3}} + \frac{\log \left (x\right )}{27 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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),x, algorithm="maxima")

[Out]

-1/27*integrate((b^3*x^5 + 9*a*b^2*x^3 + 27*a^2*c*x^2 + 27*a^2*b*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)/a^3 + 1/27*log(x)/a^3

_______________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[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),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),
x)