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

Optimal. Leaf size=69 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 a \sqrt{b^2-4 a c}}-\frac{\log \left (a+b x^3+c x^6\right )}{6 a}+\frac{\log (x)}{a} \]

[Out]

(b*ArcTanh[(b + 2*c*x^3)/Sqrt[b^2 - 4*a*c]])/(3*a*Sqrt[b^2 - 4*a*c]) + Log[x]/a
- Log[a + b*x^3 + c*x^6]/(6*a)

_______________________________________________________________________________________

Rubi [A]  time = 0.134114, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 a \sqrt{b^2-4 a c}}-\frac{\log \left (a+b x^3+c x^6\right )}{6 a}+\frac{\log (x)}{a} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^3 + c*x^6)),x]

[Out]

(b*ArcTanh[(b + 2*c*x^3)/Sqrt[b^2 - 4*a*c]])/(3*a*Sqrt[b^2 - 4*a*c]) + Log[x]/a
- Log[a + b*x^3 + c*x^6]/(6*a)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 26.8782, size = 63, normalized size = 0.91 \[ \frac{b \operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{\sqrt{- 4 a c + b^{2}}} \right )}}{3 a \sqrt{- 4 a c + b^{2}}} + \frac{\log{\left (x^{3} \right )}}{3 a} - \frac{\log{\left (a + b x^{3} + c x^{6} \right )}}{6 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(c*x**6+b*x**3+a),x)

[Out]

b*atanh((b + 2*c*x**3)/sqrt(-4*a*c + b**2))/(3*a*sqrt(-4*a*c + b**2)) + log(x**3
)/(3*a) - log(a + b*x**3 + c*x**6)/(6*a)

_______________________________________________________________________________________

Mathematica [C]  time = 0.0399524, size = 66, normalized size = 0.96 \[ \frac{\log (x)}{a}-\frac{\text{RootSum}\left [\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1}^3 c \log (x-\text{$\#$1})+b \log (x-\text{$\#$1})}{2 \text{$\#$1}^3 c+b}\&\right ]}{3 a} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a + b*x^3 + c*x^6)),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 66, normalized size = 1. \[{\frac{\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( c{x}^{6}+b{x}^{3}+a \right ) }{6\,a}}-{\frac{b}{3\,a}\arctan \left ({(2\,c{x}^{3}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(c*x^6+b*x^3+a),x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.28225, size = 1, normalized size = 0.01 \[ \left [\frac{b \log \left (\frac{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{3} + b^{3} - 4 \, a b c +{\left (2 \, c^{2} x^{6} + 2 \, b c x^{3} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{6} + b x^{3} + a}\right ) - \sqrt{b^{2} - 4 \, a c}{\left (\log \left (c x^{6} + b x^{3} + a\right ) - 6 \, \log \left (x\right )\right )}}{6 \, \sqrt{b^{2} - 4 \, a c} a}, -\frac{2 \, b \arctan \left (-\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + \sqrt{-b^{2} + 4 \, a c}{\left (\log \left (c x^{6} + b x^{3} + a\right ) - 6 \, \log \left (x\right )\right )}}{6 \, \sqrt{-b^{2} + 4 \, a c} a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(b*log((2*(b^2*c - 4*a*c^2)*x^3 + b^3 - 4*a*b*c + (2*c^2*x^6 + 2*b*c*x^3 +
b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^6 + b*x^3 + a)) - sqrt(b^2 - 4*a*c)*(log(c*
x^6 + b*x^3 + a) - 6*log(x)))/(sqrt(b^2 - 4*a*c)*a), -1/6*(2*b*arctan(-(2*c*x^3
+ b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + sqrt(-b^2 + 4*a*c)*(log(c*x^6 + b*x^3 +
a) - 6*log(x)))/(sqrt(-b^2 + 4*a*c)*a)]

_______________________________________________________________________________________

Sympy [A]  time = 12.1341, size = 253, normalized size = 3.67 \[ \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{6 a \left (4 a c - b^{2}\right )} - \frac{1}{6 a}\right ) \log{\left (x^{3} + \frac{- 12 a^{2} c \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{6 a \left (4 a c - b^{2}\right )} - \frac{1}{6 a}\right ) + 3 a b^{2} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{6 a \left (4 a c - b^{2}\right )} - \frac{1}{6 a}\right ) - 2 a c + b^{2}}{b c} \right )} + \left (\frac{b \sqrt{- 4 a c + b^{2}}}{6 a \left (4 a c - b^{2}\right )} - \frac{1}{6 a}\right ) \log{\left (x^{3} + \frac{- 12 a^{2} c \left (\frac{b \sqrt{- 4 a c + b^{2}}}{6 a \left (4 a c - b^{2}\right )} - \frac{1}{6 a}\right ) + 3 a b^{2} \left (\frac{b \sqrt{- 4 a c + b^{2}}}{6 a \left (4 a c - b^{2}\right )} - \frac{1}{6 a}\right ) - 2 a c + b^{2}}{b c} \right )} + \frac{\log{\left (x \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-b*sqrt(-4*a*c + b**2)/(6*a*(4*a*c - b**2)) - 1/(6*a))*log(x**3 + (-12*a**2*c*(
-b*sqrt(-4*a*c + b**2)/(6*a*(4*a*c - b**2)) - 1/(6*a)) + 3*a*b**2*(-b*sqrt(-4*a*
c + b**2)/(6*a*(4*a*c - b**2)) - 1/(6*a)) - 2*a*c + b**2)/(b*c)) + (b*sqrt(-4*a*
c + b**2)/(6*a*(4*a*c - b**2)) - 1/(6*a))*log(x**3 + (-12*a**2*c*(b*sqrt(-4*a*c
+ b**2)/(6*a*(4*a*c - b**2)) - 1/(6*a)) + 3*a*b**2*(b*sqrt(-4*a*c + b**2)/(6*a*(
4*a*c - b**2)) - 1/(6*a)) - 2*a*c + b**2)/(b*c)) + log(x)/a

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.263246, size = 89, normalized size = 1.29 \[ -\frac{b \arctan \left (\frac{2 \, c x^{3} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{3 \, \sqrt{-b^{2} + 4 \, a c} a} - \frac{{\rm ln}\left (c x^{6} + b x^{3} + a\right )}{6 \, a} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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