3.88 \(\int \frac{1}{x^2 \left (a x+b x^3+c x^5\right )} \, dx\)

Optimal. Leaf size=89 \[ -\frac{\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 \sqrt{b^2-4 a c}}+\frac{b \log \left (a+b x^2+c x^4\right )}{4 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{2 a x^2} \]

[Out]

-1/(2*a*x^2) - ((b^2 - 2*a*c)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^2*S
qrt[b^2 - 4*a*c]) - (b*Log[x])/a^2 + (b*Log[a + b*x^2 + c*x^4])/(4*a^2)

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Rubi [A]  time = 0.263068, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 \sqrt{b^2-4 a c}}+\frac{b \log \left (a+b x^2+c x^4\right )}{4 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a*x + b*x^3 + c*x^5)),x]

[Out]

-1/(2*a*x^2) - ((b^2 - 2*a*c)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^2*S
qrt[b^2 - 4*a*c]) - (b*Log[x])/a^2 + (b*Log[a + b*x^2 + c*x^4])/(4*a^2)

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Rubi in Sympy [A]  time = 45.2274, size = 87, normalized size = 0.98 \[ - \frac{1}{2 a x^{2}} - \frac{b \log{\left (x^{2} \right )}}{2 a^{2}} + \frac{b \log{\left (a + b x^{2} + c x^{4} \right )}}{4 a^{2}} - \frac{\left (- 2 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 a^{2} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(c*x**5+b*x**3+a*x),x)

[Out]

-1/(2*a*x**2) - b*log(x**2)/(2*a**2) + b*log(a + b*x**2 + c*x**4)/(4*a**2) - (-2
*a*c + b**2)*atanh((b + 2*c*x**2)/sqrt(-4*a*c + b**2))/(2*a**2*sqrt(-4*a*c + b**
2))

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Mathematica [A]  time = 0.235116, size = 135, normalized size = 1.52 \[ \frac{\frac{\left (b \sqrt{b^2-4 a c}-2 a c+b^2\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\sqrt{b^2-4 a c}}+\frac{\left (b \sqrt{b^2-4 a c}+2 a c-b^2\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\sqrt{b^2-4 a c}}-\frac{2 a}{x^2}-4 b \log (x)}{4 a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a*x + b*x^3 + c*x^5)),x]

[Out]

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

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Maple [A]  time = 0.01, size = 119, normalized size = 1.3 \[{\frac{b\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) }{4\,{a}^{2}}}-{\frac{c}{a}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}}{2\,{a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{1}{2\,a{x}^{2}}}-{\frac{b\ln \left ( x \right ) }{{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(c*x^5+b*x^3+a*x),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{b \log \left (x\right )}{a^{2}} + \frac{\int \frac{b c x^{3} +{\left (b^{2} - a c\right )} x}{c x^{4} + b x^{2} + a}\,{d x}}{a^{2}} - \frac{1}{2 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^5 + b*x^3 + a*x)*x^2),x, algorithm="maxima")

[Out]

-b*log(x)/a^2 + integrate((b*c*x^3 + (b^2 - a*c)*x)/(c*x^4 + b*x^2 + a), x)/a^2
- 1/2/(a*x^2)

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Fricas [A]  time = 0.30246, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b^{2} - 2 \, a c\right )} x^{2} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) -{\left (b x^{2} \log \left (c x^{4} + b x^{2} + a\right ) - 4 \, b x^{2} \log \left (x\right ) - 2 \, a\right )} \sqrt{b^{2} - 4 \, a c}}{4 \, \sqrt{b^{2} - 4 \, a c} a^{2} x^{2}}, \frac{2 \,{\left (b^{2} - 2 \, a c\right )} x^{2} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) +{\left (b x^{2} \log \left (c x^{4} + b x^{2} + a\right ) - 4 \, b x^{2} \log \left (x\right ) - 2 \, a\right )} \sqrt{-b^{2} + 4 \, a c}}{4 \, \sqrt{-b^{2} + 4 \, a c} a^{2} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^5 + b*x^3 + a*x)*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 22.5222, size = 345, normalized size = 3.88 \[ \left (\frac{b}{4 a^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{2} + \frac{- 8 a^{3} c \left (\frac{b}{4 a^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) + 2 a^{2} b^{2} \left (\frac{b}{4 a^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) + 3 a b c - b^{3}}{2 a c^{2} - b^{2} c} \right )} + \left (\frac{b}{4 a^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{2} + \frac{- 8 a^{3} c \left (\frac{b}{4 a^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) + 2 a^{2} b^{2} \left (\frac{b}{4 a^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) + 3 a b c - b^{3}}{2 a c^{2} - b^{2} c} \right )} - \frac{1}{2 a x^{2}} - \frac{b \log{\left (x \right )}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(c*x**5+b*x**3+a*x),x)

[Out]

(b/(4*a**2) - sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(4*a**2*(4*a*c - b**2)))*log(x*
*2 + (-8*a**3*c*(b/(4*a**2) - sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(4*a**2*(4*a*c
- b**2))) + 2*a**2*b**2*(b/(4*a**2) - sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(4*a**2
*(4*a*c - b**2))) + 3*a*b*c - b**3)/(2*a*c**2 - b**2*c)) + (b/(4*a**2) + sqrt(-4
*a*c + b**2)*(2*a*c - b**2)/(4*a**2*(4*a*c - b**2)))*log(x**2 + (-8*a**3*c*(b/(4
*a**2) + sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(4*a**2*(4*a*c - b**2))) + 2*a**2*b*
*2*(b/(4*a**2) + sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(4*a**2*(4*a*c - b**2))) + 3
*a*b*c - b**3)/(2*a*c**2 - b**2*c)) - 1/(2*a*x**2) - b*log(x)/a**2

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GIAC/XCAS [A]  time = 0.25636, size = 127, normalized size = 1.43 \[ \frac{b{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \, a^{2}} - \frac{b{\rm ln}\left (x^{2}\right )}{2 \, a^{2}} + \frac{{\left (b^{2} - 2 \, a c\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} a^{2}} + \frac{b x^{2} - a}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^5 + b*x^3 + a*x)*x^2),x, algorithm="giac")

[Out]

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