3.86 \(\int \frac{1}{a x+b x^3+c x^5} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.142217, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a \sqrt{b^2-4 a c}}-\frac{\log \left (a+b x^2+c x^4\right )}{4 a}+\frac{\log (x)}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.9803, size = 63, normalized size = 0.91 \[ \frac{b \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 a \sqrt{- 4 a c + b^{2}}} + \frac{\log{\left (x^{2} \right )}}{2 a} - \frac{\log{\left (a + b x^{2} + c x^{4} \right )}}{4 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.117639, size = 113, normalized size = 1.64 \[ \frac{-\left (\sqrt{b^2-4 a c}+b\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right )+\left (b-\sqrt{b^2-4 a c}\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )+4 \log (x) \sqrt{b^2-4 a c}}{4 a \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 66, normalized size = 1. \[ -{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) }{4\,a}}-{\frac{b}{2\,a}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{\ln \left ( x \right ) }{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^5+b*x^3+a*x),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.278983, size = 1, normalized size = 0.01 \[ \left [\frac{b \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) - \sqrt{b^{2} - 4 \, a c}{\left (\log \left (c x^{4} + b x^{2} + a\right ) - 4 \, \log \left (x\right )\right )}}{4 \, \sqrt{b^{2} - 4 \, a c} a}, -\frac{2 \, b \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + \sqrt{-b^{2} + 4 \, a c}{\left (\log \left (c x^{4} + b x^{2} + a\right ) - 4 \, \log \left (x\right )\right )}}{4 \, \sqrt{-b^{2} + 4 \, a c} a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.40503, size = 253, normalized size = 3.67 \[ \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )} - \frac{1}{4 a}\right ) \log{\left (x^{2} + \frac{- 8 a^{2} c \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )} - \frac{1}{4 a}\right ) + 2 a b^{2} \left (- \frac{b \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )} - \frac{1}{4 a}\right ) - 2 a c + b^{2}}{b c} \right )} + \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )} - \frac{1}{4 a}\right ) \log{\left (x^{2} + \frac{- 8 a^{2} c \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )} - \frac{1}{4 a}\right ) + 2 a b^{2} \left (\frac{b \sqrt{- 4 a c + b^{2}}}{4 a \left (4 a c - b^{2}\right )} - \frac{1}{4 a}\right ) - 2 a c + b^{2}}{b c} \right )} + \frac{\log{\left (x \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.251115, size = 92, normalized size = 1.33 \[ -\frac{b \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} a} - \frac{{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \, a} + \frac{{\rm ln}\left (x^{2}\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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