3.909 \(\int \frac{1}{x \left (a+b+2 a x^2+a x^4\right )} \, dx\)

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

[Out]

-(Sqrt[a]*ArcTan[(Sqrt[a]*(1 + x^2))/Sqrt[b]])/(2*Sqrt[b]*(a + b)) + Log[x]/(a +
b) - Log[a + b + 2*a*x^2 + a*x^4]/(4*(a + b))

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Rubi [A]  time = 0.148223, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{b} (a+b)}-\frac{\log \left (a x^4+2 a x^2+a+b\right )}{4 (a+b)}+\frac{\log (x)}{a+b} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b + 2*a*x^2 + a*x^4)),x]

[Out]

-(Sqrt[a]*ArcTan[(Sqrt[a]*(1 + x^2))/Sqrt[b]])/(2*Sqrt[b]*(a + b)) + Log[x]/(a +
b) - Log[a + b + 2*a*x^2 + a*x^4]/(4*(a + b))

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Rubi in Sympy [A]  time = 28.3211, size = 66, normalized size = 0.96 \[ - \frac{\sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{a} \left (x^{2} + 1\right )}{\sqrt{b}} \right )}}{2 \sqrt{b} \left (a + b\right )} + \frac{\log{\left (x^{2} \right )}}{2 \left (a + b\right )} - \frac{\log{\left (a x^{4} + 2 a x^{2} + a + b \right )}}{4 \left (a + b\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(a*x**4+2*a*x**2+a+b),x)

[Out]

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

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Mathematica [C]  time = 0.0908083, size = 105, normalized size = 1.52 \[ \frac{i \left (\sqrt{a}+i \sqrt{b}\right ) \log \left (\sqrt{a} \left (x^2+1\right )-i \sqrt{b}\right )+\left (-\sqrt{b}-i \sqrt{a}\right ) \log \left (\sqrt{a} \left (x^2+1\right )+i \sqrt{b}\right )+4 \sqrt{b} \log (x)}{4 \sqrt{b} (a+b)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a + b + 2*a*x^2 + a*x^4)),x]

[Out]

(4*Sqrt[b]*Log[x] + I*(Sqrt[a] + I*Sqrt[b])*Log[(-I)*Sqrt[b] + Sqrt[a]*(1 + x^2)
] + ((-I)*Sqrt[a] - Sqrt[b])*Log[I*Sqrt[b] + Sqrt[a]*(1 + x^2)])/(4*Sqrt[b]*(a +
b))

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Maple [A]  time = 0.009, size = 63, normalized size = 0.9 \[ -{\frac{\ln \left ( a{x}^{4}+2\,a{x}^{2}+a+b \right ) }{4\,a+4\,b}}-{\frac{a}{2\,a+2\,b}\arctan \left ({\frac{2\,a{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{\ln \left ( x \right ) }{a+b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a*x^4+2*a*x^2+a+b),x)

[Out]

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

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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/((a*x^4 + 2*a*x^2 + a + b)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.284861, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{-\frac{a}{b}} \log \left (\frac{a x^{4} + 2 \, a x^{2} - 2 \,{\left (b x^{2} + b\right )} \sqrt{-\frac{a}{b}} + a - b}{a x^{4} + 2 \, a x^{2} + a + b}\right ) - \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) + 4 \, \log \left (x\right )}{4 \,{\left (a + b\right )}}, \frac{2 \, \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{\frac{a}{b}}}{a x^{2} + a}\right ) - \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) + 4 \, \log \left (x\right )}{4 \,{\left (a + b\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a*x^4 + 2*a*x^2 + a + b)*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.15128, size = 194, normalized size = 2.81 \[ \left (- \frac{1}{4 \left (a + b\right )} - \frac{\sqrt{- a b}}{4 b \left (a + b\right )}\right ) \log{\left (x^{2} + \frac{- 4 a b \left (- \frac{1}{4 \left (a + b\right )} - \frac{\sqrt{- a b}}{4 b \left (a + b\right )}\right ) + a - 4 b^{2} \left (- \frac{1}{4 \left (a + b\right )} - \frac{\sqrt{- a b}}{4 b \left (a + b\right )}\right ) - b}{a} \right )} + \left (- \frac{1}{4 \left (a + b\right )} + \frac{\sqrt{- a b}}{4 b \left (a + b\right )}\right ) \log{\left (x^{2} + \frac{- 4 a b \left (- \frac{1}{4 \left (a + b\right )} + \frac{\sqrt{- a b}}{4 b \left (a + b\right )}\right ) + a - 4 b^{2} \left (- \frac{1}{4 \left (a + b\right )} + \frac{\sqrt{- a b}}{4 b \left (a + b\right )}\right ) - b}{a} \right )} + \frac{\log{\left (x \right )}}{a + b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(a*x**4+2*a*x**2+a+b),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.545411, size = 82, normalized size = 1.19 \[ -\frac{a \arctan \left (\frac{a x^{2} + a}{\sqrt{a b}}\right )}{2 \, \sqrt{a b}{\left (a + b\right )}} - \frac{{\rm ln}\left (a x^{4} + 2 \, a x^{2} + a + b\right )}{4 \,{\left (a + b\right )}} + \frac{{\rm ln}\left (x^{2}\right )}{2 \,{\left (a + b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a*x^4 + 2*a*x^2 + a + b)*x),x, algorithm="giac")

[Out]

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