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

Optimal. Leaf size=38 \[ -\frac{\log \left (a+b x^2\right )}{2 a^2}+\frac{\log (x)}{a^2}+\frac{1}{2 a \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0880225, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\log \left (a+b x^2\right )}{2 a^2}+\frac{\log (x)}{a^2}+\frac{1}{2 a \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.349, size = 34, normalized size = 0.89 \[ \frac{1}{2 a \left (a + b x^{2}\right )} + \frac{\log{\left (x^{2} \right )}}{2 a^{2}} - \frac{\log{\left (a + b x^{2} \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0224913, size = 33, normalized size = 0.87 \[ \frac{\frac{a}{a+b x^2}-\log \left (a+b x^2\right )+2 \log (x)}{2 a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 35, normalized size = 0.9 \[{\frac{1}{2\,a \left ( b{x}^{2}+a \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{2}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.685047, size = 50, normalized size = 1.32 \[ \frac{1}{2 \,{\left (a b x^{2} + a^{2}\right )}} - \frac{\log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac{\log \left (x^{2}\right )}{2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.259953, size = 63, normalized size = 1.66 \[ -\frac{{\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right ) - 2 \,{\left (b x^{2} + a\right )} \log \left (x\right ) - a}{2 \,{\left (a^{2} b x^{2} + a^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*((b*x^2 + a)*log(b*x^2 + a) - 2*(b*x^2 + a)*log(x) - a)/(a^2*b*x^2 + a^3)

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Sympy [A]  time = 1.613, size = 34, normalized size = 0.89 \[ \frac{1}{2 a^{2} + 2 a b x^{2}} + \frac{\log{\left (x \right )}}{a^{2}} - \frac{\log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.272543, size = 63, normalized size = 1.66 \[ \frac{{\rm ln}\left (x^{2}\right )}{2 \, a^{2}} - \frac{{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2}} + \frac{b x^{2} + 2 \, a}{2 \,{\left (b x^{2} + a\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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