3.236 \(\int \frac{1}{x \left (a-b x^2\right )^2} \, dx\)

Optimal. Leaf size=40 \[ -\frac{\log \left (a-b x^2\right )}{2 a^2}+\frac{\log (x)}{a^2}+\frac{1}{2 a \left (a-b x^2\right )} \]

[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 = 0.0675177, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{\log \left (a-b x^2\right )}{2 a^2}+\frac{\log (x)}{a^2}+\frac{1}{2 a \left (a-b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a - b*x^2)^2),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 = 9.78796, size = 34, normalized size = 0.85 \[ \frac{1}{2 a \left (a - b x^{2}\right )} + \frac{\log{\left (x^{2} \right )}}{2 a^{2}} - \frac{\log{\left (a - b x^{2} \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(-b*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 = 0.0327199, size = 35, normalized size = 0.88 \[ \frac{\frac{a}{a-b x^2}-\log \left (a-b x^2\right )+2 \log (x)}{2 a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a - b*x^2)^2),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 = 39, normalized size = 1. \[{\frac{\ln \left ( x \right ) }{{a}^{2}}}-{\frac{\ln \left ( b{x}^{2}-a \right ) }{2\,{a}^{2}}}-{\frac{1}{2\,a \left ( b{x}^{2}-a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-b*x^2+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.34529, size = 55, normalized size = 1.38 \[ -\frac{1}{2 \,{\left (a b x^{2} - a^{2}\right )}} - \frac{\log \left (b x^{2} - a\right )}{2 \, a^{2}} + \frac{\log \left (x^{2}\right )}{2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*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 = 0.211061, size = 72, normalized size = 1.8 \[ -\frac{{\left (b x^{2} - a\right )} \log \left (b x^{2} - a\right ) - 2 \,{\left (b x^{2} - a\right )} \log \left (x\right ) + a}{2 \,{\left (a^{2} b x^{2} - a^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*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 = 1.71236, size = 34, normalized size = 0.85 \[ - \frac{1}{- 2 a^{2} + 2 a b x^{2}} + \frac{\log{\left (x \right )}}{a^{2}} - \frac{\log{\left (- \frac{a}{b} + x^{2} \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(-b*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 = 0.218619, size = 69, normalized size = 1.72 \[ \frac{{\rm ln}\left (x^{2}\right )}{2 \, a^{2}} - \frac{{\rm ln}\left ({\left | b x^{2} - a \right |}\right )}{2 \, a^{2}} + \frac{b x^{2} - 2 \, a}{2 \,{\left (b x^{2} - a\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*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)