3.243 \(\int \frac{1}{x \left (a-b x^2\right )^3} \, dx\)

Optimal. Leaf size=57 \[ -\frac{\log \left (a-b x^2\right )}{2 a^3}+\frac{\log (x)}{a^3}+\frac{1}{2 a^2 \left (a-b x^2\right )}+\frac{1}{4 a \left (a-b x^2\right )^2} \]

[Out]

1/(4*a*(a - b*x^2)^2) + 1/(2*a^2*(a - b*x^2)) + Log[x]/a^3 - Log[a - b*x^2]/(2*a
^3)

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Rubi [A]  time = 0.0919974, antiderivative size = 57, 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 \[ -\frac{\log \left (a-b x^2\right )}{2 a^3}+\frac{\log (x)}{a^3}+\frac{1}{2 a^2 \left (a-b x^2\right )}+\frac{1}{4 a \left (a-b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a - b*x^2)^3),x]

[Out]

1/(4*a*(a - b*x^2)^2) + 1/(2*a^2*(a - b*x^2)) + Log[x]/a^3 - Log[a - b*x^2]/(2*a
^3)

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Rubi in Sympy [A]  time = 12.7262, size = 49, normalized size = 0.86 \[ \frac{1}{4 a \left (a - b x^{2}\right )^{2}} + \frac{1}{2 a^{2} \left (a - b x^{2}\right )} + \frac{\log{\left (x^{2} \right )}}{2 a^{3}} - \frac{\log{\left (a - b x^{2} \right )}}{2 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0621884, size = 45, normalized size = 0.79 \[ \frac{\frac{a \left (3 a-2 b x^2\right )}{\left (a-b x^2\right )^2}-2 \log \left (a-b x^2\right )+4 \log (x)}{4 a^3} \]

Antiderivative was successfully verified.

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

[Out]

((a*(3*a - 2*b*x^2))/(a - b*x^2)^2 + 4*Log[x] - 2*Log[a - b*x^2])/(4*a^3)

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Maple [A]  time = 0.016, size = 55, normalized size = 1. \[{\frac{\ln \left ( x \right ) }{{a}^{3}}}-{\frac{\ln \left ( b{x}^{2}-a \right ) }{2\,{a}^{3}}}+{\frac{1}{4\,a \left ( b{x}^{2}-a \right ) ^{2}}}-{\frac{1}{2\,{a}^{2} \left ( b{x}^{2}-a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-b*x^2+a)^3,x)

[Out]

ln(x)/a^3-1/2/a^3*ln(b*x^2-a)+1/4/a/(b*x^2-a)^2-1/2/a^2/(b*x^2-a)

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Maxima [A]  time = 1.33124, size = 84, normalized size = 1.47 \[ -\frac{2 \, b x^{2} - 3 \, a}{4 \,{\left (a^{2} b^{2} x^{4} - 2 \, a^{3} b x^{2} + a^{4}\right )}} - \frac{\log \left (b x^{2} - a\right )}{2 \, a^{3}} + \frac{\log \left (x^{2}\right )}{2 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.224136, size = 124, normalized size = 2.18 \[ -\frac{2 \, a b x^{2} - 3 \, a^{2} + 2 \,{\left (b^{2} x^{4} - 2 \, a b x^{2} + a^{2}\right )} \log \left (b x^{2} - a\right ) - 4 \,{\left (b^{2} x^{4} - 2 \, a b x^{2} + a^{2}\right )} \log \left (x\right )}{4 \,{\left (a^{3} b^{2} x^{4} - 2 \, a^{4} b x^{2} + a^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(2*a*b*x^2 - 3*a^2 + 2*(b^2*x^4 - 2*a*b*x^2 + a^2)*log(b*x^2 - a) - 4*(b^2*
x^4 - 2*a*b*x^2 + a^2)*log(x))/(a^3*b^2*x^4 - 2*a^4*b*x^2 + a^5)

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Sympy [A]  time = 2.28582, size = 56, normalized size = 0.98 \[ - \frac{- 3 a + 2 b x^{2}}{4 a^{4} - 8 a^{3} b x^{2} + 4 a^{2} b^{2} x^{4}} + \frac{\log{\left (x \right )}}{a^{3}} - \frac{\log{\left (- \frac{a}{b} + x^{2} \right )}}{2 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(-b*x**2+a)**3,x)

[Out]

-(-3*a + 2*b*x**2)/(4*a**4 - 8*a**3*b*x**2 + 4*a**2*b**2*x**4) + log(x)/a**3 - l
og(-a/b + x**2)/(2*a**3)

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GIAC/XCAS [A]  time = 0.212794, size = 85, normalized size = 1.49 \[ \frac{{\rm ln}\left (x^{2}\right )}{2 \, a^{3}} - \frac{{\rm ln}\left ({\left | b x^{2} - a \right |}\right )}{2 \, a^{3}} + \frac{3 \, b^{2} x^{4} - 8 \, a b x^{2} + 6 \, a^{2}}{4 \,{\left (b x^{2} - a\right )}^{2} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*ln(x^2)/a^3 - 1/2*ln(abs(b*x^2 - a))/a^3 + 1/4*(3*b^2*x^4 - 8*a*b*x^2 + 6*a^
2)/((b*x^2 - a)^2*a^3)