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

Optimal. Leaf size=35 \[ -\frac{b \log \left (a-b x^2\right )}{2 a^2}+\frac{b \log (x)}{a^2}-\frac{1}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.0606195, antiderivative size = 35, 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 \[ -\frac{b \log \left (a-b x^2\right )}{2 a^2}+\frac{b \log (x)}{a^2}-\frac{1}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a - b*x^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.48546, size = 34, normalized size = 0.97 \[ - \frac{1}{2 a x^{2}} + \frac{b \log{\left (x^{2} \right )}}{2 a^{2}} - \frac{b \log{\left (a - b x^{2} \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0142872, size = 35, normalized size = 1. \[ -\frac{b \log \left (a-b x^2\right )}{2 a^2}+\frac{b \log (x)}{a^2}-\frac{1}{2 a x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 33, normalized size = 0.9 \[ -{\frac{1}{2\,a{x}^{2}}}+{\frac{b\ln \left ( x \right ) }{{a}^{2}}}-{\frac{b\ln \left ( b{x}^{2}-a \right ) }{2\,{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(-b*x^2+a),x)

[Out]

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

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Maxima [A]  time = 1.35136, size = 47, normalized size = 1.34 \[ -\frac{b \log \left (b x^{2} - a\right )}{2 \, a^{2}} + \frac{b \log \left (x^{2}\right )}{2 \, a^{2}} - \frac{1}{2 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.206204, size = 45, normalized size = 1.29 \[ -\frac{b x^{2} \log \left (b x^{2} - a\right ) - 2 \, b x^{2} \log \left (x\right ) + a}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.65492, size = 31, normalized size = 0.89 \[ - \frac{1}{2 a x^{2}} + \frac{b \log{\left (x \right )}}{a^{2}} - \frac{b \log{\left (- \frac{a}{b} + x^{2} \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(-b*x**2+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.211988, size = 58, normalized size = 1.66 \[ \frac{b{\rm ln}\left (x^{2}\right )}{2 \, a^{2}} - \frac{b{\rm ln}\left ({\left | b x^{2} - a \right |}\right )}{2 \, a^{2}} - \frac{b x^{2} + a}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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