3.320 \(\int \frac{1}{-x^3+b x^5} \, dx\)

Optimal. Leaf size=27 \[ \frac{1}{2} b \log \left (1-b x^2\right )-b \log (x)+\frac{1}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0452162, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{1}{2} b \log \left (1-b x^2\right )-b \log (x)+\frac{1}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(-x^3 + b*x^5)^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.52057, size = 26, normalized size = 0.96 \[ - \frac{b \log{\left (x^{2} \right )}}{2} + \frac{b \log{\left (- b x^{2} + 1 \right )}}{2} + \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**5-x**3),x)

[Out]

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

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Mathematica [A]  time = 0.00814101, size = 27, normalized size = 1. \[ \frac{1}{2} b \log \left (1-b x^2\right )-b \log (x)+\frac{1}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(-x^3 + b*x^5)^(-1),x]

[Out]

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

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Maple [A]  time = 0.009, size = 23, normalized size = 0.9 \[{\frac{1}{2\,{x}^{2}}}-b\ln \left ( x \right ) +{\frac{b\ln \left ( b{x}^{2}-1 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^5-x^3),x)

[Out]

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

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Maxima [A]  time = 1.37633, size = 30, normalized size = 1.11 \[ \frac{1}{2} \, b \log \left (b x^{2} - 1\right ) - b \log \left (x\right ) + \frac{1}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x^5 - x^3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.212011, size = 38, normalized size = 1.41 \[ \frac{b x^{2} \log \left (b x^{2} - 1\right ) - 2 \, b x^{2} \log \left (x\right ) + 1}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x^5 - x^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.516603, size = 22, normalized size = 0.81 \[ - b \log{\left (x \right )} + \frac{b \log{\left (x^{2} - \frac{1}{b} \right )}}{2} + \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**5-x**3),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.219165, size = 43, normalized size = 1.59 \[ -\frac{1}{2} \, b{\rm ln}\left (x^{2}\right ) + \frac{1}{2} \, b{\rm ln}\left ({\left | b x^{2} - 1 \right |}\right ) + \frac{b x^{2} + 1}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x^5 - x^3),x, algorithm="giac")

[Out]

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