3.264 \(\int \frac{1}{x \sqrt{b x^2+c x^4}} \, dx\)

Optimal. Leaf size=23 \[ -\frac{\sqrt{b x^2+c x^4}}{b x^2} \]

[Out]

-(Sqrt[b*x^2 + c*x^4]/(b*x^2))

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Rubi [A]  time = 0.0689733, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ -\frac{\sqrt{b x^2+c x^4}}{b x^2} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[b*x^2 + c*x^4]/(b*x^2))

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Rubi in Sympy [A]  time = 8.16235, size = 19, normalized size = 0.83 \[ - \frac{\sqrt{b x^{2} + c x^{4}}}{b x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(b*x**2 + c*x**4)/(b*x**2)

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Mathematica [A]  time = 0.0229697, size = 23, normalized size = 1. \[ -\frac{\sqrt{x^2 \left (b+c x^2\right )}}{b x^2} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[x^2*(b + c*x^2)]/(b*x^2))

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Maple [A]  time = 0.005, size = 26, normalized size = 1.1 \[ -{\frac{c{x}^{2}+b}{b}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(c*x^4+b*x^2)^(1/2),x)

[Out]

-(c*x^2+b)/b/(c*x^4+b*x^2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25907, size = 28, normalized size = 1.22 \[ -\frac{\sqrt{c x^{4} + b x^{2}}}{b x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c*x^4 + b*x^2)/(b*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*sqrt(x**2*(b + c*x**2))), x)

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GIAC/XCAS [A]  time = 0.276991, size = 19, normalized size = 0.83 \[ -\frac{\sqrt{c + \frac{b}{x^{2}}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c + b/x^2)/b