3.836 \(\int \frac{1}{x^3 \sqrt{a-b x^4}} \, dx\)

Optimal. Leaf size=22 \[ -\frac{\sqrt{a-b x^4}}{2 a x^2} \]

[Out]

-Sqrt[a - b*x^4]/(2*a*x^2)

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Rubi [A]  time = 0.0209266, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{\sqrt{a-b x^4}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*Sqrt[a - b*x^4]),x]

[Out]

-Sqrt[a - b*x^4]/(2*a*x^2)

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Rubi in Sympy [A]  time = 3.0411, size = 17, normalized size = 0.77 \[ - \frac{\sqrt{a - b x^{4}}}{2 a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(-b*x**4+a)**(1/2),x)

[Out]

-sqrt(a - b*x**4)/(2*a*x**2)

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Mathematica [A]  time = 0.0173172, size = 22, normalized size = 1. \[ -\frac{\sqrt{a-b x^4}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*Sqrt[a - b*x^4]),x]

[Out]

-Sqrt[a - b*x^4]/(2*a*x^2)

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Maple [A]  time = 0.006, size = 19, normalized size = 0.9 \[ -{\frac{1}{2\,a{x}^{2}}\sqrt{-b{x}^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(-b*x^4+a)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.43681, size = 24, normalized size = 1.09 \[ -\frac{\sqrt{-b x^{4} + a}}{2 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-b*x^4 + a)*x^3),x, algorithm="maxima")

[Out]

-1/2*sqrt(-b*x^4 + a)/(a*x^2)

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Fricas [A]  time = 0.272873, size = 24, normalized size = 1.09 \[ -\frac{\sqrt{-b x^{4} + a}}{2 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-b*x^4 + a)*x^3),x, algorithm="fricas")

[Out]

-1/2*sqrt(-b*x^4 + a)/(a*x^2)

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Sympy [A]  time = 2.21865, size = 51, normalized size = 2.32 \[ \begin{cases} - \frac{\sqrt{b} \sqrt{\frac{a}{b x^{4}} - 1}}{2 a} & \text{for}\: \left |{\frac{a}{b x^{4}}}\right | > 1 \\- \frac{i \sqrt{b} \sqrt{- \frac{a}{b x^{4}} + 1}}{2 a} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(-b*x**4+a)**(1/2),x)

[Out]

Piecewise((-sqrt(b)*sqrt(a/(b*x**4) - 1)/(2*a), Abs(a/(b*x**4)) > 1), (-I*sqrt(b
)*sqrt(-a/(b*x**4) + 1)/(2*a), True))

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GIAC/XCAS [A]  time = 0.222101, size = 22, normalized size = 1. \[ -\frac{\sqrt{-b + \frac{a}{x^{4}}}}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-b*x^4 + a)*x^3),x, algorithm="giac")

[Out]

-1/2*sqrt(-b + a/x^4)/a