3.1027 \(\int \frac{1}{x^2 \sqrt{2+2 a-2 (1+a)+c x^4}} \, dx\)

Optimal. Leaf size=16 \[ -\frac{1}{3 x \sqrt{c x^4}} \]

[Out]

-1/(3*x*Sqrt[c*x^4])

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Rubi [A]  time = 0.00762296, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ -\frac{1}{3 x \sqrt{c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(3*x*Sqrt[c*x^4])

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Rubi in Sympy [A]  time = 2.13506, size = 15, normalized size = 0.94 \[ - \frac{\sqrt{c x^{4}}}{3 c x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c*x**4)/(3*c*x**5)

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Mathematica [A]  time = 0.00365389, size = 16, normalized size = 1. \[ -\frac{1}{3 x \sqrt{c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(3*x*Sqrt[c*x^4])

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Maple [A]  time = 0.004, size = 13, normalized size = 0.8 \[ -{\frac{1}{3\,x}{\frac{1}{\sqrt{c{x}^{4}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(c*x^4)^(1/2),x)

[Out]

-1/3/x/(c*x^4)^(1/2)

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Maxima [A]  time = 0.708985, size = 16, normalized size = 1. \[ -\frac{1}{3 \, \sqrt{c x^{4}} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/(sqrt(c*x^4)*x)

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Fricas [A]  time = 0.263154, size = 20, normalized size = 1.25 \[ -\frac{\sqrt{c x^{4}}}{3 \, c x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*sqrt(c*x^4)/(c*x^5)

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Sympy [A]  time = 1.67977, size = 17, normalized size = 1.06 \[ - \frac{1}{3 \sqrt{c} x \sqrt{x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(3*sqrt(c)*x*sqrt(x**4))

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GIAC/XCAS [A]  time = 0.261782, size = 11, normalized size = 0.69 \[ -\frac{1}{3 \, \sqrt{c} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/(sqrt(c)*x^3)