3.186 \(\int \frac{\left (a x^n\right )^{-1/n}}{x^2} \, dx\)

Optimal. Leaf size=18 \[ -\frac{\left (a x^n\right )^{-1/n}}{2 x} \]

[Out]

-1/(2*x*(a*x^n)^n^(-1))

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Rubi [A]  time = 0.00792918, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{\left (a x^n\right )^{-1/n}}{2 x} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a*x^n)^n^(-1)),x]

[Out]

-1/(2*x*(a*x^n)^n^(-1))

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Rubi in Sympy [A]  time = 2.41362, size = 14, normalized size = 0.78 \[ - \frac{\left (a x^{n}\right )^{- \frac{1}{n}}}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/((a*x**n)**(1/n)),x)

[Out]

-(a*x**n)**(-1/n)/(2*x)

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Mathematica [A]  time = 0.00333774, size = 18, normalized size = 1. \[ -\frac{\left (a x^n\right )^{-1/n}}{2 x} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*x*(a*x^n)^n^(-1))

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Maple [A]  time = 0.002, size = 17, normalized size = 0.9 \[ -{\frac{1}{2\,x\sqrt [n]{a{x}^{n}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/((a*x^n)^(1/n)),x)

[Out]

-1/2/x/((a*x^n)^(1/n))

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Maxima [A]  time = 1.48477, size = 16, normalized size = 0.89 \[ -\frac{a^{-\frac{1}{n}}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*a^(-1/n)/x^2

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Fricas [A]  time = 0.221657, size = 16, normalized size = 0.89 \[ -\frac{1}{2 \, a^{\left (\frac{1}{n}\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(a^(1/n)*x^2)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/((a*x**n)**(1/n)),x)

[Out]

Exception raised: RecursionError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a x^{n}\right )^{\left (\frac{1}{n}\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((a*x^n)^(1/n)*x^2), x)