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

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 = 0.0017376, 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, Rules used = {15, 30} \[ -\frac{\left (a x^n\right )^{-1/n}}{2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a x^n\right )^{-1/n}}{x^2} \, dx &=\left (x \left (a x^n\right )^{-1/n}\right ) \int \frac{1}{x^3} \, dx\\ &=-\frac{\left (a x^n\right )^{-1/n}}{2 x}\\ \end{align*}

Mathematica [A]  time = 0.0016529, 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 = 0.003, size = 17, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,x\sqrt [n]{a{x}^{n}}}} \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a x^{n}\right )^{\left (\frac{1}{n}\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((a*x^n)^(1/n)*x^2), x)

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Fricas [A]  time = 1.72035, size = 27, normalized size = 1.5 \begin{align*} -\frac{1}{2 \, a^{\left (\frac{1}{n}\right )} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.15301, size = 60, normalized size = 3.33 \begin{align*} \begin{cases} - \frac{a^{- \frac{1}{n}} \left (x^{n}\right )^{- \frac{1}{n}}}{2 x} & \text{for}\: a \neq 0^{n} \\- \frac{1}{0^{n} \tilde{\infty }^{n} x \left (0^{n}\right )^{\frac{1}{n}} \left (x^{n}\right )^{\frac{1}{n}} + x \left (0^{n}\right )^{\frac{1}{n}} \left (x^{n}\right )^{\frac{1}{n}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/((a*x**n)**(1/n)),x)

[Out]

Piecewise((-a**(-1/n)*(x**n)**(-1/n)/(2*x), Ne(a, 0**n)), (-1/(0**n*zoo**n*x*(0**n)**(1/n)*(x**n)**(1/n) + x*(
0**n)**(1/n)*(x**n)**(1/n)), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a x^{n}\right )^{\left (\frac{1}{n}\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((a*x^n)^(1/n)*x^2), x)