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

Optimal. Leaf size=18 \[ \frac{1}{2} x^3 \left (a x^n\right )^{-1/n} \]

[Out]

x^3/(2*(a*x^n)^n^(-1))

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Rubi [A]  time = 0.001827, 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} \[ \frac{1}{2} x^3 \left (a x^n\right )^{-1/n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.001945, size = 18, normalized size = 1. \[ \frac{1}{2} x^3 \left (a x^n\right )^{-1/n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^3/(2*(a*x^n)^n^(-1))

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Maple [A]  time = 0.002, size = 17, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{2\,\sqrt [n]{a{x}^{n}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/((a*x^n)^(1/n)),x)

[Out]

1/2*x^3/((a*x^n)^(1/n))

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Maxima [A]  time = 1.01288, size = 28, normalized size = 1.56 \begin{align*} \frac{x^{3}}{2 \, a^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^3/(a^(1/n)*(x^n)^(1/n))

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Fricas [A]  time = 1.82491, size = 23, normalized size = 1.28 \begin{align*} \frac{x^{2}}{2 \, a^{\left (\frac{1}{n}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 3.24361, size = 58, normalized size = 3.22 \begin{align*} \begin{cases} \frac{a^{- \frac{1}{n}} x^{3} \left (x^{n}\right )^{- \frac{1}{n}}}{2} & \text{for}\: a \neq 0^{n} \\- \frac{x^{3}}{0^{n} \tilde{\infty }^{n} \left (0^{n}\right )^{\frac{1}{n}} \left (x^{n}\right )^{\frac{1}{n}} - 3 \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(x**2/((a*x**n)**(1/n)),x)

[Out]

Piecewise((a**(-1/n)*x**3*(x**n)**(-1/n)/2, Ne(a, 0**n)), (-x**3/(0**n*zoo**n*(0**n)**(1/n)*(x**n)**(1/n) - 3*
(0**n)**(1/n)*(x**n)**(1/n)), True))

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Giac [A]  time = 1.10506, size = 16, normalized size = 0.89 \begin{align*} \frac{x^{2}}{2 \, a^{\left (\frac{1}{n}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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