3.2.82 \(\int x^2 (a x^n)^{-1/n} \, dx\) [182]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {1}{2} x^3 \left (a x^n\right )^{-1/n} \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} \frac {1}{2} x^3 \left (a x^n\right )^{-1/n} \end {gather*}

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 = 0.02, size = 17, normalized size = 0.94

method	result	size
gosper	\(\frac {x^{3} \left (a \,x^{n}\right )^{-\frac {1}{n}}}{2}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.31, size = 21, normalized size = 1.17 \begin {gather*} \frac {x^{3}}{2 \, a^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )}} \end {gather*}

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 = 0.34, size = 12, normalized size = 0.67 \begin {gather*} \frac {x^{2}}{2 \, a^{\left (\frac {1}{n}\right )}} \end {gather*}

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 = 0.41, size = 12, normalized size = 0.67 \begin {gather*} \frac {x^{3} \left (a x^{n}\right )^{- \frac {1}{n}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.95, size = 12, normalized size = 0.67 \begin {gather*} \frac {x^{2}}{2 \, a^{\left (\frac {1}{n}\right )}} \end {gather*}

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)

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Mupad [B]
time = 1.12, size = 16, normalized size = 0.89 \begin {gather*} \frac {x^3}{2\,{\left (a\,x^n\right )}^{1/n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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