\(\int x (a x^n)^{-1/n} \, dx\) [183]

Optimal result

Integrand size = 13, antiderivative size = 15 \[ \int x \left (a x^n\right )^{-1/n} \, dx=x^2 \left (a x^n\right )^{-1/n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 8} \[ \int x \left (a x^n\right )^{-1/n} \, dx=x^2 \left (a x^n\right )^{-1/n} \]

[In]

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

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int x \left (a x^n\right )^{-1/n} \, dx=x^2 \left (a x^n\right )^{-1/n} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07

method	result	size
parallelrisch	\(x^{2} \left (a \,x^{n}\right )^{-\frac {1}{n}}\)	\(16\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.60 \[ \int x \left (a x^n\right )^{-1/n} \, dx=\frac {x}{a^{\left (\frac {1}{n}\right )}} \]

[In]

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

[Out]

x/a^(1/n)

Sympy [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int x \left (a x^n\right )^{-1/n} \, dx=x^{2} \left (a x^{n}\right )^{- \frac {1}{n}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.33 \[ \int x \left (a x^n\right )^{-1/n} \, dx=\frac {x^{2}}{a^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.60 \[ \int x \left (a x^n\right )^{-1/n} \, dx=\frac {x}{a^{\left (\frac {1}{n}\right )}} \]

[In]

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

[Out]

x/a^(1/n)

Mupad [B] (verification not implemented)

Time = 5.73 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int x \left (a x^n\right )^{-1/n} \, dx=\frac {x^2}{{\left (a\,x^n\right )}^{1/n}} \]

[In]

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

[Out]

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