3.2.97 \(\int x (a (b x^m)^n)^{-\frac {1}{m n}} \, dx\) [197]

Optimal. Leaf size=22 \[ x^2 \left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1971, 8} \begin {gather*} x^2 \left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 8

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

Rule 1971

Int[(u_.)*((c_.)*((d_.)*((a_.) + (b_.)*(x_))^(n_))^(q_))^(p_), x_Symbol] :> Dist[(c*(d*(a + b*x)^n)^q)^p/(a +
b*x)^(n*p*q), Int[u*(a + b*x)^(n*p*q), x], x] /; FreeQ[{a, b, c, d, n, q, p}, x] &&  !IntegerQ[q] &&  !Integer
Q[p]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 22, normalized size = 1.00 \begin {gather*} x^2 \left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int x \left (a \left (b \,x^{m}\right )^{n}\right )^{-\frac {1}{m n}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.40, size = 37, normalized size = 1.68 \begin {gather*} \frac {x^{2}}{a^{\frac {1}{m n}} b^{\left (\frac {1}{m}\right )} {\left ({\left (x^{m}\right )}^{n}\right )}^{\frac {1}{m n}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.36, size = 18, normalized size = 0.82 \begin {gather*} x e^{\left (-\frac {n \log \left (b\right ) + \log \left (a\right )}{m n}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(-(n*log(b) + log(a))/(m*n))

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Sympy [A]
time = 1.66, size = 15, normalized size = 0.68 \begin {gather*} x^{2} \left (a \left (b x^{m}\right )^{n}\right )^{- \frac {1}{m n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.77, size = 18, normalized size = 0.82 \begin {gather*} x e^{\left (-\frac {n \log \left (b\right ) + \log \left (a\right )}{m n}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(-(n*log(b) + log(a))/(m*n))

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Mupad [B]
time = 0.98, size = 23, normalized size = 1.05 \begin {gather*} \frac {x^2}{{\left (a\,{\left (b\,x^m\right )}^n\right )}^{\frac {1}{m\,n}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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