3.2.99 \(\int \frac {(a (b x^m)^n)^{-\frac {1}{m n}}}{x} \, dx\) [199]

Optimal. Leaf size=20 \[ -\left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1971, 30} \begin {gather*} -\left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 30

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

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 \frac {\left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}}}{x} \, dx &=\left (x \left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}}\right ) \int \frac {1}{x^2} \, dx\\ &=-\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 = 20, normalized size = 1.00 \begin {gather*} -\left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.05, size = 22, normalized size = 1.10

method	result	size
gosper	\(-\left (a \left (b \,x^{m}\right )^{n}\right )^{-\frac {1}{m n}}\)	\(22\)
derivativedivides	\(-\left (a \left (b \,x^{m}\right )^{n}\right )^{-\frac {1}{m n}}\)	\(22\)
default	\(-\left (a \left (b \,x^{m}\right )^{n}\right )^{-\frac {1}{m n}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.40, size = 35, normalized size = 1.75 \begin {gather*} -\frac {1}{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(1/x/((a*(b*x^m)^n)^(1/m/n)),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.36, size = 21, normalized size = 1.05 \begin {gather*} -\frac {e^{\left (-\frac {n \log \left (b\right ) + \log \left (a\right )}{m n}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(-(n*log(b) + log(a))/(m*n))/x

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Sympy [A]
time = 1.55, size = 15, normalized size = 0.75 \begin {gather*} - \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(1/x/((a*(b*x**m)**n)**(1/m/n)),x)

[Out]

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

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Giac [A]
time = 2.34, size = 21, normalized size = 1.05 \begin {gather*} -\frac {1}{\left (\left (b x^{m}\right )^{n} a\right )^{\frac {1}{m n}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.97, size = 21, normalized size = 1.05 \begin {gather*} -\frac {1}{{\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(1/(x*(a*(b*x^m)^n)^(1/(m*n))),x)

[Out]

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

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