3.200 \(\int \frac {(a (b x^m)^n)^{-\frac {1}{m n}}}{x^2} \, dx\)

Optimal. Leaf size=25 \[ -\frac {\left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}}}{2 x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 25, 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 = {6679, 30} \[ -\frac {\left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}}}{2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*x*(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 6679

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 25, normalized size = 1.00 \[ -\frac {\left (a \left (b x^m\right )^n\right )^{-\frac {1}{m n}}}{2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.82, size = 21, normalized size = 0.84 \[ -\frac {e^{\left (-\frac {n \log \relax (b) + \log \relax (a)}{m n}\right )}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*e^(-(n*log(b) + log(a))/(m*n))/x^2

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\left (b x^{m}\right )^{n} a\right )^{\frac {1}{m n}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.00, size = 25, normalized size = 1.00 \[ -\frac {\left (a \left (b \,x^{m}\right )^{n}\right )^{-\frac {1}{m n}}}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\left (b x^{m}\right )^{n} a\right )^{\frac {1}{m n}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{x^2\,{\left (a\,{\left (b\,x^m\right )}^n\right )}^{\frac {1}{m\,n}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 11.06, size = 252, normalized size = 10.08 \[ \begin {cases} - \frac {1}{0^{m n} \tilde {\infty }^{m n} x \left (0^{m n}\right )^{\frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{\frac {1}{m n}} \left (\left (\left (0^{m n}\right )^{\frac {1}{n}}\right )^{n}\right )^{\frac {1}{m n}} + x \left (0^{m n}\right )^{\frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{\frac {1}{m n}} \left (\left (\left (0^{m n}\right )^{\frac {1}{n}}\right )^{n}\right )^{\frac {1}{m n}}} & \text {for}\: a = 0^{m n} \wedge b = \left (0^{m n}\right )^{\frac {1}{n}} \\- \frac {a^{- \frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{- \frac {1}{m n}} \left (\left (\left (0^{m n}\right )^{\frac {1}{n}}\right )^{n}\right )^{- \frac {1}{m n}}}{2 x} & \text {for}\: b = \left (0^{m n}\right )^{\frac {1}{n}} \\- \frac {1}{0^{m n} \tilde {\infty }^{m n} x \left (0^{m n}\right )^{\frac {1}{m n}} \left (b^{n}\right )^{\frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{\frac {1}{m n}} + x \left (0^{m n}\right )^{\frac {1}{m n}} \left (b^{n}\right )^{\frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{\frac {1}{m n}}} & \text {for}\: a = 0^{m n} \\- \frac {a^{- \frac {1}{m n}} \left (b^{n}\right )^{- \frac {1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{- \frac {1}{m n}}}{2 x} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-1/(0**(m*n)*zoo**(m*n)*x*(0**(m*n))**(1/(m*n))*((x**m)**n)**(1/(m*n))*(((0**(m*n))**(1/n))**n)**(1
/(m*n)) + x*(0**(m*n))**(1/(m*n))*((x**m)**n)**(1/(m*n))*(((0**(m*n))**(1/n))**n)**(1/(m*n))), Eq(a, 0**(m*n))
& Eq(b, (0**(m*n))**(1/n))), (-a**(-1/(m*n))*((x**m)**n)**(-1/(m*n))*(((0**(m*n))**(1/n))**n)**(-1/(m*n))/(2*
x), Eq(b, (0**(m*n))**(1/n))), (-1/(0**(m*n)*zoo**(m*n)*x*(0**(m*n))**(1/(m*n))*(b**n)**(1/(m*n))*((x**m)**n)*
*(1/(m*n)) + x*(0**(m*n))**(1/(m*n))*(b**n)**(1/(m*n))*((x**m)**n)**(1/(m*n))), Eq(a, 0**(m*n))), (-a**(-1/(m*
n))*(b**n)**(-1/(m*n))*((x**m)**n)**(-1/(m*n))/(2*x), True))

________________________________________________________________________________________