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

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 = 0.0287356, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {6679, 8} \[ x^2 \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]

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 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]

Rule 8

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

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*}

Mathematica [A]  time = 0.0021264, size = 22, normalized size = 1. \[ x^2 \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]

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 = 0.077, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \left ( a \left ( b{x}^{m} \right ) ^{n} \right ) ^{{\frac{1}{mn}}} \right ) ^{-1}}\, dx \end{align*}

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 = 1.4956, size = 58, normalized size = 2.64 \begin{align*} \frac{x^{2}}{a^{\frac{1}{m n}}{\left (b^{n}\right )}^{\frac{1}{m n}}{\left ({\left (x^{m}\right )}^{n}\right )}^{\frac{1}{m n}}} \end{align*}

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^n)^(1/(m*n))*((x^m)^n)^(1/(m*n)))

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Fricas [A]  time = 1.72714, size = 46, normalized size = 2.09 \begin{align*} x e^{\left (-\frac{n \log \left (b\right ) + \log \left (a\right )}{m n}\right )} \end{align*}

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 = 8.60442, size = 245, normalized size = 11.14 \begin{align*} \begin{cases} - \frac{x^{2}}{0^{m n} \tilde{\infty }^{m n} \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}} - 2 \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}} \\a^{- \frac{1}{m n}} x^{2} \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}\: b = \left (0^{m n}\right )^{\frac{1}{n}} \\- \frac{x^{2}}{0^{m n} \tilde{\infty }^{m n} \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}} - 2 \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} \\a^{- \frac{1}{m n}} x^{2} \left (b^{n}\right )^{- \frac{1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{- \frac{1}{m n}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19272, size = 24, normalized size = 1.09 \begin{align*} x e^{\left (-\frac{n \log \left (b\right ) + \log \left (a\right )}{m n}\right )} \end{align*}

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))