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

Optimal. Leaf size=25 \[ \frac{1}{2} x^3 \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]

[Out]

x^3/(2*(a*(b*x^m)^n)^(1/(m*n)))

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Rubi [A]  time = 0.0499666, antiderivative size = 25, normalized size of antiderivative = 1., 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} \[ \frac{1}{2} x^3 \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^3/(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 30

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

Rubi steps

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

Mathematica [A]  time = 0.0095051, size = 25, normalized size = 1. \[ \frac{1}{2} x^3 \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^3/(2*(a*(b*x^m)^n)^(1/(m*n)))

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Maple [A]  time = 0.003, size = 25, normalized size = 1. \begin{align*}{\frac{{x}^{3}}{2} \left ( \left ( a \left ( b{x}^{m} \right ) ^{n} \right ) ^{{\frac{1}{mn}}} \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^3/((a*(b*x^m)^n)^(1/m/n))

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Maxima [A]  time = 1.54464, size = 59, normalized size = 2.36 \begin{align*} \frac{x^{3}}{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^2/((a*(b*x^m)^n)^(1/m/n)),x, algorithm="maxima")

[Out]

1/2*x^3/(a^(1/(m*n))*(b^n)^(1/(m*n))*((x^m)^n)^(1/(m*n)))

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Fricas [A]  time = 1.68945, size = 54, normalized size = 2.16 \begin{align*} \frac{1}{2} \, x^{2} 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^2/((a*(b*x^m)^n)^(1/m/n)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 16.5999, size = 248, normalized size = 9.92 \begin{align*} \begin{cases} - \frac{x^{3}}{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}} - 3 \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}} x^{3} \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} & \text{for}\: b = \left (0^{m n}\right )^{\frac{1}{n}} \\- \frac{x^{3}}{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}} - 3 \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}} x^{3} \left (b^{n}\right )^{- \frac{1}{m n}} \left (\left (x^{m}\right )^{n}\right )^{- \frac{1}{m n}}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-x**3/(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)) - 3*(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**3*((x**m)**n)**(-1/(m*n))*(((0**(m*n))**(1/n))**n)**(-1/(m*n)
)/2, Eq(b, (0**(m*n))**(1/n))), (-x**3/(0**(m*n)*zoo**(m*n)*(0**(m*n))**(1/(m*n))*(b**n)**(1/(m*n))*((x**m)**n
)**(1/(m*n)) - 3*(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**3*(b**n)**(-1/(m*n))*((x**m)**n)**(-1/(m*n))/2, True))

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Giac [A]  time = 1.13731, size = 28, normalized size = 1.12 \begin{align*} \frac{1}{2} \, x^{2} 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^2/((a*(b*x^m)^n)^(1/m/n)),x, algorithm="giac")

[Out]

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