3.203 \(\int x^{-n p q} (a (b x^n)^p)^q \, dx\)

Optimal. Leaf size=21 \[ x^{1-n p q} \left (a \left (b x^n\right )^p\right )^q \]

[Out]

x^(1 - n*p*q)*(a*(b*x^n)^p)^q

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Rubi [A]  time = 0.0429259, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {6679, 8} \[ x^{1-n p q} \left (a \left (b x^n\right )^p\right )^q \]

Antiderivative was successfully verified.

[In]

Int[(a*(b*x^n)^p)^q/x^(n*p*q),x]

[Out]

x^(1 - n*p*q)*(a*(b*x^n)^p)^q

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^{-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx &=\left (x^{-n p q} \left (a \left (b x^n\right )^p\right )^q\right ) \int 1 \, dx\\ &=x^{1-n p q} \left (a \left (b x^n\right )^p\right )^q\\ \end{align*}

Mathematica [A]  time = 0.0030358, size = 21, normalized size = 1. \[ x^{1-n p q} \left (a \left (b x^n\right )^p\right )^q \]

Antiderivative was successfully verified.

[In]

Integrate[(a*(b*x^n)^p)^q/x^(n*p*q),x]

[Out]

x^(1 - n*p*q)*(a*(b*x^n)^p)^q

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Maple [F]  time = 0.181, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q}}{{x}^{npq}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*(b*x^n)^p)^q/(x^(n*p*q)),x)

[Out]

int((a*(b*x^n)^p)^q/(x^(n*p*q)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (b x^{n}\right )^{p} a\right )^{q}}{x^{n p q}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*(b*x^n)^p)^q/(x^(n*p*q)),x, algorithm="maxima")

[Out]

integrate(((b*x^n)^p*a)^q/x^(n*p*q), x)

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Fricas [A]  time = 1.75243, size = 39, normalized size = 1.86 \begin{align*} x e^{\left (p q \log \left (b\right ) + q \log \left (a\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*(b*x^n)^p)^q/(x^(n*p*q)),x, algorithm="fricas")

[Out]

x*e^(p*q*log(b) + q*log(a))

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Sympy [A]  time = 3.13647, size = 22, normalized size = 1.05 \begin{align*} a^{q} x x^{- n p q} \left (b^{p}\right )^{q} \left (\left (x^{n}\right )^{p}\right )^{q} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*(b*x**n)**p)**q/(x**(n*p*q)),x)

[Out]

a**q*x*x**(-n*p*q)*(b**p)**q*((x**n)**p)**q

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Giac [A]  time = 1.16183, size = 36, normalized size = 1.71 \begin{align*} \frac{x e^{\left (n p q \log \left (x\right ) + p q \log \left (b\right ) + q \log \left (a\right )\right )}}{x^{n p q}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*(b*x^n)^p)^q/(x^(n*p*q)),x, algorithm="giac")

[Out]

x*e^(n*p*q*log(x) + p*q*log(b) + q*log(a))/x^(n*p*q)