3.3.3 \(\int x^{-n p q} (a (b x^n)^p)^q \, dx\) [203]

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

[Out]

x^(-n*p*q+1)*(a*(b*x^n)^p)^q

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Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, 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 = {1971, 8} \begin {gather*} x^{1-n p q} \left (a \left (b x^n\right )^p\right )^q \end {gather*}

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 8

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

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 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*}

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Mathematica [A]
time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} x^{1-n p q} \left (a \left (b x^n\right )^p\right )^q \end {gather*}

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 = 0.16, size = 0, normalized size = 0.00 \[\int \left (a \left (b \,x^{n}\right )^{p}\right )^{q} x^{-n p q}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.36, size = 13, normalized size = 0.62 \begin {gather*} x e^{\left (p q \log \left (b\right ) + q \log \left (a\right )\right )} \end {gather*}

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 = 0.48, size = 17, normalized size = 0.81 \begin {gather*} x x^{- n p q} \left (a \left (b x^{n}\right )^{p}\right )^{q} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 2.60, size = 27, normalized size = 1.29 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 1.00, size = 21, normalized size = 1.00 \begin {gather*} x^{1-n\,p\,q}\,{\left (a\,{\left (b\,x^n\right )}^p\right )}^q \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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