3.204 \(\int x^{-1-n p q} (a (b x^n)^p)^q \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0436247, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {6679, 29} \[ \log (x) x^{-n p q} \left (a \left (b x^n\right )^p\right )^q \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - n*p*q)*(a*(b*x^n)^p)^q,x]

[Out]

((a*(b*x^n)^p)^q*Log[x])/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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

Mathematica [A]  time = 0.0050226, size = 21, normalized size = 1. \[ \log (x) x^{-n p q} \left (a \left (b x^n\right )^p\right )^q \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - n*p*q)*(a*(b*x^n)^p)^q,x]

[Out]

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

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Maple [F]  time = 0.171, size = 0, normalized size = 0. \begin{align*} \int{x}^{-npq-1} \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\left (b x^{n}\right )^{p} a\right )^{q} x^{-n p q - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-n*p*q-1)*(a*(b*x^n)^p)^q,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.87412, size = 46, normalized size = 2.19 \begin{align*} e^{\left (p q \log \left (b\right ) + q \log \left (a\right )\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-n*p*q-1)*(a*(b*x^n)^p)^q,x, algorithm="fricas")

[Out]

e^(p*q*log(b) + q*log(a))*log(x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{- n p q - 1} \left (a \left (b x^{n}\right )^{p}\right )^{q}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-n*p*q-1)*(a*(b*x**n)**p)**q,x)

[Out]

Integral(x**(-n*p*q - 1)*(a*(b*x**n)**p)**q, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\left (b x^{n}\right )^{p} a\right )^{q} x^{-n p q - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-n*p*q-1)*(a*(b*x^n)^p)^q,x, algorithm="giac")

[Out]

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