3.188 \(\int x^{-1-n p} (a x^n)^p \, dx\)

Optimal. Leaf size=16 \[ \log (x) x^{-n p} \left (a x^n\right )^p \]

[Out]

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

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Rubi [A]  time = 0.0031623, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {15, 29} \[ \log (x) x^{-n p} \left (a x^n\right )^p \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - n*p)*(a*x^n)^p,x]

[Out]

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

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 29

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

Rubi steps

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

Mathematica [A]  time = 0.0050411, size = 16, normalized size = 1. \[ \log (x) x^{-n p} \left (a x^n\right )^p \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - n*p)*(a*x^n)^p,x]

[Out]

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

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{x}^{-np-1} \left ( a{x}^{n} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*x^n)^p*x^(-n*p - 1), x)

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Fricas [A]  time = 1.75884, size = 16, normalized size = 1. \begin{align*} a^{p} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^p*log(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*x^n)^p*x^(-n*p - 1), x)