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*ln(x)/(x^(n*p))

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \[ \log (x) x^{-n p} \left (a x^n\right )^p \]

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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fricas [A]  time = 0.82, size = 6, normalized size = 0.38 \[ a^{p} \log \relax (x) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a x^{n}\right )^{p} x^{-n p - 1}\,{d x} \]

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)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[ \int x^{-n p -1} \left (a \,x^{n}\right )^{p}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a x^{n}\right )^{p} x^{-n p - 1}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {{\left (a\,x^n\right )}^p}{x^{n\,p+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^n)^p/x^(n*p + 1),x)

[Out]

int((a*x^n)^p/x^(n*p + 1), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{- n p - 1} \left (a x^{n}\right )^{p}\, dx \]

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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