3.2.88 \(\int x^{-1-n p} (a x^n)^p \, dx\) [188]

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

[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} \begin {gather*} \log (x) x^{-n p} \left (a x^n\right )^p \end {gather*}

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 = 0.00, size = 16, normalized size = 1.00 \begin {gather*} x^{-n p} \left (a x^n\right )^p \log (x) \end {gather*}

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 = 0.03, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.36, size = 6, normalized size = 0.38 \begin {gather*} a^{p} \log \left (x\right ) \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{- n p - 1} \left (a x^{n}\right )^{p}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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