3.150 \(\int \frac {\log ^m(a x^n)^p}{x} \, dx\)

Optimal. Leaf size=27 \[ \frac {\log \left (a x^n\right ) \log ^m\left (a x^n\right )^p}{n (m p+1)} \]

[Out]

ln(a*x^n)*(ln(a*x^n)^m)^p/n/(m*p+1)

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Rubi [A]  time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {15, 30} \[ \frac {\log \left (a x^n\right ) \log ^m\left (a x^n\right )^p}{n (m p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Log[a*x^n]*(Log[a*x^n]^m)^p)/(n*(1 + m*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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.00 \[ \frac {\log \left (a x^n\right ) \log ^m\left (a x^n\right )^p}{n (m p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Log[a*x^n]*(Log[a*x^n]^m)^p)/(n*(1 + m*p))

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fricas [A]  time = 0.79, size = 27, normalized size = 1.00 \[ \frac {{\left (n \log \relax (x) + \log \relax (a)\right )} {\left (n \log \relax (x) + \log \relax (a)\right )}^{m p}}{m n p + n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(a*x^n)^m)^p/x,x, algorithm="fricas")

[Out]

(n*log(x) + log(a))*(n*log(x) + log(a))^(m*p)/(m*n*p + n)

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giac [A]  time = 0.18, size = 24, normalized size = 0.89 \[ \frac {{\left (n \log \relax (x) + \log \relax (a)\right )}^{m p + 1}}{{\left (m p + 1\right )} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(a*x^n)^m)^p/x,x, algorithm="giac")

[Out]

(n*log(x) + log(a))^(m*p + 1)/((m*p + 1)*n)

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maple [C]  time = 2.00, size = 71, normalized size = 2.63 \[ \frac {\left (-\frac {i \pi \left (\mathrm {csgn}\left (i a \right )-\mathrm {csgn}\left (i a \,x^{n}\right )\right ) \left (\mathrm {csgn}\left (i x^{n}\right )-\mathrm {csgn}\left (i a \,x^{n}\right )\right ) \mathrm {csgn}\left (i a \,x^{n}\right )}{2}+\ln \relax (a )+\ln \left (x^{n}\right )\right )^{m p +1}}{\left (m p +1\right ) n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(a*x^n)^m)^p/x,x)

[Out]

1/n*(ln(a)+ln(x^n)-1/2*I*Pi*csgn(I*a*x^n)*(-csgn(I*a*x^n)+csgn(I*a))*(-csgn(I*a*x^n)+csgn(I*x^n)))^(m*p+1)/(m*
p+1)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(a*x^n)^m)^p/x,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

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mupad [B]  time = 0.38, size = 27, normalized size = 1.00 \[ \frac {\ln \left (a\,x^n\right )\,{\left ({\ln \left (a\,x^n\right )}^m\right )}^p}{n\,\left (m\,p+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(a*x^n)^m)^p/x,x)

[Out]

(log(a*x^n)*(log(a*x^n)^m)^p)/(n*(m*p + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(a*x**n)**m)**p/x,x)

[Out]

Integral((log(a*x**n)**m)**p/x, x)

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