### 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]

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

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Rubi [A]  time = 0.0347011, antiderivative size = 27, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0059178, size = 27, normalized size = 1. $\frac{\log \left (a x^n\right ) \log ^m\left (a x^n\right )^p}{n (m p+1)}$

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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Maple [C]  time = 0.245, size = 71, normalized size = 2.6 \begin{align*}{\frac{ \left ( \ln \left ( a \right ) +\ln \left ({x}^{n} \right ) -{\frac{i}{2}}\pi \,{\it csgn} \left ( ia{x}^{n} \right ) \left ( -{\it csgn} \left ( ia{x}^{n} \right ) +{\it csgn} \left ( ia \right ) \right ) \left ( -{\it csgn} \left ( ia{x}^{n} \right ) +{\it csgn} \left ( i{x}^{n} \right ) \right ) \right ) ^{mp+1}}{n \left ( mp+1 \right ) }} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

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

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Fricas [A]  time = 2.17485, size = 80, normalized size = 2.96 \begin{align*} \frac{{\left (n \log \left (x\right ) + \log \left (a\right )\right )}{\left (n \log \left (x\right ) + \log \left (a\right )\right )}^{m p}}{m n p + n} \end{align*}

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

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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Giac [A]  time = 1.36615, size = 32, normalized size = 1.19 \begin{align*} \frac{{\left (n \log \left (x\right ) + \log \left (a\right )\right )}^{m p + 1}}{{\left (m p + 1\right )} n} \end{align*}

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)