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

Optimal. Leaf size=29 \[ \frac{\log \left (a x^n\right ) \left (b \log ^m\left (a x^n\right )\right )^p}{n (m p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0376942, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {15, 30} \[ \frac{\log \left (a x^n\right ) \left (b \log ^m\left (a x^n\right )\right )^p}{n (m p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.006181, size = 29, normalized size = 1. \[ \frac{\log \left (a x^n\right ) \left (b \log ^m\left (a x^n\right )\right )^p}{n (m p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.102, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( b \left ( \ln \left ( a{x}^{n} \right ) \right ) ^{m} \right ) ^{p}}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.09417, size = 101, normalized size = 3.48 \begin{align*} \frac{{\left (n \log \left (x\right ) + \log \left (a\right )\right )} e^{\left (m p \log \left (n \log \left (x\right ) + \log \left (a\right )\right ) + p \log \left (b\right )\right )}}{m n p + n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b \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((b*ln(a*x**n)**m)**p/x,x)

[Out]

Integral((b*log(a*x**n)**m)**p/x, x)

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Giac [A]  time = 1.46244, size = 47, normalized size = 1.62 \begin{align*} \frac{{\left (n \log \left (x\right ) + \log \left (a\right )\right )} e^{\left (m p \log \left (n \log \left (x\right ) + \log \left (a\right )\right ) + p \log \left (b\right )\right )}}{{\left (m p + 1\right )} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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