3.149 \(\int \frac {\log ^2(a x^n)^p}{x} \, dx\)

Optimal. Leaf size=27 \[ \frac {\log \left (a x^n\right ) \log ^2\left (a x^n\right )^p}{n (2 p+1)} \]

[Out]

ln(a*x^n)*(ln(a*x^n)^2)^p/n/(1+2*p)

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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} \[ \frac {\log \left (a x^n\right ) \log ^2\left (a x^n\right )^p}{n (2 p+1)} \]

Antiderivative was successfully verified.

[In]

Int[(Log[a*x^n]^2)^p/x,x]

[Out]

(Log[a*x^n]*(Log[a*x^n]^2)^p)/(n*(1 + 2*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 ^2\left (a x^n\right )^p}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \left (x^2\right )^p \, dx,x,\log \left (a x^n\right )\right )}{n}\\ &=\frac {\left (\log ^{-2 p}\left (a x^n\right ) \log ^2\left (a x^n\right )^p\right ) \operatorname {Subst}\left (\int x^{2 p} \, dx,x,\log \left (a x^n\right )\right )}{n}\\ &=\frac {\log \left (a x^n\right ) \log ^2\left (a x^n\right )^p}{n (1+2 p)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.00 \[ \frac {\log \left (a x^n\right ) \log ^2\left (a x^n\right )^p}{n (2 p+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(Log[a*x^n]^2)^p/x,x]

[Out]

(Log[a*x^n]*(Log[a*x^n]^2)^p)/(n*(1 + 2*p))

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fricas [A]  time = 0.92, size = 38, normalized size = 1.41 \[ \frac {{\left (n \log \relax (x) + \log \relax (a)\right )} {\left (n^{2} \log \relax (x)^{2} + 2 \, n \log \relax (a) \log \relax (x) + \log \relax (a)^{2}\right )}^{p}}{2 \, n p + n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(a*x^n)^2)^p/x,x, algorithm="fricas")

[Out]

(n*log(x) + log(a))*(n^2*log(x)^2 + 2*n*log(a)*log(x) + log(a)^2)^p/(2*n*p + n)

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giac [B]  time = 0.36, size = 68, normalized size = 2.52 \[ \frac {{\left (n \log \relax (x) \mathrm {sgn}\left (\log \left (a x^{n}\right )\right ) + \log \relax (a) \mathrm {sgn}\left (\log \left (a x^{n}\right )\right )\right )} {\left (n \log \relax (x) \mathrm {sgn}\left (\log \left (a x^{n}\right )\right ) + \log \relax (a) \mathrm {sgn}\left (\log \left (a x^{n}\right )\right )\right )}^{2 \, p}}{n {\left (2 \, p + 1\right )} \mathrm {sgn}\left (\log \left (a x^{n}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(a*x^n)^2)^p/x,x, algorithm="giac")

[Out]

(n*log(x)*sgn(log(a*x^n)) + log(a)*sgn(log(a*x^n)))*(n*log(x)*sgn(log(a*x^n)) + log(a)*sgn(log(a*x^n)))^(2*p)/
(n*(2*p + 1)*sgn(log(a*x^n)))

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maple [F]  time = 3.56, size = 0, normalized size = 0.00 \[ \int \frac {\left (\ln \left (a \,x^{n}\right )^{2}\right )^{p}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(a*x^n)^2)^p/x,x)

[Out]

int((ln(a*x^n)^2)^p/x,x)

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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)^2)^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 = 0.40, size = 27, normalized size = 1.00 \[ \frac {\ln \left (a\,x^n\right )\,{\left ({\ln \left (a\,x^n\right )}^2\right )}^p}{n\,\left (2\,p+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(a*x^n)^2)^p/x,x)

[Out]

(log(a*x^n)*(log(a*x^n)^2)^p)/(n*(2*p + 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\log {\left (a x^{n} \right )}^{2}\right )^{p}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(a*x**n)**2)**p/x,x)

[Out]

Integral((log(a*x**n)**2)**p/x, x)

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