3.172 \(\int \frac{(a+b \log (c x^n))^p}{x^2} \, dx\)

Optimal. Leaf size=78 \[ -\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,\frac{a+b \log \left (c x^n\right )}{b n}\right )}{x} \]

[Out]

-((E^(a/(b*n))*(c*x^n)^n^(-1)*Gamma[1 + p, (a + b*Log[c*x^n])/(b*n)]*(a + b*Log[c*x^n])^p)/(x*((a + b*Log[c*x^
n])/(b*n))^p))

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Rubi [A]  time = 0.0544756, antiderivative size = 78, normalized size of antiderivative = 1., 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 = {2310, 2181} \[ -\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,\frac{a+b \log \left (c x^n\right )}{b n}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])^p/x^2,x]

[Out]

-((E^(a/(b*n))*(c*x^n)^n^(-1)*Gamma[1 + p, (a + b*Log[c*x^n])/(b*n)]*(a + b*Log[c*x^n])^p)/(x*((a + b*Log[c*x^
n])/(b*n))^p))

Rule 2310

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^p}{x^2} \, dx &=\frac{\left (c x^n\right )^{\frac{1}{n}} \operatorname{Subst}\left (\int e^{-\frac{x}{n}} (a+b x)^p \, dx,x,\log \left (c x^n\right )\right )}{n x}\\ &=-\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \Gamma \left (1+p,\frac{a+b \log \left (c x^n\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p}}{x}\\ \end{align*}

Mathematica [A]  time = 0.0729687, size = 78, normalized size = 1. \[ -\frac{e^{\frac{a}{b n}} \left (c x^n\right )^{\frac{1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,\frac{a+b \log \left (c x^n\right )}{b n}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])^p/x^2,x]

[Out]

-((E^(a/(b*n))*(c*x^n)^n^(-1)*Gamma[1 + p, (a + b*Log[c*x^n])/(b*n)]*(a + b*Log[c*x^n])^p)/(x*((a + b*Log[c*x^
n])/(b*n))^p))

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Maple [F]  time = 0.36, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{p}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a+b*ln(c*x^n))^p/x^2,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((a+b*log(c*x^n))^p/x^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b \log \left (c x^{n}\right ) + a\right )}^{p}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*log(c*x^n) + a)^p/x^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \log{\left (c x^{n} \right )}\right )^{p}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*log(c*x**n))**p/x**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )}^{p}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*log(c*x^n) + a)^p/x^2, x)