3.333 \(\int \frac{a+b \log (c x^n)}{(d+\frac{e}{x}) x} \, dx\)

Optimal. Leaf size=39 \[ \frac{b n \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{d}+\frac{\log \left (\frac{d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d} \]

[Out]

((a + b*Log[c*x^n])*Log[1 + (d*x)/e])/d + (b*n*PolyLog[2, -((d*x)/e)])/d

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Rubi [A]  time = 0.0750483, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2333, 2317, 2391} \[ \frac{b n \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{d}+\frac{\log \left (\frac{d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/((d + e/x)*x),x]

[Out]

((a + b*Log[c*x^n])*Log[1 + (d*x)/e])/d + (b*n*PolyLog[2, -((d*x)/e)])/d

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)/(x_))^(q_.)*(x_)^(m_.), x_Symbol] :> Int[(e + d*
x)^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[m, q] && IntegerQ[q]

Rule 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{\left (d+\frac{e}{x}\right ) x} \, dx &=\int \frac{a+b \log \left (c x^n\right )}{e+d x} \, dx\\ &=\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{d x}{e}\right )}{d}-\frac{(b n) \int \frac{\log \left (1+\frac{d x}{e}\right )}{x} \, dx}{d}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{d x}{e}\right )}{d}+\frac{b n \text{Li}_2\left (-\frac{d x}{e}\right )}{d}\\ \end{align*}

Mathematica [A]  time = 0.0070959, size = 37, normalized size = 0.95 \[ \frac{b n \text{PolyLog}\left (2,-\frac{d x}{e}\right )+\log \left (\frac{d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])/((d + e/x)*x),x]

[Out]

((a + b*Log[c*x^n])*Log[1 + (d*x)/e] + b*n*PolyLog[2, -((d*x)/e)])/d

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Maple [C]  time = 0.135, size = 195, normalized size = 5. \begin{align*}{\frac{b\ln \left ( dx+e \right ) \ln \left ({x}^{n} \right ) }{d}}-{\frac{bn\ln \left ( dx+e \right ) }{d}\ln \left ( -{\frac{dx}{e}} \right ) }-{\frac{bn}{d}{\it dilog} \left ( -{\frac{dx}{e}} \right ) }+{\frac{{\frac{i}{2}}\ln \left ( dx+e \right ) b\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}}{d}}-{\frac{{\frac{i}{2}}\ln \left ( dx+e \right ) b\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) }{d}}-{\frac{{\frac{i}{2}}\ln \left ( dx+e \right ) b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}}{d}}+{\frac{{\frac{i}{2}}\ln \left ( dx+e \right ) b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) }{d}}+{\frac{b\ln \left ( dx+e \right ) \ln \left ( c \right ) }{d}}+{\frac{a\ln \left ( dx+e \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))/(d+e/x)/x,x)

[Out]

b*ln(d*x+e)/d*ln(x^n)-b/d*n*ln(d*x+e)*ln(-d*x/e)-b/d*n*dilog(-d*x/e)+1/2*I*ln(d*x+e)/d*b*Pi*csgn(I*x^n)*csgn(I
*c*x^n)^2-1/2*I*ln(d*x+e)/d*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-1/2*I*ln(d*x+e)/d*b*Pi*csgn(I*c*x^n)^3+1/
2*I*ln(d*x+e)/d*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)+ln(d*x+e)/d*b*ln(c)+a*ln(d*x+e)/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/(d+e/x)/x,x, algorithm="maxima")

[Out]

b*integrate((log(c) + log(x^n))/(d*x + e), x) + a*log(d*x + e)/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*log(c*x^n) + a)/(d*x + e), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))/(d+e/x)/x,x)

[Out]

Integral((a + b*log(c*x**n))/(d*x + e), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*log(c*x^n) + a)/((d + e/x)*x), x)