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

Optimal. Leaf size=36 \[ \frac{b \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{d}+\frac{\log \left (\frac{d x}{e}+1\right ) (a+b \log (c x))}{d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0707699, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2333, 2317, 2391} \[ \frac{b \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{d}+\frac{\log \left (\frac{d x}{e}+1\right ) (a+b \log (c x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x])/((d + e/x)*x),x]

[Out]

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

Mathematica [A]  time = 0.0067497, size = 34, normalized size = 0.94 \[ \frac{b \text{PolyLog}\left (2,-\frac{d x}{e}\right )+\log \left (\frac{d x}{e}+1\right ) (a+b \log (c x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 62, normalized size = 1.7 \begin{align*}{\frac{a\ln \left ( cdx+ce \right ) }{d}}+{\frac{b}{d}{\it dilog} \left ({\frac{cdx+ce}{ce}} \right ) }+{\frac{b\ln \left ( cx \right ) }{d}\ln \left ({\frac{cdx+ce}{ce}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*ln(c*d*x+c*e)/d+b*dilog((c*d*x+c*e)/c/e)/d+b*ln(c*x)*ln((c*d*x+c*e)/c/e)/d

________________________________________________________________________________________

Maxima [A]  time = 1.35111, size = 58, normalized size = 1.61 \begin{align*} \frac{{\left (\log \left (\frac{d x}{e} + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-\frac{d x}{e}\right )\right )} b}{d} + \frac{{\left (b \log \left (c\right ) + a\right )} \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))/(d+e/x)/x,x, algorithm="maxima")

[Out]

(log(d*x/e + 1)*log(x) + dilog(-d*x/e))*b/d + (b*log(c) + a)*log(d*x + e)/d

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \log \left (c x\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))/(d+e/x)/x,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left (c x\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))/(d+e/x)/x,x, algorithm="giac")

[Out]

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