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

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

[Out]

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

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Rubi [A]  time = 0.0260456, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2317, 2391} \[ \frac{b n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e}+\frac{\log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/(d + e*x),x]

[Out]

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

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

Mathematica [A]  time = 0.0066954, size = 37, normalized size = 0.95 \[ \frac{b n \text{PolyLog}\left (2,-\frac{e x}{d}\right )+\log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.167, size = 195, normalized size = 5. \begin{align*}{\frac{b\ln \left ( ex+d \right ) \ln \left ({x}^{n} \right ) }{e}}-{\frac{bn\ln \left ( ex+d \right ) }{e}\ln \left ( -{\frac{ex}{d}} \right ) }-{\frac{bn}{e}{\it dilog} \left ( -{\frac{ex}{d}} \right ) }+{\frac{{\frac{i}{2}}\ln \left ( ex+d \right ) b\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}}{e}}-{\frac{{\frac{i}{2}}\ln \left ( ex+d \right ) b\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) }{e}}-{\frac{{\frac{i}{2}}\ln \left ( ex+d \right ) b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}}{e}}+{\frac{{\frac{i}{2}}\ln \left ( ex+d \right ) b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) }{e}}+{\frac{b\ln \left ( ex+d \right ) \ln \left ( c \right ) }{e}}+{\frac{a\ln \left ( ex+d \right ) }{e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )}{e x + d}\,{d x} + \frac{a \log \left (e x + d\right )}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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}{e x + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*log(c*x^n) + a)/(e*x + d), 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 + e x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*log(c*x**n))/(d + e*x), 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}{e x + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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