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

Optimal. Leaf size=63 \[ \frac{b n \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right )}{g}+\frac{\log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g} \]

[Out]

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

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Rubi [A]  time = 0.0453695, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2394, 2393, 2391} \[ \frac{b n \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right )}{g}+\frac{\log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2394

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

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 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 (d+e x)^n\right )}{f+g x} \, dx &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g}-\frac{(b e n) \int \frac{\log \left (\frac{e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{g}\\ &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g}-\frac{(b n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g}\\ &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g}+\frac{b n \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{g}\\ \end{align*}

Mathematica [A]  time = 0.0065298, size = 62, normalized size = 0.98 \[ \frac{b n \text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right )}{g}+\frac{\log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.115, size = 261, normalized size = 4.1 \begin{align*}{\frac{b\ln \left ( gx+f \right ) \ln \left ( \left ( ex+d \right ) ^{n} \right ) }{g}}-{\frac{bn}{g}{\it dilog} \left ({\frac{ \left ( gx+f \right ) e+dg-fe}{dg-fe}} \right ) }-{\frac{bn\ln \left ( gx+f \right ) }{g}\ln \left ({\frac{ \left ( gx+f \right ) e+dg-fe}{dg-fe}} \right ) }-{\frac{{\frac{i}{2}}\ln \left ( gx+f \right ) b\pi \,{\it csgn} \left ( ic \right ){\it csgn} \left ( i \left ( ex+d \right ) ^{n} \right ){\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) }{g}}+{\frac{{\frac{i}{2}}\ln \left ( gx+f \right ) b\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}{g}}+{\frac{{\frac{i}{2}}\ln \left ( gx+f \right ) b\pi \,{\it csgn} \left ( i \left ( ex+d \right ) ^{n} \right ) \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}{g}}-{\frac{{\frac{i}{2}}\ln \left ( gx+f \right ) b\pi \, \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) \right ) ^{3}}{g}}+{\frac{b\ln \left ( gx+f \right ) \ln \left ( c \right ) }{g}}+{\frac{a\ln \left ( gx+f \right ) }{g}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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