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

Optimal. Leaf size=217 \[ \frac{B g n \text{PolyLog}\left (2,-\frac{b (f x+g)}{a f-b g}\right )}{f^2}-\frac{B g n \text{PolyLog}\left (2,-\frac{d (f x+g)}{c f-d g}\right )}{f^2}-\frac{g \log (f x+g) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{f^2}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f}-\frac{B n (b c-a d) \log (c+d x)}{b d f}+\frac{B g n \log (f x+g) \log \left (\frac{f (a+b x)}{a f-b g}\right )}{f^2}+\frac{A x}{f}-\frac{B g n \log (f x+g) \log \left (\frac{f (c+d x)}{c f-d g}\right )}{f^2} \]

[Out]

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

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Rubi [A]  time = 0.33225, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2528, 2486, 31, 2524, 2418, 2394, 2393, 2391} \[ \frac{B g n \text{PolyLog}\left (2,-\frac{b (f x+g)}{a f-b g}\right )}{f^2}-\frac{B g n \text{PolyLog}\left (2,-\frac{d (f x+g)}{c f-d g}\right )}{f^2}-\frac{g \log (f x+g) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{f^2}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f}-\frac{B n (b c-a d) \log (c+d x)}{b d f}+\frac{B g n \log (f x+g) \log \left (\frac{f (a+b x)}{a f-b g}\right )}{f^2}+\frac{A x}{f}-\frac{B g n \log (f x+g) \log \left (\frac{f (c+d x)}{c f-d g}\right )}{f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 2486

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[((
a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + Dist[(q*r*s*(b*c - a*d))/b, Int[Log[e*(f*(a + b*x)^p*
(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] &&
EqQ[p + q, 0] && IGtQ[s, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2524

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

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

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 (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{f+\frac{g}{x}} \, dx &=\int \left (\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{f}-\frac{g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f (g+f x)}\right ) \, dx\\ &=\frac{\int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{f}-\frac{g \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{g+f x} \, dx}{f}\\ &=\frac{A x}{f}-\frac{g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^2}+\frac{B \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{f}+\frac{(B g n) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (g+f x)}{a+b x} \, dx}{f^2}\\ &=\frac{A x}{f}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f}-\frac{g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^2}-\frac{(B (b c-a d) n) \int \frac{1}{c+d x} \, dx}{b f}+\frac{(B g n) \int \left (\frac{b \log (g+f x)}{a+b x}-\frac{d \log (g+f x)}{c+d x}\right ) \, dx}{f^2}\\ &=\frac{A x}{f}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f}-\frac{B (b c-a d) n \log (c+d x)}{b d f}-\frac{g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^2}+\frac{(b B g n) \int \frac{\log (g+f x)}{a+b x} \, dx}{f^2}-\frac{(B d g n) \int \frac{\log (g+f x)}{c+d x} \, dx}{f^2}\\ &=\frac{A x}{f}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f}-\frac{B (b c-a d) n \log (c+d x)}{b d f}+\frac{B g n \log \left (\frac{f (a+b x)}{a f-b g}\right ) \log (g+f x)}{f^2}-\frac{g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^2}-\frac{B g n \log \left (\frac{f (c+d x)}{c f-d g}\right ) \log (g+f x)}{f^2}-\frac{(B g n) \int \frac{\log \left (\frac{f (a+b x)}{a f-b g}\right )}{g+f x} \, dx}{f}+\frac{(B g n) \int \frac{\log \left (\frac{f (c+d x)}{c f-d g}\right )}{g+f x} \, dx}{f}\\ &=\frac{A x}{f}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f}-\frac{B (b c-a d) n \log (c+d x)}{b d f}+\frac{B g n \log \left (\frac{f (a+b x)}{a f-b g}\right ) \log (g+f x)}{f^2}-\frac{g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^2}-\frac{B g n \log \left (\frac{f (c+d x)}{c f-d g}\right ) \log (g+f x)}{f^2}-\frac{(B g n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{a f-b g}\right )}{x} \, dx,x,g+f x\right )}{f^2}+\frac{(B g n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{c f-d g}\right )}{x} \, dx,x,g+f x\right )}{f^2}\\ &=\frac{A x}{f}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f}-\frac{B (b c-a d) n \log (c+d x)}{b d f}+\frac{B g n \log \left (\frac{f (a+b x)}{a f-b g}\right ) \log (g+f x)}{f^2}-\frac{g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^2}-\frac{B g n \log \left (\frac{f (c+d x)}{c f-d g}\right ) \log (g+f x)}{f^2}+\frac{B g n \text{Li}_2\left (-\frac{b (g+f x)}{a f-b g}\right )}{f^2}-\frac{B g n \text{Li}_2\left (-\frac{d (g+f x)}{c f-d g}\right )}{f^2}\\ \end{align*}

Mathematica [A]  time = 0.148163, size = 185, normalized size = 0.85 \[ \frac{B g n \left (\text{PolyLog}\left (2,\frac{b (f x+g)}{b g-a f}\right )-\text{PolyLog}\left (2,\frac{d (f x+g)}{d g-c f}\right )+\log (f x+g) \left (\log \left (\frac{f (a+b x)}{a f-b g}\right )-\log \left (\frac{f (c+d x)}{c f-d g}\right )\right )\right )-g \log (f x+g) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+\frac{B f (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}-\frac{B f n (b c-a d) \log (c+d x)}{b d}+A f x}{f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.199, size = 0, normalized size = 0. \begin{align*} \int{ \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) \left ( f+{\frac{g}{x}} \right ) ^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} A{\left (\frac{x}{f} - \frac{g \log \left (f x + g\right )}{f^{2}}\right )} - B \int -\frac{x \log \left ({\left (b x + a\right )}^{n}\right ) - x \log \left ({\left (d x + c\right )}^{n}\right ) + x \log \left (e\right )}{f x + g}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*(x/f - g*log(f*x + g)/f^2) - B*integrate(-(x*log((b*x + a)^n) - x*log((d*x + c)^n) + x*log(e))/(f*x + g), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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