\(\int \frac {\log (\frac {a}{a+b x}) \log ^2(\frac {c x}{a+b x})}{x (a+b x)} \, dx\) [90]

Optimal result

Integrand size = 34, antiderivative size = 82 \[ \int \frac {\log \left (\frac {a}{a+b x}\right ) \log ^2\left (\frac {c x}{a+b x}\right )}{x (a+b x)} \, dx=-\frac {\log ^2\left (\frac {c x}{a+b x}\right ) \operatorname {PolyLog}\left (2,1-\frac {a}{a+b x}\right )}{a}+\frac {2 \log \left (\frac {c x}{a+b x}\right ) \operatorname {PolyLog}\left (3,1-\frac {a}{a+b x}\right )}{a}-\frac {2 \operatorname {PolyLog}\left (4,1-\frac {a}{a+b x}\right )}{a} \]

[Out]

-ln(c*x/(b*x+a))^2*polylog(2,1-a/(b*x+a))/a+2*ln(c*x/(b*x+a))*polylog(3,1-a/(b*x+a))/a-2*polylog(4,1-a/(b*x+a)
)/a

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {2588, 2590, 6745} \[ \int \frac {\log \left (\frac {a}{a+b x}\right ) \log ^2\left (\frac {c x}{a+b x}\right )}{x (a+b x)} \, dx=-\frac {\operatorname {PolyLog}\left (2,1-\frac {a}{a+b x}\right ) \log ^2\left (\frac {c x}{a+b x}\right )}{a}+\frac {2 \operatorname {PolyLog}\left (3,1-\frac {a}{a+b x}\right ) \log \left (\frac {c x}{a+b x}\right )}{a}-\frac {2 \operatorname {PolyLog}\left (4,1-\frac {a}{a+b x}\right )}{a} \]

[In]

Int[(Log[a/(a + b*x)]*Log[(c*x)/(a + b*x)]^2)/(x*(a + b*x)),x]

[Out]

-((Log[(c*x)/(a + b*x)]^2*PolyLog[2, 1 - a/(a + b*x)])/a) + (2*Log[(c*x)/(a + b*x)]*PolyLog[3, 1 - a/(a + b*x)
])/a - (2*PolyLog[4, 1 - a/(a + b*x)])/a

Rule 2588

Int[Log[v_]*Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_), x_Symbo
l] :> With[{g = Simplify[(v - 1)*((c + d*x)/(a + b*x))], h = Simplify[u*(a + b*x)*(c + d*x)]}, Simp[(-h)*PolyL
og[2, 1 - v]*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(b*c - a*d)), x] + Dist[h*p*r*s, Int[PolyLog[2, 1 - v]*(L
og[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/((a + b*x)*(c + d*x))), x], x] /; FreeQ[{g, h}, x]] /; FreeQ[{a, b
, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0]

Rule 2590

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_)*PolyLog[n_, v_],
x_Symbol] :> With[{g = Simplify[v*((c + d*x)/(a + b*x))], h = Simplify[u*(a + b*x)*(c + d*x)]}, Simp[h*PolyLo
g[n + 1, v]*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(b*c - a*d)), x] - Dist[h*p*r*s, Int[PolyLog[n + 1, v]*(Lo
g[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/((a + b*x)*(c + d*x))), x], x] /; FreeQ[{g, h}, x]] /; FreeQ[{a, b,
c, d, e, f, n, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0]

Rule 6745

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {\log ^2\left (\frac {c x}{a+b x}\right ) \text {Li}_2\left (1-\frac {a}{a+b x}\right )}{a}+2 \int \frac {\log \left (\frac {c x}{a+b x}\right ) \text {Li}_2\left (1-\frac {a}{a+b x}\right )}{x (a+b x)} \, dx \\ & = -\frac {\log ^2\left (\frac {c x}{a+b x}\right ) \text {Li}_2\left (1-\frac {a}{a+b x}\right )}{a}+\frac {2 \log \left (\frac {c x}{a+b x}\right ) \text {Li}_3\left (1-\frac {a}{a+b x}\right )}{a}-2 \int \frac {\text {Li}_3\left (1-\frac {a}{a+b x}\right )}{x (a+b x)} \, dx \\ & = -\frac {\log ^2\left (\frac {c x}{a+b x}\right ) \text {Li}_2\left (1-\frac {a}{a+b x}\right )}{a}+\frac {2 \log \left (\frac {c x}{a+b x}\right ) \text {Li}_3\left (1-\frac {a}{a+b x}\right )}{a}-\frac {2 \text {Li}_4\left (1-\frac {a}{a+b x}\right )}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.93 \[ \int \frac {\log \left (\frac {a}{a+b x}\right ) \log ^2\left (\frac {c x}{a+b x}\right )}{x (a+b x)} \, dx=-\frac {\log ^2\left (\frac {c x}{a+b x}\right ) \operatorname {PolyLog}\left (2,\frac {b x}{a+b x}\right )}{a}+\frac {2 \log \left (\frac {c x}{a+b x}\right ) \operatorname {PolyLog}\left (3,\frac {b x}{a+b x}\right )}{a}-\frac {2 \operatorname {PolyLog}\left (4,\frac {b x}{a+b x}\right )}{a} \]

[In]

Integrate[(Log[a/(a + b*x)]*Log[(c*x)/(a + b*x)]^2)/(x*(a + b*x)),x]

[Out]

-((Log[(c*x)/(a + b*x)]^2*PolyLog[2, (b*x)/(a + b*x)])/a) + (2*Log[(c*x)/(a + b*x)]*PolyLog[3, (b*x)/(a + b*x)
])/a - (2*PolyLog[4, (b*x)/(a + b*x)])/a

Maple [F]

\[\int \frac {\ln \left (\frac {a}{b x +a}\right ) \ln \left (\frac {c x}{b x +a}\right )^{2}}{x \left (b x +a \right )}d x\]

[In]

int(ln(a/(b*x+a))*ln(c*x/(b*x+a))^2/x/(b*x+a),x)

[Out]

int(ln(a/(b*x+a))*ln(c*x/(b*x+a))^2/x/(b*x+a),x)

Fricas [F]

\[ \int \frac {\log \left (\frac {a}{a+b x}\right ) \log ^2\left (\frac {c x}{a+b x}\right )}{x (a+b x)} \, dx=\int { \frac {\log \left (\frac {c x}{b x + a}\right )^{2} \log \left (\frac {a}{b x + a}\right )}{{\left (b x + a\right )} x} \,d x } \]

[In]

integrate(log(a/(b*x+a))*log(c*x/(b*x+a))^2/x/(b*x+a),x, algorithm="fricas")

[Out]

integral(log(c*x/(b*x + a))^2*log(a/(b*x + a))/(b*x^2 + a*x), x)

Sympy [F]

\[ \int \frac {\log \left (\frac {a}{a+b x}\right ) \log ^2\left (\frac {c x}{a+b x}\right )}{x (a+b x)} \, dx=\int \frac {\log {\left (\frac {a}{a + b x} \right )} \log {\left (\frac {c x}{a + b x} \right )}^{2}}{x \left (a + b x\right )}\, dx \]

[In]

integrate(ln(a/(b*x+a))*ln(c*x/(b*x+a))**2/x/(b*x+a),x)

[Out]

Integral(log(a/(a + b*x))*log(c*x/(a + b*x))**2/(x*(a + b*x)), x)

Maxima [F]

\[ \int \frac {\log \left (\frac {a}{a+b x}\right ) \log ^2\left (\frac {c x}{a+b x}\right )}{x (a+b x)} \, dx=\int { \frac {\log \left (\frac {c x}{b x + a}\right )^{2} \log \left (\frac {a}{b x + a}\right )}{{\left (b x + a\right )} x} \,d x } \]

[In]

integrate(log(a/(b*x+a))*log(c*x/(b*x+a))^2/x/(b*x+a),x, algorithm="maxima")

[Out]

1/4*(log(b*x + a)^4 - 4*log(b*x + a)^3*log(x))/a + integrate((a*log(a)*log(c)^2 + 2*a*log(a)*log(c)*log(x) + a
*log(a)*log(x)^2 + (a*(log(a) + 2*log(c)) + (3*b*x + 2*a)*log(x))*log(b*x + a)^2 - (2*a*(log(a) + log(c))*log(
x) + a*log(x)^2 + (2*log(a)*log(c) + log(c)^2)*a)*log(b*x + a))/(a*b*x^2 + a^2*x), x)

Giac [F]

\[ \int \frac {\log \left (\frac {a}{a+b x}\right ) \log ^2\left (\frac {c x}{a+b x}\right )}{x (a+b x)} \, dx=\int { \frac {\log \left (\frac {c x}{b x + a}\right )^{2} \log \left (\frac {a}{b x + a}\right )}{{\left (b x + a\right )} x} \,d x } \]

[In]

integrate(log(a/(b*x+a))*log(c*x/(b*x+a))^2/x/(b*x+a),x, algorithm="giac")

[Out]

integrate(log(c*x/(b*x + a))^2*log(a/(b*x + a))/((b*x + a)*x), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (\frac {a}{a+b x}\right ) \log ^2\left (\frac {c x}{a+b x}\right )}{x (a+b x)} \, dx=\int \frac {{\ln \left (\frac {c\,x}{a+b\,x}\right )}^2\,\ln \left (\frac {a}{a+b\,x}\right )}{x\,\left (a+b\,x\right )} \,d x \]

[In]

int((log((c*x)/(a + b*x))^2*log(a/(a + b*x)))/(x*(a + b*x)),x)

[Out]

int((log((c*x)/(a + b*x))^2*log(a/(a + b*x)))/(x*(a + b*x)), x)