\(\int \log ^3(\frac {c (b+a x)}{x}) \, dx\) [95]

Optimal result

Integrand size = 13, antiderivative size = 97 \[ \int \log ^3\left (\frac {c (b+a x)}{x}\right ) \, dx=\frac {(b+a x) \log ^3\left (a c+\frac {b c}{x}\right )}{a}-\frac {3 b \log ^2\left (c \left (a+\frac {b}{x}\right )\right ) \log \left (-\frac {b}{a x}\right )}{a}-\frac {6 b \log \left (c \left (a+\frac {b}{x}\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {b}{a x}\right )}{a}+\frac {6 b \operatorname {PolyLog}\left (3,1+\frac {b}{a x}\right )}{a} \]

[Out]

(a*x+b)*ln(a*c+b*c/x)^3/a-3*b*ln(c*(a+b/x))^2*ln(-b/a/x)/a-6*b*ln(c*(a+b/x))*polylog(2,1+b/a/x)/a+6*b*polylog(
3,1+b/a/x)/a

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2503, 2499, 2504, 2443, 2481, 2421, 6724} \[ \int \log ^3\left (\frac {c (b+a x)}{x}\right ) \, dx=-\frac {6 b \operatorname {PolyLog}\left (2,\frac {b}{a x}+1\right ) \log \left (c \left (a+\frac {b}{x}\right )\right )}{a}+\frac {(a x+b) \log ^3\left (a c+\frac {b c}{x}\right )}{a}-\frac {3 b \log \left (-\frac {b}{a x}\right ) \log ^2\left (c \left (a+\frac {b}{x}\right )\right )}{a}+\frac {6 b \operatorname {PolyLog}\left (3,\frac {b}{a x}+1\right )}{a} \]

[In]

Int[Log[(c*(b + a*x))/x]^3,x]

[Out]

((b + a*x)*Log[a*c + (b*c)/x]^3)/a - (3*b*Log[c*(a + b/x)]^2*Log[-(b/(a*x))])/a - (6*b*Log[c*(a + b/x)]*PolyLo
g[2, 1 + b/(a*x)])/a + (6*b*PolyLog[3, 1 + b/(a*x)])/a

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2443

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

Rule 2481

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 2499

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

Rule 2503

Int[((a_.) + Log[(c_.)*(v_)^(p_.)]*(b_.))^(q_.), x_Symbol] :> Int[(a + b*Log[c*ExpandToSum[v, x]^p])^q, x] /;
FreeQ[{a, b, c, p, q}, x] && BinomialQ[v, x] &&  !BinomialMatchQ[v, x]

Rule 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps \begin{align*} \text {integral}& = \int \log ^3\left (a c+\frac {b c}{x}\right ) \, dx \\ & = \frac {(b+a x) \log ^3\left (a c+\frac {b c}{x}\right )}{a}+\frac {(3 b) \int \frac {\log ^2\left (a c+\frac {b c}{x}\right )}{x} \, dx}{a} \\ & = \frac {(b+a x) \log ^3\left (a c+\frac {b c}{x}\right )}{a}-\frac {(3 b) \text {Subst}\left (\int \frac {\log ^2(a c+b c x)}{x} \, dx,x,\frac {1}{x}\right )}{a} \\ & = \frac {(b+a x) \log ^3\left (a c+\frac {b c}{x}\right )}{a}-\frac {3 b \log ^2\left (c \left (a+\frac {b}{x}\right )\right ) \log \left (-\frac {b}{a x}\right )}{a}+\frac {\left (6 b^2 c\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right ) \log (a c+b c x)}{a c+b c x} \, dx,x,\frac {1}{x}\right )}{a} \\ & = \frac {(b+a x) \log ^3\left (a c+\frac {b c}{x}\right )}{a}-\frac {3 b \log ^2\left (c \left (a+\frac {b}{x}\right )\right ) \log \left (-\frac {b}{a x}\right )}{a}+\frac {(6 b) \text {Subst}\left (\int \frac {\log (x) \log \left (-\frac {b \left (-\frac {a}{b}+\frac {x}{b c}\right )}{a}\right )}{x} \, dx,x,a c+\frac {b c}{x}\right )}{a} \\ & = \frac {(b+a x) \log ^3\left (a c+\frac {b c}{x}\right )}{a}-\frac {3 b \log ^2\left (c \left (a+\frac {b}{x}\right )\right ) \log \left (-\frac {b}{a x}\right )}{a}-\frac {6 b \log \left (c \left (a+\frac {b}{x}\right )\right ) \text {Li}_2\left (1+\frac {b}{a x}\right )}{a}+\frac {(6 b) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{a c}\right )}{x} \, dx,x,a c+\frac {b c}{x}\right )}{a} \\ & = \frac {(b+a x) \log ^3\left (a c+\frac {b c}{x}\right )}{a}-\frac {3 b \log ^2\left (c \left (a+\frac {b}{x}\right )\right ) \log \left (-\frac {b}{a x}\right )}{a}-\frac {6 b \log \left (c \left (a+\frac {b}{x}\right )\right ) \text {Li}_2\left (1+\frac {b}{a x}\right )}{a}+\frac {6 b \text {Li}_3\left (1+\frac {b}{a x}\right )}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.94 \[ \int \log ^3\left (\frac {c (b+a x)}{x}\right ) \, dx=\frac {\log ^2\left (\frac {c (b+a x)}{x}\right ) \left (-3 b \log \left (-\frac {b}{a x}\right )+(b+a x) \log \left (\frac {c (b+a x)}{x}\right )\right )-6 b \log \left (\frac {c (b+a x)}{x}\right ) \operatorname {PolyLog}\left (2,1+\frac {b}{a x}\right )+6 b \operatorname {PolyLog}\left (3,1+\frac {b}{a x}\right )}{a} \]

[In]

Integrate[Log[(c*(b + a*x))/x]^3,x]

[Out]

(Log[(c*(b + a*x))/x]^2*(-3*b*Log[-(b/(a*x))] + (b + a*x)*Log[(c*(b + a*x))/x]) - 6*b*Log[(c*(b + a*x))/x]*Pol
yLog[2, 1 + b/(a*x)] + 6*b*PolyLog[3, 1 + b/(a*x)])/a

Maple [F]

\[\int \ln \left (\frac {c \left (a x +b \right )}{x}\right )^{3}d x\]

[In]

int(ln(c*(a*x+b)/x)^3,x)

[Out]

int(ln(c*(a*x+b)/x)^3,x)

Fricas [F]

\[ \int \log ^3\left (\frac {c (b+a x)}{x}\right ) \, dx=\int { \log \left (\frac {{\left (a x + b\right )} c}{x}\right )^{3} \,d x } \]

[In]

integrate(log(c*(a*x+b)/x)^3,x, algorithm="fricas")

[Out]

integral(log((a*c*x + b*c)/x)^3, x)

Sympy [F]

\[ \int \log ^3\left (\frac {c (b+a x)}{x}\right ) \, dx=3 b \int \frac {\log {\left (a c + \frac {b c}{x} \right )}^{2}}{a x + b}\, dx + x \log {\left (\frac {c \left (a x + b\right )}{x} \right )}^{3} \]

[In]

integrate(ln(c*(a*x+b)/x)**3,x)

[Out]

3*b*Integral(log(a*c + b*c/x)**2/(a*x + b), x) + x*log(c*(a*x + b)/x)**3

Maxima [F]

\[ \int \log ^3\left (\frac {c (b+a x)}{x}\right ) \, dx=\int { \log \left (\frac {{\left (a x + b\right )} c}{x}\right )^{3} \,d x } \]

[In]

integrate(log(c*(a*x+b)/x)^3,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \log ^3\left (\frac {c (b+a x)}{x}\right ) \, dx=\int { \log \left (\frac {{\left (a x + b\right )} c}{x}\right )^{3} \,d x } \]

[In]

integrate(log(c*(a*x+b)/x)^3,x, algorithm="giac")

[Out]

integrate(log((a*x + b)*c/x)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \log ^3\left (\frac {c (b+a x)}{x}\right ) \, dx=\int {\ln \left (\frac {c\,\left (b+a\,x\right )}{x}\right )}^3 \,d x \]

[In]

int(log((c*(b + a*x))/x)^3,x)

[Out]

int(log((c*(b + a*x))/x)^3, x)