3.1.98 \(\int \log ^3(\frac {c (b+a x)^2}{x^2}) \, dx\) [98]

3.1.98.1 Optimal result

Integrand size = 15, antiderivative size = 102 \[ \int \log ^3\left (\frac {c (b+a x)^2}{x^2}\right ) \, dx=x \log ^3\left (\frac {c (b+a x)^2}{x^2}\right )-\frac {6 b \log ^2\left (\frac {c (b+a x)^2}{x^2}\right ) \log \left (1-\frac {a x}{b+a x}\right )}{a}+\frac {24 b \log \left (\frac {c (b+a x)^2}{x^2}\right ) \operatorname {PolyLog}\left (2,\frac {a x}{b+a x}\right )}{a}+\frac {48 b \operatorname {PolyLog}\left (3,\frac {a x}{b+a x}\right )}{a} \]

output

x*ln(c*(a*x+b)^2/x^2)^3-6*b*ln(c*(a*x+b)^2/x^2)^2*ln(1-a*x/(a*x+b))/a+24*b 
*ln(c*(a*x+b)^2/x^2)*polylog(2,a*x/(a*x+b))/a+48*b*polylog(3,a*x/(a*x+b))/ 
a
 

3.1.98.2 Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.96 \[ \int \log ^3\left (\frac {c (b+a x)^2}{x^2}\right ) \, dx=-\frac {6 b \log \left (\frac {b}{b+a x}\right ) \log ^2\left (\frac {c (b+a x)^2}{x^2}\right )}{a}+x \log ^3\left (\frac {c (b+a x)^2}{x^2}\right )+\frac {24 b \log \left (\frac {c (b+a x)^2}{x^2}\right ) \operatorname {PolyLog}\left (2,\frac {a x}{b+a x}\right )}{a}+\frac {48 b \operatorname {PolyLog}\left (3,\frac {a x}{b+a x}\right )}{a} \]

input

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

output

(-6*b*Log[b/(b + a*x)]*Log[(c*(b + a*x)^2)/x^2]^2)/a + x*Log[(c*(b + a*x)^ 
2)/x^2]^3 + (24*b*Log[(c*(b + a*x)^2)/x^2]*PolyLog[2, (a*x)/(b + a*x)])/a 
+ (48*b*PolyLog[3, (a*x)/(b + a*x)])/a
 

3.1.98.3 Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2936, 2950, 2779, 2821, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2936

\(\displaystyle 6 b \int \frac {\log ^2\left (\frac {c (b+a x)^2}{x^2}\right )}{b+a x}dx+x \log ^3\left (\frac {c (a x+b)^2}{x^2}\right )\)

\(\Big \downarrow \) 2950

\(\displaystyle 6 b \int \frac {x \log ^2\left (\frac {c (b+a x)^2}{x^2}\right )}{(b+a x) \left (a-\frac {b+a x}{x}\right )}d\frac {b+a x}{x}+x \log ^3\left (\frac {c (a x+b)^2}{x^2}\right )\)

\(\Big \downarrow \) 2779

\(\displaystyle 6 b \left (\frac {4 \int \frac {x \log \left (\frac {c (b+a x)^2}{x^2}\right ) \log \left (1-\frac {a x}{b+a x}\right )}{b+a x}d\frac {b+a x}{x}}{a}-\frac {\log \left (1-\frac {a x}{a x+b}\right ) \log ^2\left (\frac {c (a x+b)^2}{x^2}\right )}{a}\right )+x \log ^3\left (\frac {c (a x+b)^2}{x^2}\right )\)

\(\Big \downarrow \) 2821

\(\displaystyle 6 b \left (\frac {4 \left (\operatorname {PolyLog}\left (2,\frac {a x}{b+a x}\right ) \log \left (\frac {c (a x+b)^2}{x^2}\right )-2 \int \frac {x \operatorname {PolyLog}\left (2,\frac {a x}{b+a x}\right )}{b+a x}d\frac {b+a x}{x}\right )}{a}-\frac {\log \left (1-\frac {a x}{a x+b}\right ) \log ^2\left (\frac {c (a x+b)^2}{x^2}\right )}{a}\right )+x \log ^3\left (\frac {c (a x+b)^2}{x^2}\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle 6 b \left (\frac {4 \left (\operatorname {PolyLog}\left (2,\frac {a x}{b+a x}\right ) \log \left (\frac {c (a x+b)^2}{x^2}\right )+2 \operatorname {PolyLog}\left (3,\frac {a x}{b+a x}\right )\right )}{a}-\frac {\log \left (1-\frac {a x}{a x+b}\right ) \log ^2\left (\frac {c (a x+b)^2}{x^2}\right )}{a}\right )+x \log ^3\left (\frac {c (a x+b)^2}{x^2}\right )\)

input

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

output

x*Log[(c*(b + a*x)^2)/x^2]^3 + 6*b*(-((Log[(c*(b + a*x)^2)/x^2]^2*Log[1 - 
(a*x)/(b + a*x)])/a) + (4*(Log[(c*(b + a*x)^2)/x^2]*PolyLog[2, (a*x)/(b + 
a*x)] + 2*PolyLog[3, (a*x)/(b + a*x)]))/a)
 

3.1.98.3.1 Defintions of rubi rules used

rule 2779

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

rule 2821

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] + Simp[b*n*(p/m)   Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[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 2936

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))^(p_.), x_Symbol] :> Simp[(a + b*x)*((A + B*Log[e*((a + b*x)^n/(c 
+ d*x)^n)])^p/b), x] - Simp[B*n*p*((b*c - a*d)/b)   Int[(A + B*Log[e*((a + 
b*x)^n/(c + d*x)^n)])^(p - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, A, 
 B, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0] && IGtQ[p, 0]
 

rule 2950

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)^( 
m + 1)*(g/b)^m   Subst[Int[x^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + 2)), x] 
, x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] & 
& EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] && E 
qQ[b*f - a*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> 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]
 

3.1.98.4 Maple [F]

\[\int \ln \left (\frac {c \left (a x +b \right )^{2}}{x^{2}}\right )^{3}d x\]

input

int(ln(c*(a*x+b)^2/x^2)^3,x)
 

output

int(ln(c*(a*x+b)^2/x^2)^3,x)
 

3.1.98.5 Fricas [F]

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

input

integrate(log(c*(a*x+b)^2/x^2)^3,x, algorithm="fricas")
 

output

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

3.1.98.6 Sympy [F]

\[ \int \log ^3\left (\frac {c (b+a x)^2}{x^2}\right ) \, dx=6 b \int \frac {\log {\left (a^{2} c + \frac {2 a b c}{x} + \frac {b^{2} c}{x^{2}} \right )}^{2}}{a x + b}\, dx + x \log {\left (\frac {c \left (a x + b\right )^{2}}{x^{2}} \right )}^{3} \]

input

integrate(ln(c*(a*x+b)**2/x**2)**3,x)
 

output

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

3.1.98.7 Maxima [F]

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

input

integrate(log(c*(a*x+b)^2/x^2)^3,x, algorithm="maxima")
 

output

4*(2*(a*x + b)*log(a*x + b)^3 + 3*(a*x*log(c) - 2*a*x*log(x))*log(a*x + b) 
^2)/a + integrate((a*x*log(c)^3 + b*log(c)^3 - 8*(a*x + b)*log(x)^3 + 12*( 
a*x*log(c) + b*log(c))*log(x)^2 + 6*((log(c)^2 - 4*log(c))*a*x + b*log(c)^ 
2 + 4*(a*x + b)*log(x)^2 - 4*(a*x*(log(c) - 2) + b*log(c))*log(x))*log(a*x 
 + b) - 6*(a*x*log(c)^2 + b*log(c)^2)*log(x))/(a*x + b), x)
 

3.1.98.8 Giac [F]

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

input

integrate(log(c*(a*x+b)^2/x^2)^3,x, algorithm="giac")
 

output

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

3.1.98.9 Mupad [F(-1)]

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

input

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

output

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