3.2.1 \(\int \log ^3(\frac {c x^2}{(b+a x)^2}) \, dx\) [101]

3.2.1.1 Optimal result

Integrand size = 15, antiderivative size = 98 \[ \int \log ^3\left (\frac {c x^2}{(b+a x)^2}\right ) \, dx=x \log ^3\left (\frac {c x^2}{(b+a x)^2}\right )+\frac {6 b \log ^2\left (\frac {c x^2}{(b+a x)^2}\right ) \log \left (\frac {b}{b+a x}\right )}{a}+\frac {24 b \log \left (\frac {c x^2}{(b+a x)^2}\right ) \operatorname {PolyLog}\left (2,\frac {a x}{b+a x}\right )}{a}-\frac {48 b \operatorname {PolyLog}\left (3,\frac {a x}{b+a x}\right )}{a} \]

output

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

3.2.1.2 Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00 \[ \int \log ^3\left (\frac {c x^2}{(b+a x)^2}\right ) \, dx=x \log ^3\left (\frac {c x^2}{(b+a x)^2}\right )+\frac {6 b \log ^2\left (\frac {c x^2}{(b+a x)^2}\right ) \log \left (\frac {b}{b+a x}\right )}{a}+\frac {24 b \log \left (\frac {c x^2}{(b+a x)^2}\right ) \operatorname {PolyLog}\left (2,\frac {a x}{b+a x}\right )}{a}-\frac {48 b \operatorname {PolyLog}\left (3,\frac {a x}{b+a x}\right )}{a} \]

input

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

output

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

3.2.1.3 Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2936, 2952, 2754, 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.

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

\(\Big \downarrow \) 2936

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

\(\Big \downarrow \) 2952

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

\(\Big \downarrow \) 2754

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 7143

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

input

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

output

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

3.2.1.3.1 Defintions of rubi rules used

rule 2754

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb 
ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) 
  Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, 
b, c, d, e, n}, 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 2952

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/d)^m   Subst[Int[(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] && EqQ[d*f 
 - c*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]
 

3.2.1.4 Maple [F]

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

input

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

output

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

3.2.1.5 Fricas [F]

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

input

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

output

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

3.2.1.6 Sympy [F]

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

input

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

output

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

3.2.1.7 Maxima [F]

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

input

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

3.2.1.8 Giac [F]

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

input

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

output

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

3.2.1.9 Mupad [F(-1)]

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

input

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

output

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