3.4.74 \(\int \frac {\log (x) \log ^2(a+b x)}{x} \, dx\) [374]

3.4.74.1 Optimal result

Integrand size = 14, antiderivative size = 519 \[ \int \frac {\log (x) \log ^2(a+b x)}{x} \, dx=\frac {1}{12} \left (\log ^4\left (-\frac {b x}{a}\right )+6 \log ^2\left (-\frac {b x}{a}\right ) \log ^2\left (-\frac {b x}{a+b x}\right )-4 \left (\log \left (-\frac {b x}{a}\right )+\log \left (\frac {a}{a+b x}\right )\right ) \log ^3\left (-\frac {b x}{a+b x}\right )+\log ^4\left (-\frac {b x}{a+b x}\right )+6 \log ^2(x) \log ^2(a+b x)+4 \left (2 \log ^3\left (-\frac {b x}{a}\right )-3 \log ^2(x) \log (a+b x)\right ) \log \left (1+\frac {b x}{a}\right )+6 \left (\log (x)-\log \left (-\frac {b x}{a}\right )\right ) \left (\log (x)+3 \log \left (-\frac {b x}{a}\right )\right ) \log ^2\left (1+\frac {b x}{a}\right )-4 \log ^2\left (-\frac {b x}{a}\right ) \log \left (-\frac {b x}{a+b x}\right ) \left (\log \left (-\frac {b x}{a}\right )+3 \log \left (1+\frac {b x}{a}\right )\right )+12 \left (\log ^2\left (-\frac {b x}{a}\right )-2 \log \left (-\frac {b x}{a}\right ) \left (\log \left (-\frac {b x}{a+b x}\right )+\log \left (1+\frac {b x}{a}\right )\right )+2 \log (x) \left (-\log (a+b x)+\log \left (1+\frac {b x}{a}\right )\right )\right ) \operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )-12 \log ^2\left (-\frac {b x}{a+b x}\right ) \operatorname {PolyLog}\left (2,\frac {b x}{a+b x}\right )+12 \left (\log \left (-\frac {b x}{a}\right )-\log \left (-\frac {b x}{a+b x}\right )\right )^2 \operatorname {PolyLog}\left (2,1+\frac {b x}{a}\right )+24 \left (\log (x)-\log \left (-\frac {b x}{a}\right )\right ) \log \left (1+\frac {b x}{a}\right ) \operatorname {PolyLog}\left (2,1+\frac {b x}{a}\right )+24 \left (\log \left (-\frac {b x}{a+b x}\right )+\log (a+b x)\right ) \operatorname {PolyLog}\left (3,-\frac {b x}{a}\right )+24 \log \left (-\frac {b x}{a+b x}\right ) \operatorname {PolyLog}\left (3,\frac {b x}{a+b x}\right )+24 \left (-\log (x)+\log \left (-\frac {b x}{a+b x}\right )\right ) \operatorname {PolyLog}\left (3,1+\frac {b x}{a}\right )-24 \left (\operatorname {PolyLog}\left (4,-\frac {b x}{a}\right )+\operatorname {PolyLog}\left (4,\frac {b x}{a+b x}\right )-\operatorname {PolyLog}\left (4,1+\frac {b x}{a}\right )\right )\right ) \]

output

1/12*ln(-b*x/a)^4+1/2*ln(-b*x/a)^2*ln(-b*x/(b*x+a))^2-1/3*(ln(-b*x/a)+ln(a 
/(b*x+a)))*ln(-b*x/(b*x+a))^3+1/12*ln(-b*x/(b*x+a))^4+1/2*ln(x)^2*ln(b*x+a 
)^2+1/3*(2*ln(-b*x/a)^3-3*ln(x)^2*ln(b*x+a))*ln(1+b*x/a)+1/2*(ln(x)-ln(-b* 
x/a))*(ln(x)+3*ln(-b*x/a))*ln(1+b*x/a)^2-1/3*ln(-b*x/a)^2*ln(-b*x/(b*x+a)) 
*(ln(-b*x/a)+3*ln(1+b*x/a))+(ln(-b*x/a)^2-2*ln(-b*x/a)*(ln(-b*x/(b*x+a))+l 
n(1+b*x/a))+2*ln(x)*(-ln(b*x+a)+ln(1+b*x/a)))*polylog(2,-b*x/a)-ln(-b*x/(b 
*x+a))^2*polylog(2,b*x/(b*x+a))+(ln(-b*x/a)-ln(-b*x/(b*x+a)))^2*polylog(2, 
1+b*x/a)+2*(ln(x)-ln(-b*x/a))*ln(1+b*x/a)*polylog(2,1+b*x/a)+2*(ln(-b*x/(b 
*x+a))+ln(b*x+a))*polylog(3,-b*x/a)+2*ln(-b*x/(b*x+a))*polylog(3,b*x/(b*x+ 
a))+2*(-ln(x)+ln(-b*x/(b*x+a)))*polylog(3,1+b*x/a)-2*polylog(4,-b*x/a)-2*p 
olylog(4,b*x/(b*x+a))+2*polylog(4,1+b*x/a)
 

3.4.74.2 Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.00 \[ \int \frac {\log (x) \log ^2(a+b x)}{x} \, dx=\frac {1}{12} \left (\log ^4\left (-\frac {b x}{a}\right )+6 \log ^2\left (-\frac {b x}{a}\right ) \log ^2\left (-\frac {b x}{a+b x}\right )-4 \left (\log \left (-\frac {b x}{a}\right )+\log \left (\frac {a}{a+b x}\right )\right ) \log ^3\left (-\frac {b x}{a+b x}\right )+\log ^4\left (-\frac {b x}{a+b x}\right )+6 \log ^2(x) \log ^2(a+b x)+4 \left (2 \log ^3\left (-\frac {b x}{a}\right )-3 \log ^2(x) \log (a+b x)\right ) \log \left (1+\frac {b x}{a}\right )+6 \left (\log (x)-\log \left (-\frac {b x}{a}\right )\right ) \left (\log (x)+3 \log \left (-\frac {b x}{a}\right )\right ) \log ^2\left (1+\frac {b x}{a}\right )-4 \log ^2\left (-\frac {b x}{a}\right ) \log \left (-\frac {b x}{a+b x}\right ) \left (\log \left (-\frac {b x}{a}\right )+3 \log \left (1+\frac {b x}{a}\right )\right )+12 \left (\log ^2\left (-\frac {b x}{a}\right )-2 \log \left (-\frac {b x}{a}\right ) \left (\log \left (-\frac {b x}{a+b x}\right )+\log \left (1+\frac {b x}{a}\right )\right )+2 \log (x) \left (-\log (a+b x)+\log \left (1+\frac {b x}{a}\right )\right )\right ) \operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )-12 \log ^2\left (-\frac {b x}{a+b x}\right ) \operatorname {PolyLog}\left (2,\frac {b x}{a+b x}\right )+12 \left (\log \left (-\frac {b x}{a}\right )-\log \left (-\frac {b x}{a+b x}\right )\right )^2 \operatorname {PolyLog}\left (2,1+\frac {b x}{a}\right )+24 \left (\log (x)-\log \left (-\frac {b x}{a}\right )\right ) \log \left (1+\frac {b x}{a}\right ) \operatorname {PolyLog}\left (2,1+\frac {b x}{a}\right )+24 \left (\log \left (-\frac {b x}{a+b x}\right )+\log (a+b x)\right ) \operatorname {PolyLog}\left (3,-\frac {b x}{a}\right )+24 \log \left (-\frac {b x}{a+b x}\right ) \operatorname {PolyLog}\left (3,\frac {b x}{a+b x}\right )+24 \left (-\log (x)+\log \left (-\frac {b x}{a+b x}\right )\right ) \operatorname {PolyLog}\left (3,1+\frac {b x}{a}\right )-24 \left (\operatorname {PolyLog}\left (4,-\frac {b x}{a}\right )+\operatorname {PolyLog}\left (4,\frac {b x}{a+b x}\right )-\operatorname {PolyLog}\left (4,1+\frac {b x}{a}\right )\right )\right ) \]

input

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

output

(Log[-((b*x)/a)]^4 + 6*Log[-((b*x)/a)]^2*Log[-((b*x)/(a + b*x))]^2 - 4*(Lo 
g[-((b*x)/a)] + Log[a/(a + b*x)])*Log[-((b*x)/(a + b*x))]^3 + Log[-((b*x)/ 
(a + b*x))]^4 + 6*Log[x]^2*Log[a + b*x]^2 + 4*(2*Log[-((b*x)/a)]^3 - 3*Log 
[x]^2*Log[a + b*x])*Log[1 + (b*x)/a] + 6*(Log[x] - Log[-((b*x)/a)])*(Log[x 
] + 3*Log[-((b*x)/a)])*Log[1 + (b*x)/a]^2 - 4*Log[-((b*x)/a)]^2*Log[-((b*x 
)/(a + b*x))]*(Log[-((b*x)/a)] + 3*Log[1 + (b*x)/a]) + 12*(Log[-((b*x)/a)] 
^2 - 2*Log[-((b*x)/a)]*(Log[-((b*x)/(a + b*x))] + Log[1 + (b*x)/a]) + 2*Lo 
g[x]*(-Log[a + b*x] + Log[1 + (b*x)/a]))*PolyLog[2, -((b*x)/a)] - 12*Log[- 
((b*x)/(a + b*x))]^2*PolyLog[2, (b*x)/(a + b*x)] + 12*(Log[-((b*x)/a)] - L 
og[-((b*x)/(a + b*x))])^2*PolyLog[2, 1 + (b*x)/a] + 24*(Log[x] - Log[-((b* 
x)/a)])*Log[1 + (b*x)/a]*PolyLog[2, 1 + (b*x)/a] + 24*(Log[-((b*x)/(a + b* 
x))] + Log[a + b*x])*PolyLog[3, -((b*x)/a)] + 24*Log[-((b*x)/(a + b*x))]*P 
olyLog[3, (b*x)/(a + b*x)] + 24*(-Log[x] + Log[-((b*x)/(a + b*x))])*PolyLo 
g[3, 1 + (b*x)/a] - 24*(PolyLog[4, -((b*x)/a)] + PolyLog[4, (b*x)/(a + b*x 
)] - PolyLog[4, 1 + (b*x)/a]))/12
 

3.4.74.3 Rubi [F]

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.

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

\(\Big \downarrow \) 2874

\(\displaystyle \frac {1}{2} \log ^2(x) \log ^2(a+b x)-b \int \frac {\log ^2(x) \log (a+b x)}{a+b x}dx\)

\(\Big \downarrow \) 2891

\(\displaystyle \frac {1}{2} \log ^2(x) \log ^2(a+b x)-b \int \frac {\log ^2(x) \log (a+b x)}{a+b x}dx\)

input

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

output

$Aborted
 

3.4.74.3.1 Defintions of rubi rules used

rule 2874

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

rule 2891

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

3.4.74.4 Maple [F]

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

input

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

output

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

3.4.74.5 Fricas [F]

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

input

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

output

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

3.4.74.6 Sympy [F]

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

input

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

output

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

3.4.74.7 Maxima [F]

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

input

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

output

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

3.4.74.8 Giac [F]

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

input

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

output

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

3.4.74.9 Mupad [F(-1)]

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

input

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

output

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