\(\int \frac {\log ^2(e (f (a+b x)^p (c+d x)^q)^r)}{g+h x} \, dx\) [39]

Optimal result

Integrand size = 31, antiderivative size = 1471 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx =\text {Too large to display} \] Output:

2*p*r*ln(b*x+a)*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*ln(b*(h*x+g)/(-a*h+b*g))/h 
-2*p*r*ln(b*x+a)*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*ln(h*x+g)/h-2*p*q*r^2*ln( 
(-a*h+b*g)*(d*x+c)/(-c*h+d*g)/(b*x+a))*polylog(2,(-a*h+b*g)*(d*x+c)/(-c*h+ 
d*g)/(b*x+a))/h+2*p*q*r^2*ln((-a*h+b*g)*(d*x+c)/(-c*h+d*g)/(b*x+a))*polylo 
g(2,b*(d*x+c)/d/(b*x+a))/h+2*q*r*ln(d*x+c)*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r) 
*ln(d*(h*x+g)/(-c*h+d*g))/h-2*q*r*ln(d*x+c)*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r 
)*ln(h*x+g)/h+ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2*ln(h*x+g)/h-2*q^2*r^2*poly 
log(3,-h*(d*x+c)/(-c*h+d*g))/h-2*p^2*r^2*polylog(3,-h*(b*x+a)/(-a*h+b*g))/ 
h-p*q*r^2*ln((-a*h+b*g)*(d*x+c)/(-c*h+d*g)/(b*x+a))^2*ln(-(-a*d+b*c)*(h*x+ 
g)/(-c*h+d*g)/(b*x+a))/h+p*q*r^2*ln(-(-a*d+b*c)/d/(b*x+a))*ln((-a*h+b*g)*( 
d*x+c)/(-c*h+d*g)/(b*x+a))^2/h-p*q*r^2*ln(-h*(d*x+c)/(-c*h+d*g))^2*ln(d*(h 
*x+g)/(-c*h+d*g))/h+p*q*r^2*ln((-a*h+b*g)*(d*x+c)/(-c*h+d*g)/(b*x+a))^2*ln 
(b*(h*x+g)/(-a*h+b*g))/h+p*q*r^2*ln(-h*(d*x+c)/(-c*h+d*g))^2*ln(b*(h*x+g)/ 
(-a*h+b*g))/h-q^2*r^2*ln(d*x+c)^2*ln(d*(h*x+g)/(-c*h+d*g))/h+q^2*r^2*ln(d* 
x+c)^2*ln(h*x+g)/h-p^2*r^2*ln(b*x+a)^2*ln(b*(h*x+g)/(-a*h+b*g))/h+p^2*r^2* 
ln(b*x+a)^2*ln(h*x+g)/h-2*p*q*r^2*ln(b*x+a)*ln(d*x+c)*ln(d*(h*x+g)/(-c*h+d 
*g))/h+2*p*q*r^2*ln(b*x+a)*ln(d*x+c)*ln(h*x+g)/h+2*p*q*r^2*ln(-h*(d*x+c)/( 
-c*h+d*g))*ln((-a*h+b*g)*(d*x+c)/(-c*h+d*g)/(b*x+a))*ln(d*(h*x+g)/(-c*h+d* 
g))/h-2*p*q*r^2*ln(-h*(d*x+c)/(-c*h+d*g))*ln((-a*h+b*g)*(d*x+c)/(-c*h+d*g) 
/(b*x+a))*ln(b*(h*x+g)/(-a*h+b*g))/h+2*p*q*r^2*ln(b*x+a)*ln(-h*(d*x+c)/...
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 1370, normalized size of antiderivative = 0.93 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx =\text {Too large to display} \] Input:

Integrate[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(g + h*x),x]
 

Output:

(p*q*r^2*Log[(-(b*c) + a*d)/(d*(a + b*x))]*Log[((b*g - a*h)*(c + d*x))/((d 
*g - c*h)*(a + b*x))]^2 + p^2*r^2*Log[a + b*x]^2*Log[g + h*x] + 2*p*q*r^2* 
Log[a + b*x]*Log[c + d*x]*Log[g + h*x] + q^2*r^2*Log[c + d*x]^2*Log[g + h* 
x] - 2*p*r*Log[a + b*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*Log[g + h*x] 
- 2*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*Log[g + h*x] + L 
og[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2*Log[g + h*x] - p^2*r^2*Log[a + b*x]^ 
2*Log[(b*(g + h*x))/(b*g - a*h)] - 2*p*q*r^2*Log[a + b*x]*Log[(h*(c + d*x) 
)/(-(d*g) + c*h)]*Log[(b*(g + h*x))/(b*g - a*h)] + p*q*r^2*Log[(h*(c + d*x 
))/(-(d*g) + c*h)]^2*Log[(b*(g + h*x))/(b*g - a*h)] - 2*p*q*r^2*Log[(h*(c 
+ d*x))/(-(d*g) + c*h)]*Log[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x) 
)]*Log[(b*(g + h*x))/(b*g - a*h)] + p*q*r^2*Log[((b*g - a*h)*(c + d*x))/(( 
d*g - c*h)*(a + b*x))]^2*Log[(b*(g + h*x))/(b*g - a*h)] + 2*p*r*Log[a + b* 
x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]*Log[(b*(g + h*x))/(b*g - a*h)] - 2 
*p*q*r^2*Log[a + b*x]*Log[c + d*x]*Log[(d*(g + h*x))/(d*g - c*h)] - q^2*r^ 
2*Log[c + d*x]^2*Log[(d*(g + h*x))/(d*g - c*h)] + 2*p*q*r^2*Log[a + b*x]*L 
og[(h*(c + d*x))/(-(d*g) + c*h)]*Log[(d*(g + h*x))/(d*g - c*h)] - p*q*r^2* 
Log[(h*(c + d*x))/(-(d*g) + c*h)]^2*Log[(d*(g + h*x))/(d*g - c*h)] + 2*p*q 
*r^2*Log[(h*(c + d*x))/(-(d*g) + c*h)]*Log[((b*g - a*h)*(c + d*x))/((d*g - 
 c*h)*(a + b*x))]*Log[(d*(g + h*x))/(d*g - c*h)] + 2*q*r*Log[c + d*x]*Log[ 
e*(f*(a + b*x)^p*(c + d*x)^q)^r]*Log[(d*(g + h*x))/(d*g - c*h)] - p*q*r...
 

Rubi [A] (verified)

Time = 5.84 (sec) , antiderivative size = 2728, normalized size of antiderivative = 1.85, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.548, Rules used = {2983, 2986, 2841, 2840, 2838, 2881, 2822, 27, 2754, 2821, 2890, 2887, 2841, 27, 2752, 2885, 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.

\(\displaystyle \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx\)

\(\Big \downarrow \) 2983

\(\displaystyle -\frac {2 b p r \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{a+b x}dx}{h}-\frac {2 d q r \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{c+d x}dx}{h}+\frac {\log (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h}\)

\(\Big \downarrow \) 2986

\(\displaystyle -\frac {2 b p r \left (-\left (\left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right ) \int \frac {\log (g+h x)}{a+b x}dx\right )+\int \frac {\log \left ((c+d x)^{q r}\right ) \log (g+h x)}{a+b x}dx+\int \frac {\log \left ((a+b x)^{p r}\right ) \log (g+h x)}{a+b x}dx\right )}{h}-\frac {2 d q r \left (-\left (\left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right ) \int \frac {\log (g+h x)}{c+d x}dx\right )+\int \frac {\log \left ((a+b x)^{p r}\right ) \log (g+h x)}{c+d x}dx+\int \frac {\log \left ((c+d x)^{q r}\right ) \log (g+h x)}{c+d x}dx\right )}{h}+\frac {\log (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h}\)

\(\Big \downarrow \) 2841

\(\displaystyle -\frac {2 b p r \left (-\left (\left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right ) \left (\frac {\log (g+h x) \log \left (-\frac {h (a+b x)}{b g-a h}\right )}{b}-\frac {h \int \frac {\log \left (-\frac {h (a+b x)}{b g-a h}\right )}{g+h x}dx}{b}\right )\right )+\int \frac {\log \left ((c+d x)^{q r}\right ) \log (g+h x)}{a+b x}dx+\int \frac {\log \left ((a+b x)^{p r}\right ) \log (g+h x)}{a+b x}dx\right )}{h}-\frac {2 d q r \left (-\left (\left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right ) \left (\frac {\log (g+h x) \log \left (-\frac {h (c+d x)}{d g-c h}\right )}{d}-\frac {h \int \frac {\log \left (-\frac {h (c+d x)}{d g-c h}\right )}{g+h x}dx}{d}\right )\right )+\int \frac {\log \left ((a+b x)^{p r}\right ) \log (g+h x)}{c+d x}dx+\int \frac {\log \left ((c+d x)^{q r}\right ) \log (g+h x)}{c+d x}dx\right )}{h}+\frac {\log (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h}\)

\(\Big \downarrow \) 2840

\(\displaystyle -\frac {2 b p r \left (-\left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right ) \left (\frac {\log (g+h x) \log \left (-\frac {h (a+b x)}{b g-a h}\right )}{b}-\frac {\int \frac {\log \left (1-\frac {b (g+h x)}{b g-a h}\right )}{g+h x}d(g+h x)}{b}\right )+\int \frac {\log \left ((c+d x)^{q r}\right ) \log (g+h x)}{a+b x}dx+\int \frac {\log \left ((a+b x)^{p r}\right ) \log (g+h x)}{a+b x}dx\right )}{h}-\frac {2 d q r \left (-\left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right ) \left (\frac {\log (g+h x) \log \left (-\frac {h (c+d x)}{d g-c h}\right )}{d}-\frac {\int \frac {\log \left (1-\frac {d (g+h x)}{d g-c h}\right )}{g+h x}d(g+h x)}{d}\right )+\int \frac {\log \left ((a+b x)^{p r}\right ) \log (g+h x)}{c+d x}dx+\int \frac {\log \left ((c+d x)^{q r}\right ) \log (g+h x)}{c+d x}dx\right )}{h}+\frac {\log (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {2 b p r \left (\int \frac {\log \left ((c+d x)^{q r}\right ) \log (g+h x)}{a+b x}dx+\int \frac {\log \left ((a+b x)^{p r}\right ) \log (g+h x)}{a+b x}dx-\left (\left (\frac {\operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{b}+\frac {\log (g+h x) \log \left (-\frac {h (a+b x)}{b g-a h}\right )}{b}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right )\right )\right )}{h}-\frac {2 d q r \left (\int \frac {\log \left ((a+b x)^{p r}\right ) \log (g+h x)}{c+d x}dx+\int \frac {\log \left ((c+d x)^{q r}\right ) \log (g+h x)}{c+d x}dx-\left (\left (\frac {\operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{d}+\frac {\log (g+h x) \log \left (-\frac {h (c+d x)}{d g-c h}\right )}{d}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right )\right )\right )}{h}+\frac {\log (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h}\)

\(\Big \downarrow \) 2881

\(\displaystyle -\frac {2 b p r \left (\int \frac {\log \left ((c+d x)^{q r}\right ) \log (g+h x)}{a+b x}dx+\frac {\int \frac {\log \left ((a+b x)^{p r}\right ) \log \left (g-\frac {a h}{b}+\frac {h (a+b x)}{b}\right )}{a+b x}d(a+b x)}{b}-\left (\left (\frac {\operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{b}+\frac {\log (g+h x) \log \left (-\frac {h (a+b x)}{b g-a h}\right )}{b}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right )\right )\right )}{h}-\frac {2 d q r \left (\int \frac {\log \left ((a+b x)^{p r}\right ) \log (g+h x)}{c+d x}dx+\frac {\int \frac {\log \left ((c+d x)^{q r}\right ) \log \left (g-\frac {c h}{d}+\frac {h (c+d x)}{d}\right )}{c+d x}d(c+d x)}{d}-\left (\left (\frac {\operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{d}+\frac {\log (g+h x) \log \left (-\frac {h (c+d x)}{d g-c h}\right )}{d}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right )\right )\right )}{h}+\frac {\log (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h}\)

\(\Big \downarrow \) 2822

\(\displaystyle -\frac {2 d q r \left (\int \frac {\log \left ((a+b x)^{p r}\right ) \log (g+h x)}{c+d x}dx+\frac {\frac {\log \left (\frac {h (c+d x)}{d}-\frac {c h}{d}+g\right ) \log ^2\left ((c+d x)^{q r}\right )}{2 q r}-\frac {h \int \frac {d \log ^2\left ((c+d x)^{q r}\right )}{d \left (g-\frac {c h}{d}\right )+h (c+d x)}d(c+d x)}{2 d q r}}{d}-\left (\left (\frac {\operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{d}+\frac {\log (g+h x) \log \left (-\frac {h (c+d x)}{d g-c h}\right )}{d}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right )\right )\right )}{h}-\frac {2 b p r \left (\int \frac {\log \left ((c+d x)^{q r}\right ) \log (g+h x)}{a+b x}dx+\frac {\frac {\log \left (\frac {h (a+b x)}{b}-\frac {a h}{b}+g\right ) \log ^2\left ((a+b x)^{p r}\right )}{2 p r}-\frac {h \int \frac {b \log ^2\left ((a+b x)^{p r}\right )}{b \left (g-\frac {a h}{b}\right )+h (a+b x)}d(a+b x)}{2 b p r}}{b}-\left (\left (\frac {\operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{b}+\frac {\log (g+h x) \log \left (-\frac {h (a+b x)}{b g-a h}\right )}{b}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right )\right )\right )}{h}+\frac {\log (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 d q r \left (\int \frac {\log \left ((a+b x)^{p r}\right ) \log (g+h x)}{c+d x}dx+\frac {\frac {\log \left (\frac {h (c+d x)}{d}-\frac {c h}{d}+g\right ) \log ^2\left ((c+d x)^{q r}\right )}{2 q r}-\frac {h \int \frac {\log ^2\left ((c+d x)^{q r}\right )}{d g-c h+h (c+d x)}d(c+d x)}{2 q r}}{d}-\left (\left (\frac {\operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{d}+\frac {\log (g+h x) \log \left (-\frac {h (c+d x)}{d g-c h}\right )}{d}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right )\right )\right )}{h}-\frac {2 b p r \left (\int \frac {\log \left ((c+d x)^{q r}\right ) \log (g+h x)}{a+b x}dx+\frac {\frac {\log \left (\frac {h (a+b x)}{b}-\frac {a h}{b}+g\right ) \log ^2\left ((a+b x)^{p r}\right )}{2 p r}-\frac {h \int \frac {\log ^2\left ((a+b x)^{p r}\right )}{b g-a h+h (a+b x)}d(a+b x)}{2 p r}}{b}-\left (\left (\frac {\operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{b}+\frac {\log (g+h x) \log \left (-\frac {h (a+b x)}{b g-a h}\right )}{b}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right )\right )\right )}{h}+\frac {\log (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h}\)

\(\Big \downarrow \) 2754

\(\displaystyle -\frac {2 b p r \left (\int \frac {\log \left ((c+d x)^{q r}\right ) \log (g+h x)}{a+b x}dx+\frac {\frac {\log \left (\frac {h (a+b x)}{b}-\frac {a h}{b}+g\right ) \log ^2\left ((a+b x)^{p r}\right )}{2 p r}-\frac {h \left (\frac {\log \left (\frac {h (a+b x)}{b g-a h}+1\right ) \log ^2\left ((a+b x)^{p r}\right )}{h}-\frac {2 p r \int \frac {\log \left ((a+b x)^{p r}\right ) \log \left (\frac {h (a+b x)}{b g-a h}+1\right )}{a+b x}d(a+b x)}{h}\right )}{2 p r}}{b}-\left (\left (\frac {\operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{b}+\frac {\log (g+h x) \log \left (-\frac {h (a+b x)}{b g-a h}\right )}{b}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right )\right )\right )}{h}-\frac {2 d q r \left (\int \frac {\log \left ((a+b x)^{p r}\right ) \log (g+h x)}{c+d x}dx+\frac {\frac {\log \left (\frac {h (c+d x)}{d}-\frac {c h}{d}+g\right ) \log ^2\left ((c+d x)^{q r}\right )}{2 q r}-\frac {h \left (\frac {\log \left (\frac {h (c+d x)}{d g-c h}+1\right ) \log ^2\left ((c+d x)^{q r}\right )}{h}-\frac {2 q r \int \frac {\log \left ((c+d x)^{q r}\right ) \log \left (\frac {h (c+d x)}{d g-c h}+1\right )}{c+d x}d(c+d x)}{h}\right )}{2 q r}}{d}-\left (\left (\frac {\operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{d}+\frac {\log (g+h x) \log \left (-\frac {h (c+d x)}{d g-c h}\right )}{d}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right )\right )\right )}{h}+\frac {\log (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h}\)

\(\Big \downarrow \) 2821

\(\displaystyle -\frac {2 b p r \left (\int \frac {\log \left ((c+d x)^{q r}\right ) \log (g+h x)}{a+b x}dx+\frac {\frac {\log \left (\frac {h (a+b x)}{b}-\frac {a h}{b}+g\right ) \log ^2\left ((a+b x)^{p r}\right )}{2 p r}-\frac {h \left (\frac {\log \left (\frac {h (a+b x)}{b g-a h}+1\right ) \log ^2\left ((a+b x)^{p r}\right )}{h}-\frac {2 p r \left (p r \int \frac {\operatorname {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right )}{a+b x}d(a+b x)-\operatorname {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right ) \log \left ((a+b x)^{p r}\right )\right )}{h}\right )}{2 p r}}{b}-\left (\left (\frac {\operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{b}+\frac {\log (g+h x) \log \left (-\frac {h (a+b x)}{b g-a h}\right )}{b}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right )\right )\right )}{h}-\frac {2 d q r \left (\int \frac {\log \left ((a+b x)^{p r}\right ) \log (g+h x)}{c+d x}dx+\frac {\frac {\log \left (\frac {h (c+d x)}{d}-\frac {c h}{d}+g\right ) \log ^2\left ((c+d x)^{q r}\right )}{2 q r}-\frac {h \left (\frac {\log \left (\frac {h (c+d x)}{d g-c h}+1\right ) \log ^2\left ((c+d x)^{q r}\right )}{h}-\frac {2 q r \left (q r \int \frac {\operatorname {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right )}{c+d x}d(c+d x)-\operatorname {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right ) \log \left ((c+d x)^{q r}\right )\right )}{h}\right )}{2 q r}}{d}-\left (\left (\frac {\operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{d}+\frac {\log (g+h x) \log \left (-\frac {h (c+d x)}{d g-c h}\right )}{d}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right )\right )\right )}{h}+\frac {\log (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h}\)

\(\Big \downarrow \) 2890

\(\displaystyle -\frac {2 b p r \left (\frac {\int \frac {\log \left (\left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right )^{q r}\right ) \log \left (g-\frac {a h}{b}+\frac {h (a+b x)}{b}\right )}{a+b x}d(a+b x)}{b}+\frac {\frac {\log \left (\frac {h (a+b x)}{b}-\frac {a h}{b}+g\right ) \log ^2\left ((a+b x)^{p r}\right )}{2 p r}-\frac {h \left (\frac {\log \left (\frac {h (a+b x)}{b g-a h}+1\right ) \log ^2\left ((a+b x)^{p r}\right )}{h}-\frac {2 p r \left (p r \int \frac {\operatorname {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right )}{a+b x}d(a+b x)-\operatorname {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right ) \log \left ((a+b x)^{p r}\right )\right )}{h}\right )}{2 p r}}{b}-\left (\left (\frac {\operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{b}+\frac {\log (g+h x) \log \left (-\frac {h (a+b x)}{b g-a h}\right )}{b}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right )\right )\right )}{h}-\frac {2 d q r \left (\frac {\int \frac {\log \left (\left (a+\frac {b (c+d x)}{d}-\frac {b c}{d}\right )^{p r}\right ) \log \left (g-\frac {c h}{d}+\frac {h (c+d x)}{d}\right )}{c+d x}d(c+d x)}{d}+\frac {\frac {\log \left (\frac {h (c+d x)}{d}-\frac {c h}{d}+g\right ) \log ^2\left ((c+d x)^{q r}\right )}{2 q r}-\frac {h \left (\frac {\log \left (\frac {h (c+d x)}{d g-c h}+1\right ) \log ^2\left ((c+d x)^{q r}\right )}{h}-\frac {2 q r \left (q r \int \frac {\operatorname {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right )}{c+d x}d(c+d x)-\operatorname {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right ) \log \left ((c+d x)^{q r}\right )\right )}{h}\right )}{2 q r}}{d}-\left (\left (\frac {\operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{d}+\frac {\log (g+h x) \log \left (-\frac {h (c+d x)}{d g-c h}\right )}{d}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right )\right )\right )}{h}+\frac {\log (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h}\)

\(\Big \downarrow \) 2887

\(\displaystyle \frac {\log (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h}-\frac {2 b p r \left (-\left (\left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \left (\frac {\log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log (g+h x)}{b}+\frac {\operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{b}\right )\right )+\frac {q r \int \frac {\log \left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right ) \log \left (g-\frac {a h}{b}+\frac {h (a+b x)}{b}\right )}{a+b x}d(a+b x)-\left (q r \log \left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right )-\log \left (\left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right )^{q r}\right )\right ) \int \frac {\log \left (g-\frac {a h}{b}+\frac {h (a+b x)}{b}\right )}{a+b x}d(a+b x)}{b}+\frac {\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left (g-\frac {a h}{b}+\frac {h (a+b x)}{b}\right )}{2 p r}-\frac {h \left (\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left (\frac {h (a+b x)}{b g-a h}+1\right )}{h}-\frac {2 p r \left (p r \int \frac {\operatorname {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right )}{a+b x}d(a+b x)-\log \left ((a+b x)^{p r}\right ) \operatorname {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right )\right )}{h}\right )}{2 p r}}{b}\right )}{h}-\frac {2 d q r \left (-\left (\left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \left (\frac {\log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x)}{d}+\frac {\operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{d}\right )\right )+\frac {p r \int \frac {\log \left (a+\frac {b (c+d x)}{d}-\frac {b c}{d}\right ) \log \left (g-\frac {c h}{d}+\frac {h (c+d x)}{d}\right )}{c+d x}d(c+d x)-\left (p r \log \left (a+\frac {b (c+d x)}{d}-\frac {b c}{d}\right )-\log \left (\left (a+\frac {b (c+d x)}{d}-\frac {b c}{d}\right )^{p r}\right )\right ) \int \frac {\log \left (g-\frac {c h}{d}+\frac {h (c+d x)}{d}\right )}{c+d x}d(c+d x)}{d}+\frac {\frac {\log ^2\left ((c+d x)^{q r}\right ) \log \left (g-\frac {c h}{d}+\frac {h (c+d x)}{d}\right )}{2 q r}-\frac {h \left (\frac {\log ^2\left ((c+d x)^{q r}\right ) \log \left (\frac {h (c+d x)}{d g-c h}+1\right )}{h}-\frac {2 q r \left (q r \int \frac {\operatorname {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right )}{c+d x}d(c+d x)-\log \left ((c+d x)^{q r}\right ) \operatorname {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right )\right )}{h}\right )}{2 q r}}{d}\right )}{h}\)

\(\Big \downarrow \) 2841

\(\displaystyle \frac {\log (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h}-\frac {2 b p r \left (-\left (\left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \left (\frac {\log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log (g+h x)}{b}+\frac {\operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{b}\right )\right )+\frac {q r \int \frac {\log \left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right ) \log \left (g-\frac {a h}{b}+\frac {h (a+b x)}{b}\right )}{a+b x}d(a+b x)-\left (q r \log \left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right )-\log \left (\left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right )^{q r}\right )\right ) \left (\log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log \left (g-\frac {a h}{b}+\frac {h (a+b x)}{b}\right )-\frac {h \int \frac {b \log \left (-\frac {h (a+b x)}{b g-a h}\right )}{b \left (g-\frac {a h}{b}\right )+h (a+b x)}d(a+b x)}{b}\right )}{b}+\frac {\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left (g-\frac {a h}{b}+\frac {h (a+b x)}{b}\right )}{2 p r}-\frac {h \left (\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left (\frac {h (a+b x)}{b g-a h}+1\right )}{h}-\frac {2 p r \left (p r \int \frac {\operatorname {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right )}{a+b x}d(a+b x)-\log \left ((a+b x)^{p r}\right ) \operatorname {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right )\right )}{h}\right )}{2 p r}}{b}\right )}{h}-\frac {2 d q r \left (-\left (\left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \left (\frac {\log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x)}{d}+\frac {\operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{d}\right )\right )+\frac {p r \int \frac {\log \left (a+\frac {b (c+d x)}{d}-\frac {b c}{d}\right ) \log \left (g-\frac {c h}{d}+\frac {h (c+d x)}{d}\right )}{c+d x}d(c+d x)-\left (p r \log \left (a+\frac {b (c+d x)}{d}-\frac {b c}{d}\right )-\log \left (\left (a+\frac {b (c+d x)}{d}-\frac {b c}{d}\right )^{p r}\right )\right ) \left (\log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log \left (g-\frac {c h}{d}+\frac {h (c+d x)}{d}\right )-\frac {h \int \frac {d \log \left (-\frac {h (c+d x)}{d g-c h}\right )}{d \left (g-\frac {c h}{d}\right )+h (c+d x)}d(c+d x)}{d}\right )}{d}+\frac {\frac {\log ^2\left ((c+d x)^{q r}\right ) \log \left (g-\frac {c h}{d}+\frac {h (c+d x)}{d}\right )}{2 q r}-\frac {h \left (\frac {\log ^2\left ((c+d x)^{q r}\right ) \log \left (\frac {h (c+d x)}{d g-c h}+1\right )}{h}-\frac {2 q r \left (q r \int \frac {\operatorname {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right )}{c+d x}d(c+d x)-\log \left ((c+d x)^{q r}\right ) \operatorname {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right )\right )}{h}\right )}{2 q r}}{d}\right )}{h}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\log (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h}-\frac {2 b p r \left (-\left (\left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \left (\frac {\log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log (g+h x)}{b}+\frac {\operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{b}\right )\right )+\frac {q r \int \frac {\log \left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right ) \log \left (g-\frac {a h}{b}+\frac {h (a+b x)}{b}\right )}{a+b x}d(a+b x)-\left (q r \log \left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right )-\log \left (\left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right )^{q r}\right )\right ) \left (\log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log \left (g-\frac {a h}{b}+\frac {h (a+b x)}{b}\right )-h \int \frac {\log \left (-\frac {h (a+b x)}{b g-a h}\right )}{b g-a h+h (a+b x)}d(a+b x)\right )}{b}+\frac {\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left (g-\frac {a h}{b}+\frac {h (a+b x)}{b}\right )}{2 p r}-\frac {h \left (\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left (\frac {h (a+b x)}{b g-a h}+1\right )}{h}-\frac {2 p r \left (p r \int \frac {\operatorname {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right )}{a+b x}d(a+b x)-\log \left ((a+b x)^{p r}\right ) \operatorname {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right )\right )}{h}\right )}{2 p r}}{b}\right )}{h}-\frac {2 d q r \left (-\left (\left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \left (\frac {\log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x)}{d}+\frac {\operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{d}\right )\right )+\frac {p r \int \frac {\log \left (a+\frac {b (c+d x)}{d}-\frac {b c}{d}\right ) \log \left (g-\frac {c h}{d}+\frac {h (c+d x)}{d}\right )}{c+d x}d(c+d x)-\left (p r \log \left (a+\frac {b (c+d x)}{d}-\frac {b c}{d}\right )-\log \left (\left (a+\frac {b (c+d x)}{d}-\frac {b c}{d}\right )^{p r}\right )\right ) \left (\log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log \left (g-\frac {c h}{d}+\frac {h (c+d x)}{d}\right )-h \int \frac {\log \left (-\frac {h (c+d x)}{d g-c h}\right )}{d g-c h+h (c+d x)}d(c+d x)\right )}{d}+\frac {\frac {\log ^2\left ((c+d x)^{q r}\right ) \log \left (g-\frac {c h}{d}+\frac {h (c+d x)}{d}\right )}{2 q r}-\frac {h \left (\frac {\log ^2\left ((c+d x)^{q r}\right ) \log \left (\frac {h (c+d x)}{d g-c h}+1\right )}{h}-\frac {2 q r \left (q r \int \frac {\operatorname {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right )}{c+d x}d(c+d x)-\log \left ((c+d x)^{q r}\right ) \operatorname {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right )\right )}{h}\right )}{2 q r}}{d}\right )}{h}\)

\(\Big \downarrow \) 2752

\(\displaystyle \frac {\log (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h}-\frac {2 b p r \left (-\left (\left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \left (\frac {\log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log (g+h x)}{b}+\frac {\operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{b}\right )\right )+\frac {q r \int \frac {\log \left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right ) \log \left (g-\frac {a h}{b}+\frac {h (a+b x)}{b}\right )}{a+b x}d(a+b x)-\left (q r \log \left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right )-\log \left (\left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right )^{q r}\right )\right ) \left (\log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log \left (g-\frac {a h}{b}+\frac {h (a+b x)}{b}\right )+\operatorname {PolyLog}\left (2,\frac {h (a+b x)}{b g-a h}+1\right )\right )}{b}+\frac {\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left (g-\frac {a h}{b}+\frac {h (a+b x)}{b}\right )}{2 p r}-\frac {h \left (\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left (\frac {h (a+b x)}{b g-a h}+1\right )}{h}-\frac {2 p r \left (p r \int \frac {\operatorname {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right )}{a+b x}d(a+b x)-\log \left ((a+b x)^{p r}\right ) \operatorname {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right )\right )}{h}\right )}{2 p r}}{b}\right )}{h}-\frac {2 d q r \left (-\left (\left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \left (\frac {\log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x)}{d}+\frac {\operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{d}\right )\right )+\frac {p r \int \frac {\log \left (a+\frac {b (c+d x)}{d}-\frac {b c}{d}\right ) \log \left (g-\frac {c h}{d}+\frac {h (c+d x)}{d}\right )}{c+d x}d(c+d x)-\left (p r \log \left (a+\frac {b (c+d x)}{d}-\frac {b c}{d}\right )-\log \left (\left (a+\frac {b (c+d x)}{d}-\frac {b c}{d}\right )^{p r}\right )\right ) \left (\log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log \left (g-\frac {c h}{d}+\frac {h (c+d x)}{d}\right )+\operatorname {PolyLog}\left (2,\frac {h (c+d x)}{d g-c h}+1\right )\right )}{d}+\frac {\frac {\log ^2\left ((c+d x)^{q r}\right ) \log \left (g-\frac {c h}{d}+\frac {h (c+d x)}{d}\right )}{2 q r}-\frac {h \left (\frac {\log ^2\left ((c+d x)^{q r}\right ) \log \left (\frac {h (c+d x)}{d g-c h}+1\right )}{h}-\frac {2 q r \left (q r \int \frac {\operatorname {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right )}{c+d x}d(c+d x)-\log \left ((c+d x)^{q r}\right ) \operatorname {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right )\right )}{h}\right )}{2 q r}}{d}\right )}{h}\)

\(\Big \downarrow \) 2885

\(\displaystyle \frac {\log (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h}-\frac {2 b p r \left (-\left (\left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \left (\frac {\log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log (g+h x)}{b}+\frac {\operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{b}\right )\right )+\frac {q r \left (\frac {1}{2} \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )+\log \left (\frac {b (d g-c h)}{d \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}\right )-\log \left (-\frac {b (d g-c h) (a+b x)}{(b c-a d) \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}\right )\right ) \log ^2\left (\frac {(b c-a d) \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}{(b g-a h) \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}\right )-\operatorname {PolyLog}\left (2,\frac {h \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}{d \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}\right ) \log \left (\frac {(b c-a d) \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}{(b g-a h) \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}\right )+\operatorname {PolyLog}\left (2,\frac {(b g-a h) \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}{(b c-a d) \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}\right ) \log \left (\frac {(b c-a d) \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}{(b g-a h) \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}\right )-\frac {1}{2} \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )-\log \left (-\frac {h (a+b x)}{b g-a h}\right )\right ) \left (\log \left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right )+\log \left (\frac {(b c-a d) \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}{(b g-a h) \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}\right )\right )^2+\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log \left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right ) \log \left (g-\frac {a h}{b}+\frac {h (a+b x)}{b}\right )-\left (\log \left (\frac {(b c-a d) \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}{(b g-a h) \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}\right )-\log \left (g-\frac {a h}{b}+\frac {h (a+b x)}{b}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b c-a d}+1\right )+\left (\log \left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right )+\log \left (\frac {(b c-a d) \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}{(b g-a h) \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}\right )\right ) \operatorname {PolyLog}\left (2,\frac {h (a+b x)}{b g-a h}+1\right )-\operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b c-a d}+1\right )-\operatorname {PolyLog}\left (3,\frac {h \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}{d \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}\right )+\operatorname {PolyLog}\left (3,\frac {(b g-a h) \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}{(b c-a d) \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}\right )-\operatorname {PolyLog}\left (3,\frac {h (a+b x)}{b g-a h}+1\right )\right )-\left (q r \log \left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right )-\log \left (\left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right )^{q r}\right )\right ) \left (\log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log \left (g-\frac {a h}{b}+\frac {h (a+b x)}{b}\right )+\operatorname {PolyLog}\left (2,\frac {h (a+b x)}{b g-a h}+1\right )\right )}{b}+\frac {\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left (g-\frac {a h}{b}+\frac {h (a+b x)}{b}\right )}{2 p r}-\frac {h \left (\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left (\frac {h (a+b x)}{b g-a h}+1\right )}{h}-\frac {2 p r \left (p r \int \frac {\operatorname {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right )}{a+b x}d(a+b x)-\log \left ((a+b x)^{p r}\right ) \operatorname {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right )\right )}{h}\right )}{2 p r}}{b}\right )}{h}-\frac {2 d q r \left (-\left (\left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \left (\frac {\log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x)}{d}+\frac {\operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{d}\right )\right )+\frac {p r \left (\frac {1}{2} \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )+\log \left (\frac {d (b g-a h)}{b \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}\right )-\log \left (\frac {d (b g-a h) (c+d x)}{(b c-a d) \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}\right )\right ) \log ^2\left (-\frac {(b c-a d) \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}{(d g-c h) \left (\left (a-\frac {b c}{d}\right ) d+b (c+d x)\right )}\right )-\operatorname {PolyLog}\left (2,\frac {h \left (\left (a-\frac {b c}{d}\right ) d+b (c+d x)\right )}{b \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}\right ) \log \left (-\frac {(b c-a d) \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}{(d g-c h) \left (\left (a-\frac {b c}{d}\right ) d+b (c+d x)\right )}\right )+\operatorname {PolyLog}\left (2,-\frac {(d g-c h) \left (\left (a-\frac {b c}{d}\right ) d+b (c+d x)\right )}{(b c-a d) \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}\right ) \log \left (-\frac {(b c-a d) \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}{(d g-c h) \left (\left (a-\frac {b c}{d}\right ) d+b (c+d x)\right )}\right )-\frac {1}{2} \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )-\log \left (-\frac {h (c+d x)}{d g-c h}\right )\right ) \left (\log \left (a+\frac {b (c+d x)}{d}-\frac {b c}{d}\right )+\log \left (-\frac {(b c-a d) \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}{(d g-c h) \left (\left (a-\frac {b c}{d}\right ) d+b (c+d x)\right )}\right )\right )^2+\log \left (\frac {b (c+d x)}{b c-a d}\right ) \log \left (a+\frac {b (c+d x)}{d}-\frac {b c}{d}\right ) \log \left (g-\frac {c h}{d}+\frac {h (c+d x)}{d}\right )-\left (\log \left (-\frac {(b c-a d) \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}{(d g-c h) \left (\left (a-\frac {b c}{d}\right ) d+b (c+d x)\right )}\right )-\log \left (g-\frac {c h}{d}+\frac {h (c+d x)}{d}\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {b (c+d x)}{b c-a d}\right )+\left (\log \left (a+\frac {b (c+d x)}{d}-\frac {b c}{d}\right )+\log \left (-\frac {(b c-a d) \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}{(d g-c h) \left (\left (a-\frac {b c}{d}\right ) d+b (c+d x)\right )}\right )\right ) \operatorname {PolyLog}\left (2,\frac {h (c+d x)}{d g-c h}+1\right )-\operatorname {PolyLog}\left (3,1-\frac {b (c+d x)}{b c-a d}\right )-\operatorname {PolyLog}\left (3,\frac {h \left (\left (a-\frac {b c}{d}\right ) d+b (c+d x)\right )}{b \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}\right )+\operatorname {PolyLog}\left (3,-\frac {(d g-c h) \left (\left (a-\frac {b c}{d}\right ) d+b (c+d x)\right )}{(b c-a d) \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}\right )-\operatorname {PolyLog}\left (3,\frac {h (c+d x)}{d g-c h}+1\right )\right )-\left (p r \log \left (a+\frac {b (c+d x)}{d}-\frac {b c}{d}\right )-\log \left (\left (a+\frac {b (c+d x)}{d}-\frac {b c}{d}\right )^{p r}\right )\right ) \left (\log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log \left (g-\frac {c h}{d}+\frac {h (c+d x)}{d}\right )+\operatorname {PolyLog}\left (2,\frac {h (c+d x)}{d g-c h}+1\right )\right )}{d}+\frac {\frac {\log ^2\left ((c+d x)^{q r}\right ) \log \left (g-\frac {c h}{d}+\frac {h (c+d x)}{d}\right )}{2 q r}-\frac {h \left (\frac {\log ^2\left ((c+d x)^{q r}\right ) \log \left (\frac {h (c+d x)}{d g-c h}+1\right )}{h}-\frac {2 q r \left (q r \int \frac {\operatorname {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right )}{c+d x}d(c+d x)-\log \left ((c+d x)^{q r}\right ) \operatorname {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right )\right )}{h}\right )}{2 q r}}{d}\right )}{h}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\log (g+h x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h}-\frac {2 b p r \left (-\left (\left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \left (\frac {\log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log (g+h x)}{b}+\frac {\operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{b}\right )\right )+\frac {\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left (g-\frac {a h}{b}+\frac {h (a+b x)}{b}\right )}{2 p r}-\frac {h \left (\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left (\frac {h (a+b x)}{b g-a h}+1\right )}{h}-\frac {2 p r \left (p r \operatorname {PolyLog}\left (3,-\frac {h (a+b x)}{b g-a h}\right )-\log \left ((a+b x)^{p r}\right ) \operatorname {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right )\right )}{h}\right )}{2 p r}}{b}+\frac {q r \left (\frac {1}{2} \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )+\log \left (\frac {b (d g-c h)}{d \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}\right )-\log \left (-\frac {b (d g-c h) (a+b x)}{(b c-a d) \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}\right )\right ) \log ^2\left (\frac {(b c-a d) \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}{(b g-a h) \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}\right )-\operatorname {PolyLog}\left (2,\frac {h \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}{d \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}\right ) \log \left (\frac {(b c-a d) \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}{(b g-a h) \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}\right )+\operatorname {PolyLog}\left (2,\frac {(b g-a h) \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}{(b c-a d) \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}\right ) \log \left (\frac {(b c-a d) \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}{(b g-a h) \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}\right )-\frac {1}{2} \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )-\log \left (-\frac {h (a+b x)}{b g-a h}\right )\right ) \left (\log \left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right )+\log \left (\frac {(b c-a d) \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}{(b g-a h) \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}\right )\right )^2+\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log \left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right ) \log \left (g-\frac {a h}{b}+\frac {h (a+b x)}{b}\right )-\left (\log \left (\frac {(b c-a d) \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}{(b g-a h) \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}\right )-\log \left (g-\frac {a h}{b}+\frac {h (a+b x)}{b}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b c-a d}+1\right )+\left (\log \left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right )+\log \left (\frac {(b c-a d) \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}{(b g-a h) \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}\right )\right ) \operatorname {PolyLog}\left (2,\frac {h (a+b x)}{b g-a h}+1\right )-\operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b c-a d}+1\right )-\operatorname {PolyLog}\left (3,\frac {h \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}{d \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}\right )+\operatorname {PolyLog}\left (3,\frac {(b g-a h) \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}{(b c-a d) \left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )}\right )-\operatorname {PolyLog}\left (3,\frac {h (a+b x)}{b g-a h}+1\right )\right )-\left (q r \log \left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right )-\log \left (\left (c-\frac {a d}{b}+\frac {d (a+b x)}{b}\right )^{q r}\right )\right ) \left (\log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log \left (g-\frac {a h}{b}+\frac {h (a+b x)}{b}\right )+\operatorname {PolyLog}\left (2,\frac {h (a+b x)}{b g-a h}+1\right )\right )}{b}\right )}{h}-\frac {2 d q r \left (-\left (\left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \left (\frac {\log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x)}{d}+\frac {\operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{d}\right )\right )+\frac {\frac {\log ^2\left ((c+d x)^{q r}\right ) \log \left (g-\frac {c h}{d}+\frac {h (c+d x)}{d}\right )}{2 q r}-\frac {h \left (\frac {\log ^2\left ((c+d x)^{q r}\right ) \log \left (\frac {h (c+d x)}{d g-c h}+1\right )}{h}-\frac {2 q r \left (q r \operatorname {PolyLog}\left (3,-\frac {h (c+d x)}{d g-c h}\right )-\log \left ((c+d x)^{q r}\right ) \operatorname {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right )\right )}{h}\right )}{2 q r}}{d}+\frac {p r \left (\frac {1}{2} \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )+\log \left (\frac {d (b g-a h)}{b \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}\right )-\log \left (\frac {d (b g-a h) (c+d x)}{(b c-a d) \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}\right )\right ) \log ^2\left (-\frac {(b c-a d) \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}{(d g-c h) \left (\left (a-\frac {b c}{d}\right ) d+b (c+d x)\right )}\right )-\operatorname {PolyLog}\left (2,\frac {h \left (\left (a-\frac {b c}{d}\right ) d+b (c+d x)\right )}{b \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}\right ) \log \left (-\frac {(b c-a d) \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}{(d g-c h) \left (\left (a-\frac {b c}{d}\right ) d+b (c+d x)\right )}\right )+\operatorname {PolyLog}\left (2,-\frac {(d g-c h) \left (\left (a-\frac {b c}{d}\right ) d+b (c+d x)\right )}{(b c-a d) \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}\right ) \log \left (-\frac {(b c-a d) \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}{(d g-c h) \left (\left (a-\frac {b c}{d}\right ) d+b (c+d x)\right )}\right )-\frac {1}{2} \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )-\log \left (-\frac {h (c+d x)}{d g-c h}\right )\right ) \left (\log \left (a+\frac {b (c+d x)}{d}-\frac {b c}{d}\right )+\log \left (-\frac {(b c-a d) \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}{(d g-c h) \left (\left (a-\frac {b c}{d}\right ) d+b (c+d x)\right )}\right )\right )^2+\log \left (\frac {b (c+d x)}{b c-a d}\right ) \log \left (a+\frac {b (c+d x)}{d}-\frac {b c}{d}\right ) \log \left (g-\frac {c h}{d}+\frac {h (c+d x)}{d}\right )-\left (\log \left (-\frac {(b c-a d) \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}{(d g-c h) \left (\left (a-\frac {b c}{d}\right ) d+b (c+d x)\right )}\right )-\log \left (g-\frac {c h}{d}+\frac {h (c+d x)}{d}\right )\right ) \operatorname {PolyLog}\left (2,1-\frac {b (c+d x)}{b c-a d}\right )+\left (\log \left (a+\frac {b (c+d x)}{d}-\frac {b c}{d}\right )+\log \left (-\frac {(b c-a d) \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}{(d g-c h) \left (\left (a-\frac {b c}{d}\right ) d+b (c+d x)\right )}\right )\right ) \operatorname {PolyLog}\left (2,\frac {h (c+d x)}{d g-c h}+1\right )-\operatorname {PolyLog}\left (3,1-\frac {b (c+d x)}{b c-a d}\right )-\operatorname {PolyLog}\left (3,\frac {h \left (\left (a-\frac {b c}{d}\right ) d+b (c+d x)\right )}{b \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}\right )+\operatorname {PolyLog}\left (3,-\frac {(d g-c h) \left (\left (a-\frac {b c}{d}\right ) d+b (c+d x)\right )}{(b c-a d) \left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )}\right )-\operatorname {PolyLog}\left (3,\frac {h (c+d x)}{d g-c h}+1\right )\right )-\left (p r \log \left (a+\frac {b (c+d x)}{d}-\frac {b c}{d}\right )-\log \left (\left (a+\frac {b (c+d x)}{d}-\frac {b c}{d}\right )^{p r}\right )\right ) \left (\log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log \left (g-\frac {c h}{d}+\frac {h (c+d x)}{d}\right )+\operatorname {PolyLog}\left (2,\frac {h (c+d x)}{d g-c h}+1\right )\right )}{d}\right )}{h}\)

Input:

Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(g + h*x),x]
 

Output:

(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2*Log[g + h*x])/h - (2*b*p*r*(-((Log 
[(a + b*x)^(p*r)] + Log[(c + d*x)^(q*r)] - Log[e*(f*(a + b*x)^p*(c + d*x)^ 
q)^r])*((Log[-((h*(a + b*x))/(b*g - a*h))]*Log[g + h*x])/b + PolyLog[2, (b 
*(g + h*x))/(b*g - a*h)]/b)) + ((Log[(a + b*x)^(p*r)]^2*Log[g - (a*h)/b + 
(h*(a + b*x))/b])/(2*p*r) - (h*((Log[(a + b*x)^(p*r)]^2*Log[1 + (h*(a + b* 
x))/(b*g - a*h)])/h - (2*p*r*(-(Log[(a + b*x)^(p*r)]*PolyLog[2, -((h*(a + 
b*x))/(b*g - a*h))]) + p*r*PolyLog[3, -((h*(a + b*x))/(b*g - a*h))]))/h))/ 
(2*p*r))/b + (-((q*r*Log[c - (a*d)/b + (d*(a + b*x))/b] - Log[(c - (a*d)/b 
 + (d*(a + b*x))/b)^(q*r)])*(Log[-((h*(a + b*x))/(b*g - a*h))]*Log[g - (a* 
h)/b + (h*(a + b*x))/b] + PolyLog[2, 1 + (h*(a + b*x))/(b*g - a*h)])) + q* 
r*(((Log[-((d*(a + b*x))/(b*c - a*d))] + Log[(b*(d*g - c*h))/(d*(b*(g - (a 
*h)/b) + h*(a + b*x)))] - Log[-((b*(d*g - c*h)*(a + b*x))/((b*c - a*d)*(b* 
(g - (a*h)/b) + h*(a + b*x))))])*Log[((b*c - a*d)*(b*(g - (a*h)/b) + h*(a 
+ b*x)))/((b*g - a*h)*(b*(c - (a*d)/b) + d*(a + b*x)))]^2)/2 - ((Log[-((d* 
(a + b*x))/(b*c - a*d))] - Log[-((h*(a + b*x))/(b*g - a*h))])*(Log[c - (a* 
d)/b + (d*(a + b*x))/b] + Log[((b*c - a*d)*(b*(g - (a*h)/b) + h*(a + b*x)) 
)/((b*g - a*h)*(b*(c - (a*d)/b) + d*(a + b*x)))])^2)/2 + Log[-((d*(a + b*x 
))/(b*c - a*d))]*Log[c - (a*d)/b + (d*(a + b*x))/b]*Log[g - (a*h)/b + (h*( 
a + b*x))/b] - (Log[((b*c - a*d)*(b*(g - (a*h)/b) + h*(a + b*x)))/((b*g - 
a*h)*(b*(c - (a*d)/b) + d*(a + b*x)))] - Log[g - (a*h)/b + (h*(a + b*x)...
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

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 2822

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

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 2840

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_ 
Symbol] :> Simp[1/g   Subst[Int[(a + b*Log[1 + c*e*(x/g)])/x, x], x, f + g* 
x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g + c 
*(e*f - d*g), 0]
 

rule 2841

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

rule 2881

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log 
[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Sym 
bol] :> Simp[1/e   Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Log[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 2885

Int[(Log[(a_) + (b_.)*(x_)]*Log[(c_) + (d_.)*(x_)])/(x_), x_Symbol] :> Simp 
[Log[(-b)*(x/a)]*Log[a + b*x]*Log[c + d*x], x] + (Simp[(1/2)*(Log[(-b)*(x/a 
)] - Log[(-(b*c - a*d))*(x/(a*(c + d*x)))] + Log[(b*c - a*d)/(b*(c + d*x))] 
)*Log[a*((c + d*x)/(c*(a + b*x)))]^2, x] - Simp[(1/2)*(Log[(-b)*(x/a)] - Lo 
g[(-d)*(x/c)])*(Log[a + b*x] + Log[a*((c + d*x)/(c*(a + b*x)))])^2, x] + Si 
mp[(Log[c + d*x] - Log[a*((c + d*x)/(c*(a + b*x)))])*PolyLog[2, 1 + b*(x/a) 
], x] + Simp[(Log[a + b*x] + Log[a*((c + d*x)/(c*(a + b*x)))])*PolyLog[2, 1 
 + d*(x/c)], x] + Simp[Log[a*((c + d*x)/(c*(a + b*x)))]*PolyLog[2, c*((a + 
b*x)/(a*(c + d*x)))], x] - Simp[Log[a*((c + d*x)/(c*(a + b*x)))]*PolyLog[2, 
 d*((a + b*x)/(b*(c + d*x)))], x] - Simp[PolyLog[3, 1 + b*(x/a)], x] - Simp 
[PolyLog[3, 1 + d*(x/c)], x] + Simp[PolyLog[3, c*((a + b*x)/(a*(c + d*x)))] 
, x] - Simp[PolyLog[3, d*((a + b*x)/(b*(c + d*x)))], x]) /; FreeQ[{a, b, c, 
 d}, x] && NeQ[b*c - a*d, 0]
 

rule 2887

Int[(Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*Log[(h_.)*((i_.) + (j_.)*(x_))^(m 
_.)])/(x_), x_Symbol] :> Simp[m   Int[Log[i + j*x]*(Log[c*(d + e*x)^n]/x), 
x], x] - Simp[(m*Log[i + j*x] - Log[h*(i + j*x)^m])   Int[Log[c*(d + e*x)^n 
]/x, x], x] /; FreeQ[{c, d, e, h, i, j, m, n}, x] && NeQ[e*i - d*j, 0] && N 
eQ[i + j*x, h*(i + j*x)^m]
 

rule 2890

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

rule 2983

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.)) 
^(r_.)]^2/((g_.) + (h_.)*(x_)), x_Symbol] :> Simp[Log[g + h*x]*(Log[e*(f*(a 
 + b*x)^p*(c + d*x)^q)^r]^2/h), x] + (-Simp[2*b*p*(r/h)   Int[Log[g + h*x]* 
(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(a + b*x)), x], x] - Simp[2*d*q*(r/h) 
   Int[Log[g + h*x]*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(c + d*x)), x], x 
]) /; FreeQ[{a, b, c, d, e, f, g, h, p, q, r}, x] && NeQ[b*c - a*d, 0]
 

rule 2986

Int[(Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.) 
)^(r_.)]*((s_.) + Log[(i_.)*((g_.) + (h_.)*(x_))^(n_.)]*(t_.)))/((j_.) + (k 
_.)*(x_)), x_Symbol] :> Simp[(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - Log[(a 
 + b*x)^(p*r)] - Log[(c + d*x)^(q*r)])   Int[(s + t*Log[i*(g + h*x)^n])/(j 
+ k*x), x], x] + (Int[(Log[(a + b*x)^(p*r)]*(s + t*Log[i*(g + h*x)^n]))/(j 
+ k*x), x] + Int[(Log[(c + d*x)^(q*r)]*(s + t*Log[i*(g + h*x)^n]))/(j + k*x 
), x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, s, t, n, p, q, r}, x] && 
NeQ[b*c - a*d, 0]
 

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]
 

Maple [F]

\[\int \frac {{\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}^{2}}{h x +g}d x\]

Input:

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(h*x+g),x)
 

Output:

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(h*x+g),x)
 

Fricas [F]

\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{h x + g} \,d x } \] Input:

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(h*x+g),x, algorithm="fricas" 
)
 

Output:

integral(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2/(h*x + g), x)
 

Sympy [F]

\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx=\int \frac {\log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}^{2}}{g + h x}\, dx \] Input:

integrate(ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)**2/(h*x+g),x)
 

Output:

Integral(log(e*(f*(a + b*x)**p*(c + d*x)**q)**r)**2/(g + h*x), x)
 

Maxima [F]

\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{h x + g} \,d x } \] Input:

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(h*x+g),x, algorithm="maxima" 
)
 

Output:

integrate(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2/(h*x + g), x)
 

Giac [F]

\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{h x + g} \,d x } \] Input:

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(h*x+g),x, algorithm="giac")
 

Output:

integrate(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2/(h*x + g), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx=\int \frac {{\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2}{g+h\,x} \,d x \] Input:

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^2/(g + h*x),x)
 

Output:

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^2/(g + h*x), x)
 

Reduce [F]

\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx=\text {too large to display} \] Input:

int(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(h*x+g),x)
                                                                                    
                                                                                    
 

Output:

(3*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)**2/(a**2*c*d*g*h*q + 
a**2*c*d*h**2*q*x + a**2*d**2*g*h*q*x + a**2*d**2*h**2*q*x**2 + a*b*c**2*g 
*h*p + a*b*c**2*h**2*p*x + a*b*c*d*g**2*p + a*b*c*d*g**2*q + 2*a*b*c*d*g*h 
*p*x + 2*a*b*c*d*g*h*q*x + a*b*c*d*h**2*p*x**2 + a*b*c*d*h**2*q*x**2 + a*b 
*d**2*g**2*p*x + a*b*d**2*g**2*q*x + a*b*d**2*g*h*p*x**2 + 2*a*b*d**2*g*h* 
q*x**2 + a*b*d**2*h**2*q*x**3 + b**2*c**2*g*h*p*x + b**2*c**2*h**2*p*x**2 
+ b**2*c*d*g**2*p*x + b**2*c*d*g**2*q*x + 2*b**2*c*d*g*h*p*x**2 + b**2*c*d 
*g*h*q*x**2 + b**2*c*d*h**2*p*x**3 + b**2*d**2*g**2*p*x**2 + b**2*d**2*g** 
2*q*x**2 + b**2*d**2*g*h*p*x**3 + b**2*d**2*g*h*q*x**3),x)*a**3*c*d**2*h** 
2*q**2*r - 3*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)**2/(a**2*c* 
d*g*h*q + a**2*c*d*h**2*q*x + a**2*d**2*g*h*q*x + a**2*d**2*h**2*q*x**2 + 
a*b*c**2*g*h*p + a*b*c**2*h**2*p*x + a*b*c*d*g**2*p + a*b*c*d*g**2*q + 2*a 
*b*c*d*g*h*p*x + 2*a*b*c*d*g*h*q*x + a*b*c*d*h**2*p*x**2 + a*b*c*d*h**2*q* 
x**2 + a*b*d**2*g**2*p*x + a*b*d**2*g**2*q*x + a*b*d**2*g*h*p*x**2 + 2*a*b 
*d**2*g*h*q*x**2 + a*b*d**2*h**2*q*x**3 + b**2*c**2*g*h*p*x + b**2*c**2*h* 
*2*p*x**2 + b**2*c*d*g**2*p*x + b**2*c*d*g**2*q*x + 2*b**2*c*d*g*h*p*x**2 
+ b**2*c*d*g*h*q*x**2 + b**2*c*d*h**2*p*x**3 + b**2*d**2*g**2*p*x**2 + b** 
2*d**2*g**2*q*x**2 + b**2*d**2*g*h*p*x**3 + b**2*d**2*g*h*q*x**3),x)*a**3* 
d**3*g*h*q**2*r + 6*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)**2/( 
a**2*c*d*g*h*q + a**2*c*d*h**2*q*x + a**2*d**2*g*h*q*x + a**2*d**2*h**2...