\(\int \frac {\log ^2(e (f (a+b x)^p (c+d x)^q)^r)}{(g+h x)^4} \, dx\) [42]

Optimal result

Integrand size = 31, antiderivative size = 1957 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^4} \, dx =\text {Too large to display} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(47127\) vs. \(2(1957)=3914\).

Time = 6.52 (sec) , antiderivative size = 47127, normalized size of antiderivative = 24.08 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^4} \, dx=\text {Result too large to show} \] Input:

Integrate[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(g + h*x)^4,x]
 

Output:

Result too large to show
 

Rubi [A] (warning: unable to verify)

Time = 4.68 (sec) , antiderivative size = 1656, normalized size of antiderivative = 0.85, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.548, Rules used = {2984, 2993, 54, 2009, 2858, 27, 2789, 2756, 54, 2009, 2789, 2751, 16, 2779, 2838, 2865, 2009}

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.

\(\displaystyle \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^4} \, dx\)

\(\Big \downarrow \) 2984

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

\(\Big \downarrow \) 2993

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2858

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2751

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 2865

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

\(\Big \downarrow \) 2009

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

Input:

Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(g + h*x)^4,x]
 

Output:

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

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

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 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2751

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x 
_Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* 
(n/d)   Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, 
x] && EqQ[r*(q + 1) + 1, 0]
 

rule 2756

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), 
x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] 
- Simp[b*n*(p/(e*(q + 1)))   Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 
 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, 
 -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] & 
& NeQ[q, 1]))
 

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 2789

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ 
(x_), x_Symbol] :> Simp[1/d   Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x 
), x], x] - Simp[e/d   Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free 
Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
 

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 2858

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ 
.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e   Subst[In 
t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + 
e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - 
d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
 

rule 2865

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Sy 
mbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, 
Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunctionQ[ 
RFx, x] && IntegerQ[p]
 

rule 2984

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

rule 2993

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.)) 
^(r_.)]*(RFx_.), x_Symbol] :> Simp[p*r   Int[RFx*Log[a + b*x], x], x] + (Si 
mp[q*r   Int[RFx*Log[c + d*x], x], x] - Simp[(p*r*Log[a + b*x] + q*r*Log[c 
+ d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])   Int[RFx, x], x]) /; FreeQ[ 
{a, b, c, d, e, f, p, q, r}, x] && RationalFunctionQ[RFx, x] && NeQ[b*c - a 
*d, 0] &&  !MatchQ[RFx, (u_.)*(a + b*x)^(m_.)*(c + d*x)^(n_.) /; IntegersQ[ 
m, n]]
 

Maple [F]

\[\int \frac {{\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}^{2}}{\left (h x +g \right )^{4}}d x\]

Input:

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(h*x+g)^4,x)
 

Output:

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(h*x+g)^4,x)
 

Fricas [F]

\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^4} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{{\left (h x + g\right )}^{4}} \,d x } \] Input:

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(h*x+g)^4,x, algorithm="frica 
s")
 

Output:

integral(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2/(h^4*x^4 + 4*g*h^3*x^3 + 6 
*g^2*h^2*x^2 + 4*g^3*h*x + g^4), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^4} \, dx=\text {Timed out} \] Input:

integrate(ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)**2/(h*x+g)**4,x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4732 vs. \(2 (1875) = 3750\).

Time = 0.66 (sec) , antiderivative size = 4732, normalized size of antiderivative = 2.42 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^4} \, dx=\text {Too large to display} \] Input:

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(h*x+g)^4,x, algorithm="maxim 
a")
 

Output:

1/3*(2*b^3*f*p*log(b*x + a)/(b^3*g^3 - 3*a*b^2*g^2*h + 3*a^2*b*g*h^2 - a^3 
*h^3) + 2*d^3*f*q*log(d*x + c)/(d^3*g^3 - 3*c*d^2*g^2*h + 3*c^2*d*g*h^2 - 
c^3*h^3) - 2*(3*a*b^2*d^3*f*g^2*h*q - 3*a^2*b*d^3*f*g*h^2*q + a^3*d^3*f*h^ 
3*q - (d^3*f*g^3*(p + q) - 3*c*d^2*f*g^2*h*p + 3*c^2*d*f*g*h^2*p - c^3*f*h 
^3*p)*b^3)*log(h*x + g)/((d^3*g^3*h^3 - 3*c*d^2*g^2*h^4 + 3*c^2*d*g*h^5 - 
c^3*h^6)*a^3 - 3*(d^3*g^4*h^2 - 3*c*d^2*g^3*h^3 + 3*c^2*d*g^2*h^4 - c^3*g* 
h^5)*a^2*b + 3*(d^3*g^5*h - 3*c*d^2*g^4*h^2 + 3*c^2*d*g^3*h^3 - c^3*g^2*h^ 
4)*a*b^2 - (d^3*g^6 - 3*c*d^2*g^5*h + 3*c^2*d*g^4*h^2 - c^3*g^3*h^3)*b^3) 
+ ((3*d^2*f*g*h^2*q - c*d*f*h^3*q)*a^2 - (d^2*f*g^2*h*(p + 6*q) - 2*c*d*f* 
g*h^2*(p + q) + c^2*f*h^3*p)*a*b - (c*d*f*g^2*h*(6*p + q) - 3*d^2*f*g^3*(p 
 + q) - 3*c^2*f*g*h^2*p)*b^2 - 2*(2*a*b*d^2*f*g*h^2*q - a^2*d^2*f*h^3*q - 
(d^2*f*g^2*h*(p + q) - 2*c*d*f*g*h^2*p + c^2*f*h^3*p)*b^2)*x)/((d^2*g^4*h^ 
2 - 2*c*d*g^3*h^3 + c^2*g^2*h^4)*a^2 - 2*(d^2*g^5*h - 2*c*d*g^4*h^2 + c^2* 
g^3*h^3)*a*b + (d^2*g^6 - 2*c*d*g^5*h + c^2*g^4*h^2)*b^2 + ((d^2*g^2*h^4 - 
 2*c*d*g*h^5 + c^2*h^6)*a^2 - 2*(d^2*g^3*h^3 - 2*c*d*g^2*h^4 + c^2*g*h^5)* 
a*b + (d^2*g^4*h^2 - 2*c*d*g^3*h^3 + c^2*g^2*h^4)*b^2)*x^2 + 2*((d^2*g^3*h 
^3 - 2*c*d*g^2*h^4 + c^2*g*h^5)*a^2 - 2*(d^2*g^4*h^2 - 2*c*d*g^3*h^3 + c^2 
*g^2*h^4)*a*b + (d^2*g^5*h - 2*c*d*g^4*h^2 + c^2*g^3*h^3)*b^2)*x))*r*log(( 
(b*x + a)^p*(d*x + c)^q*f)^r*e)/(f*h) + 1/3*(2*(3*a*b^2*d^3*f^2*g^2*h*p*q 
- 3*a^2*b*d^3*f^2*g*h^2*p*q + a^3*d^3*f^2*h^3*p*q - (3*c*d^2*f^2*g^2*h*...
 

Giac [F]

\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^4} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{{\left (h x + g\right )}^{4}} \,d x } \] Input:

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(h*x+g)^4,x, algorithm="giac" 
)
 

Output:

integrate(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2/(h*x + g)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^4} \, dx=\int \frac {{\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2}{{\left (g+h\,x\right )}^4} \,d x \] Input:

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^2/(g + h*x)^4,x)
 

Output:

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^2/(g + h*x)^4, x)
 

Reduce [F]

\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^4} \, dx=\text {too large to display} \] Input:

int(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(h*x+g)^4,x)
 

Output:

( - 18*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)/(a**2*c*d*g**4 + 
4*a**2*c*d*g**3*h*x + 6*a**2*c*d*g**2*h**2*x**2 + 4*a**2*c*d*g*h**3*x**3 + 
 a**2*c*d*h**4*x**4 + a**2*d**2*g**4*x + 4*a**2*d**2*g**3*h*x**2 + 6*a**2* 
d**2*g**2*h**2*x**3 + 4*a**2*d**2*g*h**3*x**4 + a**2*d**2*h**4*x**5 + a*b* 
c**2*g**4 + 4*a*b*c**2*g**3*h*x + 6*a*b*c**2*g**2*h**2*x**2 + 4*a*b*c**2*g 
*h**3*x**3 + a*b*c**2*h**4*x**4 + 2*a*b*c*d*g**4*x + 8*a*b*c*d*g**3*h*x**2 
 + 12*a*b*c*d*g**2*h**2*x**3 + 8*a*b*c*d*g*h**3*x**4 + 2*a*b*c*d*h**4*x**5 
 + a*b*d**2*g**4*x**2 + 4*a*b*d**2*g**3*h*x**3 + 6*a*b*d**2*g**2*h**2*x**4 
 + 4*a*b*d**2*g*h**3*x**5 + a*b*d**2*h**4*x**6 + b**2*c**2*g**4*x + 4*b**2 
*c**2*g**3*h*x**2 + 6*b**2*c**2*g**2*h**2*x**3 + 4*b**2*c**2*g*h**3*x**4 + 
 b**2*c**2*h**4*x**5 + b**2*c*d*g**4*x**2 + 4*b**2*c*d*g**3*h*x**3 + 6*b** 
2*c*d*g**2*h**2*x**4 + 4*b**2*c*d*g*h**3*x**5 + b**2*c*d*h**4*x**6),x)*a** 
7*c**5*d**2*g**6*h**8*q*r - 54*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p 
*r)*e)/(a**2*c*d*g**4 + 4*a**2*c*d*g**3*h*x + 6*a**2*c*d*g**2*h**2*x**2 + 
4*a**2*c*d*g*h**3*x**3 + a**2*c*d*h**4*x**4 + a**2*d**2*g**4*x + 4*a**2*d* 
*2*g**3*h*x**2 + 6*a**2*d**2*g**2*h**2*x**3 + 4*a**2*d**2*g*h**3*x**4 + a* 
*2*d**2*h**4*x**5 + a*b*c**2*g**4 + 4*a*b*c**2*g**3*h*x + 6*a*b*c**2*g**2* 
h**2*x**2 + 4*a*b*c**2*g*h**3*x**3 + a*b*c**2*h**4*x**4 + 2*a*b*c*d*g**4*x 
 + 8*a*b*c*d*g**3*h*x**2 + 12*a*b*c*d*g**2*h**2*x**3 + 8*a*b*c*d*g*h**3*x* 
*4 + 2*a*b*c*d*h**4*x**5 + a*b*d**2*g**4*x**2 + 4*a*b*d**2*g**3*h*x**3 ...