\(\int \frac {\log ^2(e (f (a+b x)^p (c+d x)^q)^r)}{(g+h x)^3} \, dx\) [41]

Optimal result

Integrand size = 31, antiderivative size = 1304 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^3} \, dx =\text {Too large to display} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(12086\) vs. \(2(1304)=2608\).

Time = 5.49 (sec) , antiderivative size = 12086, normalized size of antiderivative = 9.27 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^3} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 3.07 (sec) , antiderivative size = 1048, normalized size of antiderivative = 0.80, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {2984, 2993, 54, 2009, 2858, 27, 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.

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

\(\Big \downarrow \) 2984

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

\(\Big \downarrow \) 2993

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2858

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2751

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 2865

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 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]

\[\int \frac {{\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}^{2}}{\left (h x +g \right )^{3}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^3} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{{\left (h x + g\right )}^{3}} \,d x } \] Input:

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(h*x+g)^3,x, algorithm="frica 
s")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^3} \, dx=\text {Timed out} \] Input:

integrate(ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)**2/(h*x+g)**3,x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 1857, normalized size of antiderivative = 1.42 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^3} \, dx=\text {Too large to display} \] Input:

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(h*x+g)^3,x, algorithm="maxim 
a")
 

Output:

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

Giac [F]

\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^3} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{{\left (h x + g\right )}^{3}} \,d x } \] Input:

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(h*x+g)^3,x, algorithm="giac" 
)
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^3} \, dx=\int \frac {{\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2}{{\left (g+h\,x\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^3} \, dx=\text {too large to display} \] Input:

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

Output:

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