\(\int \frac {\log ^2(e (f (a+b x)^p (c+d x)^q)^r)}{(g+h x)^4} \, dx\) [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=-\frac {b^2 p^2 r^2}{3 h (b g-a h)^2 (g+h x)}-\frac {2 b d p q r^2}{3 h (b g-a h) (d g-c h) (g+h x)}-\frac {d^2 q^2 r^2}{3 h (d g-c h)^2 (g+h x)}-\frac {b^3 p^2 r^2 \log (a+b x)}{3 h (b g-a h)^3}-\frac {2 b d^2 p q r^2 \log (a+b x)}{3 h (b g-a h) (d g-c h)^2}-\frac {b^2 d p q r^2 \log (a+b x)}{3 h (b g-a h)^2 (d g-c h)}+\frac {b p^2 r^2 \log (a+b x)}{3 h (b g-a h) (g+h x)^2}+\frac {d p q r^2 \log (a+b x)}{3 h (d g-c h) (g+h x)^2}+\frac {2 d^2 p q r^2 \log (a+b x)}{3 h (d g-c h)^2 (g+h x)}-\frac {2 b^2 p^2 r^2 (a+b x) \log (a+b x)}{3 (b g-a h)^3 (g+h x)}-\frac {b d^2 p q r^2 \log (c+d x)}{3 h (b g-a h) (d g-c h)^2}-\frac {2 b^2 d p q r^2 \log (c+d x)}{3 h (b g-a h)^2 (d g-c h)}-\frac {d^3 q^2 r^2 \log (c+d x)}{3 h (d g-c h)^3}+\frac {b p q r^2 \log (c+d x)}{3 h (b g-a h) (g+h x)^2}+\frac {d q^2 r^2 \log (c+d x)}{3 h (d g-c h) (g+h x)^2}+\frac {2 b^2 p q r^2 \log (c+d x)}{3 h (b g-a h)^2 (g+h x)}-\frac {2 d^2 q^2 r^2 (c+d x) \log (c+d x)}{3 (d g-c h)^3 (g+h x)}+\frac {2 b^3 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 h (b g-a h)^3}+\frac {2 d^3 p q r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 h (d g-c h)^3}-\frac {b p r \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 )}{3 h (b g-a h) (g+h x)^2}-\frac {d q r \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 )}{3 h (d g-c h) (g+h x)^2}-\frac {2 b^2 p r \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 )}{3 h (b g-a h)^2 (g+h x)}-\frac {2 d^2 q r \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 )}{3 h (d g-c h)^2 (g+h x)}-\frac {2 b^3 p r \log (a+b x) \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 )}{3 h (b g-a h)^3}-\frac {2 d^3 q r \log (c+d x) \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 )}{3 h (d g-c h)^3}-\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 {b^3 p^2 r^2 \log (g+h x)}{h (b g-a h)^3}+\frac {b d^2 p q r^2 \log (g+h x)}{h (b g-a h) (d g-c h)^2}+\frac {b^2 d p q r^2 \log (g+h x)}{h (b g-a h)^2 (d g-c h)}+\frac {d^3 q^2 r^2 \log (g+h x)}{h (d g-c h)^3}+\frac {2 b^3 p r \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 ) \log (g+h x)}{3 h (b g-a h)^3}+\frac {2 d^3 q r \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 ) \log (g+h x)}{3 h (d g-c h)^3}-\frac {2 d^3 p q r^2 \log (a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right )}{3 h (d g-c h)^3}-\frac {2 b^3 p q r^2 \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right )}{3 h (b g-a h)^3}-\frac {2 b^3 p^2 r^2 \log (a+b x) \log \left (1+\frac {b g-a h}{h (a+b x)}\right )}{3 h (b g-a h)^3}-\frac {2 d^3 q^2 r^2 \log (c+d x) \log \left (1+\frac {d g-c h}{h (c+d x)}\right )}{3 h (d g-c h)^3}+\frac {2 b^3 p^2 r^2 \operatorname {PolyLog}\left (2,-\frac {b g-a h}{h (a+b x)}\right )}{3 h (b g-a h)^3}+\frac {2 d^3 p q r^2 \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{3 h (d g-c h)^3}-\frac {2 d^3 p q r^2 \operatorname {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right )}{3 h (d g-c h)^3}+\frac {2 d^3 q^2 r^2 \operatorname {PolyLog}\left (2,-\frac {d g-c h}{h (c+d x)}\right )}{3 h (d g-c h)^3}+\frac {2 b^3 p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{3 h (b g-a h)^3}-\frac {2 b^3 p q r^2 \operatorname {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right )}{3 h (b g-a h)^3} \]

[Out]

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

Rubi [A] (verified)

Time = 1.45 (sec) , antiderivative size = 1957, normalized size of antiderivative = 1.00, number of steps used = 57, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.484, Rules used = {2584, 2593, 2458, 2389, 2379, 2438, 2351, 31, 2356, 46, 2465, 2441, 2440, 2442, 36} \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^4} \, dx=-\frac {p^2 r^2 \log (a+b x) b^3}{3 h (b g-a h)^3}+\frac {2 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) b^3}{3 h (b g-a h)^3}-\frac {2 p r \log (a+b x) \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 ) b^3}{3 h (b g-a h)^3}+\frac {p^2 r^2 \log (g+h x) b^3}{h (b g-a h)^3}+\frac {2 p r \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 ) \log (g+h x) b^3}{3 h (b g-a h)^3}-\frac {2 p q r^2 \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right ) b^3}{3 h (b g-a h)^3}-\frac {2 p^2 r^2 \log (a+b x) \log \left (\frac {b g-a h}{h (a+b x)}+1\right ) b^3}{3 h (b g-a h)^3}+\frac {2 p^2 r^2 \operatorname {PolyLog}\left (2,-\frac {b g-a h}{h (a+b x)}\right ) b^3}{3 h (b g-a h)^3}+\frac {2 p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) b^3}{3 h (b g-a h)^3}-\frac {2 p q r^2 \operatorname {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right ) b^3}{3 h (b g-a h)^3}-\frac {d p q r^2 \log (a+b x) b^2}{3 h (b g-a h)^2 (d g-c h)}-\frac {2 p^2 r^2 (a+b x) \log (a+b x) b^2}{3 (b g-a h)^3 (g+h x)}-\frac {2 d p q r^2 \log (c+d x) b^2}{3 h (b g-a h)^2 (d g-c h)}+\frac {2 p q r^2 \log (c+d x) b^2}{3 h (b g-a h)^2 (g+h x)}-\frac {2 p r \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 ) b^2}{3 h (b g-a h)^2 (g+h x)}+\frac {d p q r^2 \log (g+h x) b^2}{h (b g-a h)^2 (d g-c h)}-\frac {p^2 r^2 b^2}{3 h (b g-a h)^2 (g+h x)}-\frac {2 d^2 p q r^2 \log (a+b x) b}{3 h (b g-a h) (d g-c h)^2}+\frac {p^2 r^2 \log (a+b x) b}{3 h (b g-a h) (g+h x)^2}-\frac {d^2 p q r^2 \log (c+d x) b}{3 h (b g-a h) (d g-c h)^2}+\frac {p q r^2 \log (c+d x) b}{3 h (b g-a h) (g+h x)^2}-\frac {p r \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 ) b}{3 h (b g-a h) (g+h x)^2}+\frac {d^2 p q r^2 \log (g+h x) b}{h (b g-a h) (d g-c h)^2}-\frac {2 d p q r^2 b}{3 h (b g-a h) (d g-c h) (g+h x)}-\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^2 p q r^2 \log (a+b x)}{3 h (d g-c h)^2 (g+h x)}+\frac {d p q r^2 \log (a+b x)}{3 h (d g-c h) (g+h x)^2}-\frac {d^3 q^2 r^2 \log (c+d x)}{3 h (d g-c h)^3}-\frac {2 d^2 q^2 r^2 (c+d x) \log (c+d x)}{3 (d g-c h)^3 (g+h x)}+\frac {d q^2 r^2 \log (c+d x)}{3 h (d g-c h) (g+h x)^2}+\frac {2 d^3 p q r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 h (d g-c h)^3}-\frac {2 d^3 q r \log (c+d x) \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 )}{3 h (d g-c h)^3}-\frac {2 d^2 q r \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 )}{3 h (d g-c h)^2 (g+h x)}-\frac {d q r \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 )}{3 h (d g-c h) (g+h x)^2}+\frac {d^3 q^2 r^2 \log (g+h x)}{h (d g-c h)^3}+\frac {2 d^3 q r \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 ) \log (g+h x)}{3 h (d g-c h)^3}-\frac {2 d^3 p q r^2 \log (a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right )}{3 h (d g-c h)^3}-\frac {2 d^3 q^2 r^2 \log (c+d x) \log \left (\frac {d g-c h}{h (c+d x)}+1\right )}{3 h (d g-c h)^3}+\frac {2 d^3 p q r^2 \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{3 h (d g-c h)^3}-\frac {2 d^3 p q r^2 \operatorname {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right )}{3 h (d g-c h)^3}+\frac {2 d^3 q^2 r^2 \operatorname {PolyLog}\left (2,-\frac {d g-c h}{h (c+d x)}\right )}{3 h (d g-c h)^3}-\frac {d^2 q^2 r^2}{3 h (d g-c h)^2 (g+h x)} \]

[In]

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

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 46

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

Rule 2351

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] - Dist[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 2356

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] - Dist[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] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2379

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] + Dist[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 2389

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

Rule 2438

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 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[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 2441

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] - Dist[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 2442

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f + g*
x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/(g*(q + 1))), x] - Dist[b*e*(n/(g*(q + 1))), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(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 2465

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> 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] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2584

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] + (-Dist[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] -
Dist[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 2593

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*(RFx_.), x_Symbol] :> Dist[
p*r, Int[RFx*Log[a + b*x], x], x] + (Dist[q*r, Int[RFx*Log[c + d*x], x], x] - Dist[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_.) /; Integ
ersQ[m, n]]

Rubi steps \begin{align*} \text {integral}& = -\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) \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}+\frac {\left (2 b p^2 r^2\right ) \int \frac {\log (a+b x)}{(a+b x) (g+h x)^3} \, dx}{3 h}+\frac {\left (2 b p q r^2\right ) \int \frac {\log (c+d x)}{(a+b x) (g+h x)^3} \, dx}{3 h}+\frac {\left (2 d p q r^2\right ) \int \frac {\log (a+b x)}{(c+d x) (g+h x)^3} \, dx}{3 h}+\frac {\left (2 d q^2 r^2\right ) \int \frac {\log (c+d x)}{(c+d x) (g+h x)^3} \, dx}{3 h}-\frac {\left (2 b p r \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 )\right ) \int \frac {1}{(a+b x) (g+h x)^3} \, dx}{3 h}-\frac {\left (2 d q r \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 )\right ) \int \frac {1}{(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}+\frac {\left (2 p^2 r^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x \left (\frac {b g-a h}{b}+\frac {h x}{b}\right )^3} \, dx,x,a+b x\right )}{3 h}+\frac {\left (2 b p q r^2\right ) \int \left (\frac {b^3 \log (c+d x)}{(b g-a h)^3 (a+b x)}-\frac {h \log (c+d x)}{(b g-a h) (g+h x)^3}-\frac {b h \log (c+d x)}{(b g-a h)^2 (g+h x)^2}-\frac {b^2 h \log (c+d x)}{(b g-a h)^3 (g+h x)}\right ) \, dx}{3 h}+\frac {\left (2 d p q r^2\right ) \int \left (\frac {d^3 \log (a+b x)}{(d g-c h)^3 (c+d x)}-\frac {h \log (a+b x)}{(d g-c h) (g+h x)^3}-\frac {d h \log (a+b x)}{(d g-c h)^2 (g+h x)^2}-\frac {d^2 h \log (a+b x)}{(d g-c h)^3 (g+h x)}\right ) \, dx}{3 h}+\frac {\left (2 q^2 r^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x \left (\frac {d g-c h}{d}+\frac {h x}{d}\right )^3} \, dx,x,c+d x\right )}{3 h}-\frac {\left (2 b p r \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 )\right ) \int \left (\frac {b^3}{(b g-a h)^3 (a+b x)}-\frac {h}{(b g-a h) (g+h x)^3}-\frac {b h}{(b g-a h)^2 (g+h x)^2}-\frac {b^2 h}{(b g-a h)^3 (g+h x)}\right ) \, dx}{3 h}-\frac {\left (2 d q r \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 )\right ) \int \left (\frac {d^3}{(d g-c h)^3 (c+d x)}-\frac {h}{(d g-c h) (g+h x)^3}-\frac {d h}{(d g-c h)^2 (g+h x)^2}-\frac {d^2 h}{(d g-c h)^3 (g+h x)}\right ) \, dx}{3 h} \\ & = -\frac {b p r \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 )}{3 h (b g-a h) (g+h x)^2}-\frac {d q r \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 )}{3 h (d g-c h) (g+h x)^2}-\frac {2 b^2 p r \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 )}{3 h (b g-a h)^2 (g+h x)}-\frac {2 d^2 q r \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 )}{3 h (d g-c h)^2 (g+h x)}-\frac {2 b^3 p r \log (a+b x) \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 )}{3 h (b g-a h)^3}-\frac {2 d^3 q r \log (c+d x) \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 )}{3 h (d g-c h)^3}-\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^3 p r \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 ) \log (g+h x)}{3 h (b g-a h)^3}+\frac {2 d^3 q r \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 ) \log (g+h x)}{3 h (d g-c h)^3}-\frac {\left (2 p^2 r^2\right ) \text {Subst}\left (\int \frac {\log (x)}{\left (\frac {b g-a h}{b}+\frac {h x}{b}\right )^3} \, dx,x,a+b x\right )}{3 (b g-a h)}+\frac {\left (2 b p^2 r^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x \left (\frac {b g-a h}{b}+\frac {h x}{b}\right )^2} \, dx,x,a+b x\right )}{3 h (b g-a h)}-\frac {\left (2 b^3 p q r^2\right ) \int \frac {\log (c+d x)}{g+h x} \, dx}{3 (b g-a h)^3}+\frac {\left (2 b^4 p q r^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3 h (b g-a h)^3}-\frac {\left (2 b^2 p q r^2\right ) \int \frac {\log (c+d x)}{(g+h x)^2} \, dx}{3 (b g-a h)^2}-\frac {\left (2 b p q r^2\right ) \int \frac {\log (c+d x)}{(g+h x)^3} \, dx}{3 (b g-a h)}-\frac {\left (2 d^3 p q r^2\right ) \int \frac {\log (a+b x)}{g+h x} \, dx}{3 (d g-c h)^3}+\frac {\left (2 d^4 p q r^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{3 h (d g-c h)^3}-\frac {\left (2 d^2 p q r^2\right ) \int \frac {\log (a+b x)}{(g+h x)^2} \, dx}{3 (d g-c h)^2}-\frac {\left (2 d p q r^2\right ) \int \frac {\log (a+b x)}{(g+h x)^3} \, dx}{3 (d g-c h)}-\frac {\left (2 q^2 r^2\right ) \text {Subst}\left (\int \frac {\log (x)}{\left (\frac {d g-c h}{d}+\frac {h x}{d}\right )^3} \, dx,x,c+d x\right )}{3 (d g-c h)}+\frac {\left (2 d q^2 r^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x \left (\frac {d g-c h}{d}+\frac {h x}{d}\right )^2} \, dx,x,c+d x\right )}{3 h (d g-c h)} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [B] (verified)

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

Time = 6.30 (sec) , antiderivative size = 47127, normalized size of antiderivative = 24.08 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^4} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Result too large to show

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

[In]

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

[Out]

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 = 0.80 (sec) , antiderivative size = 4732, normalized size of antiderivative = 2.42 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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*p*q - 3*c^2*d*f^2*g*h^2*p*q + c^3*f^2*h^3*p*q)*b^3)*(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^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) - 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*p^2 - 3*c^2*d*f^2*g*h^2*p^2 + c^3*f^2*h^3*p^2 - (p^2 + p*q)*d^3*f^2*g^3)*b^3)*(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^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) - 2*(3*a*b^2*d^3*f^2*g^2*h*q^2 - 3*a^2*b*d^3*f^2*g*h^2*q^2 + a^3*d^3*f^2*h^3*q^2 + (3*c*d^2*f^2*g^2*h*p*q
- 3*c^2*d*f^2*g*h^2*p*q + c^3*f^2*h^3*p*q - (p*q + q^2)*d^3*f^2*g^3)*b^3)*(log(d*x + c)*log((d*h*x + c*h)/(d*
g - c*h) + 1) + dilog(-(d*h*x + c*h)/(d*g - c*h)))/((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*a^
2*d^3*f^2*h^2*q^2 + (c*d^2*f^2*h^2*p*q - (p*q + 6*q^2)*d^3*f^2*g*h)*a*b - (5*c*d^2*f^2*g*h*p*q - 2*c^2*d*f^2*h
^2*p*q - 3*(p*q + q^2)*d^3*f^2*g^2)*b^2)*log(d*x + c)/((d^3*g^3*h^2 - 3*c*d^2*g^2*h^3 + 3*c^2*d*g*h^4 - c^3*h^
5)*a^2 - 2*(d^3*g^4*h - 3*c*d^2*g^3*h^2 + 3*c^2*d*g^2*h^3 - c^3*g*h^4)*a*b + (d^3*g^5 - 3*c*d^2*g^4*h + 3*c^2*
d*g^3*h^2 - c^3*g^2*h^3)*b^2) + 3*(a^3*d^3*f^2*h^3*q^2 + (c*d^2*f^2*h^3*p*q - (p*q + 3*q^2)*d^3*f^2*g*h^2)*a^2
*b - (4*c*d^2*f^2*g*h^2*p*q - c^2*d*f^2*h^3*p*q - 3*(p*q + q^2)*d^3*f^2*g^2*h)*a*b^2 + (c^3*f^2*h^3*p^2 - (p^2
+ 2*p*q + q^2)*d^3*f^2*g^3 + 3*(p^2 + p*q)*c*d^2*f^2*g^2*h - (3*p^2 + p*q)*c^2*d*f^2*g*h^2)*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) - ((d^3*f^2*g*h^3*q^2 - c*d^2*f^2*h^4*q^2)*a^3 - (2*c^2*d
*f^2*h^4*p*q + (2*p*q + 3*q^2)*d^3*f^2*g^2*h^2 - (4*p*q + 3*q^2)*c*d^2*f^2*g*h^3)*a^2*b - (c^3*f^2*h^4*p^2 - (
p^2 + 4*p*q + 3*q^2)*d^3*f^2*g^3*h + (3*p^2 + 8*p*q + 3*q^2)*c*d^2*f^2*g^2*h^2 - (3*p^2 + 4*p*q)*c^2*d*f^2*g*h
^3)*a*b^2 + (c^3*f^2*g*h^3*p^2 - (p^2 + 2*p*q + q^2)*d^3*f^2*g^4 + (3*p^2 + 4*p*q + q^2)*c*d^2*f^2*g^3*h - (3*
p^2 + 2*p*q)*c^2*d*f^2*g^2*h^2)*b^3 - ((d^3*f^2*g^3*h*p^2 - 3*c*d^2*f^2*g^2*h^2*p^2 + 3*c^2*d*f^2*g*h^3*p^2 -
c^3*f^2*h^4*p^2)*b^3*x + (d^3*f^2*g^4*p^2 - 3*c*d^2*f^2*g^3*h*p^2 + 3*c^2*d*f^2*g^2*h^2*p^2 - c^3*f^2*g*h^3*p^
2)*b^3)*log(b*x + a)^2 - 2*(b^3*d^3*f^2*g^4*p*q - 3*a*b^2*d^3*f^2*g^3*h*p*q + 3*a^2*b*d^3*f^2*g^2*h^2*p*q - a^
3*d^3*f^2*g*h^3*p*q + (b^3*d^3*f^2*g^3*h*p*q - 3*a*b^2*d^3*f^2*g^2*h^2*p*q + 3*a^2*b*d^3*f^2*g*h^3*p*q - a^3*d
^3*f^2*h^4*p*q)*x)*log(b*x + a)*log(d*x + c) - (b^3*d^3*f^2*g^4*q^2 - 3*a*b^2*d^3*f^2*g^3*h*q^2 + 3*a^2*b*d^3*
f^2*g^2*h^2*q^2 - a^3*d^3*f^2*g*h^3*q^2 + (b^3*d^3*f^2*g^3*h*q^2 - 3*a*b^2*d^3*f^2*g^2*h^2*q^2 + 3*a^2*b*d^3*f
^2*g*h^3*q^2 - a^3*d^3*f^2*h^4*q^2)*x)*log(d*x + c)^2 - (2*(d^3*f^2*g^2*h^2*p*q - c*d^2*f^2*g*h^3*p*q)*a^2*b -
(5*d^3*f^2*g^3*h*p*q - 6*c*d^2*f^2*g^2*h^2*p*q + c^2*d*f^2*g*h^3*p*q)*a*b^2 - (3*c^3*f^2*g*h^3*p^2 - 3*(p^2 +
p*q)*d^3*f^2*g^4 + (9*p^2 + 4*p*q)*c*d^2*f^2*g^3*h - (9*p^2 + p*q)*c^2*d*f^2*g^2*h^2)*b^3 + (2*(d^3*f^2*g*h^3
*p*q - c*d^2*f^2*h^4*p*q)*a^2*b - (5*d^3*f^2*g^2*h^2*p*q - 6*c*d^2*f^2*g*h^3*p*q + c^2*d*f^2*h^4*p*q)*a*b^2 -
(3*c^3*f^2*h^4*p^2 - 3*(p^2 + p*q)*d^3*f^2*g^3*h + (9*p^2 + 4*p*q)*c*d^2*f^2*g^2*h^2 - (9*p^2 + p*q)*c^2*d*f^2
*g*h^3)*b^3)*x)*log(b*x + a) - 2*((3*a*b^2*d^3*f^2*g^3*h*p*q - 3*a^2*b*d^3*f^2*g^2*h^2*p*q + a^3*d^3*f^2*g*h^3
*p*q + (3*c*d^2*f^2*g^3*h*p^2 - 3*c^2*d*f^2*g^2*h^2*p^2 + c^3*f^2*g*h^3*p^2 - (p^2 + p*q)*d^3*f^2*g^4)*b^3 + (
3*a*b^2*d^3*f^2*g^2*h^2*p*q - 3*a^2*b*d^3*f^2*g*h^3*p*q + a^3*d^3*f^2*h^4*p*q + (3*c*d^2*f^2*g^2*h^2*p^2 - 3*c
^2*d*f^2*g*h^3*p^2 + c^3*f^2*h^4*p^2 - (p^2 + p*q)*d^3*f^2*g^3*h)*b^3)*x)*log(b*x + a) + (3*a*b^2*d^3*f^2*g^3*
h*q^2 - 3*a^2*b*d^3*f^2*g^2*h^2*q^2 + a^3*d^3*f^2*g*h^3*q^2 + (3*c*d^2*f^2*g^3*h*p*q - 3*c^2*d*f^2*g^2*h^2*p*q
+ c^3*f^2*g*h^3*p*q - (p*q + q^2)*d^3*f^2*g^4)*b^3 + (3*a*b^2*d^3*f^2*g^2*h^2*q^2 - 3*a^2*b*d^3*f^2*g*h^3*q^2
+ a^3*d^3*f^2*h^4*q^2 + (3*c*d^2*f^2*g^2*h^2*p*q - 3*c^2*d*f^2*g*h^3*p*q + c^3*f^2*h^4*p*q - (p*q + q^2)*d^3*
f^2*g^3*h)*b^3)*x)*log(d*x + c))*log(h*x + g))/((d^3*g^4*h^3 - 3*c*d^2*g^3*h^4 + 3*c^2*d*g^2*h^5 - c^3*g*h^6)*
a^3 - 3*(d^3*g^5*h^2 - 3*c*d^2*g^4*h^3 + 3*c^2*d*g^3*h^4 - c^3*g^2*h^5)*a^2*b + 3*(d^3*g^6*h - 3*c*d^2*g^5*h^2
+ 3*c^2*d*g^4*h^3 - c^3*g^3*h^4)*a*b^2 - (d^3*g^7 - 3*c*d^2*g^6*h + 3*c^2*d*g^5*h^2 - c^3*g^4*h^3)*b^3 + ((d^
3*g^3*h^4 - 3*c*d^2*g^2*h^5 + 3*c^2*d*g*h^6 - c^3*h^7)*a^3 - 3*(d^3*g^4*h^3 - 3*c*d^2*g^3*h^4 + 3*c^2*d*g^2*h^
5 - c^3*g*h^6)*a^2*b + 3*(d^3*g^5*h^2 - 3*c*d^2*g^4*h^3 + 3*c^2*d*g^3*h^4 - c^3*g^2*h^5)*a*b^2 - (d^3*g^6*h -
3*c*d^2*g^5*h^2 + 3*c^2*d*g^4*h^3 - c^3*g^3*h^4)*b^3)*x))*r^2/(f^2*h) - 1/3*log(((b*x + a)^p*(d*x + c)^q*f)^r*
e)^2/((h*x + g)^3*h)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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