\(\int (g+h x)^3 \log ^2(e (f (a+b x)^p (c+d x)^q)^r) \, dx\) [35]

Optimal result

Integrand size = 31, antiderivative size = 2240 \[ \int (g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {2 (b g-a h)^3 p^2 r^2 x}{b^3}+\frac {5 (b g-a h)^3 p q r^2 x}{8 b^3}+\frac {5 (b g-a h)^2 (d g-c h) p q r^2 x}{12 b^2 d}+\frac {5 (b g-a h) (d g-c h)^2 p q r^2 x}{12 b d^2}+\frac {5 (d g-c h)^3 p q r^2 x}{8 d^3}+\frac {2 (d g-c h)^3 q^2 r^2 x}{d^3}+\frac {3 h (b g-a h)^2 p^2 r^2 (a+b x)^2}{4 b^4}+\frac {2 h^2 (b g-a h) p^2 r^2 (a+b x)^3}{9 b^4}+\frac {h^3 p^2 r^2 (a+b x)^4}{32 b^4}+\frac {3 h (d g-c h)^2 q^2 r^2 (c+d x)^2}{4 d^4}+\frac {2 h^2 (d g-c h) q^2 r^2 (c+d x)^3}{9 d^4}+\frac {h^3 q^2 r^2 (c+d x)^4}{32 d^4}+\frac {3 (b g-a h)^2 p q r^2 (g+h x)^2}{16 b^2 h}+\frac {(b g-a h) (d g-c h) p q r^2 (g+h x)^2}{6 b d h}+\frac {3 (d g-c h)^2 p q r^2 (g+h x)^2}{16 d^2 h}+\frac {7 (b g-a h) p q r^2 (g+h x)^3}{72 b h}+\frac {7 (d g-c h) p q r^2 (g+h x)^3}{72 d h}+\frac {p q r^2 (g+h x)^4}{16 h}+\frac {(b g-a h)^4 p q r^2 \log (a+b x)}{8 b^4 h}+\frac {(b g-a h)^3 (d g-c h) p q r^2 \log (a+b x)}{6 b^3 d h}+\frac {(b g-a h)^2 (d g-c h)^2 p q r^2 \log (a+b x)}{4 b^2 d^2 h}-\frac {2 (b g-a h)^3 p^2 r^2 (a+b x) \log (a+b x)}{b^4}-\frac {(d g-c h)^3 p q r^2 (a+b x) \log (a+b x)}{2 b d^3}-\frac {3 h (b g-a h)^2 p^2 r^2 (a+b x)^2 \log (a+b x)}{2 b^4}-\frac {2 h^2 (b g-a h) p^2 r^2 (a+b x)^3 \log (a+b x)}{3 b^4}-\frac {h^3 p^2 r^2 (a+b x)^4 \log (a+b x)}{8 b^4}-\frac {(d g-c h)^2 p q r^2 (g+h x)^2 \log (a+b x)}{4 d^2 h}-\frac {(d g-c h) p q r^2 (g+h x)^3 \log (a+b x)}{6 d h}-\frac {p q r^2 (g+h x)^4 \log (a+b x)}{8 h}-\frac {(b g-a h)^4 p^2 r^2 \log ^2(a+b x)}{4 b^4 h}+\frac {(b g-a h)^2 (d g-c h)^2 p q r^2 \log (c+d x)}{4 b^2 d^2 h}+\frac {(b g-a h) (d g-c h)^3 p q r^2 \log (c+d x)}{6 b d^3 h}+\frac {(d g-c h)^4 p q r^2 \log (c+d x)}{8 d^4 h}-\frac {(b g-a h)^3 p q r^2 (c+d x) \log (c+d x)}{2 b^3 d}-\frac {2 (d g-c h)^3 q^2 r^2 (c+d x) \log (c+d x)}{d^4}-\frac {3 h (d g-c h)^2 q^2 r^2 (c+d x)^2 \log (c+d x)}{2 d^4}-\frac {2 h^2 (d g-c h) q^2 r^2 (c+d x)^3 \log (c+d x)}{3 d^4}-\frac {h^3 q^2 r^2 (c+d x)^4 \log (c+d x)}{8 d^4}-\frac {(b g-a h)^2 p q r^2 (g+h x)^2 \log (c+d x)}{4 b^2 h}-\frac {(b g-a h) p q r^2 (g+h x)^3 \log (c+d x)}{6 b h}-\frac {p q r^2 (g+h x)^4 \log (c+d x)}{8 h}-\frac {(b g-a h)^4 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b^4 h}-\frac {(d g-c h)^4 q^2 r^2 \log ^2(c+d x)}{4 d^4 h}-\frac {(d g-c h)^4 p q r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 d^4 h}+\frac {(b g-a h)^3 p r 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 )}{2 b^3}+\frac {(d g-c h)^3 q r 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 )}{2 d^3}+\frac {(b g-a h)^2 p r (g+h x)^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 )}{4 b^2 h}+\frac {(d g-c h)^2 q r (g+h x)^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 )}{4 d^2 h}+\frac {(b g-a h) p r (g+h x)^3 \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 )}{6 b h}+\frac {(d g-c h) q r (g+h x)^3 \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 )}{6 d h}+\frac {p r (g+h x)^4 \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 )}{8 h}+\frac {q r (g+h x)^4 \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 )}{8 h}+\frac {(b g-a h)^4 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 )}{2 b^4 h}+\frac {(d g-c h)^4 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 )}{2 d^4 h}+\frac {(g+h x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h}-\frac {(d g-c h)^4 p q r^2 \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{2 d^4 h}-\frac {(b g-a h)^4 p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{2 b^4 h} \]

[Out]

-1/2*(-c*h+d*g)^3*p*q*r^2*(b*x+a)*ln(b*x+a)/b/d^3-1/4*(-c*h+d*g)^2*p*q*r^2*(h*x+g)^2*ln(b*x+a)/d^2/h+1/6*(-a*h
+b*g)^3*(-c*h+d*g)*p*q*r^2*ln(b*x+a)/b^3/d/h+1/4*(-a*h+b*g)^2*(-c*h+d*g)^2*p*q*r^2*ln(b*x+a)/b^2/d^2/h+1/4*(-a
*h+b*g)^2*(-c*h+d*g)^2*p*q*r^2*ln(d*x+c)/b^2/d^2/h+1/6*(-a*h+b*g)*(-c*h+d*g)^3*p*q*r^2*ln(d*x+c)/b/d^3/h+1/6*(
-a*h+b*g)*(-c*h+d*g)*p*q*r^2*(h*x+g)^2/b/d/h+1/4*(h*x+g)^4*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/h+3/16*(-c*h+d*g)
^2*p*q*r^2*(h*x+g)^2/d^2/h+7/72*(-a*h+b*g)*p*q*r^2*(h*x+g)^3/b/h+7/72*(-c*h+d*g)*p*q*r^2*(h*x+g)^3/d/h-1/2*(-c
*h+d*g)^4*p*q*r^2*polylog(2,-d*(b*x+a)/(-a*d+b*c))/d^4/h-1/2*(-a*h+b*g)^4*p*q*r^2*polylog(2,b*(d*x+c)/(-a*d+b*
c))/b^4/h-2*(-a*h+b*g)^3*p^2*r^2*(b*x+a)*ln(b*x+a)/b^4-1/8*h^3*p^2*r^2*(b*x+a)^4*ln(b*x+a)/b^4-1/8*p*q*r^2*(h*
x+g)^4*ln(b*x+a)/h-1/4*(-a*h+b*g)^4*p^2*r^2*ln(b*x+a)^2/b^4/h-2*(-c*h+d*g)^3*q^2*r^2*(d*x+c)*ln(d*x+c)/d^4-1/8
*h^3*q^2*r^2*(d*x+c)^4*ln(d*x+c)/d^4-1/6*(-c*h+d*g)*p*q*r^2*(h*x+g)^3*ln(b*x+a)/d/h-1/2*(-a*h+b*g)^3*p*q*r^2*(
d*x+c)*ln(d*x+c)/b^3/d-1/4*(-a*h+b*g)^2*p*q*r^2*(h*x+g)^2*ln(d*x+c)/b^2/h-1/6*(-a*h+b*g)*p*q*r^2*(h*x+g)^3*ln(
d*x+c)/b/h-1/2*(-a*h+b*g)^4*p*q*r^2*ln(-d*(b*x+a)/(-a*d+b*c))*ln(d*x+c)/b^4/h-1/2*(-c*h+d*g)^4*p*q*r^2*ln(b*x+
a)*ln(b*(d*x+c)/(-a*d+b*c))/d^4/h+5/12*(-a*h+b*g)^2*(-c*h+d*g)*p*q*r^2*x/b^2/d+5/12*(-a*h+b*g)*(-c*h+d*g)^2*p*
q*r^2*x/b/d^2-1/8*p*q*r^2*(h*x+g)^4*ln(d*x+c)/h-1/4*(-c*h+d*g)^4*q^2*r^2*ln(d*x+c)^2/d^4/h+1/2*(-a*h+b*g)^3*p*
r*x*(p*r*ln(b*x+a)+q*r*ln(d*x+c)-ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r))/b^3+1/2*(-c*h+d*g)^3*q*r*x*(p*r*ln(b*x+a)+q*
r*ln(d*x+c)-ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r))/d^3+3/4*h*(-c*h+d*g)^2*q^2*r^2*(d*x+c)^2/d^4+2/9*h^2*(-c*h+d*g)*q
^2*r^2*(d*x+c)^3/d^4+5/8*(-a*h+b*g)^3*p*q*r^2*x/b^3+5/8*(-c*h+d*g)^3*p*q*r^2*x/d^3+3/4*h*(-a*h+b*g)^2*p^2*r^2*
(b*x+a)^2/b^4+2/9*h^2*(-a*h+b*g)*p^2*r^2*(b*x+a)^3/b^4+1/8*p*r*(h*x+g)^4*(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+1/8*q*r*(h*x+g)^4*(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+2
*(-a*h+b*g)^3*p^2*r^2*x/b^3+2*(-c*h+d*g)^3*q^2*r^2*x/d^3+1/32*h^3*p^2*r^2*(b*x+a)^4/b^4+1/32*h^3*q^2*r^2*(d*x+
c)^4/d^4+1/16*p*q*r^2*(h*x+g)^4/h+1/8*(-a*h+b*g)^4*p*q*r^2*ln(b*x+a)/b^4/h-3/2*h*(-a*h+b*g)^2*p^2*r^2*(b*x+a)^
2*ln(b*x+a)/b^4-2/3*h^2*(-a*h+b*g)*p^2*r^2*(b*x+a)^3*ln(b*x+a)/b^4+1/8*(-c*h+d*g)^4*p*q*r^2*ln(d*x+c)/d^4/h-3/
2*h*(-c*h+d*g)^2*q^2*r^2*(d*x+c)^2*ln(d*x+c)/d^4-2/3*h^2*(-c*h+d*g)*q^2*r^2*(d*x+c)^3*ln(d*x+c)/d^4+1/4*(-a*h+
b*g)^2*p*r*(h*x+g)^2*(p*r*ln(b*x+a)+q*r*ln(d*x+c)-ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r))/b^2/h+1/4*(-c*h+d*g)^2*q*r*
(h*x+g)^2*(p*r*ln(b*x+a)+q*r*ln(d*x+c)-ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r))/d^2/h+1/6*(-a*h+b*g)*p*r*(h*x+g)^3*(p*
r*ln(b*x+a)+q*r*ln(d*x+c)-ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r))/b/h+1/6*(-c*h+d*g)*q*r*(h*x+g)^3*(p*r*ln(b*x+a)+q*r
*ln(d*x+c)-ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r))/d/h+1/2*(-a*h+b*g)^4*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))/b^4/h+1/2*(-c*h+d*g)^4*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))/d^4/h+3/16*(-a*h+b*g)^2*p*q*r^2*(h*x+g)^2/b^2/h

Rubi [A] (verified)

Time = 1.63 (sec) , antiderivative size = 2240, normalized size of antiderivative = 1.00, number of steps used = 49, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules used = {2584, 2593, 2458, 45, 2372, 12, 2338, 2465, 2436, 2332, 2441, 2440, 2438, 2442} \[ \int (g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {p^2 r^2 \log ^2(a+b x) (b g-a h)^4}{4 b^4 h}+\frac {p q r^2 \log (a+b x) (b g-a h)^4}{8 b^4 h}-\frac {p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) (b g-a h)^4}{2 b^4 h}+\frac {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 g-a h)^4}{2 b^4 h}-\frac {p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) (b g-a h)^4}{2 b^4 h}+\frac {2 p^2 r^2 x (b g-a h)^3}{b^3}+\frac {5 p q r^2 x (b g-a h)^3}{8 b^3}+\frac {(d g-c h) p q r^2 \log (a+b x) (b g-a h)^3}{6 b^3 d h}-\frac {2 p^2 r^2 (a+b x) \log (a+b x) (b g-a h)^3}{b^4}-\frac {p q r^2 (c+d x) \log (c+d x) (b g-a h)^3}{2 b^3 d}+\frac {p r 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 g-a h)^3}{2 b^3}+\frac {3 h p^2 r^2 (a+b x)^2 (b g-a h)^2}{4 b^4}+\frac {3 p q r^2 (g+h x)^2 (b g-a h)^2}{16 b^2 h}+\frac {5 (d g-c h) p q r^2 x (b g-a h)^2}{12 b^2 d}+\frac {(d g-c h)^2 p q r^2 \log (a+b x) (b g-a h)^2}{4 b^2 d^2 h}-\frac {3 h p^2 r^2 (a+b x)^2 \log (a+b x) (b g-a h)^2}{2 b^4}+\frac {(d g-c h)^2 p q r^2 \log (c+d x) (b g-a h)^2}{4 b^2 d^2 h}-\frac {p q r^2 (g+h x)^2 \log (c+d x) (b g-a h)^2}{4 b^2 h}+\frac {p r (g+h x)^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 ) (b g-a h)^2}{4 b^2 h}+\frac {2 h^2 p^2 r^2 (a+b x)^3 (b g-a h)}{9 b^4}+\frac {7 p q r^2 (g+h x)^3 (b g-a h)}{72 b h}+\frac {(d g-c h) p q r^2 (g+h x)^2 (b g-a h)}{6 b d h}+\frac {5 (d g-c h)^2 p q r^2 x (b g-a h)}{12 b d^2}-\frac {2 h^2 p^2 r^2 (a+b x)^3 \log (a+b x) (b g-a h)}{3 b^4}-\frac {p q r^2 (g+h x)^3 \log (c+d x) (b g-a h)}{6 b h}+\frac {(d g-c h)^3 p q r^2 \log (c+d x) (b g-a h)}{6 b d^3 h}+\frac {p r (g+h x)^3 \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 g-a h)}{6 b h}+\frac {h^3 p^2 r^2 (a+b x)^4}{32 b^4}+\frac {h^3 q^2 r^2 (c+d x)^4}{32 d^4}+\frac {p q r^2 (g+h x)^4}{16 h}+\frac {2 h^2 (d g-c h) q^2 r^2 (c+d x)^3}{9 d^4}+\frac {7 (d g-c h) p q r^2 (g+h x)^3}{72 d h}+\frac {3 h (d g-c h)^2 q^2 r^2 (c+d x)^2}{4 d^4}+\frac {3 (d g-c h)^2 p q r^2 (g+h x)^2}{16 d^2 h}-\frac {(d g-c h)^4 q^2 r^2 \log ^2(c+d x)}{4 d^4 h}+\frac {(g+h x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h}+\frac {2 (d g-c h)^3 q^2 r^2 x}{d^3}+\frac {5 (d g-c h)^3 p q r^2 x}{8 d^3}-\frac {h^3 p^2 r^2 (a+b x)^4 \log (a+b x)}{8 b^4}-\frac {p q r^2 (g+h x)^4 \log (a+b x)}{8 h}-\frac {(d g-c h) p q r^2 (g+h x)^3 \log (a+b x)}{6 d h}-\frac {(d g-c h)^2 p q r^2 (g+h x)^2 \log (a+b x)}{4 d^2 h}-\frac {(d g-c h)^3 p q r^2 (a+b x) \log (a+b x)}{2 b d^3}-\frac {h^3 q^2 r^2 (c+d x)^4 \log (c+d x)}{8 d^4}-\frac {p q r^2 (g+h x)^4 \log (c+d x)}{8 h}-\frac {2 h^2 (d g-c h) q^2 r^2 (c+d x)^3 \log (c+d x)}{3 d^4}+\frac {(d g-c h)^4 p q r^2 \log (c+d x)}{8 d^4 h}-\frac {3 h (d g-c h)^2 q^2 r^2 (c+d x)^2 \log (c+d x)}{2 d^4}-\frac {2 (d g-c h)^3 q^2 r^2 (c+d x) \log (c+d x)}{d^4}-\frac {(d g-c h)^4 p q r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 d^4 h}+\frac {p r (g+h x)^4 \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 )}{8 h}+\frac {q r (g+h x)^4 \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 )}{8 h}+\frac {(d g-c h) q r (g+h x)^3 \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 )}{6 d h}+\frac {(d g-c h)^2 q r (g+h x)^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 )}{4 d^2 h}+\frac {(d g-c h)^3 q r 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 )}{2 d^3}+\frac {(d g-c h)^4 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 )}{2 d^4 h}-\frac {(d g-c h)^4 p q r^2 \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{2 d^4 h} \]

[In]

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

[Out]

(2*(b*g - a*h)^3*p^2*r^2*x)/b^3 + (5*(b*g - a*h)^3*p*q*r^2*x)/(8*b^3) + (5*(b*g - a*h)^2*(d*g - c*h)*p*q*r^2*x
)/(12*b^2*d) + (5*(b*g - a*h)*(d*g - c*h)^2*p*q*r^2*x)/(12*b*d^2) + (5*(d*g - c*h)^3*p*q*r^2*x)/(8*d^3) + (2*(
d*g - c*h)^3*q^2*r^2*x)/d^3 + (3*h*(b*g - a*h)^2*p^2*r^2*(a + b*x)^2)/(4*b^4) + (2*h^2*(b*g - a*h)*p^2*r^2*(a
+ b*x)^3)/(9*b^4) + (h^3*p^2*r^2*(a + b*x)^4)/(32*b^4) + (3*h*(d*g - c*h)^2*q^2*r^2*(c + d*x)^2)/(4*d^4) + (2*
h^2*(d*g - c*h)*q^2*r^2*(c + d*x)^3)/(9*d^4) + (h^3*q^2*r^2*(c + d*x)^4)/(32*d^4) + (3*(b*g - a*h)^2*p*q*r^2*(
g + h*x)^2)/(16*b^2*h) + ((b*g - a*h)*(d*g - c*h)*p*q*r^2*(g + h*x)^2)/(6*b*d*h) + (3*(d*g - c*h)^2*p*q*r^2*(g
+ h*x)^2)/(16*d^2*h) + (7*(b*g - a*h)*p*q*r^2*(g + h*x)^3)/(72*b*h) + (7*(d*g - c*h)*p*q*r^2*(g + h*x)^3)/(72
*d*h) + (p*q*r^2*(g + h*x)^4)/(16*h) + ((b*g - a*h)^4*p*q*r^2*Log[a + b*x])/(8*b^4*h) + ((b*g - a*h)^3*(d*g -
c*h)*p*q*r^2*Log[a + b*x])/(6*b^3*d*h) + ((b*g - a*h)^2*(d*g - c*h)^2*p*q*r^2*Log[a + b*x])/(4*b^2*d^2*h) - (2
*(b*g - a*h)^3*p^2*r^2*(a + b*x)*Log[a + b*x])/b^4 - ((d*g - c*h)^3*p*q*r^2*(a + b*x)*Log[a + b*x])/(2*b*d^3)
- (3*h*(b*g - a*h)^2*p^2*r^2*(a + b*x)^2*Log[a + b*x])/(2*b^4) - (2*h^2*(b*g - a*h)*p^2*r^2*(a + b*x)^3*Log[a
+ b*x])/(3*b^4) - (h^3*p^2*r^2*(a + b*x)^4*Log[a + b*x])/(8*b^4) - ((d*g - c*h)^2*p*q*r^2*(g + h*x)^2*Log[a +
b*x])/(4*d^2*h) - ((d*g - c*h)*p*q*r^2*(g + h*x)^3*Log[a + b*x])/(6*d*h) - (p*q*r^2*(g + h*x)^4*Log[a + b*x])/
(8*h) - ((b*g - a*h)^4*p^2*r^2*Log[a + b*x]^2)/(4*b^4*h) + ((b*g - a*h)^2*(d*g - c*h)^2*p*q*r^2*Log[c + d*x])/
(4*b^2*d^2*h) + ((b*g - a*h)*(d*g - c*h)^3*p*q*r^2*Log[c + d*x])/(6*b*d^3*h) + ((d*g - c*h)^4*p*q*r^2*Log[c +
d*x])/(8*d^4*h) - ((b*g - a*h)^3*p*q*r^2*(c + d*x)*Log[c + d*x])/(2*b^3*d) - (2*(d*g - c*h)^3*q^2*r^2*(c + d*x
)*Log[c + d*x])/d^4 - (3*h*(d*g - c*h)^2*q^2*r^2*(c + d*x)^2*Log[c + d*x])/(2*d^4) - (2*h^2*(d*g - c*h)*q^2*r^
2*(c + d*x)^3*Log[c + d*x])/(3*d^4) - (h^3*q^2*r^2*(c + d*x)^4*Log[c + d*x])/(8*d^4) - ((b*g - a*h)^2*p*q*r^2*
(g + h*x)^2*Log[c + d*x])/(4*b^2*h) - ((b*g - a*h)*p*q*r^2*(g + h*x)^3*Log[c + d*x])/(6*b*h) - (p*q*r^2*(g + h
*x)^4*Log[c + d*x])/(8*h) - ((b*g - a*h)^4*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(2*b^4*h) -
((d*g - c*h)^4*q^2*r^2*Log[c + d*x]^2)/(4*d^4*h) - ((d*g - c*h)^4*p*q*r^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c
- a*d)])/(2*d^4*h) + ((b*g - a*h)^3*p*r*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]))/(2*b^3) + ((d*g - c*h)^3*q*r*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]))/(2*d^3) + ((b*g - a*h)^2*p*r*(g + h*x)^2*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p
*(c + d*x)^q)^r]))/(4*b^2*h) + ((d*g - c*h)^2*q*r*(g + h*x)^2*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*
(a + b*x)^p*(c + d*x)^q)^r]))/(4*d^2*h) + ((b*g - a*h)*p*r*(g + h*x)^3*(p*r*Log[a + b*x] + q*r*Log[c + d*x] -
Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(6*b*h) + ((d*g - c*h)*q*r*(g + h*x)^3*(p*r*Log[a + b*x] + q*r*Log[c +
d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(6*d*h) + (p*r*(g + h*x)^4*(p*r*Log[a + b*x] + q*r*Log[c + d*x]
- Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(8*h) + (q*r*(g + h*x)^4*(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e
*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(8*h) + ((b*g - a*h)^4*p*r*Log[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]))/(2*b^4*h) + ((d*g - c*h)^4*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]))/(2*d^4*h) + ((g + h*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^
q)^r]^2)/(4*h) - ((d*g - c*h)^4*p*q*r^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(2*d^4*h) - ((b*g - a*h)^4*p
*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(2*b^4*h)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2372

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]]
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

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

Mathematica [A] (verified)

Time = 1.63 (sec) , antiderivative size = 1386, normalized size of antiderivative = 0.62 \[ \int (g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {72 a d^4 \left (-4 b^3 g^3+6 a b^2 g^2 h-4 a^2 b g h^2+a^3 h^3\right ) p^2 r^2 \log ^2(a+b x)+12 p r \log (a+b x) \left (12 b^4 c \left (-4 d^3 g^3+6 c d^2 g^2 h-4 c^2 d g h^2+c^3 h^3\right ) q r \log (c+d x)-12 \left (4 a b^3 d^4 g^3-6 a^2 b^2 d^4 g^2 h+4 a^3 b d^4 g h^2-a^4 d^4 h^3+b^4 c \left (-4 d^3 g^3+6 c d^2 g^2 h-4 c^2 d g h^2+c^3 h^3\right )\right ) q r \log \left (\frac {b (c+d x)}{b c-a d}\right )+a d \left (\left (12 b^3 \left (-4 d^3 g^3+6 c d^2 g^2 h-4 c^2 d g h^2+c^3 h^3\right ) q+a^3 d^3 h^3 (25 p+3 q)-4 a^2 b d^2 h^2 (22 d g p+4 d g q-c h q)+6 a b^2 d h \left (-4 c d g h q+c^2 h^2 q+6 d^2 g^2 (3 p+q)\right )\right ) r+12 d^3 \left (4 b^3 g^3-6 a b^2 g^2 h+4 a^2 b g h^2-a^3 h^3\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right )+b \left (72 b^3 c \left (-4 d^3 g^3+6 c d^2 g^2 h-4 c^2 d g h^2+c^3 h^3\right ) q^2 r^2 \log ^2(c+d x)+12 q r \log (c+d x) \left (\left (12 a^3 c d^3 h^3 p+6 a^2 b c d^2 h^2 (-8 d g+c h) p+4 a b^2 d \left (12 d^3 g^3+18 c d^2 g^2 h-6 c^2 d g h^2+c^3 h^3\right ) p+b^3 c \left (-48 d^3 g^3 (p+q)+36 c d^2 g^2 h (p+3 q)-8 c^2 d g h^2 (2 p+11 q)+c^3 h^3 (3 p+25 q)\right )\right ) r-12 b^3 c \left (-4 d^3 g^3+6 c d^2 g^2 h-4 c^2 d g h^2+c^3 h^3\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )+d \left (r^2 \left (-60 a^3 d^3 h^3 p (5 p+3 q) x+6 a^2 b d^2 h^2 p x (-20 c h q+16 d g (11 p+8 q)+d h (13 p+9 q) x)+b^3 x \left (-60 c^3 h^3 q (3 p+5 q)+6 c^2 d h^2 q (16 g (8 p+11 q)+h (9 p+13 q) x)-4 c d^2 h q (p+q) \left (324 g^2+60 g h x+7 h^2 x^2\right )+d^3 (p+q)^2 \left (576 g^3+216 g^2 h x+64 g h^2 x^2+9 h^3 x^3\right )\right )-4 a b^2 p \left (36 c^3 h^3 q+6 c^2 d h^2 q (-24 g+5 h x)-12 c d^2 h q \left (-18 g^2+12 g h x+h^2 x^2\right )+d^3 \left (-144 g^3 q+324 g^2 h (p+q) x+60 g h^2 (p+q) x^2+7 h^3 (p+q) x^3\right )\right )\right )+12 r \left (12 a^3 d^3 h^3 p x-6 a^2 b d^3 h^2 p x (8 g+h x)+4 a b^2 d^3 p \left (-12 g^3+18 g^2 h x+6 g h^2 x^2+h^3 x^3\right )-b^3 x \left (-12 c^3 h^3 q+6 c^2 d h^2 q (8 g+h x)-4 c d^2 h q \left (18 g^2+6 g h x+h^2 x^2\right )+d^3 (p+q) \left (48 g^3+36 g^2 h x+16 g h^2 x^2+3 h^3 x^3\right )\right )\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+72 b^3 d^3 x \left (4 g^3+6 g^2 h x+4 g h^2 x^2+h^3 x^3\right ) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right )-144 \left (4 a b^3 d^4 g^3-6 a^2 b^2 d^4 g^2 h+4 a^3 b d^4 g h^2-a^4 d^4 h^3+b^4 c \left (-4 d^3 g^3+6 c d^2 g^2 h-4 c^2 d g h^2+c^3 h^3\right )\right ) p q r^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )}{288 b^4 d^4} \]

[In]

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

[Out]

(72*a*d^4*(-4*b^3*g^3 + 6*a*b^2*g^2*h - 4*a^2*b*g*h^2 + a^3*h^3)*p^2*r^2*Log[a + b*x]^2 + 12*p*r*Log[a + b*x]*
(12*b^4*c*(-4*d^3*g^3 + 6*c*d^2*g^2*h - 4*c^2*d*g*h^2 + c^3*h^3)*q*r*Log[c + d*x] - 12*(4*a*b^3*d^4*g^3 - 6*a^
2*b^2*d^4*g^2*h + 4*a^3*b*d^4*g*h^2 - a^4*d^4*h^3 + b^4*c*(-4*d^3*g^3 + 6*c*d^2*g^2*h - 4*c^2*d*g*h^2 + c^3*h^
3))*q*r*Log[(b*(c + d*x))/(b*c - a*d)] + a*d*((12*b^3*(-4*d^3*g^3 + 6*c*d^2*g^2*h - 4*c^2*d*g*h^2 + c^3*h^3)*q
+ a^3*d^3*h^3*(25*p + 3*q) - 4*a^2*b*d^2*h^2*(22*d*g*p + 4*d*g*q - c*h*q) + 6*a*b^2*d*h*(-4*c*d*g*h*q + c^2*h
^2*q + 6*d^2*g^2*(3*p + q)))*r + 12*d^3*(4*b^3*g^3 - 6*a*b^2*g^2*h + 4*a^2*b*g*h^2 - a^3*h^3)*Log[e*(f*(a + b*
x)^p*(c + d*x)^q)^r])) + b*(72*b^3*c*(-4*d^3*g^3 + 6*c*d^2*g^2*h - 4*c^2*d*g*h^2 + c^3*h^3)*q^2*r^2*Log[c + d*
x]^2 + 12*q*r*Log[c + d*x]*((12*a^3*c*d^3*h^3*p + 6*a^2*b*c*d^2*h^2*(-8*d*g + c*h)*p + 4*a*b^2*d*(12*d^3*g^3 +
18*c*d^2*g^2*h - 6*c^2*d*g*h^2 + c^3*h^3)*p + b^3*c*(-48*d^3*g^3*(p + q) + 36*c*d^2*g^2*h*(p + 3*q) - 8*c^2*d
*g*h^2*(2*p + 11*q) + c^3*h^3*(3*p + 25*q)))*r - 12*b^3*c*(-4*d^3*g^3 + 6*c*d^2*g^2*h - 4*c^2*d*g*h^2 + c^3*h^
3)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]) + d*(r^2*(-60*a^3*d^3*h^3*p*(5*p + 3*q)*x + 6*a^2*b*d^2*h^2*p*x*(-20*
c*h*q + 16*d*g*(11*p + 8*q) + d*h*(13*p + 9*q)*x) + b^3*x*(-60*c^3*h^3*q*(3*p + 5*q) + 6*c^2*d*h^2*q*(16*g*(8*
p + 11*q) + h*(9*p + 13*q)*x) - 4*c*d^2*h*q*(p + q)*(324*g^2 + 60*g*h*x + 7*h^2*x^2) + d^3*(p + q)^2*(576*g^3
+ 216*g^2*h*x + 64*g*h^2*x^2 + 9*h^3*x^3)) - 4*a*b^2*p*(36*c^3*h^3*q + 6*c^2*d*h^2*q*(-24*g + 5*h*x) - 12*c*d^
2*h*q*(-18*g^2 + 12*g*h*x + h^2*x^2) + d^3*(-144*g^3*q + 324*g^2*h*(p + q)*x + 60*g*h^2*(p + q)*x^2 + 7*h^3*(p
+ q)*x^3))) + 12*r*(12*a^3*d^3*h^3*p*x - 6*a^2*b*d^3*h^2*p*x*(8*g + h*x) + 4*a*b^2*d^3*p*(-12*g^3 + 18*g^2*h*
x + 6*g*h^2*x^2 + h^3*x^3) - b^3*x*(-12*c^3*h^3*q + 6*c^2*d*h^2*q*(8*g + h*x) - 4*c*d^2*h*q*(18*g^2 + 6*g*h*x
+ h^2*x^2) + d^3*(p + q)*(48*g^3 + 36*g^2*h*x + 16*g*h^2*x^2 + 3*h^3*x^3)))*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^
r] + 72*b^3*d^3*x*(4*g^3 + 6*g^2*h*x + 4*g*h^2*x^2 + h^3*x^3)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)) - 144*(
4*a*b^3*d^4*g^3 - 6*a^2*b^2*d^4*g^2*h + 4*a^3*b*d^4*g*h^2 - a^4*d^4*h^3 + b^4*c*(-4*d^3*g^3 + 6*c*d^2*g^2*h -
4*c^2*d*g*h^2 + c^3*h^3))*p*q*r^2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)])/(288*b^4*d^4)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 1799, normalized size of antiderivative = 0.80 \[ \int (g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/4*(h^3*x^4 + 4*g*h^2*x^3 + 6*g^2*h*x^2 + 4*g^3*x)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2 + 1/24*r*(12*(4*a*b
^3*f*g^3*p - 6*a^2*b^2*f*g^2*h*p + 4*a^3*b*f*g*h^2*p - a^4*f*h^3*p)*log(b*x + a)/b^4 + 12*(4*c*d^3*f*g^3*q - 6
*c^2*d^2*f*g^2*h*q + 4*c^3*d*f*g*h^2*q - c^4*f*h^3*q)*log(d*x + c)/d^4 - (3*b^3*d^3*f*h^3*(p + q)*x^4 - 4*(a*b
^2*d^3*f*h^3*p - (4*d^3*f*g*h^2*(p + q) - c*d^2*f*h^3*q)*b^3)*x^3 - 6*(4*a*b^2*d^3*f*g*h^2*p - a^2*b*d^3*f*h^3
*p - (6*d^3*f*g^2*h*(p + q) - 4*c*d^2*f*g*h^2*q + c^2*d*f*h^3*q)*b^3)*x^2 - 12*(6*a*b^2*d^3*f*g^2*h*p - 4*a^2*
b*d^3*f*g*h^2*p + a^3*d^3*f*h^3*p - (4*d^3*f*g^3*(p + q) - 6*c*d^2*f*g^2*h*q + 4*c^2*d*f*g*h^2*q - c^3*f*h^3*q
)*b^3)*x)/(b^3*d^3))*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/f + 1/288*r^2*(12*(12*a^3*c*d^3*f^2*h^3*p*q - 6*(8*c
*d^3*f^2*g*h^2*p*q - c^2*d^2*f^2*h^3*p*q)*a^2*b + 4*(18*c*d^3*f^2*g^2*h*p*q - 6*c^2*d^2*f^2*g*h^2*p*q + c^3*d*
f^2*h^3*p*q)*a*b^2 - (48*(p*q + q^2)*c*d^3*f^2*g^3 - 36*(p*q + 3*q^2)*c^2*d^2*f^2*g^2*h + 8*(2*p*q + 11*q^2)*c
^3*d*f^2*g*h^2 - (3*p*q + 25*q^2)*c^4*f^2*h^3)*b^3)*log(d*x + c)/(b^3*d^4) - 144*(4*a*b^3*d^4*f^2*g^3*p*q - 6*
a^2*b^2*d^4*f^2*g^2*h*p*q + 4*a^3*b*d^4*f^2*g*h^2*p*q - a^4*d^4*f^2*h^3*p*q - (4*c*d^3*f^2*g^3*p*q - 6*c^2*d^2
*f^2*g^2*h*p*q + 4*c^3*d*f^2*g*h^2*p*q - c^4*f^2*h^3*p*q)*b^4)*(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)))/(b^4*d^4) + (9*(p^2 + 2*p*q + q^2)*b^4*d^4*f^2*h^3*x^4 - 144*(4*c*d^3*f
^2*g^3*p*q - 6*c^2*d^2*f^2*g^2*h*p*q + 4*c^3*d*f^2*g*h^2*p*q - c^4*f^2*h^3*p*q)*b^4*log(b*x + a)*log(d*x + c)
- 72*(4*c*d^3*f^2*g^3*q^2 - 6*c^2*d^2*f^2*g^2*h*q^2 + 4*c^3*d*f^2*g*h^2*q^2 - c^4*f^2*h^3*q^2)*b^4*log(d*x + c
)^2 - 4*(7*(p^2 + p*q)*a*b^3*d^4*f^2*h^3 - (16*(p^2 + 2*p*q + q^2)*d^4*f^2*g*h^2 - 7*(p*q + q^2)*c*d^3*f^2*h^3
)*b^4)*x^3 + 6*((13*p^2 + 9*p*q)*a^2*b^2*d^4*f^2*h^3 + 8*(c*d^3*f^2*h^3*p*q - 5*(p^2 + p*q)*d^4*f^2*g*h^2)*a*b
^3 + (36*(p^2 + 2*p*q + q^2)*d^4*f^2*g^2*h - 40*(p*q + q^2)*c*d^3*f^2*g*h^2 + (9*p*q + 13*q^2)*c^2*d^2*f^2*h^3
)*b^4)*x^2 - 72*(4*a*b^3*d^4*f^2*g^3*p^2 - 6*a^2*b^2*d^4*f^2*g^2*h*p^2 + 4*a^3*b*d^4*f^2*g*h^2*p^2 - a^4*d^4*f
^2*h^3*p^2)*log(b*x + a)^2 - 12*(5*(5*p^2 + 3*p*q)*a^3*b*d^4*f^2*h^3 + 2*(5*c*d^3*f^2*h^3*p*q - 4*(11*p^2 + 8*
p*q)*d^4*f^2*g*h^2)*a^2*b^2 - 2*(24*c*d^3*f^2*g*h^2*p*q - 5*c^2*d^2*f^2*h^3*p*q - 54*(p^2 + p*q)*d^4*f^2*g^2*h
)*a*b^3 - (48*(p^2 + 2*p*q + q^2)*d^4*f^2*g^3 - 108*(p*q + q^2)*c*d^3*f^2*g^2*h + 8*(8*p*q + 11*q^2)*c^2*d^2*f
^2*g*h^2 - 5*(3*p*q + 5*q^2)*c^3*d*f^2*h^3)*b^4)*x + 12*((25*p^2 + 3*p*q)*a^4*d^4*f^2*h^3 + 4*(c*d^3*f^2*h^3*p
*q - 2*(11*p^2 + 2*p*q)*d^4*f^2*g*h^2)*a^3*b - 6*(4*c*d^3*f^2*g*h^2*p*q - c^2*d^2*f^2*h^3*p*q - 6*(3*p^2 + p*q
)*d^4*f^2*g^2*h)*a^2*b^2 + 12*(6*c*d^3*f^2*g^2*h*p*q - 4*c^2*d^2*f^2*g*h^2*p*q + c^3*d*f^2*h^3*p*q - 4*(p^2 +
p*q)*d^4*f^2*g^3)*a*b^3)*log(b*x + a))/(b^4*d^4))/f^2

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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