3.1.36 \(\int (g+h x)^2 \log ^2(e (f (a+b x)^p (c+d x)^q)^r) \, dx\) [36]

Optimal. Leaf size=1645 \[ \frac {2 (b g-a h)^2 p^2 r^2 x}{b^2}+\frac {8 (b g-a h)^2 p q r^2 x}{9 b^2}+\frac {2 (b g-a h) (d g-c h) p q r^2 x}{3 b d}+\frac {8 (d g-c h)^2 p q r^2 x}{9 d^2}+\frac {2 (d g-c h)^2 q^2 r^2 x}{d^2}+\frac {h (b g-a h) p^2 r^2 (a+b x)^2}{2 b^3}+\frac {2 h^2 p^2 r^2 (a+b x)^3}{27 b^3}+\frac {h (d g-c h) q^2 r^2 (c+d x)^2}{2 d^3}+\frac {2 h^2 q^2 r^2 (c+d x)^3}{27 d^3}+\frac {5 (b g-a h) p q r^2 (g+h x)^2}{18 b h}+\frac {5 (d g-c h) p q r^2 (g+h x)^2}{18 d h}+\frac {4 p q r^2 (g+h x)^3}{27 h}+\frac {2 (b g-a h)^3 p q r^2 \log (a+b x)}{9 b^3 h}+\frac {(b g-a h)^2 (d g-c h) p q r^2 \log (a+b x)}{3 b^2 d h}-\frac {2 (b g-a h)^2 p^2 r^2 (a+b x) \log (a+b x)}{b^3}-\frac {2 (d g-c h)^2 p q r^2 (a+b x) \log (a+b x)}{3 b d^2}-\frac {h (b g-a h) p^2 r^2 (a+b x)^2 \log (a+b x)}{b^3}-\frac {2 h^2 p^2 r^2 (a+b x)^3 \log (a+b x)}{9 b^3}-\frac {(d g-c h) p q r^2 (g+h x)^2 \log (a+b x)}{3 d h}-\frac {2 p q r^2 (g+h x)^3 \log (a+b x)}{9 h}-\frac {(b g-a h)^3 p^2 r^2 \log ^2(a+b x)}{3 b^3 h}+\frac {(b g-a h) (d g-c h)^2 p q r^2 \log (c+d x)}{3 b d^2 h}+\frac {2 (d g-c h)^3 p q r^2 \log (c+d x)}{9 d^3 h}-\frac {2 (b g-a h)^2 p q r^2 (c+d x) \log (c+d x)}{3 b^2 d}-\frac {2 (d g-c h)^2 q^2 r^2 (c+d x) \log (c+d x)}{d^3}-\frac {h (d g-c h) q^2 r^2 (c+d x)^2 \log (c+d x)}{d^3}-\frac {2 h^2 q^2 r^2 (c+d x)^3 \log (c+d x)}{9 d^3}-\frac {(b g-a h) p q r^2 (g+h x)^2 \log (c+d x)}{3 b h}-\frac {2 p q r^2 (g+h x)^3 \log (c+d x)}{9 h}-\frac {2 (b g-a h)^3 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 h}-\frac {(d g-c h)^3 q^2 r^2 \log ^2(c+d x)}{3 d^3 h}-\frac {2 (d g-c h)^3 p q r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 d^3 h}+\frac {2 (b g-a h)^2 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 )}{3 b^2}+\frac {2 (d g-c h)^2 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 )}{3 d^2}+\frac {(b g-a h) 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 )}{3 b h}+\frac {(d g-c h) 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 )}{3 d h}+\frac {2 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 )}{9 h}+\frac {2 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 )}{9 h}+\frac {2 (b g-a h)^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 b^3 h}+\frac {2 (d g-c h)^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 d^3 h}+\frac {(g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}-\frac {2 (d g-c h)^3 p q r^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{3 d^3 h}-\frac {2 (b g-a h)^3 p q r^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^3 h} \]

[Out]

8/9*(-a*h+b*g)^2*p*q*r^2*x/b^2+8/9*(-c*h+d*g)^2*p*q*r^2*x/d^2+1/2*h*(-a*h+b*g)*p^2*r^2*(b*x+a)^2/b^3+1/2*h*(-c
*h+d*g)*q^2*r^2*(d*x+c)^2/d^3+2/9*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))/
h+2/9*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))/h+1/3*(h*x+g)^3*ln(e*(f*(b*x
+a)^p*(d*x+c)^q)^r)^2/h+5/18*(-a*h+b*g)*p*q*r^2*(h*x+g)^2/b/h+5/18*(-c*h+d*g)*p*q*r^2*(h*x+g)^2/d/h-2/3*(-c*h+
d*g)^3*p*q*r^2*polylog(2,-d*(b*x+a)/(-a*d+b*c))/d^3/h-2/3*(-a*h+b*g)^3*p*q*r^2*polylog(2,b*(d*x+c)/(-a*d+b*c))
/b^3/h+2*(-a*h+b*g)^2*p^2*r^2*x/b^2+2*(-c*h+d*g)^2*q^2*r^2*x/d^2+2/27*h^2*p^2*r^2*(b*x+a)^3/b^3+2/27*h^2*q^2*r
^2*(d*x+c)^3/d^3+4/27*p*q*r^2*(h*x+g)^3/h+2/9*(-a*h+b*g)^3*p*q*r^2*ln(b*x+a)/b^3/h+2/9*(-c*h+d*g)^3*p*q*r^2*ln
(d*x+c)/d^3/h+1/3*(-a*h+b*g)*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/h+1
/3*(-c*h+d*g)*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/h+2/3*(-a*h+b*g)^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))/b^3/h+2/3*(-c*h+d*g)^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))/d^3/h-h*(-a*h+b*g)*p^2*r^2*(b*x+a)^2*ln(b*x+
a)/b^3-h*(-c*h+d*g)*q^2*r^2*(d*x+c)^2*ln(d*x+c)/d^3+1/3*(-a*h+b*g)^2*(-c*h+d*g)*p*q*r^2*ln(b*x+a)/b^2/d/h+1/3*
(-a*h+b*g)*(-c*h+d*g)^2*p*q*r^2*ln(d*x+c)/b/d^2/h-2*(-a*h+b*g)^2*p^2*r^2*(b*x+a)*ln(b*x+a)/b^3-2/9*h^2*p^2*r^2
*(b*x+a)^3*ln(b*x+a)/b^3-2/9*p*q*r^2*(h*x+g)^3*ln(b*x+a)/h-1/3*(-a*h+b*g)^3*p^2*r^2*ln(b*x+a)^2/b^3/h-2*(-c*h+
d*g)^2*q^2*r^2*(d*x+c)*ln(d*x+c)/d^3-2/9*h^2*q^2*r^2*(d*x+c)^3*ln(d*x+c)/d^3-2/9*p*q*r^2*(h*x+g)^3*ln(d*x+c)/h
-1/3*(-c*h+d*g)^3*q^2*r^2*ln(d*x+c)^2/d^3/h+2/3*(-a*h+b*g)^2*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^2+2/3*(-c*h+d*g)^2*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^
2-2/3*(-c*h+d*g)^2*p*q*r^2*(b*x+a)*ln(b*x+a)/b/d^2-1/3*(-c*h+d*g)*p*q*r^2*(h*x+g)^2*ln(b*x+a)/d/h-2/3*(-a*h+b*
g)^2*p*q*r^2*(d*x+c)*ln(d*x+c)/b^2/d-1/3*(-a*h+b*g)*p*q*r^2*(h*x+g)^2*ln(d*x+c)/b/h-2/3*(-a*h+b*g)^3*p*q*r^2*l
n(-d*(b*x+a)/(-a*d+b*c))*ln(d*x+c)/b^3/h-2/3*(-c*h+d*g)^3*p*q*r^2*ln(b*x+a)*ln(b*(d*x+c)/(-a*d+b*c))/d^3/h+2/3
*(-a*h+b*g)*(-c*h+d*g)*p*q*r^2*x/b/d

________________________________________________________________________________________

Rubi [A]
time = 1.17, antiderivative size = 1645, normalized size of antiderivative = 1.00, number of steps used = 47, number of rules used = 15, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.484, Rules used = {2584, 2593, 2458, 45, 2372, 12, 14, 2338, 2465, 2436, 2332, 2441, 2440, 2438, 2442} \begin {gather*} -\frac {p^2 r^2 \log ^2(a+b x) (b g-a h)^3}{3 b^3 h}+\frac {2 p q r^2 \log (a+b x) (b g-a h)^3}{9 b^3 h}-\frac {2 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) (b g-a h)^3}{3 b^3 h}+\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 g-a h)^3}{3 b^3 h}-\frac {2 p q r^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) (b g-a h)^3}{3 b^3 h}+\frac {2 p^2 r^2 x (b g-a h)^2}{b^2}+\frac {8 p q r^2 x (b g-a h)^2}{9 b^2}+\frac {(d g-c h) p q r^2 \log (a+b x) (b g-a h)^2}{3 b^2 d h}-\frac {2 p^2 r^2 (a+b x) \log (a+b x) (b g-a h)^2}{b^3}-\frac {2 p q r^2 (c+d x) \log (c+d x) (b g-a h)^2}{3 b^2 d}+\frac {2 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)^2}{3 b^2}+\frac {h p^2 r^2 (a+b x)^2 (b g-a h)}{2 b^3}+\frac {5 p q r^2 (g+h x)^2 (b g-a h)}{18 b h}+\frac {2 (d g-c h) p q r^2 x (b g-a h)}{3 b d}-\frac {h p^2 r^2 (a+b x)^2 \log (a+b x) (b g-a h)}{b^3}+\frac {(d g-c h)^2 p q r^2 \log (c+d x) (b g-a h)}{3 b d^2 h}-\frac {p q r^2 (g+h x)^2 \log (c+d x) (b g-a h)}{3 b 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)}{3 b h}+\frac {2 h^2 p^2 r^2 (a+b x)^3}{27 b^3}+\frac {2 h^2 q^2 r^2 (c+d x)^3}{27 d^3}+\frac {4 p q r^2 (g+h x)^3}{27 h}+\frac {h (d g-c h) q^2 r^2 (c+d x)^2}{2 d^3}+\frac {5 (d g-c h) p q r^2 (g+h x)^2}{18 d h}-\frac {(d g-c h)^3 q^2 r^2 \log ^2(c+d x)}{3 d^3 h}+\frac {(g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}+\frac {2 (d g-c h)^2 q^2 r^2 x}{d^2}+\frac {8 (d g-c h)^2 p q r^2 x}{9 d^2}-\frac {2 h^2 p^2 r^2 (a+b x)^3 \log (a+b x)}{9 b^3}-\frac {2 p q r^2 (g+h x)^3 \log (a+b x)}{9 h}-\frac {(d g-c h) p q r^2 (g+h x)^2 \log (a+b x)}{3 d h}-\frac {2 (d g-c h)^2 p q r^2 (a+b x) \log (a+b x)}{3 b d^2}-\frac {2 h^2 q^2 r^2 (c+d x)^3 \log (c+d x)}{9 d^3}-\frac {2 p q r^2 (g+h x)^3 \log (c+d x)}{9 h}+\frac {2 (d g-c h)^3 p q r^2 \log (c+d x)}{9 d^3 h}-\frac {h (d g-c h) q^2 r^2 (c+d x)^2 \log (c+d x)}{d^3}-\frac {2 (d g-c h)^2 q^2 r^2 (c+d x) \log (c+d x)}{d^3}-\frac {2 (d g-c h)^3 p q r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 d^3 h}+\frac {2 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 )}{9 h}+\frac {2 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 )}{9 h}+\frac {(d g-c h) 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 )}{3 d h}+\frac {2 (d g-c h)^2 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 )}{3 d^2}+\frac {2 (d g-c h)^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 d^3 h}-\frac {2 (d g-c h)^3 p q r^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{3 d^3 h} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*g - a*h)^2*p^2*r^2*x)/b^2 + (8*(b*g - a*h)^2*p*q*r^2*x)/(9*b^2) + (2*(b*g - a*h)*(d*g - c*h)*p*q*r^2*x)/
(3*b*d) + (8*(d*g - c*h)^2*p*q*r^2*x)/(9*d^2) + (2*(d*g - c*h)^2*q^2*r^2*x)/d^2 + (h*(b*g - a*h)*p^2*r^2*(a +
b*x)^2)/(2*b^3) + (2*h^2*p^2*r^2*(a + b*x)^3)/(27*b^3) + (h*(d*g - c*h)*q^2*r^2*(c + d*x)^2)/(2*d^3) + (2*h^2*
q^2*r^2*(c + d*x)^3)/(27*d^3) + (5*(b*g - a*h)*p*q*r^2*(g + h*x)^2)/(18*b*h) + (5*(d*g - c*h)*p*q*r^2*(g + h*x
)^2)/(18*d*h) + (4*p*q*r^2*(g + h*x)^3)/(27*h) + (2*(b*g - a*h)^3*p*q*r^2*Log[a + b*x])/(9*b^3*h) + ((b*g - a*
h)^2*(d*g - c*h)*p*q*r^2*Log[a + b*x])/(3*b^2*d*h) - (2*(b*g - a*h)^2*p^2*r^2*(a + b*x)*Log[a + b*x])/b^3 - (2
*(d*g - c*h)^2*p*q*r^2*(a + b*x)*Log[a + b*x])/(3*b*d^2) - (h*(b*g - a*h)*p^2*r^2*(a + b*x)^2*Log[a + b*x])/b^
3 - (2*h^2*p^2*r^2*(a + b*x)^3*Log[a + b*x])/(9*b^3) - ((d*g - c*h)*p*q*r^2*(g + h*x)^2*Log[a + b*x])/(3*d*h)
- (2*p*q*r^2*(g + h*x)^3*Log[a + b*x])/(9*h) - ((b*g - a*h)^3*p^2*r^2*Log[a + b*x]^2)/(3*b^3*h) + ((b*g - a*h)
*(d*g - c*h)^2*p*q*r^2*Log[c + d*x])/(3*b*d^2*h) + (2*(d*g - c*h)^3*p*q*r^2*Log[c + d*x])/(9*d^3*h) - (2*(b*g
- a*h)^2*p*q*r^2*(c + d*x)*Log[c + d*x])/(3*b^2*d) - (2*(d*g - c*h)^2*q^2*r^2*(c + d*x)*Log[c + d*x])/d^3 - (h
*(d*g - c*h)*q^2*r^2*(c + d*x)^2*Log[c + d*x])/d^3 - (2*h^2*q^2*r^2*(c + d*x)^3*Log[c + d*x])/(9*d^3) - ((b*g
- a*h)*p*q*r^2*(g + h*x)^2*Log[c + d*x])/(3*b*h) - (2*p*q*r^2*(g + h*x)^3*Log[c + d*x])/(9*h) - (2*(b*g - a*h)
^3*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(3*b^3*h) - ((d*g - c*h)^3*q^2*r^2*Log[c + d*x]^2)/
(3*d^3*h) - (2*(d*g - c*h)^3*p*q*r^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(3*d^3*h) + (2*(b*g - a*h)^2
*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]))/(3*b^2) + (2*(d*g - c*h)^
2*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]))/(3*d^2) + ((b*g - a*h)*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]))/(3*b*h) + ((d*g -
c*h)*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]))/(3*d*h) +
(2*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]))/(9*h) + (2*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]))/(9*h) + (2*(b*g -
a*h)^3*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]))/(3*b^3*h
) + (2*(d*g - c*h)^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*d^3*h) + ((g + h*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)/(3*h) - (2*(d*g - c*h)^3*p*q*r^2*PolyLo
g[2, -((d*(a + b*x))/(b*c - a*d))])/(3*d^3*h) - (2*(b*g - a*h)^3*p*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]
)/(3*b^3*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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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*} \int (g+h x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx &=\frac {(g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}-\frac {(2 b p r) \int \frac {(g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{a+b x} \, dx}{3 h}-\frac {(2 d q r) \int \frac {(g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x} \, dx}{3 h}\\ &=\frac {(g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}-\frac {\left (2 b p^2 r^2\right ) \int \frac {(g+h x)^3 \log (a+b x)}{a+b x} \, dx}{3 h}-\frac {\left (2 b p q r^2\right ) \int \frac {(g+h x)^3 \log (c+d x)}{a+b x} \, dx}{3 h}-\frac {\left (2 d p q r^2\right ) \int \frac {(g+h x)^3 \log (a+b x)}{c+d x} \, dx}{3 h}-\frac {\left (2 d q^2 r^2\right ) \int \frac {(g+h x)^3 \log (c+d x)}{c+d x} \, 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 {(g+h x)^3}{a+b x} \, 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 {(g+h x)^3}{c+d x} \, dx}{3 h}\\ &=\frac {(g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}-\frac {\left (2 p^2 r^2\right ) \text {Subst}\left (\int \frac {\left (\frac {b g-a h}{b}+\frac {h x}{b}\right )^3 \log (x)}{x} \, dx,x,a+b x\right )}{3 h}-\frac {\left (2 b p q r^2\right ) \int \left (\frac {h (b g-a h)^2 \log (c+d x)}{b^3}+\frac {(b g-a h)^3 \log (c+d x)}{b^3 (a+b x)}+\frac {h (b g-a h) (g+h x) \log (c+d x)}{b^2}+\frac {h (g+h x)^2 \log (c+d x)}{b}\right ) \, dx}{3 h}-\frac {\left (2 d p q r^2\right ) \int \left (\frac {h (d g-c h)^2 \log (a+b x)}{d^3}+\frac {(d g-c h)^3 \log (a+b x)}{d^3 (c+d x)}+\frac {h (d g-c h) (g+h x) \log (a+b x)}{d^2}+\frac {h (g+h x)^2 \log (a+b x)}{d}\right ) \, dx}{3 h}-\frac {\left (2 q^2 r^2\right ) \text {Subst}\left (\int \frac {\left (\frac {d g-c h}{d}+\frac {h x}{d}\right )^3 \log (x)}{x} \, 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 {h (b g-a h)^2}{b^3}+\frac {(b g-a h)^3}{b^3 (a+b x)}+\frac {h (b g-a h) (g+h x)}{b^2}+\frac {h (g+h x)^2}{b}\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 {h (d g-c h)^2}{d^3}+\frac {(d g-c h)^3}{d^3 (c+d x)}+\frac {h (d g-c h) (g+h x)}{d^2}+\frac {h (g+h x)^2}{d}\right ) \, dx}{3 h}\\ &=-\frac {p^2 r^2 \log (a+b x) \left (\frac {18 h (b g-a h)^2 (a+b x)}{b^3}+\frac {9 h^2 (b g-a h) (a+b x)^2}{b^3}+\frac {2 h^3 (a+b x)^3}{b^3}+\frac {6 (b g-a h)^3 \log (a+b x)}{b^3}\right )}{9 h}-\frac {q^2 r^2 \log (c+d x) \left (\frac {18 h (d g-c h)^2 (c+d x)}{d^3}+\frac {9 h^2 (d g-c h) (c+d x)^2}{d^3}+\frac {2 h^3 (c+d x)^3}{d^3}+\frac {6 (d g-c h)^3 \log (c+d x)}{d^3}\right )}{9 h}+\frac {2 (b g-a h)^2 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 )}{3 b^2}+\frac {2 (d g-c h)^2 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 )}{3 d^2}+\frac {(b g-a h) 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 )}{3 b h}+\frac {(d g-c h) 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 )}{3 d h}+\frac {2 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 )}{9 h}+\frac {2 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 )}{9 h}+\frac {2 (b g-a h)^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 b^3 h}+\frac {2 (d g-c h)^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 d^3 h}+\frac {(g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}+\frac {\left (2 p^2 r^2\right ) \text {Subst}\left (\int \frac {h x \left (18 b^2 g^2+9 b g h (-4 a+x)+h^2 \left (18 a^2-9 a x+2 x^2\right )\right )+6 (b g-a h)^3 \log (x)}{6 b^3 x} \, dx,x,a+b x\right )}{3 h}-\frac {1}{3} \left (2 p q r^2\right ) \int (g+h x)^2 \log (a+b x) \, dx-\frac {1}{3} \left (2 p q r^2\right ) \int (g+h x)^2 \log (c+d x) \, dx-\frac {\left (2 (b g-a h) p q r^2\right ) \int (g+h x) \log (c+d x) \, dx}{3 b}-\frac {\left (2 (b g-a h)^2 p q r^2\right ) \int \log (c+d x) \, dx}{3 b^2}-\frac {\left (2 (b g-a h)^3 p q r^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3 b^2 h}-\frac {\left (2 (d g-c h) p q r^2\right ) \int (g+h x) \log (a+b x) \, dx}{3 d}-\frac {\left (2 (d g-c h)^2 p q r^2\right ) \int \log (a+b x) \, dx}{3 d^2}-\frac {\left (2 (d g-c h)^3 p q r^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{3 d^2 h}+\frac {\left (2 q^2 r^2\right ) \text {Subst}\left (\int \frac {h x \left (18 d^2 g^2+9 d g h (-4 c+x)+h^2 \left (18 c^2-9 c x+2 x^2\right )\right )+6 (d g-c h)^3 \log (x)}{6 d^3 x} \, dx,x,c+d x\right )}{3 h}\\ &=-\frac {(d g-c h) p q r^2 (g+h x)^2 \log (a+b x)}{3 d h}-\frac {2 p q r^2 (g+h x)^3 \log (a+b x)}{9 h}-\frac {p^2 r^2 \log (a+b x) \left (\frac {18 h (b g-a h)^2 (a+b x)}{b^3}+\frac {9 h^2 (b g-a h) (a+b x)^2}{b^3}+\frac {2 h^3 (a+b x)^3}{b^3}+\frac {6 (b g-a h)^3 \log (a+b x)}{b^3}\right )}{9 h}-\frac {(b g-a h) p q r^2 (g+h x)^2 \log (c+d x)}{3 b h}-\frac {2 p q r^2 (g+h x)^3 \log (c+d x)}{9 h}-\frac {2 (b g-a h)^3 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 h}-\frac {q^2 r^2 \log (c+d x) \left (\frac {18 h (d g-c h)^2 (c+d x)}{d^3}+\frac {9 h^2 (d g-c h) (c+d x)^2}{d^3}+\frac {2 h^3 (c+d x)^3}{d^3}+\frac {6 (d g-c h)^3 \log (c+d x)}{d^3}\right )}{9 h}-\frac {2 (d g-c h)^3 p q r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 d^3 h}+\frac {2 (b g-a h)^2 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 )}{3 b^2}+\frac {2 (d g-c h)^2 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 )}{3 d^2}+\frac {(b g-a h) 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 )}{3 b h}+\frac {(d g-c h) 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 )}{3 d h}+\frac {2 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 )}{9 h}+\frac {2 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 )}{9 h}+\frac {2 (b g-a h)^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 b^3 h}+\frac {2 (d g-c h)^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 d^3 h}+\frac {(g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}+\frac {\left (p^2 r^2\right ) \text {Subst}\left (\int \frac {h x \left (18 b^2 g^2+9 b g h (-4 a+x)+h^2 \left (18 a^2-9 a x+2 x^2\right )\right )+6 (b g-a h)^3 \log (x)}{x} \, dx,x,a+b x\right )}{9 b^3 h}+\frac {\left (2 b p q r^2\right ) \int \frac {(g+h x)^3}{a+b x} \, dx}{9 h}+\frac {\left (2 d p q r^2\right ) \int \frac {(g+h x)^3}{c+d x} \, dx}{9 h}+\frac {\left (d (b g-a h) p q r^2\right ) \int \frac {(g+h x)^2}{c+d x} \, dx}{3 b h}-\frac {\left (2 (b g-a h)^2 p q r^2\right ) \text {Subst}(\int \log (x) \, dx,x,c+d x)}{3 b^2 d}+\frac {\left (2 d (b g-a h)^3 p q r^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 b^3 h}+\frac {\left (b (d g-c h) p q r^2\right ) \int \frac {(g+h x)^2}{a+b x} \, dx}{3 d h}-\frac {\left (2 (d g-c h)^2 p q r^2\right ) \text {Subst}(\int \log (x) \, dx,x,a+b x)}{3 b d^2}+\frac {\left (2 b (d g-c h)^3 p q r^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3 d^3 h}+\frac {\left (q^2 r^2\right ) \text {Subst}\left (\int \frac {h x \left (18 d^2 g^2+9 d g h (-4 c+x)+h^2 \left (18 c^2-9 c x+2 x^2\right )\right )+6 (d g-c h)^3 \log (x)}{x} \, dx,x,c+d x\right )}{9 d^3 h}\\ &=\frac {2 (b g-a h)^2 p q r^2 x}{3 b^2}+\frac {2 (d g-c h)^2 p q r^2 x}{3 d^2}-\frac {2 (d g-c h)^2 p q r^2 (a+b x) \log (a+b x)}{3 b d^2}-\frac {(d g-c h) p q r^2 (g+h x)^2 \log (a+b x)}{3 d h}-\frac {2 p q r^2 (g+h x)^3 \log (a+b x)}{9 h}-\frac {p^2 r^2 \log (a+b x) \left (\frac {18 h (b g-a h)^2 (a+b x)}{b^3}+\frac {9 h^2 (b g-a h) (a+b x)^2}{b^3}+\frac {2 h^3 (a+b x)^3}{b^3}+\frac {6 (b g-a h)^3 \log (a+b x)}{b^3}\right )}{9 h}-\frac {2 (b g-a h)^2 p q r^2 (c+d x) \log (c+d x)}{3 b^2 d}-\frac {(b g-a h) p q r^2 (g+h x)^2 \log (c+d x)}{3 b h}-\frac {2 p q r^2 (g+h x)^3 \log (c+d x)}{9 h}-\frac {2 (b g-a h)^3 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 h}-\frac {q^2 r^2 \log (c+d x) \left (\frac {18 h (d g-c h)^2 (c+d x)}{d^3}+\frac {9 h^2 (d g-c h) (c+d x)^2}{d^3}+\frac {2 h^3 (c+d x)^3}{d^3}+\frac {6 (d g-c h)^3 \log (c+d x)}{d^3}\right )}{9 h}-\frac {2 (d g-c h)^3 p q r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 d^3 h}+\frac {2 (b g-a h)^2 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 )}{3 b^2}+\frac {2 (d g-c h)^2 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 )}{3 d^2}+\frac {(b g-a h) 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 )}{3 b h}+\frac {(d g-c h) 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 )}{3 d h}+\frac {2 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 )}{9 h}+\frac {2 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 )}{9 h}+\frac {2 (b g-a h)^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 b^3 h}+\frac {2 (d g-c h)^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 d^3 h}+\frac {(g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}+\frac {\left (p^2 r^2\right ) \text {Subst}\left (\int \left (h \left (18 (b g-a h)^2+9 h (b g-a h) x+2 h^2 x^2\right )+\frac {6 (b g-a h)^3 \log (x)}{x}\right ) \, dx,x,a+b x\right )}{9 b^3 h}+\frac {\left (2 b p q r^2\right ) \int \left (\frac {h (b g-a h)^2}{b^3}+\frac {(b g-a h)^3}{b^3 (a+b x)}+\frac {h (b g-a h) (g+h x)}{b^2}+\frac {h (g+h x)^2}{b}\right ) \, dx}{9 h}+\frac {\left (2 d p q r^2\right ) \int \left (\frac {h (d g-c h)^2}{d^3}+\frac {(d g-c h)^3}{d^3 (c+d x)}+\frac {h (d g-c h) (g+h x)}{d^2}+\frac {h (g+h x)^2}{d}\right ) \, dx}{9 h}+\frac {\left (d (b g-a h) p q r^2\right ) \int \left (\frac {h (d g-c h)}{d^2}+\frac {(d g-c h)^2}{d^2 (c+d x)}+\frac {h (g+h x)}{d}\right ) \, dx}{3 b h}+\frac {\left (2 (b g-a h)^3 p q r^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3 b^3 h}+\frac {\left (b (d g-c h) p q r^2\right ) \int \left (\frac {h (b g-a h)}{b^2}+\frac {(b g-a h)^2}{b^2 (a+b x)}+\frac {h (g+h x)}{b}\right ) \, dx}{3 d h}+\frac {\left (2 (d g-c h)^3 p q r^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{3 d^3 h}+\frac {\left (q^2 r^2\right ) \text {Subst}\left (\int \left (h \left (18 (d g-c h)^2+9 h (d g-c h) x+2 h^2 x^2\right )+\frac {6 (d g-c h)^3 \log (x)}{x}\right ) \, dx,x,c+d x\right )}{9 d^3 h}\\ &=\frac {8 (b g-a h)^2 p q r^2 x}{9 b^2}+\frac {2 (b g-a h) (d g-c h) p q r^2 x}{3 b d}+\frac {8 (d g-c h)^2 p q r^2 x}{9 d^2}+\frac {5 (b g-a h) p q r^2 (g+h x)^2}{18 b h}+\frac {5 (d g-c h) p q r^2 (g+h x)^2}{18 d h}+\frac {4 p q r^2 (g+h x)^3}{27 h}+\frac {2 (b g-a h)^3 p q r^2 \log (a+b x)}{9 b^3 h}+\frac {(b g-a h)^2 (d g-c h) p q r^2 \log (a+b x)}{3 b^2 d h}-\frac {2 (d g-c h)^2 p q r^2 (a+b x) \log (a+b x)}{3 b d^2}-\frac {(d g-c h) p q r^2 (g+h x)^2 \log (a+b x)}{3 d h}-\frac {2 p q r^2 (g+h x)^3 \log (a+b x)}{9 h}-\frac {p^2 r^2 \log (a+b x) \left (\frac {18 h (b g-a h)^2 (a+b x)}{b^3}+\frac {9 h^2 (b g-a h) (a+b x)^2}{b^3}+\frac {2 h^3 (a+b x)^3}{b^3}+\frac {6 (b g-a h)^3 \log (a+b x)}{b^3}\right )}{9 h}+\frac {(b g-a h) (d g-c h)^2 p q r^2 \log (c+d x)}{3 b d^2 h}+\frac {2 (d g-c h)^3 p q r^2 \log (c+d x)}{9 d^3 h}-\frac {2 (b g-a h)^2 p q r^2 (c+d x) \log (c+d x)}{3 b^2 d}-\frac {(b g-a h) p q r^2 (g+h x)^2 \log (c+d x)}{3 b h}-\frac {2 p q r^2 (g+h x)^3 \log (c+d x)}{9 h}-\frac {2 (b g-a h)^3 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 h}-\frac {q^2 r^2 \log (c+d x) \left (\frac {18 h (d g-c h)^2 (c+d x)}{d^3}+\frac {9 h^2 (d g-c h) (c+d x)^2}{d^3}+\frac {2 h^3 (c+d x)^3}{d^3}+\frac {6 (d g-c h)^3 \log (c+d x)}{d^3}\right )}{9 h}-\frac {2 (d g-c h)^3 p q r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 d^3 h}+\frac {2 (b g-a h)^2 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 )}{3 b^2}+\frac {2 (d g-c h)^2 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 )}{3 d^2}+\frac {(b g-a h) 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 )}{3 b h}+\frac {(d g-c h) 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 )}{3 d h}+\frac {2 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 )}{9 h}+\frac {2 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 )}{9 h}+\frac {2 (b g-a h)^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 b^3 h}+\frac {2 (d g-c h)^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 d^3 h}+\frac {(g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}-\frac {2 (d g-c h)^3 p q r^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{3 d^3 h}-\frac {2 (b g-a h)^3 p q r^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^3 h}+\frac {\left (p^2 r^2\right ) \text {Subst}\left (\int \left (18 (b g-a h)^2+9 h (b g-a h) x+2 h^2 x^2\right ) \, dx,x,a+b x\right )}{9 b^3}+\frac {\left (2 (b g-a h)^3 p^2 r^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{3 b^3 h}+\frac {\left (q^2 r^2\right ) \text {Subst}\left (\int \left (18 (d g-c h)^2+9 h (d g-c h) x+2 h^2 x^2\right ) \, dx,x,c+d x\right )}{9 d^3}+\frac {\left (2 (d g-c h)^3 q^2 r^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3 d^3 h}\\ &=\frac {2 (b g-a h)^2 p^2 r^2 x}{b^2}+\frac {8 (b g-a h)^2 p q r^2 x}{9 b^2}+\frac {2 (b g-a h) (d g-c h) p q r^2 x}{3 b d}+\frac {8 (d g-c h)^2 p q r^2 x}{9 d^2}+\frac {2 (d g-c h)^2 q^2 r^2 x}{d^2}+\frac {h (b g-a h) p^2 r^2 (a+b x)^2}{2 b^3}+\frac {2 h^2 p^2 r^2 (a+b x)^3}{27 b^3}+\frac {h (d g-c h) q^2 r^2 (c+d x)^2}{2 d^3}+\frac {2 h^2 q^2 r^2 (c+d x)^3}{27 d^3}+\frac {5 (b g-a h) p q r^2 (g+h x)^2}{18 b h}+\frac {5 (d g-c h) p q r^2 (g+h x)^2}{18 d h}+\frac {4 p q r^2 (g+h x)^3}{27 h}+\frac {2 (b g-a h)^3 p q r^2 \log (a+b x)}{9 b^3 h}+\frac {(b g-a h)^2 (d g-c h) p q r^2 \log (a+b x)}{3 b^2 d h}-\frac {2 (d g-c h)^2 p q r^2 (a+b x) \log (a+b x)}{3 b d^2}-\frac {(d g-c h) p q r^2 (g+h x)^2 \log (a+b x)}{3 d h}-\frac {2 p q r^2 (g+h x)^3 \log (a+b x)}{9 h}+\frac {(b g-a h)^3 p^2 r^2 \log ^2(a+b x)}{3 b^3 h}-\frac {p^2 r^2 \log (a+b x) \left (\frac {18 h (b g-a h)^2 (a+b x)}{b^3}+\frac {9 h^2 (b g-a h) (a+b x)^2}{b^3}+\frac {2 h^3 (a+b x)^3}{b^3}+\frac {6 (b g-a h)^3 \log (a+b x)}{b^3}\right )}{9 h}+\frac {(b g-a h) (d g-c h)^2 p q r^2 \log (c+d x)}{3 b d^2 h}+\frac {2 (d g-c h)^3 p q r^2 \log (c+d x)}{9 d^3 h}-\frac {2 (b g-a h)^2 p q r^2 (c+d x) \log (c+d x)}{3 b^2 d}-\frac {(b g-a h) p q r^2 (g+h x)^2 \log (c+d x)}{3 b h}-\frac {2 p q r^2 (g+h x)^3 \log (c+d x)}{9 h}-\frac {2 (b g-a h)^3 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 h}+\frac {(d g-c h)^3 q^2 r^2 \log ^2(c+d x)}{3 d^3 h}-\frac {q^2 r^2 \log (c+d x) \left (\frac {18 h (d g-c h)^2 (c+d x)}{d^3}+\frac {9 h^2 (d g-c h) (c+d x)^2}{d^3}+\frac {2 h^3 (c+d x)^3}{d^3}+\frac {6 (d g-c h)^3 \log (c+d x)}{d^3}\right )}{9 h}-\frac {2 (d g-c h)^3 p q r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 d^3 h}+\frac {2 (b g-a h)^2 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 )}{3 b^2}+\frac {2 (d g-c h)^2 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 )}{3 d^2}+\frac {(b g-a h) 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 )}{3 b h}+\frac {(d g-c h) 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 )}{3 d h}+\frac {2 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 )}{9 h}+\frac {2 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 )}{9 h}+\frac {2 (b g-a h)^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 b^3 h}+\frac {2 (d g-c h)^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 d^3 h}+\frac {(g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}-\frac {2 (d g-c h)^3 p q r^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{3 d^3 h}-\frac {2 (b g-a h)^3 p q r^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^3 h}\\ \end {align*}

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Mathematica [A]
time = 1.21, size = 899, normalized size = 0.55 \begin {gather*} \frac {-18 a d^3 \left (3 b^2 g^2-3 a b g h+a^2 h^2\right ) p^2 r^2 \log ^2(a+b x)-6 p r \log (a+b x) \left (6 b^3 c \left (3 d^2 g^2-3 c d g h+c^2 h^2\right ) q r \log (c+d x)-6 (b c-a d) \left (a^2 d^2 h^2+a b d h (-3 d g+c h)+b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) q r \log \left (\frac {b (c+d x)}{b c-a d}\right )+a d \left (\left (6 b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right ) q+a^2 d^2 h^2 (11 p+2 q)-3 a b d h (-c h q+3 d g (3 p+q))\right ) r-6 d^2 \left (3 b^2 g^2-3 a b g h+a^2 h^2\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right )+b \left (-18 b^2 c \left (3 d^2 g^2-3 c d g h+c^2 h^2\right ) q^2 r^2 \log ^2(c+d x)-6 q r \log (c+d x) \left (\left (6 a^2 c d^2 h^2 p-3 a b d \left (6 d^2 g^2+6 c d g h-c^2 h^2\right ) p+b^2 c \left (18 d^2 g^2 (p+q)-9 c d g h (p+3 q)+c^2 h^2 (2 p+11 q)\right )\right ) r-6 b^2 c \left (3 d^2 g^2-3 c d g h+c^2 h^2\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )+d \left (r^2 \left (6 a^2 d^2 h^2 p (11 p+8 q) x+b^2 x \left (6 c^2 h^2 q (8 p+11 q)-3 c d h q (p+q) (54 g+5 h x)+d^2 (p+q)^2 \left (108 g^2+27 g h x+4 h^2 x^2\right )\right )-3 a b p \left (-12 c^2 h^2 q-12 c d h q (-3 g+h x)+d^2 \left (-36 g^2 q+54 g h (p+q) x+5 h^2 (p+q) x^2\right )\right )\right )-6 r \left (6 a^2 d^2 h^2 p x+3 a b d^2 p \left (6 g^2-6 g h x-h^2 x^2\right )+b^2 x \left (6 c^2 h^2 q-3 c d h q (6 g+h x)+d^2 (p+q) \left (18 g^2+9 g h x+2 h^2 x^2\right )\right )\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+18 b^2 d^2 x \left (3 g^2+3 g h x+h^2 x^2\right ) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right )+36 (b c-a d) \left (a^2 d^2 h^2+a b d h (-3 d g+c h)+b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) p q r^2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )}{54 b^3 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-18*a*d^3*(3*b^2*g^2 - 3*a*b*g*h + a^2*h^2)*p^2*r^2*Log[a + b*x]^2 - 6*p*r*Log[a + b*x]*(6*b^3*c*(3*d^2*g^2 -
3*c*d*g*h + c^2*h^2)*q*r*Log[c + d*x] - 6*(b*c - a*d)*(a^2*d^2*h^2 + a*b*d*h*(-3*d*g + c*h) + b^2*(3*d^2*g^2
- 3*c*d*g*h + c^2*h^2))*q*r*Log[(b*(c + d*x))/(b*c - a*d)] + a*d*((6*b^2*(3*d^2*g^2 - 3*c*d*g*h + c^2*h^2)*q +
a^2*d^2*h^2*(11*p + 2*q) - 3*a*b*d*h*(-(c*h*q) + 3*d*g*(3*p + q)))*r - 6*d^2*(3*b^2*g^2 - 3*a*b*g*h + a^2*h^2
)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])) + b*(-18*b^2*c*(3*d^2*g^2 - 3*c*d*g*h + c^2*h^2)*q^2*r^2*Log[c + d*x]
^2 - 6*q*r*Log[c + d*x]*((6*a^2*c*d^2*h^2*p - 3*a*b*d*(6*d^2*g^2 + 6*c*d*g*h - c^2*h^2)*p + b^2*c*(18*d^2*g^2*
(p + q) - 9*c*d*g*h*(p + 3*q) + c^2*h^2*(2*p + 11*q)))*r - 6*b^2*c*(3*d^2*g^2 - 3*c*d*g*h + c^2*h^2)*Log[e*(f*
(a + b*x)^p*(c + d*x)^q)^r]) + d*(r^2*(6*a^2*d^2*h^2*p*(11*p + 8*q)*x + b^2*x*(6*c^2*h^2*q*(8*p + 11*q) - 3*c*
d*h*q*(p + q)*(54*g + 5*h*x) + d^2*(p + q)^2*(108*g^2 + 27*g*h*x + 4*h^2*x^2)) - 3*a*b*p*(-12*c^2*h^2*q - 12*c
*d*h*q*(-3*g + h*x) + d^2*(-36*g^2*q + 54*g*h*(p + q)*x + 5*h^2*(p + q)*x^2))) - 6*r*(6*a^2*d^2*h^2*p*x + 3*a*
b*d^2*p*(6*g^2 - 6*g*h*x - h^2*x^2) + b^2*x*(6*c^2*h^2*q - 3*c*d*h*q*(6*g + h*x) + d^2*(p + q)*(18*g^2 + 9*g*h
*x + 2*h^2*x^2)))*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + 18*b^2*d^2*x*(3*g^2 + 3*g*h*x + h^2*x^2)*Log[e*(f*(a
+ b*x)^p*(c + d*x)^q)^r]^2)) + 36*(b*c - a*d)*(a^2*d^2*h^2 + a*b*d*h*(-3*d*g + c*h) + b^2*(3*d^2*g^2 - 3*c*d*g
*h + c^2*h^2))*p*q*r^2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)])/(54*b^3*d^3)

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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \left (h x +g \right )^{2} \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.34, size = 1125, normalized size = 0.68 \begin {gather*} \frac {1}{3} \, {\left (h^{2} x^{3} + 3 \, g h x^{2} + 3 \, g^{2} x\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} + \frac {r {\left (\frac {6 \, {\left (3 \, a b^{2} f g^{2} p - 3 \, a^{2} b f g h p + a^{3} f h^{2} p\right )} \log \left (b x + a\right )}{b^{3}} + \frac {6 \, {\left (3 \, c d^{2} f g^{2} q - 3 \, c^{2} d f g h q + c^{3} f h^{2} q\right )} \log \left (d x + c\right )}{d^{3}} - \frac {2 \, b^{2} d^{2} f h^{2} {\left (p + q\right )} x^{3} - 3 \, {\left (a b d^{2} f h^{2} p - {\left (3 \, d^{2} f g h {\left (p + q\right )} - c d f h^{2} q\right )} b^{2}\right )} x^{2} - 6 \, {\left (3 \, a b d^{2} f g h p - a^{2} d^{2} f h^{2} p - {\left (3 \, d^{2} f g^{2} {\left (p + q\right )} - 3 \, c d f g h q + c^{2} f h^{2} q\right )} b^{2}\right )} x}{b^{2} d^{2}}\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{9 \, f} - \frac {r^{2} {\left (\frac {6 \, {\left (6 \, a^{2} c d^{2} f^{2} h^{2} p q - 3 \, {\left (6 \, c d^{2} f^{2} g h p q - c^{2} d f^{2} h^{2} p q\right )} a b + {\left (18 \, {\left (p q + q^{2}\right )} c d^{2} f^{2} g^{2} - 9 \, {\left (p q + 3 \, q^{2}\right )} c^{2} d f^{2} g h + {\left (2 \, p q + 11 \, q^{2}\right )} c^{3} f^{2} h^{2}\right )} b^{2}\right )} \log \left (d x + c\right )}{b^{2} d^{3}} + \frac {36 \, {\left (3 \, a b^{2} d^{3} f^{2} g^{2} p q - 3 \, a^{2} b d^{3} f^{2} g h p q + a^{3} d^{3} f^{2} h^{2} p q - {\left (3 \, c d^{2} f^{2} g^{2} p q - 3 \, c^{2} d f^{2} g h p q + c^{3} f^{2} h^{2} p q\right )} b^{3}\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )}}{b^{3} d^{3}} - \frac {4 \, {\left (p^{2} + 2 \, p q + q^{2}\right )} b^{3} d^{3} f^{2} h^{2} x^{3} - 36 \, {\left (3 \, c d^{2} f^{2} g^{2} p q - 3 \, c^{2} d f^{2} g h p q + c^{3} f^{2} h^{2} p q\right )} b^{3} \log \left (b x + a\right ) \log \left (d x + c\right ) - 18 \, {\left (3 \, c d^{2} f^{2} g^{2} q^{2} - 3 \, c^{2} d f^{2} g h q^{2} + c^{3} f^{2} h^{2} q^{2}\right )} b^{3} \log \left (d x + c\right )^{2} - 3 \, {\left (5 \, {\left (p^{2} + p q\right )} a b^{2} d^{3} f^{2} h^{2} - {\left (9 \, {\left (p^{2} + 2 \, p q + q^{2}\right )} d^{3} f^{2} g h - 5 \, {\left (p q + q^{2}\right )} c d^{2} f^{2} h^{2}\right )} b^{3}\right )} x^{2} - 18 \, {\left (3 \, a b^{2} d^{3} f^{2} g^{2} p^{2} - 3 \, a^{2} b d^{3} f^{2} g h p^{2} + a^{3} d^{3} f^{2} h^{2} p^{2}\right )} \log \left (b x + a\right )^{2} + 6 \, {\left ({\left (11 \, p^{2} + 8 \, p q\right )} a^{2} b d^{3} f^{2} h^{2} + 3 \, {\left (2 \, c d^{2} f^{2} h^{2} p q - 9 \, {\left (p^{2} + p q\right )} d^{3} f^{2} g h\right )} a b^{2} + {\left (18 \, {\left (p^{2} + 2 \, p q + q^{2}\right )} d^{3} f^{2} g^{2} - 27 \, {\left (p q + q^{2}\right )} c d^{2} f^{2} g h + {\left (8 \, p q + 11 \, q^{2}\right )} c^{2} d f^{2} h^{2}\right )} b^{3}\right )} x - 6 \, {\left ({\left (11 \, p^{2} + 2 \, p q\right )} a^{3} d^{3} f^{2} h^{2} + 3 \, {\left (c d^{2} f^{2} h^{2} p q - 3 \, {\left (3 \, p^{2} + p q\right )} d^{3} f^{2} g h\right )} a^{2} b - 6 \, {\left (3 \, c d^{2} f^{2} g h p q - c^{2} d f^{2} h^{2} p q - 3 \, {\left (p^{2} + p q\right )} d^{3} f^{2} g^{2}\right )} a b^{2}\right )} \log \left (b x + a\right )}{b^{3} d^{3}}\right )}}{54 \, f^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(h^2*x^3 + 3*g*h*x^2 + 3*g^2*x)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2 + 1/9*r*(6*(3*a*b^2*f*g^2*p - 3*a^2
*b*f*g*h*p + a^3*f*h^2*p)*log(b*x + a)/b^3 + 6*(3*c*d^2*f*g^2*q - 3*c^2*d*f*g*h*q + c^3*f*h^2*q)*log(d*x + c)/
d^3 - (2*b^2*d^2*f*h^2*(p + q)*x^3 - 3*(a*b*d^2*f*h^2*p - (3*d^2*f*g*h*(p + q) - c*d*f*h^2*q)*b^2)*x^2 - 6*(3*
a*b*d^2*f*g*h*p - a^2*d^2*f*h^2*p - (3*d^2*f*g^2*(p + q) - 3*c*d*f*g*h*q + c^2*f*h^2*q)*b^2)*x)/(b^2*d^2))*log
(((b*x + a)^p*(d*x + c)^q*f)^r*e)/f - 1/54*r^2*(6*(6*a^2*c*d^2*f^2*h^2*p*q - 3*(6*c*d^2*f^2*g*h*p*q - c^2*d*f^
2*h^2*p*q)*a*b + (18*(p*q + q^2)*c*d^2*f^2*g^2 - 9*(p*q + 3*q^2)*c^2*d*f^2*g*h + (2*p*q + 11*q^2)*c^3*f^2*h^2)
*b^2)*log(d*x + c)/(b^2*d^3) + 36*(3*a*b^2*d^3*f^2*g^2*p*q - 3*a^2*b*d^3*f^2*g*h*p*q + a^3*d^3*f^2*h^2*p*q - (
3*c*d^2*f^2*g^2*p*q - 3*c^2*d*f^2*g*h*p*q + c^3*f^2*h^2*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)))/(b^3*d^3) - (4*(p^2 + 2*p*q + q^2)*b^3*d^3*f^2*h^2*x^3 - 36*(3*c*d^2
*f^2*g^2*p*q - 3*c^2*d*f^2*g*h*p*q + c^3*f^2*h^2*p*q)*b^3*log(b*x + a)*log(d*x + c) - 18*(3*c*d^2*f^2*g^2*q^2
- 3*c^2*d*f^2*g*h*q^2 + c^3*f^2*h^2*q^2)*b^3*log(d*x + c)^2 - 3*(5*(p^2 + p*q)*a*b^2*d^3*f^2*h^2 - (9*(p^2 + 2
*p*q + q^2)*d^3*f^2*g*h - 5*(p*q + q^2)*c*d^2*f^2*h^2)*b^3)*x^2 - 18*(3*a*b^2*d^3*f^2*g^2*p^2 - 3*a^2*b*d^3*f^
2*g*h*p^2 + a^3*d^3*f^2*h^2*p^2)*log(b*x + a)^2 + 6*((11*p^2 + 8*p*q)*a^2*b*d^3*f^2*h^2 + 3*(2*c*d^2*f^2*h^2*p
*q - 9*(p^2 + p*q)*d^3*f^2*g*h)*a*b^2 + (18*(p^2 + 2*p*q + q^2)*d^3*f^2*g^2 - 27*(p*q + q^2)*c*d^2*f^2*g*h + (
8*p*q + 11*q^2)*c^2*d*f^2*h^2)*b^3)*x - 6*((11*p^2 + 2*p*q)*a^3*d^3*f^2*h^2 + 3*(c*d^2*f^2*h^2*p*q - 3*(3*p^2
+ p*q)*d^3*f^2*g*h)*a^2*b - 6*(3*c*d^2*f^2*g*h*p*q - c^2*d*f^2*h^2*p*q - 3*(p^2 + p*q)*d^3*f^2*g^2)*a*b^2)*log
(b*x + a))/(b^3*d^3))/f^2

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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