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

Optimal result

Integrand size = 31, antiderivative size = 1645 \[ \int (g+h x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx =\text {Too large to display} \] Output:

-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+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+2/9*(-c*h+d*g)^3*p 
*q*r^2*ln(d*x+c)/d^3/h+2/3*(-a*h+b*g)^3*p*r*ln(b*x+a)*(p*r*ln(b*x+a)+q*r*l 
n(d*x+c)-ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r))/b^3/h+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*(-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*l 
n(d*x+c)/h-1/3*(-c*h+d*g)^3*q^2*r^2*ln(d*x+c)^2/d^3/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+2/3*(- 
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+1/3*(h*x+g)^3*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/h+ 
2/3*(-a*h+b*g)*(-c*h+d*g)*p*q*r^2*x/b/d-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)^3*p*q*r^2*ln(-d*(b*x+a)/(-a 
*d+b*c))*ln(d*x+c)/b^3/h-1/3*(-c*h+d*g)*p*q*r^2*(h*x+g)^2*ln(b*x+a)/d/h-2/ 
3*(-c*h+d*g)^2*p*q*r^2*(b*x+a)*ln(b*x+a)/b/d^2-2/3*(-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+1/3*( 
-a*h+b*g)*(-c*h+d*g)^2*p*q*r^2*ln(d*x+c)/b/d^2/h+1/3*(-a*h+b*g)^2*(-c*h...
 

Mathematica [A] (warning: unable to verify)

Time = 1.71 (sec) , antiderivative size = 899, normalized size of antiderivative = 0.55 \[ \int (g+h x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {-18 a d^3 \left (3 b^2 g^2-3 a b g h+a^2 h^2\right ) p^2 r^2 \log ^2(a+b x)-6 p r \log (a+b x) \left (6 b^3 c \left (3 d^2 g^2-3 c d g h+c^2 h^2\right ) q r \log (c+d x)-6 (b c-a d) \left (a^2 d^2 h^2+a b d h (-3 d g+c h)+b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) q r \log \left (\frac {b (c+d x)}{b c-a d}\right )+a d \left (\left (6 b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right ) q+a^2 d^2 h^2 (11 p+2 q)-3 a b d h (-c h q+3 d g (3 p+q))\right ) r-6 d^2 \left (3 b^2 g^2-3 a b g h+a^2 h^2\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right )+b \left (-18 b^2 c \left (3 d^2 g^2-3 c d g h+c^2 h^2\right ) q^2 r^2 \log ^2(c+d x)-6 q r \log (c+d x) \left (\left (6 a^2 c d^2 h^2 p-3 a b d \left (6 d^2 g^2+6 c d g h-c^2 h^2\right ) p+b^2 c \left (18 d^2 g^2 (p+q)-9 c d g h (p+3 q)+c^2 h^2 (2 p+11 q)\right )\right ) r-6 b^2 c \left (3 d^2 g^2-3 c d g h+c^2 h^2\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )+d \left (r^2 \left (6 a^2 d^2 h^2 p (11 p+8 q) x+b^2 x \left (6 c^2 h^2 q (8 p+11 q)-3 c d h q (p+q) (54 g+5 h x)+d^2 (p+q)^2 \left (108 g^2+27 g h x+4 h^2 x^2\right )\right )-3 a b p \left (-12 c^2 h^2 q-12 c d h q (-3 g+h x)+d^2 \left (-36 g^2 q+54 g h (p+q) x+5 h^2 (p+q) x^2\right )\right )\right )-6 r \left (6 a^2 d^2 h^2 p x+3 a b d^2 p \left (6 g^2-6 g h x-h^2 x^2\right )+b^2 x \left (6 c^2 h^2 q-3 c d h q (6 g+h x)+d^2 (p+q) \left (18 g^2+9 g h x+2 h^2 x^2\right )\right )\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+18 b^2 d^2 x \left (3 g^2+3 g h x+h^2 x^2\right ) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right )+36 (b c-a d) \left (a^2 d^2 h^2+a b d h (-3 d g+c h)+b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) p q r^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )}{54 b^3 d^3} \] Input:

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

Output:

(-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)*Lo 
g[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))...
 

Rubi [A] (verified)

Time = 2.62 (sec) , antiderivative size = 1286, normalized size of antiderivative = 0.78, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {2984, 2993, 49, 2009, 2858, 27, 2772, 2009, 2865, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2984

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

\(\Big \downarrow \) 2993

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 b p r \left (q r \int \frac {(g+h x)^3 \log (c+d x)}{a+b x}dx+p r \int \frac {(g+h x)^3 \log (a+b x)}{a+b x}dx-\left (\left (\frac {(b g-a h)^3 \log (a+b x)}{b^4}+\frac {h x (b g-a h)^2}{b^3}+\frac {(g+h x)^2 (b g-a h)}{2 b^2}+\frac {(g+h x)^3}{3 b}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+p r \log (a+b x)+q r \log (c+d x)\right )\right )\right )}{3 h}-\frac {2 d q r \left (p r \int \frac {(g+h x)^3 \log (a+b x)}{c+d x}dx+q r \int \frac {(g+h x)^3 \log (c+d x)}{c+d x}dx-\left (\left (\frac {(d g-c h)^3 \log (c+d x)}{d^4}+\frac {h x (d g-c h)^2}{d^3}+\frac {(g+h x)^2 (d g-c h)}{2 d^2}+\frac {(g+h x)^3}{3 d}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+p r \log (a+b x)+q r \log (c+d x)\right )\right )\right )}{3 h}+\frac {(g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}\)

\(\Big \downarrow \) 2858

\(\displaystyle -\frac {2 b p r \left (\frac {p r \int \frac {\left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )^3 \log (a+b x)}{b^3 (a+b x)}d(a+b x)}{b}+q r \int \frac {(g+h x)^3 \log (c+d x)}{a+b x}dx-\left (\left (\frac {(b g-a h)^3 \log (a+b x)}{b^4}+\frac {h x (b g-a h)^2}{b^3}+\frac {(g+h x)^2 (b g-a h)}{2 b^2}+\frac {(g+h x)^3}{3 b}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+p r \log (a+b x)+q r \log (c+d x)\right )\right )\right )}{3 h}-\frac {2 d q r \left (p r \int \frac {(g+h x)^3 \log (a+b x)}{c+d x}dx+\frac {q r \int \frac {\left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )^3 \log (c+d x)}{d^3 (c+d x)}d(c+d x)}{d}-\left (\left (\frac {(d g-c h)^3 \log (c+d x)}{d^4}+\frac {h x (d g-c h)^2}{d^3}+\frac {(g+h x)^2 (d g-c h)}{2 d^2}+\frac {(g+h x)^3}{3 d}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+p r \log (a+b x)+q r \log (c+d x)\right )\right )\right )}{3 h}+\frac {(g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 b p r \left (\frac {p r \int \frac {(b g-a h+h (a+b x))^3 \log (a+b x)}{a+b x}d(a+b x)}{b^4}+q r \int \frac {(g+h x)^3 \log (c+d x)}{a+b x}dx-\left (\left (\frac {(b g-a h)^3 \log (a+b x)}{b^4}+\frac {h x (b g-a h)^2}{b^3}+\frac {(g+h x)^2 (b g-a h)}{2 b^2}+\frac {(g+h x)^3}{3 b}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+p r \log (a+b x)+q r \log (c+d x)\right )\right )\right )}{3 h}-\frac {2 d q r \left (p r \int \frac {(g+h x)^3 \log (a+b x)}{c+d x}dx+\frac {q r \int \frac {(d g-c h+h (c+d x))^3 \log (c+d x)}{c+d x}d(c+d x)}{d^4}-\left (\left (\frac {(d g-c h)^3 \log (c+d x)}{d^4}+\frac {h x (d g-c h)^2}{d^3}+\frac {(g+h x)^2 (d g-c h)}{2 d^2}+\frac {(g+h x)^3}{3 d}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+p r \log (a+b x)+q r \log (c+d x)\right )\right )\right )}{3 h}+\frac {(g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}\)

\(\Big \downarrow \) 2772

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 b p r \left (q r \int \frac {(g+h x)^3 \log (c+d x)}{a+b x}dx+\frac {p r \left (-\frac {3}{4} h^2 (a+b x)^2 (b g-a h)+\frac {3}{2} h^2 (a+b x)^2 (b g-a h) \log (a+b x)-3 h (a+b x) (b g-a h)^2+\frac {1}{2} (b g-a h)^3 \log ^2(a+b x)+3 h (a+b x) (b g-a h)^2 \log (a+b x)-\frac {1}{9} h^3 (a+b x)^3+\frac {1}{3} h^3 (a+b x)^3 \log (a+b x)\right )}{b^4}-\left (\frac {(b g-a h)^3 \log (a+b x)}{b^4}+\frac {h x (b g-a h)^2}{b^3}+\frac {(g+h x)^2 (b g-a h)}{2 b^2}+\frac {(g+h x)^3}{3 b}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+p r \log (a+b x)+q r \log (c+d x)\right )\right )}{3 h}-\frac {2 d q r \left (p r \int \frac {(g+h x)^3 \log (a+b x)}{c+d x}dx-\left (\frac {(d g-c h)^3 \log (c+d x)}{d^4}+\frac {h x (d g-c h)^2}{d^3}+\frac {(g+h x)^2 (d g-c h)}{2 d^2}+\frac {(g+h x)^3}{3 d}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+p r \log (a+b x)+q r \log (c+d x)\right )+\frac {q r \left (-\frac {3}{4} h^2 (c+d x)^2 (d g-c h)+\frac {3}{2} h^2 (c+d x)^2 (d g-c h) \log (c+d x)-3 h (c+d x) (d g-c h)^2+\frac {1}{2} (d g-c h)^3 \log ^2(c+d x)+3 h (c+d x) (d g-c h)^2 \log (c+d x)-\frac {1}{9} h^3 (c+d x)^3+\frac {1}{3} h^3 (c+d x)^3 \log (c+d x)\right )}{d^4}\right )}{3 h}+\frac {(g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}\)

\(\Big \downarrow \) 2865

\(\displaystyle \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (g+h x)^3}{3 h}-\frac {2 d q r \left (\frac {q r \left (\frac {1}{2} \log ^2(c+d x) (d g-c h)^3-3 h (c+d x) (d g-c h)^2+3 h (c+d x) \log (c+d x) (d g-c h)^2-\frac {3}{4} h^2 (c+d x)^2 (d g-c h)+\frac {3}{2} h^2 (c+d x)^2 \log (c+d x) (d g-c h)-\frac {1}{9} h^3 (c+d x)^3+\frac {1}{3} h^3 (c+d x)^3 \log (c+d x)\right )}{d^4}-\left (\frac {\log (c+d x) (d g-c h)^3}{d^4}+\frac {h x (d g-c h)^2}{d^3}+\frac {(g+h x)^2 (d g-c h)}{2 d^2}+\frac {(g+h x)^3}{3 d}\right ) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )+p r \int \left (\frac {\log (a+b x) (d g-c h)^3}{d^3 (c+d x)}+\frac {h \log (a+b x) (d g-c h)^2}{d^3}+\frac {h (g+h x) \log (a+b x) (d g-c h)}{d^2}+\frac {h (g+h x)^2 \log (a+b x)}{d}\right )dx\right )}{3 h}-\frac {2 b p r \left (\frac {p r \left (\frac {1}{2} \log ^2(a+b x) (b g-a h)^3-3 h (a+b x) (b g-a h)^2+3 h (a+b x) \log (a+b x) (b g-a h)^2-\frac {3}{4} h^2 (a+b x)^2 (b g-a h)+\frac {3}{2} h^2 (a+b x)^2 \log (a+b x) (b g-a h)-\frac {1}{9} h^3 (a+b x)^3+\frac {1}{3} h^3 (a+b x)^3 \log (a+b x)\right )}{b^4}-\left (\frac {\log (a+b x) (b g-a h)^3}{b^4}+\frac {h x (b g-a h)^2}{b^3}+\frac {(g+h x)^2 (b g-a h)}{2 b^2}+\frac {(g+h x)^3}{3 b}\right ) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )+q r \int \left (\frac {\log (c+d x) (b g-a h)^3}{b^3 (a+b x)}+\frac {h \log (c+d x) (b g-a h)^2}{b^3}+\frac {h (g+h x) \log (c+d x) (b g-a h)}{b^2}+\frac {h (g+h x)^2 \log (c+d x)}{b}\right )dx\right )}{3 h}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (g+h x)^3}{3 h}-\frac {2 d q r \left (\frac {q r \left (\frac {1}{2} \log ^2(c+d x) (d g-c h)^3-3 h (c+d x) (d g-c h)^2+3 h (c+d x) \log (c+d x) (d g-c h)^2-\frac {3}{4} h^2 (c+d x)^2 (d g-c h)+\frac {3}{2} h^2 (c+d x)^2 \log (c+d x) (d g-c h)-\frac {1}{9} h^3 (c+d x)^3+\frac {1}{3} h^3 (c+d x)^3 \log (c+d x)\right )}{d^4}-\left (\frac {\log (c+d x) (d g-c h)^3}{d^4}+\frac {h x (d g-c h)^2}{d^3}+\frac {(g+h x)^2 (d g-c h)}{2 d^2}+\frac {(g+h x)^3}{3 d}\right ) \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 )+p r \left (-\frac {\log (a+b x) (b g-a h)^3}{3 b^3 d}-\frac {h x (b g-a h)^2}{3 b^2 d}-\frac {(d g-c h) \log (a+b x) (b g-a h)^2}{2 b^2 d^2}-\frac {(g+h x)^2 (b g-a h)}{6 b d}-\frac {h (d g-c h) x (b g-a h)}{2 b d^2}-\frac {(g+h x)^3}{9 d}-\frac {(d g-c h) (g+h x)^2}{4 d^2}-\frac {h (d g-c h)^2 x}{d^3}+\frac {(g+h x)^3 \log (a+b x)}{3 d}+\frac {(d g-c h) (g+h x)^2 \log (a+b x)}{2 d^2}+\frac {h (d g-c h)^2 (a+b x) \log (a+b x)}{b d^3}+\frac {(d g-c h)^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d^4}+\frac {(d g-c h)^3 \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{d^4}\right )\right )}{3 h}-\frac {2 b p r \left (\frac {p r \left (\frac {1}{2} \log ^2(a+b x) (b g-a h)^3-3 h (a+b x) (b g-a h)^2+3 h (a+b x) \log (a+b x) (b g-a h)^2-\frac {3}{4} h^2 (a+b x)^2 (b g-a h)+\frac {3}{2} h^2 (a+b x)^2 \log (a+b x) (b g-a h)-\frac {1}{9} h^3 (a+b x)^3+\frac {1}{3} h^3 (a+b x)^3 \log (a+b x)\right )}{b^4}-\left (\frac {\log (a+b x) (b g-a h)^3}{b^4}+\frac {h x (b g-a h)^2}{b^3}+\frac {(g+h x)^2 (b g-a h)}{2 b^2}+\frac {(g+h x)^3}{3 b}\right ) \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 )+q r \left (\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) (b g-a h)^3}{b^4}+\frac {\operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) (b g-a h)^3}{b^4}-\frac {h x (b g-a h)^2}{b^3}+\frac {h (c+d x) \log (c+d x) (b g-a h)^2}{b^3 d}-\frac {(g+h x)^2 (b g-a h)}{4 b^2}-\frac {h (d g-c h) x (b g-a h)}{2 b^2 d}-\frac {(d g-c h)^2 \log (c+d x) (b g-a h)}{2 b^2 d^2}+\frac {(g+h x)^2 \log (c+d x) (b g-a h)}{2 b^2}-\frac {(g+h x)^3}{9 b}-\frac {(d g-c h) (g+h x)^2}{6 b d}-\frac {h (d g-c h)^2 x}{3 b d^2}-\frac {(d g-c h)^3 \log (c+d x)}{3 b d^3}+\frac {(g+h x)^3 \log (c+d x)}{3 b}\right )\right )}{3 h}\)

Input:

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

Output:

((g + h*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)/(3*h) - (2*d*q*r*((q* 
r*(-3*h*(d*g - c*h)^2*(c + d*x) - (3*h^2*(d*g - c*h)*(c + d*x)^2)/4 - (h^3 
*(c + d*x)^3)/9 + 3*h*(d*g - c*h)^2*(c + d*x)*Log[c + d*x] + (3*h^2*(d*g - 
 c*h)*(c + d*x)^2*Log[c + d*x])/2 + (h^3*(c + d*x)^3*Log[c + d*x])/3 + ((d 
*g - c*h)^3*Log[c + d*x]^2)/2))/d^4 - ((h*(d*g - c*h)^2*x)/d^3 + ((d*g - c 
*h)*(g + h*x)^2)/(2*d^2) + (g + h*x)^3/(3*d) + ((d*g - c*h)^3*Log[c + d*x] 
)/d^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]) + p*r*(-1/3*(h*(b*g - a*h)^2*x)/(b^2*d) - (h*(b*g - a*h)*(d*g - 
c*h)*x)/(2*b*d^2) - (h*(d*g - c*h)^2*x)/d^3 - ((b*g - a*h)*(g + h*x)^2)/(6 
*b*d) - ((d*g - c*h)*(g + h*x)^2)/(4*d^2) - (g + h*x)^3/(9*d) - ((b*g - a* 
h)^3*Log[a + b*x])/(3*b^3*d) - ((b*g - a*h)^2*(d*g - c*h)*Log[a + b*x])/(2 
*b^2*d^2) + (h*(d*g - c*h)^2*(a + b*x)*Log[a + b*x])/(b*d^3) + ((d*g - c*h 
)*(g + h*x)^2*Log[a + b*x])/(2*d^2) + ((g + h*x)^3*Log[a + b*x])/(3*d) + ( 
(d*g - c*h)^3*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/d^4 + ((d*g - c 
*h)^3*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/d^4)))/(3*h) - (2*b*p*r*(( 
p*r*(-3*h*(b*g - a*h)^2*(a + b*x) - (3*h^2*(b*g - a*h)*(a + b*x)^2)/4 - (h 
^3*(a + b*x)^3)/9 + 3*h*(b*g - a*h)^2*(a + b*x)*Log[a + b*x] + (3*h^2*(b*g 
 - a*h)*(a + b*x)^2*Log[a + b*x])/2 + (h^3*(a + b*x)^3*Log[a + b*x])/3 + ( 
(b*g - a*h)^3*Log[a + b*x]^2)/2))/b^4 - ((h*(b*g - a*h)^2*x)/b^3 + ((b*g - 
 a*h)*(g + h*x)^2)/(2*b^2) + (g + h*x)^3/(3*b) + ((b*g - a*h)^3*Log[a +...
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 49

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] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 2009

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

rule 2772

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ 
.))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[(a + 
 b*Log[c*x^n])   u, x] - Simp[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 2858

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

rule 2865

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

rule 2984

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

rule 2993

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 1123, normalized size of antiderivative = 0.68 \[ \int (g+h x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

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) + d 
ilog(-(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^...
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 36*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)/(a*c*p + a*c*q + 
a*d*p*x + a*d*q*x + b*c*p*x + b*c*q*x + b*d*p*x**2 + b*d*q*x**2),x)*a**4*d 
**4*h**2*p**2*q*r - 36*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)/( 
a*c*p + a*c*q + a*d*p*x + a*d*q*x + b*c*p*x + b*c*q*x + b*d*p*x**2 + b*d*q 
*x**2),x)*a**4*d**4*h**2*p*q**2*r + 36*int(log(f**r*(c + d*x)**(q*r)*(a + 
b*x)**(p*r)*e)/(a*c*p + a*c*q + a*d*p*x + a*d*q*x + b*c*p*x + b*c*q*x + b* 
d*p*x**2 + b*d*q*x**2),x)*a**3*b*c*d**3*h**2*p**2*q*r + 36*int(log(f**r*(c 
 + d*x)**(q*r)*(a + b*x)**(p*r)*e)/(a*c*p + a*c*q + a*d*p*x + a*d*q*x + b* 
c*p*x + b*c*q*x + b*d*p*x**2 + b*d*q*x**2),x)*a**3*b*c*d**3*h**2*p*q**2*r 
+ 108*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)/(a*c*p + a*c*q + a 
*d*p*x + a*d*q*x + b*c*p*x + b*c*q*x + b*d*p*x**2 + b*d*q*x**2),x)*a**3*b* 
d**4*g*h*p**2*q*r + 108*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)/ 
(a*c*p + a*c*q + a*d*p*x + a*d*q*x + b*c*p*x + b*c*q*x + b*d*p*x**2 + b*d* 
q*x**2),x)*a**3*b*d**4*g*h*p*q**2*r - 108*int(log(f**r*(c + d*x)**(q*r)*(a 
 + b*x)**(p*r)*e)/(a*c*p + a*c*q + a*d*p*x + a*d*q*x + b*c*p*x + b*c*q*x + 
 b*d*p*x**2 + b*d*q*x**2),x)*a**2*b**2*c*d**3*g*h*p**2*q*r - 108*int(log(f 
**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)/(a*c*p + a*c*q + a*d*p*x + a*d*q* 
x + b*c*p*x + b*c*q*x + b*d*p*x**2 + b*d*q*x**2),x)*a**2*b**2*c*d**3*g*h*p 
*q**2*r - 108*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)/(a*c*p + a 
*c*q + a*d*p*x + a*d*q*x + b*c*p*x + b*c*q*x + b*d*p*x**2 + b*d*q*x**2)...