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

Optimal result

Integrand size = 31, antiderivative size = 2240 \[ \int (g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx =\text {Too large to display} \] Output:

3/16*(-a*h+b*g)^2*p*q*r^2*(h*x+g)^2/b^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-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+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+1/4*(h*x+g)^4*ln(e 
*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/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^2-1/2*(-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)^3*p*q*r^2*(b*x+a)*l 
n(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*(-c* 
h+d*g)*p*q*r^2*(h*x+g)^3*ln(b*x+a)/d/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+5/8*(-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+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^4-2*(-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^4-1/8*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+1/6*(-a*h+b*g)*(-c*h+d...
 

Mathematica [A] (warning: unable to verify)

Time = 2.86 (sec) , antiderivative size = 1386, normalized size of antiderivative = 0.62 \[ \int (g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

(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*L 
og[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*...
 

Rubi [A] (verified)

Time = 3.22 (sec) , antiderivative size = 1802, normalized size of antiderivative = 0.80, 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.

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

\(\Big \downarrow \) 2984

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

\(\Big \downarrow \) 2993

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b p r \left (q r \int \frac {(g+h x)^4 \log (c+d x)}{a+b x}dx+p r \int \frac {(g+h x)^4 \log (a+b x)}{a+b x}dx-\left (\left (\frac {(b g-a h)^4 \log (a+b x)}{b^5}+\frac {h x (b g-a h)^3}{b^4}+\frac {(g+h x)^2 (b g-a h)^2}{2 b^3}+\frac {(g+h x)^3 (b g-a h)}{3 b^2}+\frac {(g+h x)^4}{4 b}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+p r \log (a+b x)+q r \log (c+d x)\right )\right )\right )}{2 h}-\frac {d q r \left (p r \int \frac {(g+h x)^4 \log (a+b x)}{c+d x}dx+q r \int \frac {(g+h x)^4 \log (c+d x)}{c+d x}dx-\left (\left (\frac {(d g-c h)^4 \log (c+d x)}{d^5}+\frac {h x (d g-c h)^3}{d^4}+\frac {(g+h x)^2 (d g-c h)^2}{2 d^3}+\frac {(g+h x)^3 (d g-c h)}{3 d^2}+\frac {(g+h x)^4}{4 d}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+p r \log (a+b x)+q r \log (c+d x)\right )\right )\right )}{2 h}+\frac {(g+h x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h}\)

\(\Big \downarrow \) 2858

\(\displaystyle -\frac {b p r \left (\frac {p r \int \frac {\left (b \left (g-\frac {a h}{b}\right )+h (a+b x)\right )^4 \log (a+b x)}{b^4 (a+b x)}d(a+b x)}{b}+q r \int \frac {(g+h x)^4 \log (c+d x)}{a+b x}dx-\left (\left (\frac {(b g-a h)^4 \log (a+b x)}{b^5}+\frac {h x (b g-a h)^3}{b^4}+\frac {(g+h x)^2 (b g-a h)^2}{2 b^3}+\frac {(g+h x)^3 (b g-a h)}{3 b^2}+\frac {(g+h x)^4}{4 b}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+p r \log (a+b x)+q r \log (c+d x)\right )\right )\right )}{2 h}-\frac {d q r \left (p r \int \frac {(g+h x)^4 \log (a+b x)}{c+d x}dx+\frac {q r \int \frac {\left (d \left (g-\frac {c h}{d}\right )+h (c+d x)\right )^4 \log (c+d x)}{d^4 (c+d x)}d(c+d x)}{d}-\left (\left (\frac {(d g-c h)^4 \log (c+d x)}{d^5}+\frac {h x (d g-c h)^3}{d^4}+\frac {(g+h x)^2 (d g-c h)^2}{2 d^3}+\frac {(g+h x)^3 (d g-c h)}{3 d^2}+\frac {(g+h x)^4}{4 d}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+p r \log (a+b x)+q r \log (c+d x)\right )\right )\right )}{2 h}+\frac {(g+h x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b p r \left (\frac {p r \int \frac {(b g-a h+h (a+b x))^4 \log (a+b x)}{a+b x}d(a+b x)}{b^5}+q r \int \frac {(g+h x)^4 \log (c+d x)}{a+b x}dx-\left (\left (\frac {(b g-a h)^4 \log (a+b x)}{b^5}+\frac {h x (b g-a h)^3}{b^4}+\frac {(g+h x)^2 (b g-a h)^2}{2 b^3}+\frac {(g+h x)^3 (b g-a h)}{3 b^2}+\frac {(g+h x)^4}{4 b}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+p r \log (a+b x)+q r \log (c+d x)\right )\right )\right )}{2 h}-\frac {d q r \left (p r \int \frac {(g+h x)^4 \log (a+b x)}{c+d x}dx+\frac {q r \int \frac {(d g-c h+h (c+d x))^4 \log (c+d x)}{c+d x}d(c+d x)}{d^5}-\left (\left (\frac {(d g-c h)^4 \log (c+d x)}{d^5}+\frac {h x (d g-c h)^3}{d^4}+\frac {(g+h x)^2 (d g-c h)^2}{2 d^3}+\frac {(g+h x)^3 (d g-c h)}{3 d^2}+\frac {(g+h x)^4}{4 d}\right ) \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+p r \log (a+b x)+q r \log (c+d x)\right )\right )\right )}{2 h}+\frac {(g+h x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 h}\)

\(\Big \downarrow \) 2772

\(\displaystyle \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (g+h x)^4}{4 h}-\frac {b p r \left (-\left (\left (\frac {\log (a+b x) (b g-a h)^4}{b^5}+\frac {h x (b g-a h)^3}{b^4}+\frac {(g+h x)^2 (b g-a h)^2}{2 b^3}+\frac {(g+h x)^3 (b g-a h)}{3 b^2}+\frac {(g+h x)^4}{4 b}\right ) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right )+\frac {p r \left (\log ^2(a+b x) (b g-a h)^4+4 h (a+b x) \log (a+b x) (b g-a h)^3+3 h^2 (a+b x)^2 \log (a+b x) (b g-a h)^2+\frac {4}{3} h^3 (a+b x)^3 \log (a+b x) (b g-a h)+\frac {1}{4} h^4 (a+b x)^4 \log (a+b x)-\int \left (\frac {1}{4} (a+b x)^3 h^4+\frac {4}{3} (b g-a h) (a+b x)^2 h^3+3 (b g-a h)^2 (a+b x) h^2+4 (b g-a h)^3 h+\frac {(b g-a h)^4 \log (a+b x)}{a+b x}\right )d(a+b x)\right )}{b^5}+q r \int \frac {(g+h x)^4 \log (c+d x)}{a+b x}dx\right )}{2 h}-\frac {d q r \left (-\left (\left (\frac {\log (c+d x) (d g-c h)^4}{d^5}+\frac {h x (d g-c h)^3}{d^4}+\frac {(g+h x)^2 (d g-c h)^2}{2 d^3}+\frac {(g+h x)^3 (d g-c h)}{3 d^2}+\frac {(g+h x)^4}{4 d}\right ) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right )+p r \int \frac {(g+h x)^4 \log (a+b x)}{c+d x}dx+\frac {q r \left (\log ^2(c+d x) (d g-c h)^4+4 h (c+d x) \log (c+d x) (d g-c h)^3+3 h^2 (c+d x)^2 \log (c+d x) (d g-c h)^2+\frac {4}{3} h^3 (c+d x)^3 \log (c+d x) (d g-c h)+\frac {1}{4} h^4 (c+d x)^4 \log (c+d x)-\int \left (\frac {1}{4} (c+d x)^3 h^4+\frac {4}{3} (d g-c h) (c+d x)^2 h^3+3 (d g-c h)^2 (c+d x) h^2+4 (d g-c h)^3 h+\frac {(d g-c h)^4 \log (c+d x)}{c+d x}\right )d(c+d x)\right )}{d^5}\right )}{2 h}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (g+h x)^4}{4 h}-\frac {d q r \left (\frac {q r \left (\frac {1}{2} \log ^2(c+d x) (d g-c h)^4-4 h (c+d x) (d g-c h)^3+4 h (c+d x) \log (c+d x) (d g-c h)^3-\frac {3}{2} h^2 (c+d x)^2 (d g-c h)^2+3 h^2 (c+d x)^2 \log (c+d x) (d g-c h)^2-\frac {4}{9} h^3 (c+d x)^3 (d g-c h)+\frac {4}{3} h^3 (c+d x)^3 \log (c+d x) (d g-c h)-\frac {1}{16} h^4 (c+d x)^4+\frac {1}{4} h^4 (c+d x)^4 \log (c+d x)\right )}{d^5}-\left (\frac {\log (c+d x) (d g-c h)^4}{d^5}+\frac {h x (d g-c h)^3}{d^4}+\frac {(g+h x)^2 (d g-c h)^2}{2 d^3}+\frac {(g+h x)^3 (d g-c h)}{3 d^2}+\frac {(g+h x)^4}{4 d}\right ) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )+p r \int \frac {(g+h x)^4 \log (a+b x)}{c+d x}dx\right )}{2 h}-\frac {b p r \left (\frac {p r \left (\frac {1}{2} \log ^2(a+b x) (b g-a h)^4-4 h (a+b x) (b g-a h)^3+4 h (a+b x) \log (a+b x) (b g-a h)^3-\frac {3}{2} h^2 (a+b x)^2 (b g-a h)^2+3 h^2 (a+b x)^2 \log (a+b x) (b g-a h)^2-\frac {4}{9} h^3 (a+b x)^3 (b g-a h)+\frac {4}{3} h^3 (a+b x)^3 \log (a+b x) (b g-a h)-\frac {1}{16} h^4 (a+b x)^4+\frac {1}{4} h^4 (a+b x)^4 \log (a+b x)\right )}{b^5}-\left (\frac {\log (a+b x) (b g-a h)^4}{b^5}+\frac {h x (b g-a h)^3}{b^4}+\frac {(g+h x)^2 (b g-a h)^2}{2 b^3}+\frac {(g+h x)^3 (b g-a h)}{3 b^2}+\frac {(g+h x)^4}{4 b}\right ) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )+q r \int \frac {(g+h x)^4 \log (c+d x)}{a+b x}dx\right )}{2 h}\)

\(\Big \downarrow \) 2865

\(\displaystyle \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (g+h x)^4}{4 h}-\frac {d q r \left (\frac {q r \left (\frac {1}{2} \log ^2(c+d x) (d g-c h)^4-4 h (c+d x) (d g-c h)^3+4 h (c+d x) \log (c+d x) (d g-c h)^3-\frac {3}{2} h^2 (c+d x)^2 (d g-c h)^2+3 h^2 (c+d x)^2 \log (c+d x) (d g-c h)^2-\frac {4}{9} h^3 (c+d x)^3 (d g-c h)+\frac {4}{3} h^3 (c+d x)^3 \log (c+d x) (d g-c h)-\frac {1}{16} h^4 (c+d x)^4+\frac {1}{4} h^4 (c+d x)^4 \log (c+d x)\right )}{d^5}-\left (\frac {\log (c+d x) (d g-c h)^4}{d^5}+\frac {h x (d g-c h)^3}{d^4}+\frac {(g+h x)^2 (d g-c h)^2}{2 d^3}+\frac {(g+h x)^3 (d g-c h)}{3 d^2}+\frac {(g+h x)^4}{4 d}\right ) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )+p r \int \left (\frac {\log (a+b x) (d g-c h)^4}{d^4 (c+d x)}+\frac {h \log (a+b x) (d g-c h)^3}{d^4}+\frac {h (g+h x) \log (a+b x) (d g-c h)^2}{d^3}+\frac {h (g+h x)^2 \log (a+b x) (d g-c h)}{d^2}+\frac {h (g+h x)^3 \log (a+b x)}{d}\right )dx\right )}{2 h}-\frac {b p r \left (\frac {p r \left (\frac {1}{2} \log ^2(a+b x) (b g-a h)^4-4 h (a+b x) (b g-a h)^3+4 h (a+b x) \log (a+b x) (b g-a h)^3-\frac {3}{2} h^2 (a+b x)^2 (b g-a h)^2+3 h^2 (a+b x)^2 \log (a+b x) (b g-a h)^2-\frac {4}{9} h^3 (a+b x)^3 (b g-a h)+\frac {4}{3} h^3 (a+b x)^3 \log (a+b x) (b g-a h)-\frac {1}{16} h^4 (a+b x)^4+\frac {1}{4} h^4 (a+b x)^4 \log (a+b x)\right )}{b^5}-\left (\frac {\log (a+b x) (b g-a h)^4}{b^5}+\frac {h x (b g-a h)^3}{b^4}+\frac {(g+h x)^2 (b g-a h)^2}{2 b^3}+\frac {(g+h x)^3 (b g-a h)}{3 b^2}+\frac {(g+h x)^4}{4 b}\right ) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )+q r \int \left (\frac {\log (c+d x) (b g-a h)^4}{b^4 (a+b x)}+\frac {h \log (c+d x) (b g-a h)^3}{b^4}+\frac {h (g+h x) \log (c+d x) (b g-a h)^2}{b^3}+\frac {h (g+h x)^2 \log (c+d x) (b g-a h)}{b^2}+\frac {h (g+h x)^3 \log (c+d x)}{b}\right )dx\right )}{2 h}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (g+h x)^4}{4 h}-\frac {d q r \left (\frac {q r \left (\frac {1}{2} \log ^2(c+d x) (d g-c h)^4-4 h (c+d x) (d g-c h)^3+4 h (c+d x) \log (c+d x) (d g-c h)^3-\frac {3}{2} h^2 (c+d x)^2 (d g-c h)^2+3 h^2 (c+d x)^2 \log (c+d x) (d g-c h)^2-\frac {4}{9} h^3 (c+d x)^3 (d g-c h)+\frac {4}{3} h^3 (c+d x)^3 \log (c+d x) (d g-c h)-\frac {1}{16} h^4 (c+d x)^4+\frac {1}{4} h^4 (c+d x)^4 \log (c+d x)\right )}{d^5}-\left (\frac {\log (c+d x) (d g-c h)^4}{d^5}+\frac {h x (d g-c h)^3}{d^4}+\frac {(g+h x)^2 (d g-c h)^2}{2 d^3}+\frac {(g+h x)^3 (d g-c h)}{3 d^2}+\frac {(g+h x)^4}{4 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)^4}{4 b^4 d}-\frac {h x (b g-a h)^3}{4 b^3 d}-\frac {(d g-c h) \log (a+b x) (b g-a h)^3}{3 b^3 d^2}-\frac {(g+h x)^2 (b g-a h)^2}{8 b^2 d}-\frac {h (d g-c h) x (b g-a h)^2}{3 b^2 d^2}-\frac {(d g-c h)^2 \log (a+b x) (b g-a h)^2}{2 b^2 d^3}-\frac {(g+h x)^3 (b g-a h)}{12 b d}-\frac {(d g-c h) (g+h x)^2 (b g-a h)}{6 b d^2}-\frac {h (d g-c h)^2 x (b g-a h)}{2 b d^3}-\frac {(g+h x)^4}{16 d}-\frac {(d g-c h) (g+h x)^3}{9 d^2}-\frac {(d g-c h)^2 (g+h x)^2}{4 d^3}-\frac {h (d g-c h)^3 x}{d^4}+\frac {(g+h x)^4 \log (a+b x)}{4 d}+\frac {(d g-c h) (g+h x)^3 \log (a+b x)}{3 d^2}+\frac {(d g-c h)^2 (g+h x)^2 \log (a+b x)}{2 d^3}+\frac {h (d g-c h)^3 (a+b x) \log (a+b x)}{b d^4}+\frac {(d g-c h)^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d^5}+\frac {(d g-c h)^4 \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{d^5}\right )\right )}{2 h}-\frac {b p r \left (\frac {p r \left (\frac {1}{2} \log ^2(a+b x) (b g-a h)^4-4 h (a+b x) (b g-a h)^3+4 h (a+b x) \log (a+b x) (b g-a h)^3-\frac {3}{2} h^2 (a+b x)^2 (b g-a h)^2+3 h^2 (a+b x)^2 \log (a+b x) (b g-a h)^2-\frac {4}{9} h^3 (a+b x)^3 (b g-a h)+\frac {4}{3} h^3 (a+b x)^3 \log (a+b x) (b g-a h)-\frac {1}{16} h^4 (a+b x)^4+\frac {1}{4} h^4 (a+b x)^4 \log (a+b x)\right )}{b^5}-\left (\frac {\log (a+b x) (b g-a h)^4}{b^5}+\frac {h x (b g-a h)^3}{b^4}+\frac {(g+h x)^2 (b g-a h)^2}{2 b^3}+\frac {(g+h x)^3 (b g-a h)}{3 b^2}+\frac {(g+h x)^4}{4 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)^4}{b^5}+\frac {\operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) (b g-a h)^4}{b^5}-\frac {h x (b g-a h)^3}{b^4}+\frac {h (c+d x) \log (c+d x) (b g-a h)^3}{b^4 d}-\frac {(g+h x)^2 (b g-a h)^2}{4 b^3}-\frac {h (d g-c h) x (b g-a h)^2}{2 b^3 d}-\frac {(d g-c h)^2 \log (c+d x) (b g-a h)^2}{2 b^3 d^2}+\frac {(g+h x)^2 \log (c+d x) (b g-a h)^2}{2 b^3}-\frac {(g+h x)^3 (b g-a h)}{9 b^2}-\frac {(d g-c h) (g+h x)^2 (b g-a h)}{6 b^2 d}-\frac {h (d g-c h)^2 x (b g-a h)}{3 b^2 d^2}-\frac {(d g-c h)^3 \log (c+d x) (b g-a h)}{3 b^2 d^3}+\frac {(g+h x)^3 \log (c+d x) (b g-a h)}{3 b^2}-\frac {(g+h x)^4}{16 b}-\frac {(d g-c h) (g+h x)^3}{12 b d}-\frac {(d g-c h)^2 (g+h x)^2}{8 b d^2}-\frac {h (d g-c h)^3 x}{4 b d^3}-\frac {(d g-c h)^4 \log (c+d x)}{4 b d^4}+\frac {(g+h x)^4 \log (c+d x)}{4 b}\right )\right )}{2 h}\)

Input:

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

Output:

((g + h*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)/(4*h) - (d*q*r*((q*r* 
(-4*h*(d*g - c*h)^3*(c + d*x) - (3*h^2*(d*g - c*h)^2*(c + d*x)^2)/2 - (4*h 
^3*(d*g - c*h)*(c + d*x)^3)/9 - (h^4*(c + d*x)^4)/16 + 4*h*(d*g - c*h)^3*( 
c + d*x)*Log[c + d*x] + 3*h^2*(d*g - c*h)^2*(c + d*x)^2*Log[c + d*x] + (4* 
h^3*(d*g - c*h)*(c + d*x)^3*Log[c + d*x])/3 + (h^4*(c + d*x)^4*Log[c + d*x 
])/4 + ((d*g - c*h)^4*Log[c + d*x]^2)/2))/d^5 - ((h*(d*g - c*h)^3*x)/d^4 + 
 ((d*g - c*h)^2*(g + h*x)^2)/(2*d^3) + ((d*g - c*h)*(g + h*x)^3)/(3*d^2) + 
 (g + h*x)^4/(4*d) + ((d*g - c*h)^4*Log[c + d*x])/d^5)*(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*(-1/4*(h*( 
b*g - a*h)^3*x)/(b^3*d) - (h*(b*g - a*h)^2*(d*g - c*h)*x)/(3*b^2*d^2) - (h 
*(b*g - a*h)*(d*g - c*h)^2*x)/(2*b*d^3) - (h*(d*g - c*h)^3*x)/d^4 - ((b*g 
- a*h)^2*(g + h*x)^2)/(8*b^2*d) - ((b*g - a*h)*(d*g - c*h)*(g + h*x)^2)/(6 
*b*d^2) - ((d*g - c*h)^2*(g + h*x)^2)/(4*d^3) - ((b*g - a*h)*(g + h*x)^3)/ 
(12*b*d) - ((d*g - c*h)*(g + h*x)^3)/(9*d^2) - (g + h*x)^4/(16*d) - ((b*g 
- a*h)^4*Log[a + b*x])/(4*b^4*d) - ((b*g - a*h)^3*(d*g - c*h)*Log[a + b*x] 
)/(3*b^3*d^2) - ((b*g - a*h)^2*(d*g - c*h)^2*Log[a + b*x])/(2*b^2*d^3) + ( 
h*(d*g - c*h)^3*(a + b*x)*Log[a + b*x])/(b*d^4) + ((d*g - c*h)^2*(g + h*x) 
^2*Log[a + b*x])/(2*d^3) + ((d*g - c*h)*(g + h*x)^3*Log[a + b*x])/(3*d^2) 
+ ((g + h*x)^4*Log[a + b*x])/(4*d) + ((d*g - c*h)^4*Log[a + b*x]*Log[(b*(c 
 + d*x))/(b*c - a*d)])/d^5 + ((d*g - c*h)^4*PolyLog[2, -((d*(a + b*x))/...
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 1799, normalized size of antiderivative = 0.80 \[ \int (g+h x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

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*...
 

Giac [F]

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

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

Output:

sage0*x
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(144*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**5*d** 
5*h**3*p**2*q*r + 144*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**5*d**5*h**3*p*q**2*r - 144*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*b*c*d**4*h**3*p**2*q*r - 144*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*b*c*d**4*h**3*p*q**2*r 
 - 576*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*b 
*d**5*g*h**2*p**2*q*r - 576*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*b*d**5*g*h**2*p*q**2*r + 576*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**2*c*d**4*g*h**2*p**2*q*r + 576 
*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**2*c* 
d**4*g*h**2*p*q**2*r + 864*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 ...