\(\int \log ^2(e (f (a+b x)^p (c+d x)^q)^r) \, dx\) [38]

Optimal result

Integrand size = 23, antiderivative size = 269 \[ \int \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=2 (p+q)^2 r^2 x-\frac {2 (b c-a d) q (p+q) r^2 \log (c+d x)}{b d}-\frac {2 (b c-a d) p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d}-\frac {(b c-a d) q^2 r^2 \log ^2(c+d x)}{b d}-\frac {2 (p+q) r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}+\frac {2 (b c-a d) q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}+\frac {(a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}-\frac {2 (b c-a d) p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b d} \] Output:

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

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.45 \[ \int \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {2 a d p q r^2+2 b d p^2 r^2 x+4 b d p q r^2 x+2 b d q^2 r^2 x-a d p^2 r^2 \log ^2(a+b x)-2 b c p q r^2 \log (c+d x)+2 a d p q r^2 \log (c+d x)-2 b c q^2 r^2 \log (c+d x)-b c q^2 r^2 \log ^2(c+d x)-2 p r \log (a+b x) \left (b c q r \log (c+d x)+(-b c+a d) q r \log \left (\frac {b (c+d x)}{b c-a d}\right )+a d \left (q r-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right )-2 a d p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 b d p r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 b d q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 b c q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+b d x \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 (b c-a d) p q r^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )}{b d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.84, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {2979, 2979, 16, 24, 2980, 2837, 2738, 2841, 2840, 2838}

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.

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

\(\Big \downarrow \) 2979

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

\(\Big \downarrow \) 2979

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 2980

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

\(\Big \downarrow \) 2837

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

\(\Big \downarrow \) 2738

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

\(\Big \downarrow \) 2841

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

\(\Big \downarrow \) 2840

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

\(\Big \downarrow \) 2838

\(\displaystyle \frac {2 q r (b c-a d) \left (\frac {\log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d}-\frac {b p r \left (\frac {\operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b}+\frac {\log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{b}\right )}{d}-\frac {q r \log ^2(c+d x)}{2 d}\right )}{b}+\frac {(a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}-2 r (p+q) \left (\frac {(a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}+\frac {q r (b c-a d) \log (c+d x)}{b d}-(r x (p+q))\right )\)

Input:

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

Output:

((a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)/b - 2*(p + q)*r*(-((p + 
 q)*r*x) + ((b*c - a*d)*q*r*Log[c + d*x])/(b*d) + ((a + b*x)*Log[e*(f*(a + 
 b*x)^p*(c + d*x)^q)^r])/b) + (2*(b*c - a*d)*q*r*(-1/2*(q*r*Log[c + d*x]^2 
)/d + (Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d - (b*p*r*((Log 
[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/b + PolyLog[2, (b*(c + d*x))/ 
(b*c - a*d)]/b))/d))/b
 

Defintions of rubi rules used

rule 16

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

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 2738

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

rule 2837

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

rule 2838

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 2840

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_ 
Symbol] :> Simp[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 2841

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] - Simp[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 2979

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.)) 
^(r_.)]^(s_.), x_Symbol] :> Simp[(a + b*x)*(Log[e*(f*(a + b*x)^p*(c + d*x)^ 
q)^r]^s/b), x] + (Simp[q*r*s*((b*c - a*d)/b)   Int[Log[e*(f*(a + b*x)^p*(c 
+ d*x)^q)^r]^(s - 1)/(c + d*x), x], x] - Simp[r*s*(p + q)   Int[Log[e*(f*(a 
 + b*x)^p*(c + d*x)^q)^r]^(s - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p, q 
, r, s}, x] && NeQ[b*c - a*d, 0] && NeQ[p + q, 0] && IGtQ[s, 0] && LtQ[s, 4 
]
 

rule 2980

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.11 \[ \int \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=x \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} - \frac {2 \, {\left (f {\left (p + q\right )} x - \frac {a f p \log \left (b x + a\right )}{b} - \frac {c f q \log \left (d x + c\right )}{d}\right )} r \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{f} - \frac {{\left (\frac {2 \, {\left (p q + q^{2}\right )} c f^{2} \log \left (d x + c\right )}{d} - \frac {2 \, {\left (b c f^{2} p q - a d f^{2} p q\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )}}{b d} + \frac {a d f^{2} p^{2} \log \left (b x + a\right )^{2} + 2 \, b c f^{2} p q \log \left (b x + a\right ) \log \left (d x + c\right ) + b c f^{2} q^{2} \log \left (d x + c\right )^{2} - 2 \, {\left (p^{2} + 2 \, p q + q^{2}\right )} b d f^{2} x + 2 \, {\left (p^{2} + p q\right )} a d f^{2} \log \left (b x + a\right )}{b d}\right )} r^{2}}{f^{2}} \] Input:

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

Output:

x*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2 - 2*(f*(p + q)*x - a*f*p*log(b*x 
+ a)/b - c*f*q*log(d*x + c)/d)*r*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/f - 
(2*(p*q + q^2)*c*f^2*log(d*x + c)/d - 2*(b*c*f^2*p*q - a*d*f^2*p*q)*(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*d) + (a*d*f^2*p^2*log(b*x + a)^2 + 2*b*c*f^2*p*q*log(b*x + a)*log 
(d*x + c) + b*c*f^2*q^2*log(d*x + c)^2 - 2*(p^2 + 2*p*q + q^2)*b*d*f^2*x + 
 2*(p^2 + p*q)*a*d*f^2*log(b*x + a))/(b*d))*r^2/f^2
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 2*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*d* 
*2*p**2*q*r - 2*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*d**2*p*q**2*r + 4*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*b*c*d*p**2*q*r + 4*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*b*c*d*p*q**2*r - 2*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)*b**2*c**2*p**2*q*r - 2*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)*b**2*c**2*p*q**2*r + 2*log(c + d*x) 
*a*d*p**2*q*r**2 + 4*log(c + d*x)*a*d*p*q**2*r**2 + 2*log(c + d*x)*a*d*q** 
3*r**2 - 2*log(c + d*x)*b*c*p**2*q*r**2 - 4*log(c + d*x)*b*c*p*q**2*r**2 - 
 2*log(c + d*x)*b*c*q**3*r**2 + log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r) 
*e)**2*a*d*p + log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)**2*b*c*q + lo 
g(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)**2*b*d*p*x + log(f**r*(c + d*x 
)**(q*r)*(a + b*x)**(p*r)*e)**2*b*d*q*x - 2*log(f**r*(c + d*x)**(q*r)*(a + 
 b*x)**(p*r)*e)*a*d*p**2*r - 4*log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*...