\(\int \frac {\log ^2(e (f (a+b x)^p (c+d x)^q)^r)}{a+b x} \, dx\) [20]

Optimal result

Integrand size = 31, antiderivative size = 430 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{a+b x} \, dx=\frac {\log ^3\left ((a+b x)^{p r}\right )}{3 b p r}-\frac {q \log ^2\left ((a+b x)^{p r}\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b p}+\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left ((c+d x)^{q r}\right )}{b p r}+\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{q r}\right )}{b}-\frac {2 q r \log \left ((a+b x)^{p r}\right ) \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b}+\frac {2 q r \log \left ((c+d x)^{q r}\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b}-\frac {1}{4} \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \left (\frac {8 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log \left ((c+d x)^{q r}\right )}{b}+\frac {\left (\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )^2}{b p r}+\frac {8 q r \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b}\right )+\frac {2 p q r^2 \operatorname {PolyLog}\left (3,-\frac {d (a+b x)}{b c-a d}\right )}{b}-\frac {2 q^2 r^2 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{b c-a d}\right )}{b} \] Output:

1/3*ln((b*x+a)^(p*r))^3/b/p/r-q*ln((b*x+a)^(p*r))^2*ln(b*(d*x+c)/(-a*d+b*c 
))/b/p+ln((b*x+a)^(p*r))^2*ln((d*x+c)^(q*r))/b/p/r+ln(-d*(b*x+a)/(-a*d+b*c 
))*ln((d*x+c)^(q*r))^2/b-2*q*r*ln((b*x+a)^(p*r))*polylog(2,-d*(b*x+a)/(-a* 
d+b*c))/b+2*q*r*ln((d*x+c)^(q*r))*polylog(2,b*(d*x+c)/(-a*d+b*c))/b-1/4*(l 
n((b*x+a)^(p*r))+ln((d*x+c)^(q*r))-ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r))*(8*ln( 
-d*(b*x+a)/(-a*d+b*c))*ln((d*x+c)^(q*r))/b+(ln((b*x+a)^(p*r))-ln((d*x+c)^( 
q*r))+ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r))^2/b/p/r+8*q*r*polylog(2,b*(d*x+c)/( 
-a*d+b*c))/b)+2*p*q*r^2*polylog(3,-d*(b*x+a)/(-a*d+b*c))/b-2*q^2*r^2*polyl 
og(3,b*(d*x+c)/(-a*d+b*c))/b
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.07 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{a+b x} \, dx=\frac {p^2 r^2 \log ^3(a+b x)+6 p q r^2 \log ^2(a+b x) \log (c+d x)-6 p q r^2 \log (a+b x) \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)+3 q^2 r^2 \log (a+b x) \log ^2(c+d x)-3 q^2 r^2 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log ^2(c+d x)-3 p q r^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )-3 p r \log ^2(a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-6 q r \log (a+b x) \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+6 q r \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+3 \log (a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-6 p q r^2 \log (a+b x) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )+6 q r \left (-p r \log (a+b x)+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )+6 p q r^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{-b c+a d}\right )-6 q^2 r^2 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{b c-a d}\right )}{3 b} \] Input:

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

Output:

(p^2*r^2*Log[a + b*x]^3 + 6*p*q*r^2*Log[a + b*x]^2*Log[c + d*x] - 6*p*q*r^ 
2*Log[a + b*x]*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x] + 3*q^2*r^2* 
Log[a + b*x]*Log[c + d*x]^2 - 3*q^2*r^2*Log[(d*(a + b*x))/(-(b*c) + a*d)]* 
Log[c + d*x]^2 - 3*p*q*r^2*Log[a + b*x]^2*Log[(b*(c + d*x))/(b*c - a*d)] - 
 3*p*r*Log[a + b*x]^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - 6*q*r*Log[a + 
 b*x]*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + 6*q*r*Log[(d*(a 
+ b*x))/(-(b*c) + a*d)]*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] 
+ 3*Log[a + b*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2 - 6*p*q*r^2*Log[a 
+ b*x]*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] + 6*q*r*(-(p*r*Log[a + b*x 
]) + Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])*PolyLog[2, (b*(c + d*x))/(b*c - 
 a*d)] + 6*p*q*r^2*PolyLog[3, (d*(a + b*x))/(-(b*c) + a*d)] - 6*q^2*r^2*Po 
lyLog[3, (b*(c + d*x))/(b*c - a*d)])/(3*b)
 

Rubi [A] (verified)

Time = 1.60 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {2982, 2841, 2840, 2838, 7237, 7293, 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.

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

\(\Big \downarrow \) 2982

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

\(\Big \downarrow \) 2841

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

\(\Big \downarrow \) 2840

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7237

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

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 2982

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.)) 
^(r_.)]^2/((g_.) + (h_.)*(x_)), x_Symbol] :> Int[(Log[(a + b*x)^(p*r)] + Lo 
g[(c + d*x)^(q*r)])^2/(g + h*x), x] + Simp[(Log[e*(f*(a + b*x)^p*(c + d*x)^ 
q)^r] - Log[(a + b*x)^(p*r)] - Log[(c + d*x)^(q*r)])*(2   Int[Log[(c + d*x) 
^(q*r)]/(g + h*x), x] + Int[(Log[(a + b*x)^(p*r)] - Log[(c + d*x)^(q*r)] + 
Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(g + h*x), x]), x] /; FreeQ[{a, b, c, 
 d, e, f, g, h, p, q, r}, x] && NeQ[b*c - a*d, 0] && EqQ[b*g - a*h, 0]
 

rule 7237

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si 
mp[q*(y^(m + 1)/(m + 1)), x] /;  !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 3*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)**2/(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*d* 
p*q*r - 3*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)**2/(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*d*q**2*r + 3*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)**2/(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*c*p*q*r + 3*int(log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)**2/(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*c*q**2*r + log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)**3)/(3 
*b*r*(p + q))