3.39 \(\int \frac {\log ^2(e (f (a+b x)^p (c+d x)^q)^r)}{g+h x} \, dx\)

Optimal. Leaf size=1471 \[ \frac {p q \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) r^2}{h}+\frac {p^2 \log ^2(a+b x) \log (g+h x) r^2}{h}+\frac {q^2 \log ^2(c+d x) \log (g+h x) r^2}{h}+\frac {2 p q \log (a+b x) \log (c+d x) \log (g+h x) r^2}{h}-\frac {p^2 \log ^2(a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right ) r^2}{h}+\frac {p q \log ^2\left (-\frac {h (c+d x)}{d g-c h}\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right ) r^2}{h}+\frac {p q \log ^2\left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right ) r^2}{h}-\frac {2 p q \log (a+b x) \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right ) r^2}{h}-\frac {2 p q \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right ) r^2}{h}-\frac {q^2 \log ^2(c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right ) r^2}{h}-\frac {p q \log ^2\left (-\frac {h (c+d x)}{d g-c h}\right ) \log \left (\frac {d (g+h x)}{d g-c h}\right ) r^2}{h}-\frac {2 p q \log (a+b x) \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right ) r^2}{h}+\frac {2 p q \log (a+b x) \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log \left (\frac {d (g+h x)}{d g-c h}\right ) r^2}{h}+\frac {2 p q \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \log \left (\frac {d (g+h x)}{d g-c h}\right ) r^2}{h}-\frac {p q \log ^2\left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \log \left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right ) r^2}{h}+\frac {2 p q \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right ) r^2}{h}-\frac {2 p q \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \text {Li}_2\left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) r^2}{h}-\frac {2 p^2 \text {Li}_3\left (-\frac {h (a+b x)}{b g-a h}\right ) r^2}{h}-\frac {2 p q \text {Li}_3\left (-\frac {h (a+b x)}{b g-a h}\right ) r^2}{h}-\frac {2 q^2 \text {Li}_3\left (-\frac {h (c+d x)}{d g-c h}\right ) r^2}{h}-\frac {2 p q \text {Li}_3\left (-\frac {h (c+d x)}{d g-c h}\right ) r^2}{h}-\frac {2 p q \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right ) r^2}{h}+\frac {2 p q \text {Li}_3\left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) r^2}{h}-\frac {2 p \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x) r}{h}-\frac {2 q \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x) r}{h}+\frac {2 p \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right ) r}{h}+\frac {2 q \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log \left (\frac {d (g+h x)}{d g-c h}\right ) r}{h}-\frac {2 p \left (q r \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \text {Li}_2\left (-\frac {h (a+b x)}{b g-a h}\right ) r}{h}+\frac {2 q \left (p r \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \text {Li}_2\left (-\frac {h (c+d x)}{d g-c h}\right ) r}{h}+\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.93, antiderivative size = 2096, normalized size of antiderivative = 1.42, number of steps used = 29, number of rules used = 14, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules used = {2497, 2500, 2394, 2393, 2391, 2433, 2375, 2317, 2374, 6589, 2440, 2437, 2435, 2315} \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2317

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

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2375

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

Rule 2391

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 2393

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 2394

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 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 2435

Int[(Log[(a_) + (b_.)*(x_)]*Log[(c_) + (d_.)*(x_)])/(x_), x_Symbol] :> Simp[Log[-((b*x)/a)]*Log[a + b*x]*Log[c
+ d*x], x] + (Simp[(1*(Log[-((b*x)/a)] - Log[-(((b*c - a*d)*x)/(a*(c + d*x)))] + Log[(b*c - a*d)/(b*(c + d*x)
)])*Log[(a*(c + d*x))/(c*(a + b*x))]^2)/2, x] - Simp[(1*(Log[-((b*x)/a)] - Log[-((d*x)/c)])*(Log[a + b*x] + Lo
g[(a*(c + d*x))/(c*(a + b*x))])^2)/2, x] + Simp[(Log[c + d*x] - Log[(a*(c + d*x))/(c*(a + b*x))])*PolyLog[2, 1
+ (b*x)/a], x] + Simp[(Log[a + b*x] + Log[(a*(c + d*x))/(c*(a + b*x))])*PolyLog[2, 1 + (d*x)/c], x] + Simp[Lo
g[(a*(c + d*x))/(c*(a + b*x))]*PolyLog[2, (c*(a + b*x))/(a*(c + d*x))], x] - Simp[Log[(a*(c + d*x))/(c*(a + b*
x))]*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))], x] - Simp[PolyLog[3, 1 + (b*x)/a], x] - Simp[PolyLog[3, 1 + (d*x
)/c], x] + Simp[PolyLog[3, (c*(a + b*x))/(a*(c + d*x))], x] - Simp[PolyLog[3, (d*(a + b*x))/(b*(c + d*x))], x]
) /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 2437

Int[(Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)])/(x_), x_Symbol] :> Dist[m, In
t[(Log[i + j*x]*Log[c*(d + e*x)^n])/x, x], x] - Dist[m*Log[i + j*x] - Log[h*(i + j*x)^m], Int[Log[c*(d + e*x)^
n]/x, x], x] /; FreeQ[{c, d, e, h, i, j, m, n}, x] && NeQ[e*i - d*j, 0] && NeQ[i + j*x, h*(i + j*x)^m]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))
*((k_) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/l, Subst[Int[x^r*(a + b*Log[c*(-((e*k - d*l)/l) + (e*x)/l)^n])
*(f + g*Log[h*(-((j*k - i*l)/l) + (j*x)/l)^m]), x], x, k + l*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k,
l, m, n}, x] && IntegerQ[r]

Rule 2497

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

Rule 2500

Int[(Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*((s_.) + Log[(i_.)*((g_.)
+ (h_.)*(x_))^(n_.)]*(t_.)))/((j_.) + (k_.)*(x_)), x_Symbol] :> Dist[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - Lo
g[(a + b*x)^(p*r)] - Log[(c + d*x)^(q*r)], Int[(s + t*Log[i*(g + h*x)^n])/(j + k*x), x], x] + (Int[(Log[(a + b
*x)^(p*r)]*(s + t*Log[i*(g + h*x)^n]))/(j + k*x), x] + Int[(Log[(c + d*x)^(q*r)]*(s + t*Log[i*(g + h*x)^n]))/(
j + k*x), x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, s, t, n, p, q, r}, x] && NeQ[b*c - a*d, 0]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx &=\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}-\frac {(2 b p r) \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{a+b x} \, dx}{h}-\frac {(2 d q r) \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{c+d x} \, dx}{h}\\ &=\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}-\frac {(2 b p r) \int \frac {\log \left ((a+b x)^{p r}\right ) \log (g+h x)}{a+b x} \, dx}{h}-\frac {(2 b p r) \int \frac {\log \left ((c+d x)^{q r}\right ) \log (g+h x)}{a+b x} \, dx}{h}-\frac {(2 d q r) \int \frac {\log \left ((a+b x)^{p r}\right ) \log (g+h x)}{c+d x} \, dx}{h}-\frac {(2 d q r) \int \frac {\log \left ((c+d x)^{q r}\right ) \log (g+h x)}{c+d x} \, dx}{h}-\frac {\left (2 b p r \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 )\right ) \int \frac {\log (g+h x)}{a+b x} \, dx}{h}-\frac {\left (2 d q r \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 )\right ) \int \frac {\log (g+h x)}{c+d x} \, dx}{h}\\ &=\frac {2 p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \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 ) \log (g+h x)}{h}+\frac {2 q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \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 ) \log (g+h x)}{h}+\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}-\frac {(2 p r) \operatorname {Subst}\left (\int \frac {\log \left (x^{p r}\right ) \log \left (\frac {b g-a h}{b}+\frac {h x}{b}\right )}{x} \, dx,x,a+b x\right )}{h}-\frac {(2 p r) \operatorname {Subst}\left (\int \frac {\log \left (\left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )^{q r}\right ) \log \left (-\frac {-b g+a h}{b}+\frac {h x}{b}\right )}{x} \, dx,x,a+b x\right )}{h}-\frac {(2 q r) \operatorname {Subst}\left (\int \frac {\log \left (x^{q r}\right ) \log \left (\frac {d g-c h}{d}+\frac {h x}{d}\right )}{x} \, dx,x,c+d x\right )}{h}-\frac {(2 q r) \operatorname {Subst}\left (\int \frac {\log \left (\left (-\frac {b c-a d}{d}+\frac {b x}{d}\right )^{p r}\right ) \log \left (-\frac {-d g+c h}{d}+\frac {h x}{d}\right )}{x} \, dx,x,c+d x\right )}{h}+\left (2 p r \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 )\right ) \int \frac {\log \left (\frac {h (a+b x)}{-b g+a h}\right )}{g+h x} \, dx+\left (2 q r \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 )\right ) \int \frac {\log \left (\frac {h (c+d x)}{-d g+c h}\right )}{g+h x} \, dx\\ &=-\frac {\log ^2\left ((a+b x)^{p r}\right ) \log (g+h x)}{h}-\frac {\log ^2\left ((c+d x)^{q r}\right ) \log (g+h x)}{h}+\frac {2 p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \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 ) \log (g+h x)}{h}+\frac {2 q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \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 ) \log (g+h x)}{h}+\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}+\frac {\operatorname {Subst}\left (\int \frac {\log ^2\left (x^{p r}\right )}{\frac {b g-a h}{b}+\frac {h x}{b}} \, dx,x,a+b x\right )}{b}+\frac {\operatorname {Subst}\left (\int \frac {\log ^2\left (x^{q r}\right )}{\frac {d g-c h}{d}+\frac {h x}{d}} \, dx,x,c+d x\right )}{d}-\frac {\left (2 p q r^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right ) \log \left (-\frac {-b g+a h}{b}+\frac {h x}{b}\right )}{x} \, dx,x,a+b x\right )}{h}-\frac {\left (2 p q r^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {b c-a d}{d}+\frac {b x}{d}\right ) \log \left (-\frac {-d g+c h}{d}+\frac {h x}{d}\right )}{x} \, dx,x,c+d x\right )}{h}+\frac {\left (2 q r \left (p r \log (a+b x)-\log \left ((a+b x)^{p r}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {-d g+c h}{d}+\frac {h x}{d}\right )}{x} \, dx,x,c+d x\right )}{h}+\frac {\left (2 p r \left (q r \log (c+d x)-\log \left ((c+d x)^{q r}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {-b g+a h}{b}+\frac {h x}{b}\right )}{x} \, dx,x,a+b x\right )}{h}+\frac {\left (2 p r \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 )\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b g+a h}\right )}{x} \, dx,x,g+h x\right )}{h}+\frac {\left (2 q r \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 )\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{-d g+c h}\right )}{x} \, dx,x,g+h x\right )}{h}\\ &=-\frac {\log ^2\left ((a+b x)^{p r}\right ) \log (g+h x)}{h}-\frac {2 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \log (g+h x)}{h}-\frac {2 p q r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (g+h x)}{h}+\frac {2 q r \left (p r \log (a+b x)-\log \left ((a+b x)^{p r}\right )\right ) \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac {2 p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (q r \log (c+d x)-\log \left ((c+d x)^{q r}\right )\right ) \log (g+h x)}{h}-\frac {\log ^2\left ((c+d x)^{q r}\right ) \log (g+h x)}{h}+\frac {2 p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \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 ) \log (g+h x)}{h}+\frac {2 q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \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 ) \log (g+h x)}{h}+\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}+\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right )}{h}+\frac {\log ^2\left ((c+d x)^{q r}\right ) \log \left (\frac {d (g+h x)}{d g-c h}\right )}{h}-\frac {p q r^2 \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )+\log \left (\frac {b g-a h}{b (g+h x)}\right )-\log \left (\frac {(b g-a h) (c+d x)}{(b c-a d) (g+h x)}\right )\right ) \log ^2\left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )}{h}+\frac {p q r^2 \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )-\log \left (-\frac {h (c+d x)}{d g-c h}\right )\right ) \left (\log (a+b x)+\log \left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )\right )^2}{h}-\frac {p q r^2 \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )+\log \left (\frac {d g-c h}{d (g+h x)}\right )-\log \left (-\frac {(d g-c h) (a+b x)}{(b c-a d) (g+h x)}\right )\right ) \log ^2\left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac {p q r^2 \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )-\log \left (-\frac {h (a+b x)}{b g-a h}\right )\right ) \left (\log (c+d x)+\log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )\right )^2}{h}-\frac {2 p q r^2 \left (\log (g+h x)-\log \left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{h}-\frac {2 p q r^2 \left (\log (g+h x)-\log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{h}+\frac {2 p q r^2 \log \left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right ) \text {Li}_2\left (\frac {h (a+b x)}{b (g+h x)}\right )}{h}-\frac {2 p q r^2 \log \left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right ) \text {Li}_2\left (-\frac {(d g-c h) (a+b x)}{(b c-a d) (g+h x)}\right )}{h}+\frac {2 p q r^2 \log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right ) \text {Li}_2\left (\frac {h (c+d x)}{d (g+h x)}\right )}{h}-\frac {2 p q r^2 \log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right ) \text {Li}_2\left (\frac {(b g-a h) (c+d x)}{(b c-a d) (g+h x)}\right )}{h}+\frac {2 p r \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 ) \text {Li}_2\left (\frac {b (g+h x)}{b g-a h}\right )}{h}-\frac {2 p q r^2 \left (\log (c+d x)+\log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )\right ) \text {Li}_2\left (\frac {b (g+h x)}{b g-a h}\right )}{h}+\frac {2 q r \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 ) \text {Li}_2\left (\frac {d (g+h x)}{d g-c h}\right )}{h}-\frac {2 p q r^2 \left (\log (a+b x)+\log \left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )\right ) \text {Li}_2\left (\frac {d (g+h x)}{d g-c h}\right )}{h}+\frac {2 p q r^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{h}+\frac {2 p q r^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{h}+\frac {2 p q r^2 \text {Li}_3\left (\frac {h (a+b x)}{b (g+h x)}\right )}{h}-\frac {2 p q r^2 \text {Li}_3\left (-\frac {(d g-c h) (a+b x)}{(b c-a d) (g+h x)}\right )}{h}+\frac {2 p q r^2 \text {Li}_3\left (\frac {h (c+d x)}{d (g+h x)}\right )}{h}-\frac {2 p q r^2 \text {Li}_3\left (\frac {(b g-a h) (c+d x)}{(b c-a d) (g+h x)}\right )}{h}+\frac {2 p q r^2 \text {Li}_3\left (\frac {b (g+h x)}{b g-a h}\right )}{h}+\frac {2 p q r^2 \text {Li}_3\left (\frac {d (g+h x)}{d g-c h}\right )}{h}-\frac {(2 p r) \operatorname {Subst}\left (\int \frac {\log \left (x^{p r}\right ) \log \left (1+\frac {h x}{b g-a h}\right )}{x} \, dx,x,a+b x\right )}{h}-\frac {(2 q r) \operatorname {Subst}\left (\int \frac {\log \left (x^{q r}\right ) \log \left (1+\frac {h x}{d g-c h}\right )}{x} \, dx,x,c+d x\right )}{h}-\frac {\left (2 q r \left (p r \log (a+b x)-\log \left ((a+b x)^{p r}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {h x}{-d g+c h}\right )}{-\frac {-d g+c h}{d}+\frac {h x}{d}} \, dx,x,c+d x\right )}{d}-\frac {\left (2 p r \left (q r \log (c+d x)-\log \left ((c+d x)^{q r}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {h x}{-b g+a h}\right )}{-\frac {-b g+a h}{b}+\frac {h x}{b}} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\log ^2\left ((a+b x)^{p r}\right ) \log (g+h x)}{h}-\frac {2 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \log (g+h x)}{h}-\frac {2 p q r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (g+h x)}{h}+\frac {2 q r \left (p r \log (a+b x)-\log \left ((a+b x)^{p r}\right )\right ) \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac {2 p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (q r \log (c+d x)-\log \left ((c+d x)^{q r}\right )\right ) \log (g+h x)}{h}-\frac {\log ^2\left ((c+d x)^{q r}\right ) \log (g+h x)}{h}+\frac {2 p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \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 ) \log (g+h x)}{h}+\frac {2 q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \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 ) \log (g+h x)}{h}+\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}+\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right )}{h}+\frac {\log ^2\left ((c+d x)^{q r}\right ) \log \left (\frac {d (g+h x)}{d g-c h}\right )}{h}-\frac {p q r^2 \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )+\log \left (\frac {b g-a h}{b (g+h x)}\right )-\log \left (\frac {(b g-a h) (c+d x)}{(b c-a d) (g+h x)}\right )\right ) \log ^2\left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )}{h}+\frac {p q r^2 \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )-\log \left (-\frac {h (c+d x)}{d g-c h}\right )\right ) \left (\log (a+b x)+\log \left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )\right )^2}{h}-\frac {p q r^2 \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )+\log \left (\frac {d g-c h}{d (g+h x)}\right )-\log \left (-\frac {(d g-c h) (a+b x)}{(b c-a d) (g+h x)}\right )\right ) \log ^2\left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac {p q r^2 \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )-\log \left (-\frac {h (a+b x)}{b g-a h}\right )\right ) \left (\log (c+d x)+\log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )\right )^2}{h}-\frac {2 p q r^2 \left (\log (g+h x)-\log \left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{h}+\frac {2 p r \log \left ((a+b x)^{p r}\right ) \text {Li}_2\left (-\frac {h (a+b x)}{b g-a h}\right )}{h}-\frac {2 p q r^2 \left (\log (g+h x)-\log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{h}+\frac {2 q r \log \left ((c+d x)^{q r}\right ) \text {Li}_2\left (-\frac {h (c+d x)}{d g-c h}\right )}{h}+\frac {2 p q r^2 \log \left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right ) \text {Li}_2\left (\frac {h (a+b x)}{b (g+h x)}\right )}{h}-\frac {2 p q r^2 \log \left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right ) \text {Li}_2\left (-\frac {(d g-c h) (a+b x)}{(b c-a d) (g+h x)}\right )}{h}+\frac {2 p q r^2 \log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right ) \text {Li}_2\left (\frac {h (c+d x)}{d (g+h x)}\right )}{h}-\frac {2 p q r^2 \log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right ) \text {Li}_2\left (\frac {(b g-a h) (c+d x)}{(b c-a d) (g+h x)}\right )}{h}+\frac {2 p r \left (q r \log (c+d x)-\log \left ((c+d x)^{q r}\right )\right ) \text {Li}_2\left (\frac {b (g+h x)}{b g-a h}\right )}{h}+\frac {2 p r \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 ) \text {Li}_2\left (\frac {b (g+h x)}{b g-a h}\right )}{h}-\frac {2 p q r^2 \left (\log (c+d x)+\log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )\right ) \text {Li}_2\left (\frac {b (g+h x)}{b g-a h}\right )}{h}+\frac {2 q r \left (p r \log (a+b x)-\log \left ((a+b x)^{p r}\right )\right ) \text {Li}_2\left (\frac {d (g+h x)}{d g-c h}\right )}{h}+\frac {2 q r \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 ) \text {Li}_2\left (\frac {d (g+h x)}{d g-c h}\right )}{h}-\frac {2 p q r^2 \left (\log (a+b x)+\log \left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )\right ) \text {Li}_2\left (\frac {d (g+h x)}{d g-c h}\right )}{h}+\frac {2 p q r^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{h}+\frac {2 p q r^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{h}+\frac {2 p q r^2 \text {Li}_3\left (\frac {h (a+b x)}{b (g+h x)}\right )}{h}-\frac {2 p q r^2 \text {Li}_3\left (-\frac {(d g-c h) (a+b x)}{(b c-a d) (g+h x)}\right )}{h}+\frac {2 p q r^2 \text {Li}_3\left (\frac {h (c+d x)}{d (g+h x)}\right )}{h}-\frac {2 p q r^2 \text {Li}_3\left (\frac {(b g-a h) (c+d x)}{(b c-a d) (g+h x)}\right )}{h}+\frac {2 p q r^2 \text {Li}_3\left (\frac {b (g+h x)}{b g-a h}\right )}{h}+\frac {2 p q r^2 \text {Li}_3\left (\frac {d (g+h x)}{d g-c h}\right )}{h}-\frac {\left (2 p^2 r^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {h x}{b g-a h}\right )}{x} \, dx,x,a+b x\right )}{h}-\frac {\left (2 q^2 r^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {h x}{d g-c h}\right )}{x} \, dx,x,c+d x\right )}{h}\\ &=-\frac {\log ^2\left ((a+b x)^{p r}\right ) \log (g+h x)}{h}-\frac {2 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \log (g+h x)}{h}-\frac {2 p q r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (g+h x)}{h}+\frac {2 q r \left (p r \log (a+b x)-\log \left ((a+b x)^{p r}\right )\right ) \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac {2 p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (q r \log (c+d x)-\log \left ((c+d x)^{q r}\right )\right ) \log (g+h x)}{h}-\frac {\log ^2\left ((c+d x)^{q r}\right ) \log (g+h x)}{h}+\frac {2 p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \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 ) \log (g+h x)}{h}+\frac {2 q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \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 ) \log (g+h x)}{h}+\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}+\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right )}{h}+\frac {\log ^2\left ((c+d x)^{q r}\right ) \log \left (\frac {d (g+h x)}{d g-c h}\right )}{h}-\frac {p q r^2 \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )+\log \left (\frac {b g-a h}{b (g+h x)}\right )-\log \left (\frac {(b g-a h) (c+d x)}{(b c-a d) (g+h x)}\right )\right ) \log ^2\left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )}{h}+\frac {p q r^2 \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )-\log \left (-\frac {h (c+d x)}{d g-c h}\right )\right ) \left (\log (a+b x)+\log \left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )\right )^2}{h}-\frac {p q r^2 \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )+\log \left (\frac {d g-c h}{d (g+h x)}\right )-\log \left (-\frac {(d g-c h) (a+b x)}{(b c-a d) (g+h x)}\right )\right ) \log ^2\left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac {p q r^2 \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )-\log \left (-\frac {h (a+b x)}{b g-a h}\right )\right ) \left (\log (c+d x)+\log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )\right )^2}{h}-\frac {2 p q r^2 \left (\log (g+h x)-\log \left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{h}+\frac {2 p r \log \left ((a+b x)^{p r}\right ) \text {Li}_2\left (-\frac {h (a+b x)}{b g-a h}\right )}{h}-\frac {2 p q r^2 \left (\log (g+h x)-\log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{h}+\frac {2 q r \log \left ((c+d x)^{q r}\right ) \text {Li}_2\left (-\frac {h (c+d x)}{d g-c h}\right )}{h}+\frac {2 p q r^2 \log \left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right ) \text {Li}_2\left (\frac {h (a+b x)}{b (g+h x)}\right )}{h}-\frac {2 p q r^2 \log \left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right ) \text {Li}_2\left (-\frac {(d g-c h) (a+b x)}{(b c-a d) (g+h x)}\right )}{h}+\frac {2 p q r^2 \log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right ) \text {Li}_2\left (\frac {h (c+d x)}{d (g+h x)}\right )}{h}-\frac {2 p q r^2 \log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right ) \text {Li}_2\left (\frac {(b g-a h) (c+d x)}{(b c-a d) (g+h x)}\right )}{h}+\frac {2 p r \left (q r \log (c+d x)-\log \left ((c+d x)^{q r}\right )\right ) \text {Li}_2\left (\frac {b (g+h x)}{b g-a h}\right )}{h}+\frac {2 p r \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 ) \text {Li}_2\left (\frac {b (g+h x)}{b g-a h}\right )}{h}-\frac {2 p q r^2 \left (\log (c+d x)+\log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )\right ) \text {Li}_2\left (\frac {b (g+h x)}{b g-a h}\right )}{h}+\frac {2 q r \left (p r \log (a+b x)-\log \left ((a+b x)^{p r}\right )\right ) \text {Li}_2\left (\frac {d (g+h x)}{d g-c h}\right )}{h}+\frac {2 q r \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 ) \text {Li}_2\left (\frac {d (g+h x)}{d g-c h}\right )}{h}-\frac {2 p q r^2 \left (\log (a+b x)+\log \left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )\right ) \text {Li}_2\left (\frac {d (g+h x)}{d g-c h}\right )}{h}+\frac {2 p q r^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{h}-\frac {2 p^2 r^2 \text {Li}_3\left (-\frac {h (a+b x)}{b g-a h}\right )}{h}+\frac {2 p q r^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{h}-\frac {2 q^2 r^2 \text {Li}_3\left (-\frac {h (c+d x)}{d g-c h}\right )}{h}+\frac {2 p q r^2 \text {Li}_3\left (\frac {h (a+b x)}{b (g+h x)}\right )}{h}-\frac {2 p q r^2 \text {Li}_3\left (-\frac {(d g-c h) (a+b x)}{(b c-a d) (g+h x)}\right )}{h}+\frac {2 p q r^2 \text {Li}_3\left (\frac {h (c+d x)}{d (g+h x)}\right )}{h}-\frac {2 p q r^2 \text {Li}_3\left (\frac {(b g-a h) (c+d x)}{(b c-a d) (g+h x)}\right )}{h}+\frac {2 p q r^2 \text {Li}_3\left (\frac {b (g+h x)}{b g-a h}\right )}{h}+\frac {2 p q r^2 \text {Li}_3\left (\frac {d (g+h x)}{d g-c h}\right )}{h}\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 1370, normalized size = 0.93 \[ \frac {p q \log \left (\frac {a d-b c}{d (a+b x)}\right ) \log ^2\left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) r^2+p^2 \log ^2(a+b x) \log (g+h x) r^2+q^2 \log ^2(c+d x) \log (g+h x) r^2+2 p q \log (a+b x) \log (c+d x) \log (g+h x) r^2-p^2 \log ^2(a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right ) r^2+p q \log ^2\left (\frac {h (c+d x)}{c h-d g}\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right ) r^2+p q \log ^2\left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right ) r^2-2 p q \log (a+b x) \log \left (\frac {h (c+d x)}{c h-d g}\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right ) r^2-2 p q \log \left (\frac {h (c+d x)}{c h-d g}\right ) \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right ) r^2-q^2 \log ^2(c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right ) r^2-p q \log ^2\left (\frac {h (c+d x)}{c h-d g}\right ) \log \left (\frac {d (g+h x)}{d g-c h}\right ) r^2-2 p q \log (a+b x) \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right ) r^2+2 p q \log (a+b x) \log \left (\frac {h (c+d x)}{c h-d g}\right ) \log \left (\frac {d (g+h x)}{d g-c h}\right ) r^2+2 p q \log \left (\frac {h (c+d x)}{c h-d g}\right ) \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \log \left (\frac {d (g+h x)}{d g-c h}\right ) r^2-p q \log ^2\left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \log \left (\frac {(a d-b c) (g+h x)}{(d g-c h) (a+b x)}\right ) r^2+2 p q \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right ) r^2-2 p q \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \text {Li}_2\left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) r^2-2 p^2 \text {Li}_3\left (\frac {h (a+b x)}{a h-b g}\right ) r^2-2 p q \text {Li}_3\left (\frac {h (a+b x)}{a h-b g}\right ) r^2-2 q^2 \text {Li}_3\left (\frac {h (c+d x)}{c h-d g}\right ) r^2-2 p q \text {Li}_3\left (\frac {h (c+d x)}{c h-d g}\right ) r^2-2 p q \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right ) r^2+2 p q \text {Li}_3\left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) r^2-2 p \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x) r-2 q \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x) r+2 p \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right ) r+2 q \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log \left (\frac {d (g+h x)}{d g-c h}\right ) r+2 p \left (\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-q r \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right )\right ) \text {Li}_2\left (\frac {h (a+b x)}{a h-b g}\right ) r+2 q \left (p r \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \text {Li}_2\left (\frac {h (c+d x)}{c h-d g}\right ) r+\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{h x + g}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.32, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )^{2}}{h x +g}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2}{g+h\,x} \,d x \]

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),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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