3.1.39 \(\int \frac {\log ^2(e (f (a+b x)^p (c+d x)^q)^r)}{g+h x} \, dx\) [39]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 1.39, 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 = {2583, 2586, 2441, 2440, 2438, 2481, 2422, 2354, 2421, 6724, 2490, 2487, 2485, 2352} \begin {gather*} \text {Too large to display} \end {gather*}

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 2352

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

Rule 2354

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 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2422

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 2438

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 2440

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 2441

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 2481

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 2485

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[(1/2)*(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, x] - Simp[(1/2)*(Log[(-b)*(x/a)] - Log[(-d)*(x/c)])*(Log[a + b*x] +
Log[a*((c + d*x)/(c*(a + b*x)))])^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 2487

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 2490

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 2583

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 2586

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 6724

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) \text {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) \text {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) \text {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) \text {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 {\text {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 {\text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.19, size = 1370, normalized size = 0.93 \begin {gather*} \frac {p q r^2 \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 )+p^2 r^2 \log ^2(a+b x) \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 ^2(c+d x) \log (g+h x)-2 p r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)-2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)+\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)-p^2 r^2 \log ^2(a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right )-2 p q r^2 \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 )+p q r^2 \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 )-2 p q r^2 \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 )+p q r^2 \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 )+2 p r \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 )-2 p q r^2 \log (a+b x) \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right )-q^2 r^2 \log ^2(c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right )+2 p q r^2 \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 )-p q r^2 \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 )+2 p q r^2 \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 )+2 q r \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 )-p q r^2 \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 )+2 p r \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 )+2 q r \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 )+2 p q r^2 \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 )-2 p q r^2 \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 )-2 p^2 r^2 \text {Li}_3\left (\frac {h (a+b x)}{-b g+a h}\right )-2 p q r^2 \text {Li}_3\left (\frac {h (a+b x)}{-b g+a h}\right )-2 p q r^2 \text {Li}_3\left (\frac {h (c+d x)}{-d g+c h}\right )-2 q^2 r^2 \text {Li}_3\left (\frac {h (c+d x)}{-d g+c h}\right )-2 p q r^2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )+2 p q r^2 \text {Li}_3\left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right )}{h} \end {gather*}

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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Maple [F]
time = 0.16, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

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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