\(\int \frac {\log ^2(e (f (a+b x)^p (c+d x)^q)^r)}{(a+b x)^4} \, dx\) [23]

Optimal result

Integrand size = 31, antiderivative size = 764 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4} \, dx=-\frac {2 p^2 r^2}{27 b (a+b x)^3}-\frac {5 d p q r^2}{18 b (b c-a d) (a+b x)^2}+\frac {8 d^2 p q r^2}{9 b (b c-a d)^2 (a+b x)}-\frac {d^2 q^2 r^2}{3 b (b c-a d)^2 (a+b x)}+\frac {2 d^3 p q r^2 \log (a+b x)}{9 b (b c-a d)^3}-\frac {d^3 q^2 r^2 \log (a+b x)}{b (b c-a d)^3}-\frac {d^3 p q r^2 \log ^2(a+b x)}{3 b (b c-a d)^3}-\frac {2 d^3 p q r^2 \log (c+d x)}{9 b (b c-a d)^3}+\frac {d^3 q^2 r^2 \log (c+d x)}{b (b c-a d)^3}+\frac {2 d^3 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b (b c-a d)^3}+\frac {d^3 q^2 r^2 \log ^2(c+d x)}{3 b (b c-a d)^3}-\frac {2 d^3 q^2 r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3}-\frac {2 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b (a+b x)^3}-\frac {d q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d) (a+b x)^2}+\frac {2 d^2 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d)^2 (a+b x)}+\frac {2 d^3 q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d)^3}-\frac {2 d^3 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d)^3}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}-\frac {2 d^3 q^2 r^2 \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{3 b (b c-a d)^3}+\frac {2 d^3 p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3} \]

[Out]

-2/27*p^2*r^2/b/(b*x+a)^3-5/18*d*p*q*r^2/b/(-a*d+b*c)/(b*x+a)^2+8/9*d^2*p*q*r^2/b/(-a*d+b*c)^2/(b*x+a)-1/3*d^2
*q^2*r^2/b/(-a*d+b*c)^2/(b*x+a)+2/9*d^3*p*q*r^2*ln(b*x+a)/b/(-a*d+b*c)^3-d^3*q^2*r^2*ln(b*x+a)/b/(-a*d+b*c)^3-
1/3*d^3*p*q*r^2*ln(b*x+a)^2/b/(-a*d+b*c)^3-2/9*d^3*p*q*r^2*ln(d*x+c)/b/(-a*d+b*c)^3+d^3*q^2*r^2*ln(d*x+c)/b/(-
a*d+b*c)^3+2/3*d^3*p*q*r^2*ln(-d*(b*x+a)/(-a*d+b*c))*ln(d*x+c)/b/(-a*d+b*c)^3+1/3*d^3*q^2*r^2*ln(d*x+c)^2/b/(-
a*d+b*c)^3-2/3*d^3*q^2*r^2*ln(b*x+a)*ln(b*(d*x+c)/(-a*d+b*c))/b/(-a*d+b*c)^3-2/9*p*r*ln(e*(f*(b*x+a)^p*(d*x+c)
^q)^r)/b/(b*x+a)^3-1/3*d*q*r*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/(-a*d+b*c)/(b*x+a)^2+2/3*d^2*q*r*ln(e*(f*(b*x+a
)^p*(d*x+c)^q)^r)/b/(-a*d+b*c)^2/(b*x+a)+2/3*d^3*q*r*ln(b*x+a)*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/(-a*d+b*c)^3-
2/3*d^3*q*r*ln(d*x+c)*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/(-a*d+b*c)^3-1/3*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/b/(
b*x+a)^3-2/3*d^3*q^2*r^2*polylog(2,-d*(b*x+a)/(-a*d+b*c))/b/(-a*d+b*c)^3+2/3*d^3*p*q*r^2*polylog(2,b*(d*x+c)/(
-a*d+b*c))/b/(-a*d+b*c)^3

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 764, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {2584, 2581, 32, 46, 2594, 36, 31, 2580, 2437, 2338, 2441, 2440, 2438} \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4} \, dx=\frac {2 d^3 q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d)^3}-\frac {2 d^3 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d)^3}+\frac {2 d^3 p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3}-\frac {d^3 p q r^2 \log ^2(a+b x)}{3 b (b c-a d)^3}+\frac {2 d^3 p q r^2 \log (a+b x)}{9 b (b c-a d)^3}-\frac {2 d^3 p q r^2 \log (c+d x)}{9 b (b c-a d)^3}+\frac {2 d^3 p q r^2 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{3 b (b c-a d)^3}-\frac {2 d^3 q^2 r^2 \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{3 b (b c-a d)^3}+\frac {d^3 q^2 r^2 \log ^2(c+d x)}{3 b (b c-a d)^3}-\frac {d^3 q^2 r^2 \log (a+b x)}{b (b c-a d)^3}+\frac {d^3 q^2 r^2 \log (c+d x)}{b (b c-a d)^3}-\frac {2 d^3 q^2 r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3}+\frac {2 d^2 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x) (b c-a d)^2}+\frac {8 d^2 p q r^2}{9 b (a+b x) (b c-a d)^2}-\frac {d^2 q^2 r^2}{3 b (a+b x) (b c-a d)^2}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}-\frac {d q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^2 (b c-a d)}-\frac {2 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b (a+b x)^3}-\frac {5 d p q r^2}{18 b (a+b x)^2 (b c-a d)}-\frac {2 p^2 r^2}{27 b (a+b x)^3} \]

[In]

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

[Out]

(-2*p^2*r^2)/(27*b*(a + b*x)^3) - (5*d*p*q*r^2)/(18*b*(b*c - a*d)*(a + b*x)^2) + (8*d^2*p*q*r^2)/(9*b*(b*c - a
*d)^2*(a + b*x)) - (d^2*q^2*r^2)/(3*b*(b*c - a*d)^2*(a + b*x)) + (2*d^3*p*q*r^2*Log[a + b*x])/(9*b*(b*c - a*d)
^3) - (d^3*q^2*r^2*Log[a + b*x])/(b*(b*c - a*d)^3) - (d^3*p*q*r^2*Log[a + b*x]^2)/(3*b*(b*c - a*d)^3) - (2*d^3
*p*q*r^2*Log[c + d*x])/(9*b*(b*c - a*d)^3) + (d^3*q^2*r^2*Log[c + d*x])/(b*(b*c - a*d)^3) + (2*d^3*p*q*r^2*Log
[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(3*b*(b*c - a*d)^3) + (d^3*q^2*r^2*Log[c + d*x]^2)/(3*b*(b*c - a*
d)^3) - (2*d^3*q^2*r^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(3*b*(b*c - a*d)^3) - (2*p*r*Log[e*(f*(a +
b*x)^p*(c + d*x)^q)^r])/(9*b*(a + b*x)^3) - (d*q*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3*b*(b*c - a*d)*(a
+ b*x)^2) + (2*d^2*q*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3*b*(b*c - a*d)^2*(a + b*x)) + (2*d^3*q*r*Log[a
+ b*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3*b*(b*c - a*d)^3) - (2*d^3*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^
p*(c + d*x)^q)^r])/(3*b*(b*c - a*d)^3) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(3*b*(a + b*x)^3) - (2*d^3*q^2
*r^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(3*b*(b*c - a*d)^3) + (2*d^3*p*q*r^2*PolyLog[2, (b*(c + d*x))/(
b*c - a*d)])/(3*b*(b*c - a*d)^3)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 2338

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

Rule 2437

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

Rule 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 2580

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

Rule 2581

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

Rule 2584

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

Rule 2594

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(RFx_), x_Symbol] :>
With[{u = ExpandIntegrand[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a,
b, c, d, e, f, p, q, r, s}, x] && RationalFunctionQ[RFx, x] && IGtQ[s, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}+\frac {1}{3} (2 p r) \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4} \, dx+\frac {(2 d q r) \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^3 (c+d x)} \, dx}{3 b} \\ & = -\frac {2 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b (a+b x)^3}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}+\frac {(2 d q r) \int \left (\frac {b \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(b c-a d) (a+b x)^3}-\frac {b d \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(b c-a d)^3 (a+b x)}-\frac {d^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{3 b}+\frac {1}{9} \left (2 p^2 r^2\right ) \int \frac {1}{(a+b x)^4} \, dx+\frac {\left (2 d p q r^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{9 b} \\ & = -\frac {2 p^2 r^2}{27 b (a+b x)^3}-\frac {2 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b (a+b x)^3}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}+\frac {\left (2 d^3 q r\right ) \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{a+b x} \, dx}{3 (b c-a d)^3}-\frac {\left (2 d^4 q r\right ) \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x} \, dx}{3 b (b c-a d)^3}-\frac {\left (2 d^2 q r\right ) \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^2} \, dx}{3 (b c-a d)^2}+\frac {(2 d q r) \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^3} \, dx}{3 (b c-a d)}+\frac {\left (2 d p q r^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{9 b} \\ & = -\frac {2 p^2 r^2}{27 b (a+b x)^3}-\frac {d p q r^2}{9 b (b c-a d) (a+b x)^2}+\frac {2 d^2 p q r^2}{9 b (b c-a d)^2 (a+b x)}+\frac {2 d^3 p q r^2 \log (a+b x)}{9 b (b c-a d)^3}-\frac {2 d^3 p q r^2 \log (c+d x)}{9 b (b c-a d)^3}-\frac {2 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b (a+b x)^3}-\frac {d q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d) (a+b x)^2}+\frac {2 d^2 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d)^2 (a+b x)}+\frac {2 d^3 q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d)^3}-\frac {2 d^3 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d)^3}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}-\frac {\left (2 d^3 p q r^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{3 (b c-a d)^3}+\frac {\left (2 d^3 p q r^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3 (b c-a d)^3}-\frac {\left (2 d^2 p q r^2\right ) \int \frac {1}{(a+b x)^2} \, dx}{3 (b c-a d)^2}+\frac {\left (d p q r^2\right ) \int \frac {1}{(a+b x)^3} \, dx}{3 (b c-a d)}-\frac {\left (2 d^4 q^2 r^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{3 b (b c-a d)^3}+\frac {\left (2 d^4 q^2 r^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3 b (b c-a d)^3}-\frac {\left (2 d^3 q^2 r^2\right ) \int \frac {1}{(a+b x) (c+d x)} \, dx}{3 b (b c-a d)^2}+\frac {\left (d^2 q^2 r^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{3 b (b c-a d)} \\ & = -\frac {2 p^2 r^2}{27 b (a+b x)^3}-\frac {5 d p q r^2}{18 b (b c-a d) (a+b x)^2}+\frac {8 d^2 p q r^2}{9 b (b c-a d)^2 (a+b x)}+\frac {2 d^3 p q r^2 \log (a+b x)}{9 b (b c-a d)^3}-\frac {2 d^3 p q r^2 \log (c+d x)}{9 b (b c-a d)^3}+\frac {2 d^3 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b (b c-a d)^3}-\frac {2 d^3 q^2 r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3}-\frac {2 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b (a+b x)^3}-\frac {d q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d) (a+b x)^2}+\frac {2 d^2 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d)^2 (a+b x)}+\frac {2 d^3 q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d)^3}-\frac {2 d^3 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d)^3}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}-\frac {\left (2 d^3 p q r^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{3 b (b c-a d)^3}-\frac {\left (2 d^4 p q r^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 b (b c-a d)^3}-\frac {\left (2 d^3 q^2 r^2\right ) \int \frac {1}{a+b x} \, dx}{3 (b c-a d)^3}+\frac {\left (2 d^3 q^2 r^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3 (b c-a d)^3}+\frac {\left (2 d^3 q^2 r^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3 b (b c-a d)^3}+\frac {\left (2 d^4 q^2 r^2\right ) \int \frac {1}{c+d x} \, dx}{3 b (b c-a d)^3}+\frac {\left (d^2 q^2 r^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{3 b (b c-a d)} \\ & = -\frac {2 p^2 r^2}{27 b (a+b x)^3}-\frac {5 d p q r^2}{18 b (b c-a d) (a+b x)^2}+\frac {8 d^2 p q r^2}{9 b (b c-a d)^2 (a+b x)}-\frac {d^2 q^2 r^2}{3 b (b c-a d)^2 (a+b x)}+\frac {2 d^3 p q r^2 \log (a+b x)}{9 b (b c-a d)^3}-\frac {d^3 q^2 r^2 \log (a+b x)}{b (b c-a d)^3}-\frac {d^3 p q r^2 \log ^2(a+b x)}{3 b (b c-a d)^3}-\frac {2 d^3 p q r^2 \log (c+d x)}{9 b (b c-a d)^3}+\frac {d^3 q^2 r^2 \log (c+d x)}{b (b c-a d)^3}+\frac {2 d^3 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b (b c-a d)^3}+\frac {d^3 q^2 r^2 \log ^2(c+d x)}{3 b (b c-a d)^3}-\frac {2 d^3 q^2 r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3}-\frac {2 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b (a+b x)^3}-\frac {d q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d) (a+b x)^2}+\frac {2 d^2 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d)^2 (a+b x)}+\frac {2 d^3 q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d)^3}-\frac {2 d^3 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d)^3}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}-\frac {\left (2 d^3 p q r^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3 b (b c-a d)^3}+\frac {\left (2 d^3 q^2 r^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{3 b (b c-a d)^3} \\ & = -\frac {2 p^2 r^2}{27 b (a+b x)^3}-\frac {5 d p q r^2}{18 b (b c-a d) (a+b x)^2}+\frac {8 d^2 p q r^2}{9 b (b c-a d)^2 (a+b x)}-\frac {d^2 q^2 r^2}{3 b (b c-a d)^2 (a+b x)}+\frac {2 d^3 p q r^2 \log (a+b x)}{9 b (b c-a d)^3}-\frac {d^3 q^2 r^2 \log (a+b x)}{b (b c-a d)^3}-\frac {d^3 p q r^2 \log ^2(a+b x)}{3 b (b c-a d)^3}-\frac {2 d^3 p q r^2 \log (c+d x)}{9 b (b c-a d)^3}+\frac {d^3 q^2 r^2 \log (c+d x)}{b (b c-a d)^3}+\frac {2 d^3 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b (b c-a d)^3}+\frac {d^3 q^2 r^2 \log ^2(c+d x)}{3 b (b c-a d)^3}-\frac {2 d^3 q^2 r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3}-\frac {2 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b (a+b x)^3}-\frac {d q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d) (a+b x)^2}+\frac {2 d^2 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d)^2 (a+b x)}+\frac {2 d^3 q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d)^3}-\frac {2 d^3 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (b c-a d)^3}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}-\frac {2 d^3 q^2 r^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{3 b (b c-a d)^3}+\frac {2 d^3 p q r^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.94 (sec) , antiderivative size = 1407, normalized size of antiderivative = 1.84 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4} \, dx=-\frac {4 b^3 c^3 p^2 r^2-12 a b^2 c^2 d p^2 r^2+12 a^2 b c d^2 p^2 r^2-4 a^3 d^3 p^2 r^2+15 a b^2 c^2 d p q r^2-78 a^2 b c d^2 p q r^2+63 a^3 d^3 p q r^2+18 a^2 b c d^2 q^2 r^2-18 a^3 d^3 q^2 r^2+15 b^3 c^2 d p q r^2 x-126 a b^2 c d^2 p q r^2 x+111 a^2 b d^3 p q r^2 x+36 a b^2 c d^2 q^2 r^2 x-36 a^2 b d^3 q^2 r^2 x-48 b^3 c d^2 p q r^2 x^2+48 a b^2 d^3 p q r^2 x^2+18 b^3 c d^2 q^2 r^2 x^2-18 a b^2 d^3 q^2 r^2 x^2+18 d^3 p q r^2 (a+b x)^3 \log ^2(a+b x)+12 a^3 d^3 p q r^2 \log (c+d x)-54 a^3 d^3 q^2 r^2 \log (c+d x)+36 a^2 b d^3 p q r^2 x \log (c+d x)-162 a^2 b d^3 q^2 r^2 x \log (c+d x)+36 a b^2 d^3 p q r^2 x^2 \log (c+d x)-162 a b^2 d^3 q^2 r^2 x^2 \log (c+d x)+12 b^3 d^3 p q r^2 x^3 \log (c+d x)-54 b^3 d^3 q^2 r^2 x^3 \log (c+d x)-18 a^3 d^3 q^2 r^2 \log ^2(c+d x)-54 a^2 b d^3 q^2 r^2 x \log ^2(c+d x)-54 a b^2 d^3 q^2 r^2 x^2 \log ^2(c+d x)-18 b^3 d^3 q^2 r^2 x^3 \log ^2(c+d x)+12 b^3 c^3 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-36 a b^2 c^2 d p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+36 a^2 b c d^2 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-12 a^3 d^3 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+18 a b^2 c^2 d q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-72 a^2 b c d^2 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+54 a^3 d^3 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+18 b^3 c^2 d q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-108 a b^2 c d^2 q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+90 a^2 b d^3 q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-36 b^3 c d^2 q r x^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+36 a b^2 d^3 q r x^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+36 a^3 d^3 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+108 a^2 b d^3 q r x \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+108 a b^2 d^3 q r x^2 \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+36 b^3 d^3 q r x^3 \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+18 b^3 c^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-54 a b^2 c^2 d \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+54 a^2 b c d^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-18 a^3 d^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-6 d^3 q r (a+b x)^3 \log (a+b x) \left (2 p r-9 q r+6 p r \log (c+d x)-6 (p+q) r \log \left (\frac {b (c+d x)}{b c-a d}\right )+6 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )+36 d^3 q (p+q) r^2 (a+b x)^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )}{54 b (b c-a d)^3 (a+b x)^3} \]

[In]

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

[Out]

-1/54*(4*b^3*c^3*p^2*r^2 - 12*a*b^2*c^2*d*p^2*r^2 + 12*a^2*b*c*d^2*p^2*r^2 - 4*a^3*d^3*p^2*r^2 + 15*a*b^2*c^2*
d*p*q*r^2 - 78*a^2*b*c*d^2*p*q*r^2 + 63*a^3*d^3*p*q*r^2 + 18*a^2*b*c*d^2*q^2*r^2 - 18*a^3*d^3*q^2*r^2 + 15*b^3
*c^2*d*p*q*r^2*x - 126*a*b^2*c*d^2*p*q*r^2*x + 111*a^2*b*d^3*p*q*r^2*x + 36*a*b^2*c*d^2*q^2*r^2*x - 36*a^2*b*d
^3*q^2*r^2*x - 48*b^3*c*d^2*p*q*r^2*x^2 + 48*a*b^2*d^3*p*q*r^2*x^2 + 18*b^3*c*d^2*q^2*r^2*x^2 - 18*a*b^2*d^3*q
^2*r^2*x^2 + 18*d^3*p*q*r^2*(a + b*x)^3*Log[a + b*x]^2 + 12*a^3*d^3*p*q*r^2*Log[c + d*x] - 54*a^3*d^3*q^2*r^2*
Log[c + d*x] + 36*a^2*b*d^3*p*q*r^2*x*Log[c + d*x] - 162*a^2*b*d^3*q^2*r^2*x*Log[c + d*x] + 36*a*b^2*d^3*p*q*r
^2*x^2*Log[c + d*x] - 162*a*b^2*d^3*q^2*r^2*x^2*Log[c + d*x] + 12*b^3*d^3*p*q*r^2*x^3*Log[c + d*x] - 54*b^3*d^
3*q^2*r^2*x^3*Log[c + d*x] - 18*a^3*d^3*q^2*r^2*Log[c + d*x]^2 - 54*a^2*b*d^3*q^2*r^2*x*Log[c + d*x]^2 - 54*a*
b^2*d^3*q^2*r^2*x^2*Log[c + d*x]^2 - 18*b^3*d^3*q^2*r^2*x^3*Log[c + d*x]^2 + 12*b^3*c^3*p*r*Log[e*(f*(a + b*x)
^p*(c + d*x)^q)^r] - 36*a*b^2*c^2*d*p*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + 36*a^2*b*c*d^2*p*r*Log[e*(f*(a
+ b*x)^p*(c + d*x)^q)^r] - 12*a^3*d^3*p*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + 18*a*b^2*c^2*d*q*r*Log[e*(f*(
a + b*x)^p*(c + d*x)^q)^r] - 72*a^2*b*c*d^2*q*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + 54*a^3*d^3*q*r*Log[e*(f
*(a + b*x)^p*(c + d*x)^q)^r] + 18*b^3*c^2*d*q*r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - 108*a*b^2*c*d^2*q*r*x
*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + 90*a^2*b*d^3*q*r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - 36*b^3*c*d^2
*q*r*x^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + 36*a*b^2*d^3*q*r*x^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + 36
*a^3*d^3*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + 108*a^2*b*d^3*q*r*x*Log[c + d*x]*Log[e*(f*(a
+ b*x)^p*(c + d*x)^q)^r] + 108*a*b^2*d^3*q*r*x^2*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + 36*b^3*d^
3*q*r*x^3*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + 18*b^3*c^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^
2 - 54*a*b^2*c^2*d*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2 + 54*a^2*b*c*d^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r
]^2 - 18*a^3*d^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2 - 6*d^3*q*r*(a + b*x)^3*Log[a + b*x]*(2*p*r - 9*q*r +
6*p*r*Log[c + d*x] - 6*(p + q)*r*Log[(b*(c + d*x))/(b*c - a*d)] + 6*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]) + 36
*d^3*q*(p + q)*r^2*(a + b*x)^3*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)])/(b*(b*c - a*d)^3*(a + b*x)^3)

Maple [F]

\[\int \frac {{\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}^{2}}{\left (b x +a \right )^{4}}d x\]

[In]

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(b*x+a)^4,x)

[Out]

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(b*x+a)^4,x)

Fricas [F]

\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{{\left (b x + a\right )}^{4}} \,d x } \]

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(b*x+a)^4,x, algorithm="fricas")

[Out]

integral(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2/(b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x + a^4), x)

Sympy [F]

\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4} \, dx=\int \frac {\log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}^{2}}{\left (a + b x\right )^{4}}\, dx \]

[In]

integrate(ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)**2/(b*x+a)**4,x)

[Out]

Integral(log(e*(f*(a + b*x)**p*(c + d*x)**q)**r)**2/(a + b*x)**4, x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 1252, normalized size of antiderivative = 1.64 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4} \, dx=\text {Too large to display} \]

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(b*x+a)^4,x, algorithm="maxima")

[Out]

1/9*(6*d^3*f*q*log(b*x + a)/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) - 6*d^3*f*q*log(d*x + c)/(b^3*
c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) + (6*b^2*d^2*f*q*x^2 + a*b*c*d*f*(4*p - 3*q) - a^2*d^2*f*(2*p -
9*q) - 2*b^2*c^2*f*p - 3*(b^2*c*d*f*q - 5*a*b*d^2*f*q)*x)/(a^3*b^2*c^2 - 2*a^4*b*c*d + a^5*d^2 + (b^5*c^2 - 2
*a*b^4*c*d + a^2*b^3*d^2)*x^3 + 3*(a*b^4*c^2 - 2*a^2*b^3*c*d + a^3*b^2*d^2)*x^2 + 3*(a^2*b^3*c^2 - 2*a^3*b^2*c
*d + a^4*b*d^2)*x))*r*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/(b*f) - 1/54*(36*(p*q + q^2)*(log(b*x + a)*log((b*d
*x + a*d)/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d)))*d^3*f^2/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d
^2 - a^3*d^3) + 6*(2*p*q - 9*q^2)*d^3*f^2*log(d*x + c)/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) + (
4*b^3*c^3*f^2*p^2 - 3*(4*p^2 - 5*p*q)*a*b^2*c^2*d*f^2 + 6*(2*p^2 - 13*p*q + 3*q^2)*a^2*b*c*d^2*f^2 - (4*p^2 -
63*p*q + 18*q^2)*a^3*d^3*f^2 - 6*((8*p*q - 3*q^2)*b^3*c*d^2*f^2 - (8*p*q - 3*q^2)*a*b^2*d^3*f^2)*x^2 + 18*(b^3
*d^3*f^2*p*q*x^3 + 3*a*b^2*d^3*f^2*p*q*x^2 + 3*a^2*b*d^3*f^2*p*q*x + a^3*d^3*f^2*p*q)*log(b*x + a)^2 - 36*(b^3
*d^3*f^2*p*q*x^3 + 3*a*b^2*d^3*f^2*p*q*x^2 + 3*a^2*b*d^3*f^2*p*q*x + a^3*d^3*f^2*p*q)*log(b*x + a)*log(d*x + c
) - 18*(b^3*d^3*f^2*q^2*x^3 + 3*a*b^2*d^3*f^2*q^2*x^2 + 3*a^2*b*d^3*f^2*q^2*x + a^3*d^3*f^2*q^2)*log(d*x + c)^
2 + 3*(5*b^3*c^2*d*f^2*p*q - 6*(7*p*q - 2*q^2)*a*b^2*c*d^2*f^2 + (37*p*q - 12*q^2)*a^2*b*d^3*f^2)*x - 6*((2*p*
q - 9*q^2)*b^3*d^3*f^2*x^3 + 3*(2*p*q - 9*q^2)*a*b^2*d^3*f^2*x^2 + 3*(2*p*q - 9*q^2)*a^2*b*d^3*f^2*x + (2*p*q
- 9*q^2)*a^3*d^3*f^2)*log(b*x + a))/(a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3 + (b^6*c^3 - 3*a*
b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*x^3 + 3*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^4*b^2*d^
3)*x^2 + 3*(a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 3*a^4*b^2*c*d^2 - a^5*b*d^3)*x))*r^2/(b*f^2) - 1/3*log(((b*x + a)^
p*(d*x + c)^q*f)^r*e)^2/((b*x + a)^3*b)

Giac [F]

\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{{\left (b x + a\right )}^{4}} \,d x } \]

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(b*x+a)^4,x, algorithm="giac")

[Out]

integrate(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2/(b*x + a)^4, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4} \, dx=\int \frac {{\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2}{{\left (a+b\,x\right )}^4} \,d x \]

[In]

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^2/(a + b*x)^4,x)

[Out]

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^2/(a + b*x)^4, x)