\(\int \frac {\log ^2(e (f (a+b x)^p (c+d x)^q)^r)}{(a+b x)^3} \, dx\) [22]

Optimal result

Integrand size = 31, antiderivative size = 632 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^3} \, dx=-\frac {p^2 r^2}{4 b (a+b x)^2}-\frac {3 d p q r^2}{2 b (b c-a d) (a+b x)}-\frac {d^2 p q r^2 \log (a+b x)}{2 b (b c-a d)^2}+\frac {d^2 q^2 r^2 \log (a+b x)}{b (b c-a d)^2}+\frac {d^2 p q r^2 \log ^2(a+b x)}{2 b (b c-a d)^2}+\frac {d^2 p q r^2 \log (c+d x)}{2 b (b c-a d)^2}-\frac {d^2 q^2 r^2 \log (c+d x)}{b (b c-a d)^2}-\frac {d^2 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d)^2}-\frac {d^2 q^2 r^2 \log ^2(c+d x)}{2 b (b c-a d)^2}+\frac {d^2 q^2 r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2}-\frac {p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (a+b x)^2}-\frac {d q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (b c-a d) (a+b x)}-\frac {d^2 q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (b c-a d)^2}+\frac {d^2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (b c-a d)^2}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (a+b x)^2}+\frac {d^2 q^2 r^2 \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b (b c-a d)^2}-\frac {d^2 p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 632, normalized size of antiderivative = 1.00, number of steps used = 24, 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} \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^3} \, dx=-\frac {d^2 q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (b c-a d)^2}+\frac {d^2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (b c-a d)^2}-\frac {d^2 p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2}+\frac {d^2 p q r^2 \log ^2(a+b x)}{2 b (b c-a d)^2}-\frac {d^2 p q r^2 \log (a+b x)}{2 b (b c-a d)^2}+\frac {d^2 p q r^2 \log (c+d x)}{2 b (b c-a d)^2}-\frac {d^2 p q r^2 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{b (b c-a d)^2}+\frac {d^2 q^2 r^2 \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b (b c-a d)^2}-\frac {d^2 q^2 r^2 \log ^2(c+d x)}{2 b (b c-a d)^2}+\frac {d^2 q^2 r^2 \log (a+b x)}{b (b c-a d)^2}-\frac {d^2 q^2 r^2 \log (c+d x)}{b (b c-a d)^2}+\frac {d^2 q^2 r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (a+b x)^2}-\frac {d q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x) (b c-a d)}-\frac {p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (a+b x)^2}-\frac {3 d p q r^2}{2 b (a+b x) (b c-a d)}-\frac {p^2 r^2}{4 b (a+b x)^2} \]

[In]

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

[Out]

-1/4*(p^2*r^2)/(b*(a + b*x)^2) - (3*d*p*q*r^2)/(2*b*(b*c - a*d)*(a + b*x)) - (d^2*p*q*r^2*Log[a + b*x])/(2*b*(
b*c - a*d)^2) + (d^2*q^2*r^2*Log[a + b*x])/(b*(b*c - a*d)^2) + (d^2*p*q*r^2*Log[a + b*x]^2)/(2*b*(b*c - a*d)^2
) + (d^2*p*q*r^2*Log[c + d*x])/(2*b*(b*c - a*d)^2) - (d^2*q^2*r^2*Log[c + d*x])/(b*(b*c - a*d)^2) - (d^2*p*q*r
^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(b*(b*c - a*d)^2) - (d^2*q^2*r^2*Log[c + d*x]^2)/(2*b*(b*c
- a*d)^2) + (d^2*q^2*r^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(b*(b*c - a*d)^2) - (p*r*Log[e*(f*(a + b
*x)^p*(c + d*x)^q)^r])/(2*b*(a + b*x)^2) - (d*q*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(b*(b*c - a*d)*(a + b*
x)) - (d^2*q*r*Log[a + b*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(b*(b*c - a*d)^2) + (d^2*q*r*Log[c + d*x]*Lo
g[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(b*(b*c - a*d)^2) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(2*b*(a + b*x)^
2) + (d^2*q^2*r^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(b*(b*c - a*d)^2) - (d^2*p*q*r^2*PolyLog[2, (b*(c
+ d*x))/(b*c - a*d)])/(b*(b*c - a*d)^2)

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

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 872, normalized size of antiderivative = 1.38 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^3} \, dx=-\frac {b^2 c^2 p^2 r^2-2 a b c d p^2 r^2+a^2 d^2 p^2 r^2+6 a b c d p q r^2-6 a^2 d^2 p q r^2+6 b^2 c d p q r^2 x-6 a b d^2 p q r^2 x-2 d^2 p q r^2 (a+b x)^2 \log ^2(a+b x)-2 a^2 d^2 p q r^2 \log (c+d x)+4 a^2 d^2 q^2 r^2 \log (c+d x)-4 a b d^2 p q r^2 x \log (c+d x)+8 a b d^2 q^2 r^2 x \log (c+d x)-2 b^2 d^2 p q r^2 x^2 \log (c+d x)+4 b^2 d^2 q^2 r^2 x^2 \log (c+d x)+2 a^2 d^2 q^2 r^2 \log ^2(c+d x)+4 a b d^2 q^2 r^2 x \log ^2(c+d x)+2 b^2 d^2 q^2 r^2 x^2 \log ^2(c+d x)-2 d^2 q r (a+b x)^2 \log (a+b x) \left (-p r+2 q r-2 p r \log (c+d x)+2 (p+q) r \log \left (\frac {b (c+d x)}{b c-a d}\right )-2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )+2 b^2 c^2 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-4 a b c d p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 a^2 d^2 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+4 a b c d q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-4 a^2 d^2 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+4 b^2 c d q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-4 a b d^2 q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-4 a^2 d^2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-8 a b d^2 q r x \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-4 b^2 d^2 q r x^2 \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 b^2 c^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-4 a b c d \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 a^2 d^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-4 d^2 q (p+q) r^2 (a+b x)^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )}{4 b (b c-a d)^2 (a+b x)^2} \]

[In]

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

[Out]

-1/4*(b^2*c^2*p^2*r^2 - 2*a*b*c*d*p^2*r^2 + a^2*d^2*p^2*r^2 + 6*a*b*c*d*p*q*r^2 - 6*a^2*d^2*p*q*r^2 + 6*b^2*c*
d*p*q*r^2*x - 6*a*b*d^2*p*q*r^2*x - 2*d^2*p*q*r^2*(a + b*x)^2*Log[a + b*x]^2 - 2*a^2*d^2*p*q*r^2*Log[c + d*x]
+ 4*a^2*d^2*q^2*r^2*Log[c + d*x] - 4*a*b*d^2*p*q*r^2*x*Log[c + d*x] + 8*a*b*d^2*q^2*r^2*x*Log[c + d*x] - 2*b^2
*d^2*p*q*r^2*x^2*Log[c + d*x] + 4*b^2*d^2*q^2*r^2*x^2*Log[c + d*x] + 2*a^2*d^2*q^2*r^2*Log[c + d*x]^2 + 4*a*b*
d^2*q^2*r^2*x*Log[c + d*x]^2 + 2*b^2*d^2*q^2*r^2*x^2*Log[c + d*x]^2 - 2*d^2*q*r*(a + b*x)^2*Log[a + b*x]*(-(p*
r) + 2*q*r - 2*p*r*Log[c + d*x] + 2*(p + q)*r*Log[(b*(c + d*x))/(b*c - a*d)] - 2*Log[e*(f*(a + b*x)^p*(c + d*x
)^q)^r]) + 2*b^2*c^2*p*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - 4*a*b*c*d*p*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q
)^r] + 2*a^2*d^2*p*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + 4*a*b*c*d*q*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]
- 4*a^2*d^2*q*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + 4*b^2*c*d*q*r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] -
4*a*b*d^2*q*r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - 4*a^2*d^2*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d
*x)^q)^r] - 8*a*b*d^2*q*r*x*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - 4*b^2*d^2*q*r*x^2*Log[c + d*x]
*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + 2*b^2*c^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2 - 4*a*b*c*d*Log[e*(f*
(a + b*x)^p*(c + d*x)^q)^r]^2 + 2*a^2*d^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2 - 4*d^2*q*(p + q)*r^2*(a + b*
x)^2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)])/(b*(b*c - a*d)^2*(a + b*x)^2)

Maple [F]

\[\int \frac {{\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}^{2}}{\left (b x +a \right )^{3}}d x\]

[In]

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(b*x+a)^3,x)

[Out]

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(b*x+a)^3,x)

Fricas [F]

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

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(b*x+a)^3,x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate(ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)**2/(b*x+a)**3,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 755, normalized size of antiderivative = 1.19 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^3} \, dx=-\frac {{\left (\frac {2 \, d^{2} f q \log \left (b x + a\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} - \frac {2 \, d^{2} f q \log \left (d x + c\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} + \frac {2 \, b d f q x - a d f {\left (p - 2 \, q\right )} + b c f p}{a^{2} b c - a^{3} d + {\left (b^{3} c - a b^{2} d\right )} x^{2} + 2 \, {\left (a b^{2} c - a^{2} b d\right )} x}\right )} r \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{2 \, b f} + \frac {{\left (\frac {4 \, {\left (p q + q^{2}\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} d^{2} f^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} + \frac {2 \, {\left (p q - 2 \, q^{2}\right )} d^{2} f^{2} \log \left (d x + c\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} - \frac {b^{2} c^{2} f^{2} p^{2} - 2 \, {\left (p^{2} - 3 \, p q\right )} a b c d f^{2} + {\left (p^{2} - 6 \, p q\right )} a^{2} d^{2} f^{2} - 2 \, {\left (b^{2} d^{2} f^{2} p q x^{2} + 2 \, a b d^{2} f^{2} p q x + a^{2} d^{2} f^{2} p q\right )} \log \left (b x + a\right )^{2} + 4 \, {\left (b^{2} d^{2} f^{2} p q x^{2} + 2 \, a b d^{2} f^{2} p q x + a^{2} d^{2} f^{2} p q\right )} \log \left (b x + a\right ) \log \left (d x + c\right ) + 2 \, {\left (b^{2} d^{2} f^{2} q^{2} x^{2} + 2 \, a b d^{2} f^{2} q^{2} x + a^{2} d^{2} f^{2} q^{2}\right )} \log \left (d x + c\right )^{2} + 6 \, {\left (b^{2} c d f^{2} p q - a b d^{2} f^{2} p q\right )} x + 2 \, {\left ({\left (p q - 2 \, q^{2}\right )} b^{2} d^{2} f^{2} x^{2} + 2 \, {\left (p q - 2 \, q^{2}\right )} a b d^{2} f^{2} x + {\left (p q - 2 \, q^{2}\right )} a^{2} d^{2} f^{2}\right )} \log \left (b x + a\right )}{a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x}\right )} r^{2}}{4 \, b f^{2}} - \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{2 \, {\left (b x + a\right )}^{2} b} \]

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/2*(2*d^2*f*q*log(b*x + a)/(b^2*c^2 - 2*a*b*c*d + a^2*d^2) - 2*d^2*f*q*log(d*x + c)/(b^2*c^2 - 2*a*b*c*d + a
^2*d^2) + (2*b*d*f*q*x - a*d*f*(p - 2*q) + b*c*f*p)/(a^2*b*c - a^3*d + (b^3*c - a*b^2*d)*x^2 + 2*(a*b^2*c - a^
2*b*d)*x))*r*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/(b*f) + 1/4*(4*(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^2*f^2/(b^2*c^2 - 2*a*b*c*d + a^2*d^2) + 2*(p*q - 2*q^2)
*d^2*f^2*log(d*x + c)/(b^2*c^2 - 2*a*b*c*d + a^2*d^2) - (b^2*c^2*f^2*p^2 - 2*(p^2 - 3*p*q)*a*b*c*d*f^2 + (p^2
- 6*p*q)*a^2*d^2*f^2 - 2*(b^2*d^2*f^2*p*q*x^2 + 2*a*b*d^2*f^2*p*q*x + a^2*d^2*f^2*p*q)*log(b*x + a)^2 + 4*(b^2
*d^2*f^2*p*q*x^2 + 2*a*b*d^2*f^2*p*q*x + a^2*d^2*f^2*p*q)*log(b*x + a)*log(d*x + c) + 2*(b^2*d^2*f^2*q^2*x^2 +
2*a*b*d^2*f^2*q^2*x + a^2*d^2*f^2*q^2)*log(d*x + c)^2 + 6*(b^2*c*d*f^2*p*q - a*b*d^2*f^2*p*q)*x + 2*((p*q - 2
*q^2)*b^2*d^2*f^2*x^2 + 2*(p*q - 2*q^2)*a*b*d^2*f^2*x + (p*q - 2*q^2)*a^2*d^2*f^2)*log(b*x + a))/(a^2*b^2*c^2
- 2*a^3*b*c*d + a^4*d^2 + (b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^2 + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2
)*x))*r^2/(b*f^2) - 1/2*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2/((b*x + a)^2*b)

Giac [F]

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

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(b*x+a)^3,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^3} \, dx=\int \frac {{\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2}{{\left (a+b\,x\right )}^3} \,d x \]

[In]

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

[Out]

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