\(\int \frac {\log ^2(e (f (a+b x)^p (c+d x)^q)^r)}{(a+b x)^2} \, dx\) [21]

Optimal result

Integrand size = 31, antiderivative size = 465 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^2} \, dx=-\frac {2 p^2 r^2}{b (a+b x)}+\frac {2 d p q r^2 \log (a+b x)}{b (b c-a d)}-\frac {d p q r^2 \log ^2(a+b x)}{b (b c-a d)}-\frac {2 d p q r^2 \log (c+d x)}{b (b c-a d)}+\frac {2 d p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d)}+\frac {d q^2 r^2 \log ^2(c+d x)}{b (b c-a d)}-\frac {2 d q^2 r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)}-\frac {2 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}+\frac {2 d q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (b c-a d)}-\frac {2 d q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (b c-a d)}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}-\frac {2 d q^2 r^2 \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b (b c-a d)}+\frac {2 d p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {2584, 2581, 32, 36, 31, 2594, 2580, 2437, 2338, 2441, 2440, 2438} \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^2} \, dx=-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}+\frac {2 d q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (b c-a d)}-\frac {2 d q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (b c-a d)}-\frac {2 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}+\frac {2 d p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)}-\frac {d p q r^2 \log ^2(a+b x)}{b (b c-a d)}+\frac {2 d p q r^2 \log (a+b x)}{b (b c-a d)}-\frac {2 d p q r^2 \log (c+d x)}{b (b c-a d)}+\frac {2 d p q r^2 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{b (b c-a d)}-\frac {2 d q^2 r^2 \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b (b c-a d)}+\frac {d q^2 r^2 \log ^2(c+d x)}{b (b c-a d)}-\frac {2 d q^2 r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)}-\frac {2 p^2 r^2}{b (a+b x)} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 411, normalized size of antiderivative = 0.88 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^2} \, dx=\frac {-2 b c p^2 r^2+2 a d p^2 r^2-d p q r^2 (a+b x) \log ^2(a+b x)-2 a d p q r^2 \log (c+d x)-2 b d p q r^2 x \log (c+d x)+a d q^2 r^2 \log ^2(c+d x)+b d q^2 r^2 x \log ^2(c+d x)-2 b c p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 a d p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 a d q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 b d q r x \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-b c \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+a d \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 d q r (a+b x) \log (a+b x) \left (p r+p r \log (c+d x)-(p+q) r \log \left (\frac {b (c+d x)}{b c-a d}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )-2 d q (p+q) r^2 (a+b x) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )}{b (b c-a d) (a+b x)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.84 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^2} \, dx=\frac {2 \, {\left (\frac {d f q \log \left (b x + a\right )}{b c - a d} - \frac {d f q \log \left (d x + c\right )}{b c - a d} - \frac {f p}{b x + a}\right )} r \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{b f} - \frac {{\left (\frac {2 \, d f^{2} p q \log \left (d x + c\right )}{b c - a d} + \frac {2 \, {\left (p q + q^{2}\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} d f^{2}}{b c - a d} + \frac {2 \, b c f^{2} p^{2} - 2 \, a d f^{2} p^{2} + {\left (b d f^{2} p q x + a d f^{2} p q\right )} \log \left (b x + a\right )^{2} - 2 \, {\left (b d f^{2} p q x + a d f^{2} p q\right )} \log \left (b x + a\right ) \log \left (d x + c\right ) - {\left (b d f^{2} q^{2} x + a d f^{2} q^{2}\right )} \log \left (d x + c\right )^{2} - 2 \, {\left (b d f^{2} p q x + a d f^{2} p q\right )} \log \left (b x + a\right )}{a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x}\right )} r^{2}}{b f^{2}} - \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{{\left (b x + a\right )} b} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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