\(\int \frac {\log ^2(e (f (a+b x)^p (c+d x)^q)^r)}{a+b x} \, dx\) [20]

Optimal result

Integrand size = 31, antiderivative size = 431 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{a+b x} \, dx=\frac {\log ^3\left ((a+b x)^{p r}\right )}{3 b p r}-\frac {q \log ^2\left ((a+b x)^{p r}\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b p}+\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left ((c+d x)^{q r}\right )}{b p r}+\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{q r}\right )}{b}-\frac {2 q r \log \left ((a+b x)^{p r}\right ) \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b}+\frac {2 q r \log \left ((c+d x)^{q r}\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b}-\frac {1}{4} \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \left (\frac {\left (\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )^2}{b p r}+8 \left (\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log \left ((c+d x)^{q r}\right )}{b}+\frac {q r \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b}\right )\right )+\frac {2 p q r^2 \operatorname {PolyLog}\left (3,-\frac {d (a+b x)}{b c-a d}\right )}{b}-\frac {2 q^2 r^2 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{b c-a d}\right )}{b} \]

[Out]

1/3*ln((b*x+a)^(p*r))^3/b/p/r-q*ln((b*x+a)^(p*r))^2*ln(b*(d*x+c)/(-a*d+b*c))/b/p+ln((b*x+a)^(p*r))^2*ln((d*x+c
)^(q*r))/b/p/r+ln(-d*(b*x+a)/(-a*d+b*c))*ln((d*x+c)^(q*r))^2/b-2*q*r*ln((b*x+a)^(p*r))*polylog(2,-d*(b*x+a)/(-
a*d+b*c))/b+2*q*r*ln((d*x+c)^(q*r))*polylog(2,b*(d*x+c)/(-a*d+b*c))/b-1/4*(ln((b*x+a)^(p*r))+ln((d*x+c)^(q*r))
-ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r))*((ln((b*x+a)^(p*r))-ln((d*x+c)^(q*r))+ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r))^2/b/p
/r+8*ln(-d*(b*x+a)/(-a*d+b*c))*ln((d*x+c)^(q*r))/b+8*q*r*polylog(2,b*(d*x+c)/(-a*d+b*c))/b)+2*p*q*r^2*polylog(
3,-d*(b*x+a)/(-a*d+b*c))/b-2*q^2*r^2*polylog(3,b*(d*x+c)/(-a*d+b*c))/b

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.484, Rules used = {2582, 6874, 2437, 2339, 30, 2481, 2422, 2354, 2421, 6724, 2443, 2441, 2440, 2438, 6818} \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{a+b x} \, dx=-\frac {1}{4} \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right ) \left (\frac {\left (\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )\right )^2}{b p r}+8 \left (\frac {q r \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b}+\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log \left ((c+d x)^{q r}\right )}{b}\right )\right )+\frac {2 p q r^2 \operatorname {PolyLog}\left (3,-\frac {d (a+b x)}{b c-a d}\right )}{b}-\frac {2 q r \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right ) \log \left ((a+b x)^{p r}\right )}{b}-\frac {q \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2\left ((a+b x)^{p r}\right )}{b p}+\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left ((c+d x)^{q r}\right )}{b p r}-\frac {2 q^2 r^2 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{b c-a d}\right )}{b}+\frac {2 q r \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) \log \left ((c+d x)^{q r}\right )}{b}+\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{q r}\right )}{b}+\frac {\log ^3\left ((a+b x)^{p r}\right )}{3 b p r} \]

[In]

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

[Out]

Log[(a + b*x)^(p*r)]^3/(3*b*p*r) - (q*Log[(a + b*x)^(p*r)]^2*Log[(b*(c + d*x))/(b*c - a*d)])/(b*p) + (Log[(a +
b*x)^(p*r)]^2*Log[(c + d*x)^(q*r)])/(b*p*r) + (Log[-((d*(a + b*x))/(b*c - a*d))]*Log[(c + d*x)^(q*r)]^2)/b -
(2*q*r*Log[(a + b*x)^(p*r)]*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/b + (2*q*r*Log[(c + d*x)^(q*r)]*PolyLog[
2, (b*(c + d*x))/(b*c - a*d)])/b - ((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[(a + b*x)^(p*r)] - Log[(c + d*x)^(q*r)] + Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])^2/(b*p*r) + 8
*((Log[-((d*(a + b*x))/(b*c - a*d))]*Log[(c + d*x)^(q*r)])/b + (q*r*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/b))
)/4 + (2*p*q*r^2*PolyLog[3, -((d*(a + b*x))/(b*c - a*d))])/b - (2*q^2*r^2*PolyLog[3, (b*(c + d*x))/(b*c - a*d)
])/b

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

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 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 2443

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

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 2582

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

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right )^2}{a+b x} \, dx-\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 ) \left (2 \int \frac {\log \left ((c+d x)^{q r}\right )}{a+b x} \, dx+\int \frac {\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 )}{a+b x} \, dx\right ) \\ & = -\left (\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 ) \left (\frac {\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 )^2}{4 b p r}+2 \left (\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log \left ((c+d x)^{q r}\right )}{b}-\frac {(d q r) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b}\right )\right )\right )+\int \left (\frac {\log ^2\left ((a+b x)^{p r}\right )}{a+b x}+\frac {2 \log \left ((a+b x)^{p r}\right ) \log \left ((c+d x)^{q r}\right )}{a+b x}+\frac {\log ^2\left ((c+d x)^{q r}\right )}{a+b x}\right ) \, dx \\ & = 2 \int \frac {\log \left ((a+b x)^{p r}\right ) \log \left ((c+d x)^{q r}\right )}{a+b x} \, dx-\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 ) \left (\frac {\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 )^2}{4 b p r}+2 \left (\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log \left ((c+d x)^{q r}\right )}{b}-\frac {(q r) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b}\right )\right )+\int \frac {\log ^2\left ((a+b x)^{p r}\right )}{a+b x} \, dx+\int \frac {\log ^2\left ((c+d x)^{q r}\right )}{a+b x} \, dx \\ & = \frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{q r}\right )}{b}-\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 ) \left (\frac {\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 )^2}{4 b p r}+2 \left (\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log \left ((c+d x)^{q r}\right )}{b}+\frac {q r \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b}\right )\right )+\frac {\text {Subst}\left (\int \frac {\log ^2\left (x^{p r}\right )}{x} \, dx,x,a+b x\right )}{b}+\frac {2 \text {Subst}\left (\int \frac {\log \left (x^{p r}\right ) \log \left (\left (\frac {b c-a d}{b}+\frac {d x}{b}\right )^{q r}\right )}{x} \, dx,x,a+b x\right )}{b}-\frac {(2 d q r) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log \left ((c+d x)^{q r}\right )}{c+d x} \, dx}{b} \\ & = \frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left ((c+d x)^{q r}\right )}{b p r}+\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{q r}\right )}{b}-\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 ) \left (\frac {\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 )^2}{4 b p r}+2 \left (\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log \left ((c+d x)^{q r}\right )}{b}+\frac {q r \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b}\right )\right )-\frac {(d q) \text {Subst}\left (\int \frac {\log ^2\left (x^{p r}\right )}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{b^2 p}+\frac {\text {Subst}\left (\int x^2 \, dx,x,\log \left ((a+b x)^{p r}\right )\right )}{b p r}-\frac {(2 q r) \text {Subst}\left (\int \frac {\log \left (x^{q r}\right ) \log \left (\frac {d \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b} \\ & = \frac {\log ^3\left ((a+b x)^{p r}\right )}{3 b p r}-\frac {q \log ^2\left ((a+b x)^{p r}\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b p}+\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left ((c+d x)^{q r}\right )}{b p r}+\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{q r}\right )}{b}+\frac {2 q r \log \left ((c+d x)^{q r}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b}-\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 ) \left (\frac {\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 )^2}{4 b p r}+2 \left (\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log \left ((c+d x)^{q r}\right )}{b}+\frac {q r \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b}\right )\right )+\frac {(2 q r) \text {Subst}\left (\int \frac {\log \left (x^{p r}\right ) \log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b}-\frac {\left (2 q^2 r^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b} \\ & = \frac {\log ^3\left ((a+b x)^{p r}\right )}{3 b p r}-\frac {q \log ^2\left ((a+b x)^{p r}\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b p}+\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left ((c+d x)^{q r}\right )}{b p r}+\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{q r}\right )}{b}-\frac {2 q r \log \left ((a+b x)^{p r}\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b}+\frac {2 q r \log \left ((c+d x)^{q r}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b}-\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 ) \left (\frac {\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 )^2}{4 b p r}+2 \left (\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log \left ((c+d x)^{q r}\right )}{b}+\frac {q r \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b}\right )\right )-\frac {2 q^2 r^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{b}+\frac {\left (2 p q r^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b} \\ & = \frac {\log ^3\left ((a+b x)^{p r}\right )}{3 b p r}-\frac {q \log ^2\left ((a+b x)^{p r}\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b p}+\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left ((c+d x)^{q r}\right )}{b p r}+\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{q r}\right )}{b}-\frac {2 q r \log \left ((a+b x)^{p r}\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b}+\frac {2 q r \log \left ((c+d x)^{q r}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b}-\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 ) \left (\frac {\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 )^2}{4 b p r}+2 \left (\frac {\log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log \left ((c+d x)^{q r}\right )}{b}+\frac {q r \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b}\right )\right )+\frac {2 p q r^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{b}-\frac {2 q^2 r^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.07 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{a+b x} \, dx=\frac {p^2 r^2 \log ^3(a+b x)+6 p q r^2 \log ^2(a+b x) \log (c+d x)-6 p q r^2 \log (a+b x) \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)+3 q^2 r^2 \log (a+b x) \log ^2(c+d x)-3 q^2 r^2 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log ^2(c+d x)-3 p q r^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )-3 p r \log ^2(a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-6 q r \log (a+b x) \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+6 q r \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+3 \log (a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-6 p q r^2 \log (a+b x) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )+6 q r \left (-p r \log (a+b x)+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )+6 p q r^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{-b c+a d}\right )-6 q^2 r^2 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{b c-a d}\right )}{3 b} \]

[In]

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

[Out]

(p^2*r^2*Log[a + b*x]^3 + 6*p*q*r^2*Log[a + b*x]^2*Log[c + d*x] - 6*p*q*r^2*Log[a + b*x]*Log[(d*(a + b*x))/(-(
b*c) + a*d)]*Log[c + d*x] + 3*q^2*r^2*Log[a + b*x]*Log[c + d*x]^2 - 3*q^2*r^2*Log[(d*(a + b*x))/(-(b*c) + a*d)
]*Log[c + d*x]^2 - 3*p*q*r^2*Log[a + b*x]^2*Log[(b*(c + d*x))/(b*c - a*d)] - 3*p*r*Log[a + b*x]^2*Log[e*(f*(a
+ b*x)^p*(c + d*x)^q)^r] - 6*q*r*Log[a + b*x]*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + 6*q*r*Log[(d
*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + 3*Log[a + b*x]*Log[e*(f*(a + b
*x)^p*(c + d*x)^q)^r]^2 - 6*p*q*r^2*Log[a + b*x]*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] + 6*q*r*(-(p*r*Log[a
+ b*x]) + Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])*PolyLog[2, (b*(c + d*x))/(b*c - a*d)] + 6*p*q*r^2*PolyLog[3,
(d*(a + b*x))/(-(b*c) + a*d)] - 6*q^2*r^2*PolyLog[3, (b*(c + d*x))/(b*c - a*d)])/(3*b)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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