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

Optimal. Leaf size=431 \[ -\frac{1}{4} \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right ) \left (8 \left (\frac{q r \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b}+\frac{\log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log \left ((c+d x)^{q r}\right )}{b}\right )+\frac{\left (\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )\right )^2}{b p r}\right )+\frac{2 p q r^2 \text{PolyLog}\left (3,-\frac{d (a+b x)}{b c-a d}\right )}{b}-\frac{2 q r \log \left ((a+b x)^{p r}\right ) \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b}-\frac{2 q^2 r^2 \text{PolyLog}\left (3,\frac{b (c+d x)}{b c-a d}\right )}{b}+\frac{2 q r \log \left ((c+d x)^{q r}\right ) \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b}-\frac{q \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2\left ((a+b x)^{p r}\right )}{b p}+\frac{\log ^2\left ((a+b x)^{p r}\right ) \log \left ((c+d x)^{q r}\right )}{b p r}+\frac{\log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{q r}\right )}{b}+\frac{\log ^3\left ((a+b x)^{p r}\right )}{3 b p r} \]

[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

________________________________________________________________________________________

Rubi [A]  time = 0.491518, antiderivative size = 431, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 15, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.484, Rules used = {2496, 6742, 2390, 2302, 30, 2433, 2375, 2317, 2374, 6589, 2396, 2394, 2393, 2391, 6686} \[ -\frac{1}{4} \left (-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )\right ) \left (8 \left (\frac{q r \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b}+\frac{\log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log \left ((c+d x)^{q r}\right )}{b}\right )+\frac{\left (\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )\right )^2}{b p r}\right )+\frac{2 p q r^2 \text{PolyLog}\left (3,-\frac{d (a+b x)}{b c-a d}\right )}{b}-\frac{2 q r \log \left ((a+b x)^{p r}\right ) \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b}-\frac{2 q^2 r^2 \text{PolyLog}\left (3,\frac{b (c+d x)}{b c-a d}\right )}{b}+\frac{2 q r \log \left ((c+d x)^{q r}\right ) \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b}-\frac{q \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2\left ((a+b x)^{p r}\right )}{b p}+\frac{\log ^2\left ((a+b x)^{p r}\right ) \log \left ((c+d x)^{q r}\right )}{b p r}+\frac{\log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{q r}\right )}{b}+\frac{\log ^3\left ((a+b x)^{p r}\right )}{3 b p r} \]

Antiderivative was successfully verified.

[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 2496

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 6742

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

Rule 2390

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 2302

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 30

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

Rule 2433

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 2375

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 2317

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 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(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*Log
[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 6589

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 2396

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 2394

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 2393

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 2391

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 6686

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]

Rubi steps

\begin{align*} \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{\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 (\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 )+\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) \operatorname{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{\operatorname{Subst}\left (\int \frac{\log ^2\left (x^{p r}\right )}{x} \, dx,x,a+b x\right )}{b}+\frac{2 \operatorname{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) \operatorname{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{\operatorname{Subst}\left (\int x^2 \, dx,x,\log \left ((a+b x)^{p r}\right )\right )}{b p r}-\frac{(2 q r) \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 0.17125, size = 460, normalized size = 1.07 \[ \frac{6 q r \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) \left (\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-p r \log (a+b x)\right )+6 p q r^2 \text{PolyLog}\left (3,\frac{d (a+b x)}{a d-b c}\right )-6 p q r^2 \log (a+b x) \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )-6 q^2 r^2 \text{PolyLog}\left (3,\frac{b (c+d x)}{b c-a d}\right )-3 p r \log ^2(a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+3 \log (a+b x) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-6 q r \log (a+b x) \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+6 q r \log (c+d x) \log \left (\frac{d (a+b x)}{a d-b c}\right ) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+6 p q r^2 \log ^2(a+b x) \log (c+d x)-3 p q r^2 \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )-6 p q r^2 \log (a+b x) \log (c+d x) \log \left (\frac{d (a+b x)}{a d-b c}\right )+3 q^2 r^2 \log (a+b x) \log ^2(c+d x)-3 q^2 r^2 \log ^2(c+d x) \log \left (\frac{d (a+b x)}{a d-b c}\right )+p^2 r^2 \log ^3(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[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)

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Maple [F]  time = 0.416, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) \right ) ^{2}}{bx+a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\log \left (b x + a\right ) \log \left ({\left ({\left (d x + c\right )}^{q}\right )}^{r}\right )^{2}}{b} + \int \frac{{\left (\log \left (e\right )^{2} + 2 \, \log \left (e\right ) \log \left (f^{r}\right ) + \log \left (f^{r}\right )^{2}\right )} b d x +{\left (\log \left (e\right )^{2} + 2 \, \log \left (e\right ) \log \left (f^{r}\right ) + \log \left (f^{r}\right )^{2}\right )} b c +{\left (b d x + b c\right )} \log \left ({\left ({\left (b x + a\right )}^{p}\right )}^{r}\right )^{2} + 2 \,{\left (b d x{\left (\log \left (e\right ) + \log \left (f^{r}\right )\right )} + b c{\left (\log \left (e\right ) + \log \left (f^{r}\right )\right )}\right )} \log \left ({\left ({\left (b x + a\right )}^{p}\right )}^{r}\right ) + 2 \,{\left (b d x{\left (\log \left (e\right ) + \log \left (f^{r}\right )\right )} + b c{\left (\log \left (e\right ) + \log \left (f^{r}\right )\right )} -{\left (b d q r x + a d q r\right )} \log \left (b x + a\right ) +{\left (b d x + b c\right )} \log \left ({\left ({\left (b x + a\right )}^{p}\right )}^{r}\right )\right )} \log \left ({\left ({\left (d x + c\right )}^{q}\right )}^{r}\right )}{b^{2} d x^{2} + a b c +{\left (b^{2} c + a b d\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\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}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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