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

Optimal. Leaf size=465 \[ \frac{2 d p q r^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d)}-\frac{2 d q^2 r^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b (b c-a d)}-\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}+\frac{2 d q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (b c-a d)}-\frac{2 d q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (b c-a d)}-\frac{2 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}-\frac{d p q r^2 \log ^2(a+b x)}{b (b c-a d)}+\frac{2 d p q r^2 \log (a+b x)}{b (b c-a d)}-\frac{2 d p q r^2 \log (c+d x)}{b (b c-a d)}+\frac{2 d p q r^2 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b (b c-a d)}+\frac{d q^2 r^2 \log ^2(c+d x)}{b (b c-a d)}-\frac{2 d q^2 r^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d)}-\frac{2 p^2 r^2}{b (a+b x)} \]

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.387124, antiderivative size = 465, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 12, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.387, Rules used = {2498, 2495, 32, 36, 31, 2514, 2494, 2390, 2301, 2394, 2393, 2391} \[ \frac{2 d p q r^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d)}-\frac{2 d q^2 r^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b (b c-a d)}-\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}+\frac{2 d q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (b c-a d)}-\frac{2 d q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (b c-a d)}-\frac{2 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}-\frac{d p q r^2 \log ^2(a+b x)}{b (b c-a d)}+\frac{2 d p q r^2 \log (a+b x)}{b (b c-a d)}-\frac{2 d p q r^2 \log (c+d x)}{b (b c-a d)}+\frac{2 d p q r^2 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b (b c-a d)}+\frac{d q^2 r^2 \log ^2(c+d x)}{b (b c-a d)}-\frac{2 d q^2 r^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d)}-\frac{2 p^2 r^2}{b (a+b x)} \]

Antiderivative was successfully verified.

[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 2498

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 2495

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 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 31

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

Rule 2514

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]

Rule 2494

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 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 2301

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 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]

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)^2} \, dx &=-\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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 0.836383, size = 411, normalized size = 0.88 \[ \frac{-2 d q r^2 (p+q) (a+b x) \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )-b c \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+a d \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 b c p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 a d p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 a d q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 b d q r x \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 d q r (a+b x) \log (a+b x) \left (\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-r (p+q) \log \left (\frac{b (c+d x)}{b c-a d}\right )+p r \log (c+d x)+p r\right )-d p q r^2 (a+b x) \log ^2(a+b x)-2 a d p q r^2 \log (c+d x)+a d q^2 r^2 \log ^2(c+d x)+2 a d p^2 r^2-2 b d p q r^2 x \log (c+d x)+b d q^2 r^2 x \log ^2(c+d x)-2 b c p^2 r^2}{b (a+b x) (b c-a d)} \]

Antiderivative was successfully verified.

[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))

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Maple [F]  time = 0.41, 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}}{ \left ( bx+a \right ) ^{2}}}\, 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)^2,x)

[Out]

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

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Maxima [A]  time = 1.34225, size = 529, normalized size = 1.14 \begin{align*} \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} \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)^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)

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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^{2} x^{2} + 2 \, a b x + a^{2}}, 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)^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)

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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)**2,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}}{{\left (b x + a\right )}^{2}}\,{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)^2,x, algorithm="giac")

[Out]

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