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

Optimal. Leaf size=632 \[ -\frac{d^2 p q r^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2}+\frac{d^2 q^2 r^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b (b c-a d)^2}-\frac{d^2 q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (b c-a d)^2}+\frac{d^2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (b c-a d)^2}+\frac{d^2 p q r^2 \log ^2(a+b x)}{2 b (b c-a d)^2}-\frac{d^2 p q r^2 \log (a+b x)}{2 b (b c-a d)^2}+\frac{d^2 p q r^2 \log (c+d x)}{2 b (b c-a d)^2}-\frac{d^2 p q r^2 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b (b c-a d)^2}-\frac{d^2 q^2 r^2 \log ^2(c+d x)}{2 b (b c-a d)^2}+\frac{d^2 q^2 r^2 \log (a+b x)}{b (b c-a d)^2}-\frac{d^2 q^2 r^2 \log (c+d x)}{b (b c-a d)^2}+\frac{d^2 q^2 r^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2}-\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (a+b x)^2}-\frac{d q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x) (b c-a d)}-\frac{p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (a+b x)^2}-\frac{3 d p q r^2}{2 b (a+b x) (b c-a d)}-\frac{p^2 r^2}{4 b (a+b x)^2} \]

[Out]

-(p^2*r^2)/(4*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]*Log[
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)

________________________________________________________________________________________

Rubi [A]  time = 0.48971, antiderivative size = 632, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 13, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.419, Rules used = {2498, 2495, 32, 44, 2514, 36, 31, 2494, 2390, 2301, 2394, 2393, 2391} \[ -\frac{d^2 p q r^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2}+\frac{d^2 q^2 r^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b (b c-a d)^2}-\frac{d^2 q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (b c-a d)^2}+\frac{d^2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (b c-a d)^2}+\frac{d^2 p q r^2 \log ^2(a+b x)}{2 b (b c-a d)^2}-\frac{d^2 p q r^2 \log (a+b x)}{2 b (b c-a d)^2}+\frac{d^2 p q r^2 \log (c+d x)}{2 b (b c-a d)^2}-\frac{d^2 p q r^2 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b (b c-a d)^2}-\frac{d^2 q^2 r^2 \log ^2(c+d x)}{2 b (b c-a d)^2}+\frac{d^2 q^2 r^2 \log (a+b x)}{b (b c-a d)^2}-\frac{d^2 q^2 r^2 \log (c+d x)}{b (b c-a d)^2}+\frac{d^2 q^2 r^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2}-\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (a+b x)^2}-\frac{d q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x) (b c-a d)}-\frac{p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (a+b x)^2}-\frac{3 d p q r^2}{2 b (a+b x) (b c-a d)}-\frac{p^2 r^2}{4 b (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(p^2*r^2)/(4*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]*Log[
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 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 44

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] && L
tQ[m + n + 2, 0])

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 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 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)^3} \, dx &=-\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 ) \operatorname{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 ) \operatorname{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 ) \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)^2}-\frac{\left (d^2 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)^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]  time = 1.60547, size = 872, normalized size = 1.38 \[ -\frac{c^2 p^2 r^2 b^2+2 d^2 q^2 r^2 x^2 \log ^2(c+d x) b^2+2 c^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) b^2+6 c d p q r^2 x b^2+4 d^2 q^2 r^2 x^2 \log (c+d x) b^2-2 d^2 p q r^2 x^2 \log (c+d x) b^2+2 c^2 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) b^2+4 c d q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) b^2-4 d^2 q r x^2 \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) b^2-2 a c d p^2 r^2 b+6 a c d p q r^2 b+4 a d^2 q^2 r^2 x \log ^2(c+d x) b-4 a c d \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) b-6 a d^2 p q r^2 x b+8 a d^2 q^2 r^2 x \log (c+d x) b-4 a d^2 p q r^2 x \log (c+d x) b-4 a c d p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) b+4 a c d q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) b-4 a d^2 q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) b-8 a d^2 q r x \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) b+a^2 d^2 p^2 r^2-6 a^2 d^2 p q r^2-2 d^2 p q r^2 (a+b x)^2 \log ^2(a+b x)+2 a^2 d^2 q^2 r^2 \log ^2(c+d x)+2 a^2 d^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+4 a^2 d^2 q^2 r^2 \log (c+d x)-2 a^2 d^2 p q r^2 \log (c+d x)-2 d^2 q r (a+b x)^2 \log (a+b x) \left (-p r+2 q r-2 p \log (c+d x) r+2 (p+q) \log \left (\frac{b (c+d x)}{b c-a d}\right ) r-2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )+2 a^2 d^2 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-4 a^2 d^2 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-4 a^2 d^2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-4 d^2 q (p+q) r^2 (a+b x)^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )}{4 b (b c-a d)^2 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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)])/(4*b*(b*c - a*d)^2*(a + b*x)^2)

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Maple [F]  time = 0.412, 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 ) ^{3}}}\, 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)^3,x)

[Out]

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

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Maxima [A]  time = 1.48079, size = 1019, normalized size = 1.61 \begin{align*} \text{result too large to display} \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)^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)

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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^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}, 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)^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)

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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)**3,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 )}^{3}}\,{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)^3,x, algorithm="giac")

[Out]

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