3.24 \(\int \frac{\log ^2(e (f (a+b x)^p (c+d x)^q)^r)}{(a+b x)^5} \, dx\)

Optimal. Leaf size=884 \[ \frac{p q r^2 \log ^2(a+b x) d^4}{4 b (b c-a d)^4}-\frac{q^2 r^2 \log ^2(c+d x) d^4}{4 b (b c-a d)^4}+\frac{11 q^2 r^2 \log (a+b x) d^4}{12 b (b c-a d)^4}-\frac{p q r^2 \log (a+b x) d^4}{8 b (b c-a d)^4}-\frac{11 q^2 r^2 \log (c+d x) d^4}{12 b (b c-a d)^4}+\frac{p q r^2 \log (c+d x) d^4}{8 b (b c-a d)^4}-\frac{p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) d^4}{2 b (b c-a d)^4}+\frac{q^2 r^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) d^4}{2 b (b c-a d)^4}-\frac{q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d^4}{2 b (b c-a d)^4}+\frac{q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d^4}{2 b (b c-a d)^4}+\frac{q^2 r^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) d^4}{2 b (b c-a d)^4}-\frac{p q r^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) d^4}{2 b (b c-a d)^4}-\frac{q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d^3}{2 b (b c-a d)^3 (a+b x)}+\frac{5 q^2 r^2 d^3}{12 b (b c-a d)^3 (a+b x)}-\frac{5 p q r^2 d^3}{8 b (b c-a d)^3 (a+b x)}+\frac{q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d^2}{4 b (b c-a d)^2 (a+b x)^2}-\frac{q^2 r^2 d^2}{12 b (b c-a d)^2 (a+b x)^2}+\frac{3 p q r^2 d^2}{16 b (b c-a d)^2 (a+b x)^2}-\frac{q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d}{6 b (b c-a d) (a+b x)^3}-\frac{7 p q r^2 d}{72 b (b c-a d) (a+b x)^3}-\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}-\frac{p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b (a+b x)^4}-\frac{p^2 r^2}{32 b (a+b x)^4} \]

[Out]

-(p^2*r^2)/(32*b*(a + b*x)^4) - (7*d*p*q*r^2)/(72*b*(b*c - a*d)*(a + b*x)^3) + (3*d^2*p*q*r^2)/(16*b*(b*c - a*
d)^2*(a + b*x)^2) - (d^2*q^2*r^2)/(12*b*(b*c - a*d)^2*(a + b*x)^2) - (5*d^3*p*q*r^2)/(8*b*(b*c - a*d)^3*(a + b
*x)) + (5*d^3*q^2*r^2)/(12*b*(b*c - a*d)^3*(a + b*x)) - (d^4*p*q*r^2*Log[a + b*x])/(8*b*(b*c - a*d)^4) + (11*d
^4*q^2*r^2*Log[a + b*x])/(12*b*(b*c - a*d)^4) + (d^4*p*q*r^2*Log[a + b*x]^2)/(4*b*(b*c - a*d)^4) + (d^4*p*q*r^
2*Log[c + d*x])/(8*b*(b*c - a*d)^4) - (11*d^4*q^2*r^2*Log[c + d*x])/(12*b*(b*c - a*d)^4) - (d^4*p*q*r^2*Log[-(
(d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(2*b*(b*c - a*d)^4) - (d^4*q^2*r^2*Log[c + d*x]^2)/(4*b*(b*c - a*d)^
4) + (d^4*q^2*r^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(2*b*(b*c - a*d)^4) - (p*r*Log[e*(f*(a + b*x)^p
*(c + d*x)^q)^r])/(8*b*(a + b*x)^4) - (d*q*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(6*b*(b*c - a*d)*(a + b*x)^
3) + (d^2*q*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(4*b*(b*c - a*d)^2*(a + b*x)^2) - (d^3*q*r*Log[e*(f*(a + b
*x)^p*(c + d*x)^q)^r])/(2*b*(b*c - a*d)^3*(a + b*x)) - (d^4*q*r*Log[a + b*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)
^r])/(2*b*(b*c - a*d)^4) + (d^4*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*b*(b*c - a*d)^4) - L
og[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(4*b*(a + b*x)^4) + (d^4*q^2*r^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))
])/(2*b*(b*c - a*d)^4) - (d^4*p*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(2*b*(b*c - a*d)^4)

________________________________________________________________________________________

Rubi [A]  time = 0.737891, antiderivative size = 884, normalized size of antiderivative = 1., number of steps used = 32, 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} \[ \frac{p q r^2 \log ^2(a+b x) d^4}{4 b (b c-a d)^4}-\frac{q^2 r^2 \log ^2(c+d x) d^4}{4 b (b c-a d)^4}+\frac{11 q^2 r^2 \log (a+b x) d^4}{12 b (b c-a d)^4}-\frac{p q r^2 \log (a+b x) d^4}{8 b (b c-a d)^4}-\frac{11 q^2 r^2 \log (c+d x) d^4}{12 b (b c-a d)^4}+\frac{p q r^2 \log (c+d x) d^4}{8 b (b c-a d)^4}-\frac{p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) d^4}{2 b (b c-a d)^4}+\frac{q^2 r^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) d^4}{2 b (b c-a d)^4}-\frac{q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d^4}{2 b (b c-a d)^4}+\frac{q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d^4}{2 b (b c-a d)^4}+\frac{q^2 r^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) d^4}{2 b (b c-a d)^4}-\frac{p q r^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) d^4}{2 b (b c-a d)^4}-\frac{q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d^3}{2 b (b c-a d)^3 (a+b x)}+\frac{5 q^2 r^2 d^3}{12 b (b c-a d)^3 (a+b x)}-\frac{5 p q r^2 d^3}{8 b (b c-a d)^3 (a+b x)}+\frac{q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d^2}{4 b (b c-a d)^2 (a+b x)^2}-\frac{q^2 r^2 d^2}{12 b (b c-a d)^2 (a+b x)^2}+\frac{3 p q r^2 d^2}{16 b (b c-a d)^2 (a+b x)^2}-\frac{q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d}{6 b (b c-a d) (a+b x)^3}-\frac{7 p q r^2 d}{72 b (b c-a d) (a+b x)^3}-\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}-\frac{p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b (a+b x)^4}-\frac{p^2 r^2}{32 b (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(a + b*x)^5,x]

[Out]

-(p^2*r^2)/(32*b*(a + b*x)^4) - (7*d*p*q*r^2)/(72*b*(b*c - a*d)*(a + b*x)^3) + (3*d^2*p*q*r^2)/(16*b*(b*c - a*
d)^2*(a + b*x)^2) - (d^2*q^2*r^2)/(12*b*(b*c - a*d)^2*(a + b*x)^2) - (5*d^3*p*q*r^2)/(8*b*(b*c - a*d)^3*(a + b
*x)) + (5*d^3*q^2*r^2)/(12*b*(b*c - a*d)^3*(a + b*x)) - (d^4*p*q*r^2*Log[a + b*x])/(8*b*(b*c - a*d)^4) + (11*d
^4*q^2*r^2*Log[a + b*x])/(12*b*(b*c - a*d)^4) + (d^4*p*q*r^2*Log[a + b*x]^2)/(4*b*(b*c - a*d)^4) + (d^4*p*q*r^
2*Log[c + d*x])/(8*b*(b*c - a*d)^4) - (11*d^4*q^2*r^2*Log[c + d*x])/(12*b*(b*c - a*d)^4) - (d^4*p*q*r^2*Log[-(
(d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(2*b*(b*c - a*d)^4) - (d^4*q^2*r^2*Log[c + d*x]^2)/(4*b*(b*c - a*d)^
4) + (d^4*q^2*r^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(2*b*(b*c - a*d)^4) - (p*r*Log[e*(f*(a + b*x)^p
*(c + d*x)^q)^r])/(8*b*(a + b*x)^4) - (d*q*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(6*b*(b*c - a*d)*(a + b*x)^
3) + (d^2*q*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(4*b*(b*c - a*d)^2*(a + b*x)^2) - (d^3*q*r*Log[e*(f*(a + b
*x)^p*(c + d*x)^q)^r])/(2*b*(b*c - a*d)^3*(a + b*x)) - (d^4*q*r*Log[a + b*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)
^r])/(2*b*(b*c - a*d)^4) + (d^4*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*b*(b*c - a*d)^4) - L
og[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(4*b*(a + b*x)^4) + (d^4*q^2*r^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))
])/(2*b*(b*c - a*d)^4) - (d^4*p*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(2*b*(b*c - a*d)^4)

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)^5} \, dx &=-\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}+\frac{1}{2} (p r) \int \frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, 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)^4 (c+d x)} \, dx}{2 b}\\ &=-\frac{p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b (a+b x)^4}-\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}+\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)^4}-\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)^3}+\frac{b d^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(b c-a d)^4 (a+b x)}+\frac{d^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{2 b}+\frac{1}{8} \left (p^2 r^2\right ) \int \frac{1}{(a+b x)^5} \, dx+\frac{\left (d p q r^2\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{8 b}\\ &=-\frac{p^2 r^2}{32 b (a+b x)^4}-\frac{p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b (a+b x)^4}-\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}-\frac{\left (d^4 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}{2 (b c-a d)^4}+\frac{\left (d^5 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}{2 b (b c-a d)^4}+\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 )}{(a+b x)^2} \, dx}{2 (b c-a d)^3}-\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)^3} \, dx}{2 (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)^4} \, dx}{2 (b c-a d)}+\frac{\left (d p q r^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^4}-\frac{b d}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3}{(b c-a d)^4 (a+b x)}+\frac{d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{8 b}\\ &=-\frac{p^2 r^2}{32 b (a+b x)^4}-\frac{d p q r^2}{24 b (b c-a d) (a+b x)^3}+\frac{d^2 p q r^2}{16 b (b c-a d)^2 (a+b x)^2}-\frac{d^3 p q r^2}{8 b (b c-a d)^3 (a+b x)}-\frac{d^4 p q r^2 \log (a+b x)}{8 b (b c-a d)^4}+\frac{d^4 p q r^2 \log (c+d x)}{8 b (b c-a d)^4}-\frac{p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b (a+b x)^4}-\frac{d q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{6 b (b c-a d) (a+b x)^3}+\frac{d^2 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (b c-a d)^2 (a+b x)^2}-\frac{d^3 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (b c-a d)^3 (a+b x)}-\frac{d^4 q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (b c-a d)^4}+\frac{d^4 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (b c-a d)^4}-\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}+\frac{\left (d^4 p q r^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{2 (b c-a d)^4}-\frac{\left (d^4 p q r^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{2 (b c-a d)^4}+\frac{\left (d^3 p q r^2\right ) \int \frac{1}{(a+b x)^2} \, dx}{2 (b c-a d)^3}-\frac{\left (d^2 p q r^2\right ) \int \frac{1}{(a+b x)^3} \, dx}{4 (b c-a d)^2}+\frac{\left (d p q r^2\right ) \int \frac{1}{(a+b x)^4} \, dx}{6 (b c-a d)}+\frac{\left (d^5 q^2 r^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{2 b (b c-a d)^4}-\frac{\left (d^5 q^2 r^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{2 b (b c-a d)^4}+\frac{\left (d^4 q^2 r^2\right ) \int \frac{1}{(a+b x) (c+d x)} \, dx}{2 b (b c-a d)^3}-\frac{\left (d^3 q^2 r^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{4 b (b c-a d)^2}+\frac{\left (d^2 q^2 r^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{6 b (b c-a d)}\\ &=-\frac{p^2 r^2}{32 b (a+b x)^4}-\frac{7 d p q r^2}{72 b (b c-a d) (a+b x)^3}+\frac{3 d^2 p q r^2}{16 b (b c-a d)^2 (a+b x)^2}-\frac{5 d^3 p q r^2}{8 b (b c-a d)^3 (a+b x)}-\frac{d^4 p q r^2 \log (a+b x)}{8 b (b c-a d)^4}+\frac{d^4 p q r^2 \log (c+d x)}{8 b (b c-a d)^4}-\frac{d^4 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b (b c-a d)^4}+\frac{d^4 q^2 r^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4}-\frac{p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b (a+b x)^4}-\frac{d q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{6 b (b c-a d) (a+b x)^3}+\frac{d^2 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (b c-a d)^2 (a+b x)^2}-\frac{d^3 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (b c-a d)^3 (a+b x)}-\frac{d^4 q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (b c-a d)^4}+\frac{d^4 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (b c-a d)^4}-\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}+\frac{\left (d^4 p q r^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{2 b (b c-a d)^4}+\frac{\left (d^5 p q r^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{2 b (b c-a d)^4}+\frac{\left (d^4 q^2 r^2\right ) \int \frac{1}{a+b x} \, dx}{2 (b c-a d)^4}-\frac{\left (d^4 q^2 r^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{2 (b c-a d)^4}-\frac{\left (d^4 q^2 r^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{2 b (b c-a d)^4}-\frac{\left (d^5 q^2 r^2\right ) \int \frac{1}{c+d x} \, dx}{2 b (b c-a d)^4}-\frac{\left (d^3 q^2 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}{4 b (b c-a d)^2}+\frac{\left (d^2 q^2 r^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{6 b (b c-a d)}\\ &=-\frac{p^2 r^2}{32 b (a+b x)^4}-\frac{7 d p q r^2}{72 b (b c-a d) (a+b x)^3}+\frac{3 d^2 p q r^2}{16 b (b c-a d)^2 (a+b x)^2}-\frac{d^2 q^2 r^2}{12 b (b c-a d)^2 (a+b x)^2}-\frac{5 d^3 p q r^2}{8 b (b c-a d)^3 (a+b x)}+\frac{5 d^3 q^2 r^2}{12 b (b c-a d)^3 (a+b x)}-\frac{d^4 p q r^2 \log (a+b x)}{8 b (b c-a d)^4}+\frac{11 d^4 q^2 r^2 \log (a+b x)}{12 b (b c-a d)^4}+\frac{d^4 p q r^2 \log ^2(a+b x)}{4 b (b c-a d)^4}+\frac{d^4 p q r^2 \log (c+d x)}{8 b (b c-a d)^4}-\frac{11 d^4 q^2 r^2 \log (c+d x)}{12 b (b c-a d)^4}-\frac{d^4 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b (b c-a d)^4}-\frac{d^4 q^2 r^2 \log ^2(c+d x)}{4 b (b c-a d)^4}+\frac{d^4 q^2 r^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4}-\frac{p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b (a+b x)^4}-\frac{d q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{6 b (b c-a d) (a+b x)^3}+\frac{d^2 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (b c-a d)^2 (a+b x)^2}-\frac{d^3 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (b c-a d)^3 (a+b x)}-\frac{d^4 q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (b c-a d)^4}+\frac{d^4 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (b c-a d)^4}-\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}+\frac{\left (d^4 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 )}{2 b (b c-a d)^4}-\frac{\left (d^4 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 )}{2 b (b c-a d)^4}\\ &=-\frac{p^2 r^2}{32 b (a+b x)^4}-\frac{7 d p q r^2}{72 b (b c-a d) (a+b x)^3}+\frac{3 d^2 p q r^2}{16 b (b c-a d)^2 (a+b x)^2}-\frac{d^2 q^2 r^2}{12 b (b c-a d)^2 (a+b x)^2}-\frac{5 d^3 p q r^2}{8 b (b c-a d)^3 (a+b x)}+\frac{5 d^3 q^2 r^2}{12 b (b c-a d)^3 (a+b x)}-\frac{d^4 p q r^2 \log (a+b x)}{8 b (b c-a d)^4}+\frac{11 d^4 q^2 r^2 \log (a+b x)}{12 b (b c-a d)^4}+\frac{d^4 p q r^2 \log ^2(a+b x)}{4 b (b c-a d)^4}+\frac{d^4 p q r^2 \log (c+d x)}{8 b (b c-a d)^4}-\frac{11 d^4 q^2 r^2 \log (c+d x)}{12 b (b c-a d)^4}-\frac{d^4 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b (b c-a d)^4}-\frac{d^4 q^2 r^2 \log ^2(c+d x)}{4 b (b c-a d)^4}+\frac{d^4 q^2 r^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4}-\frac{p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b (a+b x)^4}-\frac{d q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{6 b (b c-a d) (a+b x)^3}+\frac{d^2 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (b c-a d)^2 (a+b x)^2}-\frac{d^3 q r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (b c-a d)^3 (a+b x)}-\frac{d^4 q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (b c-a d)^4}+\frac{d^4 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (b c-a d)^4}-\frac{\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}+\frac{d^4 q^2 r^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{2 b (b c-a d)^4}-\frac{d^4 p q r^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4}\\ \end{align*}

Mathematica [B]  time = 7.02508, size = 14573, normalized size = 16.49 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(a + b*x)^5,x]

[Out]

Result too large to show

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Maple [F]  time = 0.418, 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 ) ^{5}}}\, 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)^5,x)

[Out]

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/(b*x+a)^5,x)

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Maxima [B]  time = 2.1752, size = 2452, normalized size = 2.77 \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)^5,x, algorithm="maxima")

[Out]

-1/24*(12*d^4*f*q*log(b*x + a)/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) - 12*d^
4*f*q*log(d*x + c)/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) + (12*b^3*d^3*f*q*x
^3 - a*b^2*c^2*d*f*(9*p - 4*q) + a^2*b*c*d^2*f*(9*p - 14*q) - a^3*d^3*f*(3*p - 22*q) + 3*b^3*c^3*f*p - 6*(b^3*
c*d^2*f*q - 7*a*b^2*d^3*f*q)*x^2 + 4*(b^3*c^2*d*f*q - 5*a*b^2*c*d^2*f*q + 13*a^2*b*d^3*f*q)*x)/(a^4*b^3*c^3 -
3*a^5*b^2*c^2*d + 3*a^6*b*c*d^2 - a^7*d^3 + (b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*x^4 + 4*
(a*b^6*c^3 - 3*a^2*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*x^3 + 6*(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b
^3*c*d^2 - a^5*b^2*d^3)*x^2 + 4*(a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3)*x))*r*log(((b*x
+ a)^p*(d*x + c)^q*f)^r*e)/(b*f) + 1/288*(144*(p*q + q^2)*(log(b*x + a)*log((b*d*x + a*d)/(b*c - a*d) + 1) + d
ilog(-(b*d*x + a*d)/(b*c - a*d)))*d^4*f^2/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d
^4) + 12*(3*p*q - 22*q^2)*d^4*f^2*log(d*x + c)/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 +
a^4*d^4) - (9*b^4*c^4*f^2*p^2 - 4*(9*p^2 - 7*p*q)*a*b^3*c^3*d*f^2 + 6*(9*p^2 - 23*p*q + 4*q^2)*a^2*b^2*c^2*d^2
*f^2 - 12*(3*p^2 - 31*p*q + 14*q^2)*a^3*b*c*d^3*f^2 + (9*p^2 - 262*p*q + 144*q^2)*a^4*d^4*f^2 + 60*((3*p*q - 2
*q^2)*b^4*c*d^3*f^2 - (3*p*q - 2*q^2)*a*b^3*d^4*f^2)*x^3 - 6*((9*p*q - 4*q^2)*b^4*c^2*d^2*f^2 - 4*(27*p*q - 17
*q^2)*a*b^3*c*d^3*f^2 + (99*p*q - 64*q^2)*a^2*b^2*d^4*f^2)*x^2 - 72*(b^4*d^4*f^2*p*q*x^4 + 4*a*b^3*d^4*f^2*p*q
*x^3 + 6*a^2*b^2*d^4*f^2*p*q*x^2 + 4*a^3*b*d^4*f^2*p*q*x + a^4*d^4*f^2*p*q)*log(b*x + a)^2 + 144*(b^4*d^4*f^2*
p*q*x^4 + 4*a*b^3*d^4*f^2*p*q*x^3 + 6*a^2*b^2*d^4*f^2*p*q*x^2 + 4*a^3*b*d^4*f^2*p*q*x + a^4*d^4*f^2*p*q)*log(b
*x + a)*log(d*x + c) + 72*(b^4*d^4*f^2*q^2*x^4 + 4*a*b^3*d^4*f^2*q^2*x^3 + 6*a^2*b^2*d^4*f^2*q^2*x^2 + 4*a^3*b
*d^4*f^2*q^2*x + a^4*d^4*f^2*q^2)*log(d*x + c)^2 + 4*(7*b^4*c^3*d*f^2*p*q - 12*(4*p*q - q^2)*a*b^3*c^2*d^2*f^2
+ 6*(35*p*q - 19*q^2)*a^2*b^2*c*d^3*f^2 - (169*p*q - 102*q^2)*a^3*b*d^4*f^2)*x + 12*((3*p*q - 22*q^2)*b^4*d^4
*f^2*x^4 + 4*(3*p*q - 22*q^2)*a*b^3*d^4*f^2*x^3 + 6*(3*p*q - 22*q^2)*a^2*b^2*d^4*f^2*x^2 + 4*(3*p*q - 22*q^2)*
a^3*b*d^4*f^2*x + (3*p*q - 22*q^2)*a^4*d^4*f^2)*log(b*x + a))/(a^4*b^4*c^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d
^2 - 4*a^7*b*c*d^3 + a^8*d^4 + (b^8*c^4 - 4*a*b^7*c^3*d + 6*a^2*b^6*c^2*d^2 - 4*a^3*b^5*c*d^3 + a^4*b^4*d^4)*x
^4 + 4*(a*b^7*c^4 - 4*a^2*b^6*c^3*d + 6*a^3*b^5*c^2*d^2 - 4*a^4*b^4*c*d^3 + a^5*b^3*d^4)*x^3 + 6*(a^2*b^6*c^4
- 4*a^3*b^5*c^3*d + 6*a^4*b^4*c^2*d^2 - 4*a^5*b^3*c*d^3 + a^6*b^2*d^4)*x^2 + 4*(a^3*b^5*c^4 - 4*a^4*b^4*c^3*d
+ 6*a^5*b^3*c^2*d^2 - 4*a^6*b^2*c*d^3 + a^7*b*d^4)*x))*r^2/(b*f^2) - 1/4*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^
2/((b*x + a)^4*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^{5} x^{5} + 5 \, a b^{4} x^{4} + 10 \, a^{2} b^{3} x^{3} + 10 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x + a^{5}}, 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)^5,x, algorithm="fricas")

[Out]

integral(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2/(b^5*x^5 + 5*a*b^4*x^4 + 10*a^2*b^3*x^3 + 10*a^3*b^2*x^2 + 5*a
^4*b*x + a^5), 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)**5,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 )}^{5}}\,{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)^5,x, algorithm="giac")

[Out]

integrate(log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2/(b*x + a)^5, x)