3.16 \(\int (a+b x)^4 \log ^2(e (f (a+b x)^p (c+d x)^q)^r) \, dx\)

Optimal. Leaf size=920 \[ -\frac{q^2 r^2 \log ^2(c+d x) (b c-a d)^5}{5 b d^5}-\frac{137 q^2 r^2 \log (c+d x) (b c-a d)^5}{150 b d^5}-\frac{2 p q r^2 \log (c+d x) (b c-a d)^5}{25 b d^5}-\frac{2 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) (b c-a d)^5}{5 b d^5}+\frac{2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^5}{5 b d^5}-\frac{2 p q r^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) (b c-a d)^5}{5 b d^5}+\frac{77 q^2 r^2 x (b c-a d)^4}{150 d^4}+\frac{2 p q r^2 x (b c-a d)^4}{25 d^4}+\frac{2 q (p+q) r^2 x (b c-a d)^4}{5 d^4}-\frac{2 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^4}{5 b d^4}-\frac{b p q r^2 x^2 (b c-a d)^3}{10 d^3}-\frac{77 q^2 r^2 (a+b x)^2 (b c-a d)^3}{300 b d^3}-\frac{p q r^2 (a+b x)^2 (b c-a d)^3}{25 b d^3}-\frac{a p q r^2 x (b c-a d)^3}{5 d^3}+\frac{q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^3}{5 b d^3}+\frac{47 q^2 r^2 (a+b x)^3 (b c-a d)^2}{450 b d^2}+\frac{16 p q r^2 (a+b x)^3 (b c-a d)^2}{225 b d^2}-\frac{2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^2}{15 b d^2}-\frac{9 q^2 r^2 (a+b x)^4 (b c-a d)}{200 b d}-\frac{9 p q r^2 (a+b x)^4 (b c-a d)}{200 b d}+\frac{q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)}{10 b d}+\frac{2 p^2 r^2 (a+b x)^5}{125 b}+\frac{2 q^2 r^2 (a+b x)^5}{125 b}+\frac{4 p q r^2 (a+b x)^5}{125 b}+\frac{(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}-\frac{2 p r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}-\frac{2 q r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b} \]

[Out]

-(a*(b*c - a*d)^3*p*q*r^2*x)/(5*d^3) + (2*(b*c - a*d)^4*p*q*r^2*x)/(25*d^4) + (77*(b*c - a*d)^4*q^2*r^2*x)/(15
0*d^4) + (2*(b*c - a*d)^4*q*(p + q)*r^2*x)/(5*d^4) - (b*(b*c - a*d)^3*p*q*r^2*x^2)/(10*d^3) - ((b*c - a*d)^3*p
*q*r^2*(a + b*x)^2)/(25*b*d^3) - (77*(b*c - a*d)^3*q^2*r^2*(a + b*x)^2)/(300*b*d^3) + (16*(b*c - a*d)^2*p*q*r^
2*(a + b*x)^3)/(225*b*d^2) + (47*(b*c - a*d)^2*q^2*r^2*(a + b*x)^3)/(450*b*d^2) - (9*(b*c - a*d)*p*q*r^2*(a +
b*x)^4)/(200*b*d) - (9*(b*c - a*d)*q^2*r^2*(a + b*x)^4)/(200*b*d) + (2*p^2*r^2*(a + b*x)^5)/(125*b) + (4*p*q*r
^2*(a + b*x)^5)/(125*b) + (2*q^2*r^2*(a + b*x)^5)/(125*b) - (2*(b*c - a*d)^5*p*q*r^2*Log[c + d*x])/(25*b*d^5)
- (137*(b*c - a*d)^5*q^2*r^2*Log[c + d*x])/(150*b*d^5) - (2*(b*c - a*d)^5*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a
*d))]*Log[c + d*x])/(5*b*d^5) - ((b*c - a*d)^5*q^2*r^2*Log[c + d*x]^2)/(5*b*d^5) - (2*(b*c - a*d)^4*q*r*(a + b
*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(5*b*d^4) + ((b*c - a*d)^3*q*r*(a + b*x)^2*Log[e*(f*(a + b*x)^p*(c +
d*x)^q)^r])/(5*b*d^3) - (2*(b*c - a*d)^2*q*r*(a + b*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(15*b*d^2) + (
(b*c - a*d)*q*r*(a + b*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(10*b*d) - (2*p*r*(a + b*x)^5*Log[e*(f*(a +
b*x)^p*(c + d*x)^q)^r])/(25*b) - (2*q*r*(a + b*x)^5*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(25*b) + (2*(b*c - a
*d)^5*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(5*b*d^5) + ((a + b*x)^5*Log[e*(f*(a + b*x)^p*(c
+ d*x)^q)^r]^2)/(5*b) - (2*(b*c - a*d)^5*p*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(5*b*d^5)

________________________________________________________________________________________

Rubi [A]  time = 0.844897, antiderivative size = 920, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 14, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.452, Rules used = {2498, 2495, 32, 43, 2514, 2487, 31, 8, 2494, 2394, 2393, 2391, 2390, 2301} \[ -\frac{q^2 r^2 \log ^2(c+d x) (b c-a d)^5}{5 b d^5}-\frac{137 q^2 r^2 \log (c+d x) (b c-a d)^5}{150 b d^5}-\frac{2 p q r^2 \log (c+d x) (b c-a d)^5}{25 b d^5}-\frac{2 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) (b c-a d)^5}{5 b d^5}+\frac{2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^5}{5 b d^5}-\frac{2 p q r^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) (b c-a d)^5}{5 b d^5}+\frac{77 q^2 r^2 x (b c-a d)^4}{150 d^4}+\frac{2 p q r^2 x (b c-a d)^4}{25 d^4}+\frac{2 q (p+q) r^2 x (b c-a d)^4}{5 d^4}-\frac{2 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^4}{5 b d^4}-\frac{b p q r^2 x^2 (b c-a d)^3}{10 d^3}-\frac{77 q^2 r^2 (a+b x)^2 (b c-a d)^3}{300 b d^3}-\frac{p q r^2 (a+b x)^2 (b c-a d)^3}{25 b d^3}-\frac{a p q r^2 x (b c-a d)^3}{5 d^3}+\frac{q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^3}{5 b d^3}+\frac{47 q^2 r^2 (a+b x)^3 (b c-a d)^2}{450 b d^2}+\frac{16 p q r^2 (a+b x)^3 (b c-a d)^2}{225 b d^2}-\frac{2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^2}{15 b d^2}-\frac{9 q^2 r^2 (a+b x)^4 (b c-a d)}{200 b d}-\frac{9 p q r^2 (a+b x)^4 (b c-a d)}{200 b d}+\frac{q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)}{10 b d}+\frac{2 p^2 r^2 (a+b x)^5}{125 b}+\frac{2 q^2 r^2 (a+b x)^5}{125 b}+\frac{4 p q r^2 (a+b x)^5}{125 b}+\frac{(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}-\frac{2 p r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}-\frac{2 q r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2,x]

[Out]

-(a*(b*c - a*d)^3*p*q*r^2*x)/(5*d^3) + (2*(b*c - a*d)^4*p*q*r^2*x)/(25*d^4) + (77*(b*c - a*d)^4*q^2*r^2*x)/(15
0*d^4) + (2*(b*c - a*d)^4*q*(p + q)*r^2*x)/(5*d^4) - (b*(b*c - a*d)^3*p*q*r^2*x^2)/(10*d^3) - ((b*c - a*d)^3*p
*q*r^2*(a + b*x)^2)/(25*b*d^3) - (77*(b*c - a*d)^3*q^2*r^2*(a + b*x)^2)/(300*b*d^3) + (16*(b*c - a*d)^2*p*q*r^
2*(a + b*x)^3)/(225*b*d^2) + (47*(b*c - a*d)^2*q^2*r^2*(a + b*x)^3)/(450*b*d^2) - (9*(b*c - a*d)*p*q*r^2*(a +
b*x)^4)/(200*b*d) - (9*(b*c - a*d)*q^2*r^2*(a + b*x)^4)/(200*b*d) + (2*p^2*r^2*(a + b*x)^5)/(125*b) + (4*p*q*r
^2*(a + b*x)^5)/(125*b) + (2*q^2*r^2*(a + b*x)^5)/(125*b) - (2*(b*c - a*d)^5*p*q*r^2*Log[c + d*x])/(25*b*d^5)
- (137*(b*c - a*d)^5*q^2*r^2*Log[c + d*x])/(150*b*d^5) - (2*(b*c - a*d)^5*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a
*d))]*Log[c + d*x])/(5*b*d^5) - ((b*c - a*d)^5*q^2*r^2*Log[c + d*x]^2)/(5*b*d^5) - (2*(b*c - a*d)^4*q*r*(a + b
*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(5*b*d^4) + ((b*c - a*d)^3*q*r*(a + b*x)^2*Log[e*(f*(a + b*x)^p*(c +
d*x)^q)^r])/(5*b*d^3) - (2*(b*c - a*d)^2*q*r*(a + b*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(15*b*d^2) + (
(b*c - a*d)*q*r*(a + b*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(10*b*d) - (2*p*r*(a + b*x)^5*Log[e*(f*(a +
b*x)^p*(c + d*x)^q)^r])/(25*b) - (2*q*r*(a + b*x)^5*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(25*b) + (2*(b*c - a
*d)^5*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(5*b*d^5) + ((a + b*x)^5*Log[e*(f*(a + b*x)^p*(c
+ d*x)^q)^r]^2)/(5*b) - (2*(b*c - a*d)^5*p*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(5*b*d^5)

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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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 2487

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[((
a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + (Dist[(q*r*s*(b*c - a*d))/b, Int[Log[e*(f*(a + b*x)^p
*(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] - Dist[r*s*(p + q), Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1
), x], x]) /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && NeQ[p + q, 0] && IGtQ[s, 0] &&
LtQ[s, 4]

Rule 31

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 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 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]

Rubi steps

\begin{align*} \int (a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx &=\frac{(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}-\frac{1}{5} (2 p r) \int (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx-\frac{(2 d q r) \int \frac{(a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x} \, dx}{5 b}\\ &=-\frac{2 p r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}+\frac{(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}-\frac{(2 d q r) \int \left (\frac{b (b c-a d)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^5}-\frac{b (b c-a d)^3 (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^4}+\frac{b (b c-a d)^2 (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^3}-\frac{b (b c-a d) (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^2}+\frac{b (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d}+\frac{(-b c+a d)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^5 (c+d x)}\right ) \, dx}{5 b}+\frac{1}{25} \left (2 p^2 r^2\right ) \int (a+b x)^4 \, dx+\frac{\left (2 d p q r^2\right ) \int \frac{(a+b x)^5}{c+d x} \, dx}{25 b}\\ &=\frac{2 p^2 r^2 (a+b x)^5}{125 b}-\frac{2 p r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}+\frac{(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}-\frac{1}{5} (2 q r) \int (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx+\frac{(2 (b c-a d) q r) \int (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{5 d}-\frac{\left (2 (b c-a d)^2 q r\right ) \int (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{5 d^2}+\frac{\left (2 (b c-a d)^3 q r\right ) \int (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{5 d^3}-\frac{\left (2 (b c-a d)^4 q r\right ) \int \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{5 d^4}+\frac{\left (2 (b c-a 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}{5 b d^4}+\frac{\left (2 d p q r^2\right ) \int \left (\frac{b (b c-a d)^4}{d^5}-\frac{b (b c-a d)^3 (a+b x)}{d^4}+\frac{b (b c-a d)^2 (a+b x)^2}{d^3}-\frac{b (b c-a d) (a+b x)^3}{d^2}+\frac{b (a+b x)^4}{d}+\frac{(-b c+a d)^5}{d^5 (c+d x)}\right ) \, dx}{25 b}\\ &=\frac{2 (b c-a d)^4 p q r^2 x}{25 d^4}-\frac{(b c-a d)^3 p q r^2 (a+b x)^2}{25 b d^3}+\frac{2 (b c-a d)^2 p q r^2 (a+b x)^3}{75 b d^2}-\frac{(b c-a d) p q r^2 (a+b x)^4}{50 b d}+\frac{2 p^2 r^2 (a+b x)^5}{125 b}+\frac{2 p q r^2 (a+b x)^5}{125 b}-\frac{2 (b c-a d)^5 p q r^2 \log (c+d x)}{25 b d^5}-\frac{2 (b c-a d)^4 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^4}+\frac{(b c-a d)^3 q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^3}-\frac{2 (b c-a d)^2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{15 b d^2}+\frac{(b c-a d) q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{10 b d}-\frac{2 p r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}-\frac{2 q r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}+\frac{2 (b c-a d)^5 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^5}+\frac{(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}+\frac{1}{25} \left (2 p q r^2\right ) \int (a+b x)^4 \, dx-\frac{\left ((b c-a d) p q r^2\right ) \int (a+b x)^3 \, dx}{10 d}+\frac{\left (2 (b c-a d)^2 p q r^2\right ) \int (a+b x)^2 \, dx}{15 d^2}-\frac{\left ((b c-a d)^3 p q r^2\right ) \int (a+b x) \, dx}{5 d^3}-\frac{\left (2 (b c-a d)^5 p q r^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{5 d^5}+\frac{\left (2 d q^2 r^2\right ) \int \frac{(a+b x)^5}{c+d x} \, dx}{25 b}-\frac{\left ((b c-a d) q^2 r^2\right ) \int \frac{(a+b x)^4}{c+d x} \, dx}{10 b}+\frac{\left (2 (b c-a d)^2 q^2 r^2\right ) \int \frac{(a+b x)^3}{c+d x} \, dx}{15 b d}-\frac{\left ((b c-a d)^3 q^2 r^2\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{5 b d^2}-\frac{\left (2 (b c-a d)^5 q^2 r^2\right ) \int \frac{1}{c+d x} \, dx}{5 b d^4}-\frac{\left (2 (b c-a d)^5 q^2 r^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{5 b d^4}+\frac{\left (2 (b c-a d)^4 q (p+q) r^2\right ) \int 1 \, dx}{5 d^4}\\ &=-\frac{a (b c-a d)^3 p q r^2 x}{5 d^3}+\frac{2 (b c-a d)^4 p q r^2 x}{25 d^4}+\frac{2 (b c-a d)^4 q (p+q) r^2 x}{5 d^4}-\frac{b (b c-a d)^3 p q r^2 x^2}{10 d^3}-\frac{(b c-a d)^3 p q r^2 (a+b x)^2}{25 b d^3}+\frac{16 (b c-a d)^2 p q r^2 (a+b x)^3}{225 b d^2}-\frac{9 (b c-a d) p q r^2 (a+b x)^4}{200 b d}+\frac{2 p^2 r^2 (a+b x)^5}{125 b}+\frac{4 p q r^2 (a+b x)^5}{125 b}-\frac{2 (b c-a d)^5 p q r^2 \log (c+d x)}{25 b d^5}-\frac{2 (b c-a d)^5 q^2 r^2 \log (c+d x)}{5 b d^5}-\frac{2 (b c-a d)^5 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b d^5}-\frac{2 (b c-a d)^4 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^4}+\frac{(b c-a d)^3 q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^3}-\frac{2 (b c-a d)^2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{15 b d^2}+\frac{(b c-a d) q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{10 b d}-\frac{2 p r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}-\frac{2 q r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}+\frac{2 (b c-a d)^5 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^5}+\frac{(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}+\frac{\left (2 (b c-a 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}{5 b d^4}+\frac{\left (2 d q^2 r^2\right ) \int \left (\frac{b (b c-a d)^4}{d^5}-\frac{b (b c-a d)^3 (a+b x)}{d^4}+\frac{b (b c-a d)^2 (a+b x)^2}{d^3}-\frac{b (b c-a d) (a+b x)^3}{d^2}+\frac{b (a+b x)^4}{d}+\frac{(-b c+a d)^5}{d^5 (c+d x)}\right ) \, dx}{25 b}-\frac{\left ((b c-a d) q^2 r^2\right ) \int \left (-\frac{b (b c-a d)^3}{d^4}+\frac{b (b c-a d)^2 (a+b x)}{d^3}-\frac{b (b c-a d) (a+b x)^2}{d^2}+\frac{b (a+b x)^3}{d}+\frac{(-b c+a d)^4}{d^4 (c+d x)}\right ) \, dx}{10 b}+\frac{\left (2 (b c-a d)^2 q^2 r^2\right ) \int \left (\frac{b (b c-a d)^2}{d^3}-\frac{b (b c-a d) (a+b x)}{d^2}+\frac{b (a+b x)^2}{d}+\frac{(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{15 b d}-\frac{\left ((b c-a d)^3 q^2 r^2\right ) \int \left (-\frac{b (b c-a d)}{d^2}+\frac{b (a+b x)}{d}+\frac{(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{5 b d^2}-\frac{\left (2 (b c-a d)^5 q^2 r^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{5 b d^5}\\ &=-\frac{a (b c-a d)^3 p q r^2 x}{5 d^3}+\frac{2 (b c-a d)^4 p q r^2 x}{25 d^4}+\frac{77 (b c-a d)^4 q^2 r^2 x}{150 d^4}+\frac{2 (b c-a d)^4 q (p+q) r^2 x}{5 d^4}-\frac{b (b c-a d)^3 p q r^2 x^2}{10 d^3}-\frac{(b c-a d)^3 p q r^2 (a+b x)^2}{25 b d^3}-\frac{77 (b c-a d)^3 q^2 r^2 (a+b x)^2}{300 b d^3}+\frac{16 (b c-a d)^2 p q r^2 (a+b x)^3}{225 b d^2}+\frac{47 (b c-a d)^2 q^2 r^2 (a+b x)^3}{450 b d^2}-\frac{9 (b c-a d) p q r^2 (a+b x)^4}{200 b d}-\frac{9 (b c-a d) q^2 r^2 (a+b x)^4}{200 b d}+\frac{2 p^2 r^2 (a+b x)^5}{125 b}+\frac{4 p q r^2 (a+b x)^5}{125 b}+\frac{2 q^2 r^2 (a+b x)^5}{125 b}-\frac{2 (b c-a d)^5 p q r^2 \log (c+d x)}{25 b d^5}-\frac{137 (b c-a d)^5 q^2 r^2 \log (c+d x)}{150 b d^5}-\frac{2 (b c-a d)^5 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b d^5}-\frac{(b c-a d)^5 q^2 r^2 \log ^2(c+d x)}{5 b d^5}-\frac{2 (b c-a d)^4 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^4}+\frac{(b c-a d)^3 q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^3}-\frac{2 (b c-a d)^2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{15 b d^2}+\frac{(b c-a d) q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{10 b d}-\frac{2 p r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}-\frac{2 q r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}+\frac{2 (b c-a d)^5 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^5}+\frac{(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}+\frac{\left (2 (b c-a d)^5 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 )}{5 b d^5}\\ &=-\frac{a (b c-a d)^3 p q r^2 x}{5 d^3}+\frac{2 (b c-a d)^4 p q r^2 x}{25 d^4}+\frac{77 (b c-a d)^4 q^2 r^2 x}{150 d^4}+\frac{2 (b c-a d)^4 q (p+q) r^2 x}{5 d^4}-\frac{b (b c-a d)^3 p q r^2 x^2}{10 d^3}-\frac{(b c-a d)^3 p q r^2 (a+b x)^2}{25 b d^3}-\frac{77 (b c-a d)^3 q^2 r^2 (a+b x)^2}{300 b d^3}+\frac{16 (b c-a d)^2 p q r^2 (a+b x)^3}{225 b d^2}+\frac{47 (b c-a d)^2 q^2 r^2 (a+b x)^3}{450 b d^2}-\frac{9 (b c-a d) p q r^2 (a+b x)^4}{200 b d}-\frac{9 (b c-a d) q^2 r^2 (a+b x)^4}{200 b d}+\frac{2 p^2 r^2 (a+b x)^5}{125 b}+\frac{4 p q r^2 (a+b x)^5}{125 b}+\frac{2 q^2 r^2 (a+b x)^5}{125 b}-\frac{2 (b c-a d)^5 p q r^2 \log (c+d x)}{25 b d^5}-\frac{137 (b c-a d)^5 q^2 r^2 \log (c+d x)}{150 b d^5}-\frac{2 (b c-a d)^5 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b d^5}-\frac{(b c-a d)^5 q^2 r^2 \log ^2(c+d x)}{5 b d^5}-\frac{2 (b c-a d)^4 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^4}+\frac{(b c-a d)^3 q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^3}-\frac{2 (b c-a d)^2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{15 b d^2}+\frac{(b c-a d) q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{10 b d}-\frac{2 p r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}-\frac{2 q r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}+\frac{2 (b c-a d)^5 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^5}+\frac{(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}-\frac{2 (b c-a d)^5 p q r^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{5 b d^5}\\ \end{align*}

Mathematica [B]  time = 2.68567, size = 2508, normalized size = 2.73 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2,x]

[Out]

(2*a^5*p*q*r^2)/b + (2*a*b^3*c^4*p*q*r^2)/(5*d^4) - (2*a^2*b^2*c^3*p*q*r^2)/d^3 + (4*a^3*b*c^2*p*q*r^2)/d^2 -
(4*a^4*c*p*q*r^2)/d + (2*a^4*p^2*r^2*x)/25 + (197*a^4*p*q*r^2*x)/150 + (12*b^4*c^4*p*q*r^2*x)/(25*d^4) - (11*a
*b^3*c^3*p*q*r^2*x)/(5*d^3) + (59*a^2*b^2*c^2*p*q*r^2*x)/(15*d^2) - (101*a^3*b*c*p*q*r^2*x)/(30*d) + 2*a^4*q^2
*r^2*x + (137*b^4*c^4*q^2*r^2*x)/(150*d^4) - (25*a*b^3*c^3*q^2*r^2*x)/(6*d^3) + (22*a^2*b^2*c^2*q^2*r^2*x)/(3*
d^2) - (6*a^3*b*c*q^2*r^2*x)/d + (4*a^3*b*p^2*r^2*x^2)/25 + (283*a^3*b*p*q*r^2*x^2)/300 - (7*b^4*c^3*p*q*r^2*x
^2)/(50*d^3) + (19*a*b^3*c^2*p*q*r^2*x^2)/(30*d^2) - (67*a^2*b^2*c*p*q*r^2*x^2)/(60*d) + a^3*b*q^2*r^2*x^2 - (
77*b^4*c^3*q^2*r^2*x^2)/(300*d^3) + (13*a*b^3*c^2*q^2*r^2*x^2)/(12*d^2) - (5*a^2*b^2*c*q^2*r^2*x^2)/(3*d) + (4
*a^2*b^2*p^2*r^2*x^3)/25 + (257*a^2*b^2*p*q*r^2*x^3)/450 + (16*b^4*c^2*p*q*r^2*x^3)/(225*d^2) - (29*a*b^3*c*p*
q*r^2*x^3)/(90*d) + (4*a^2*b^2*q^2*r^2*x^3)/9 + (47*b^4*c^2*q^2*r^2*x^3)/(450*d^2) - (7*a*b^3*c*q^2*r^2*x^3)/(
18*d) + (2*a*b^3*p^2*r^2*x^4)/25 + (41*a*b^3*p*q*r^2*x^4)/200 - (9*b^4*c*p*q*r^2*x^4)/(200*d) + (a*b^3*q^2*r^2
*x^4)/8 - (9*b^4*c*q^2*r^2*x^4)/(200*d) + (2*b^4*p^2*r^2*x^5)/125 + (4*b^4*p*q*r^2*x^5)/125 + (2*b^4*q^2*r^2*x
^5)/125 - (a^5*p^2*r^2*Log[a + b*x]^2)/(5*b) + (2*a^5*p*q*r^2*Log[c + d*x])/b - (2*b^4*c^5*p*q*r^2*Log[c + d*x
])/(25*d^5) + (2*a*b^3*c^4*p*q*r^2*Log[c + d*x])/(5*d^4) - (4*a^2*b^2*c^3*p*q*r^2*Log[c + d*x])/(5*d^3) + (4*a
^3*b*c^2*p*q*r^2*Log[c + d*x])/(5*d^2) - (2*a^4*c*p*q*r^2*Log[c + d*x])/(5*d) - (137*b^4*c^5*q^2*r^2*Log[c + d
*x])/(150*d^5) + (25*a*b^3*c^4*q^2*r^2*Log[c + d*x])/(6*d^4) - (22*a^2*b^2*c^3*q^2*r^2*Log[c + d*x])/(3*d^3) +
(6*a^3*b*c^2*q^2*r^2*Log[c + d*x])/d^2 - (2*a^4*c*q^2*r^2*Log[c + d*x])/d - (b^4*c^5*q^2*r^2*Log[c + d*x]^2)/
(5*d^5) + (a*b^3*c^4*q^2*r^2*Log[c + d*x]^2)/d^4 - (2*a^2*b^2*c^3*q^2*r^2*Log[c + d*x]^2)/d^3 + (2*a^3*b*c^2*q
^2*r^2*Log[c + d*x]^2)/d^2 - (a^4*c*q^2*r^2*Log[c + d*x]^2)/d - (2*a^5*p*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r
])/b - (2*a^4*p*r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/5 - 2*a^4*q*r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]
- (2*b^4*c^4*q*r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(5*d^4) + (2*a*b^3*c^3*q*r*x*Log[e*(f*(a + b*x)^p*(c
+ d*x)^q)^r])/d^3 - (4*a^2*b^2*c^2*q*r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d^2 + (4*a^3*b*c*q*r*x*Log[e*(
f*(a + b*x)^p*(c + d*x)^q)^r])/d - (4*a^3*b*p*r*x^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/5 - 2*a^3*b*q*r*x^2*
Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + (b^4*c^3*q*r*x^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(5*d^3) - (a*b^3
*c^2*q*r*x^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d^2 + (2*a^2*b^2*c*q*r*x^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q
)^r])/d - (4*a^2*b^2*p*r*x^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/5 - (4*a^2*b^2*q*r*x^3*Log[e*(f*(a + b*x)^p
*(c + d*x)^q)^r])/3 - (2*b^4*c^2*q*r*x^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(15*d^2) + (2*a*b^3*c*q*r*x^3*L
og[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3*d) - (2*a*b^3*p*r*x^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/5 - (a*b^3
*q*r*x^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/2 + (b^4*c*q*r*x^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(10*d)
- (2*b^4*p*r*x^5*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/25 - (2*b^4*q*r*x^5*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^
r])/25 + (2*b^4*c^5*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(5*d^5) - (2*a*b^3*c^4*q*r*Log[c +
d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d^4 + (4*a^2*b^2*c^3*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x
)^q)^r])/d^3 - (4*a^3*b*c^2*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d^2 + (2*a^4*c*q*r*Log[c +
d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d + a^4*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2 + 2*a^3*b*x^2*Log[
e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2 + 2*a^2*b^2*x^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2 + a*b^3*x^4*Log[e*(f
*(a + b*x)^p*(c + d*x)^q)^r]^2 + (b^4*x^5*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)/5 + (p*r*Log[a + b*x]*(a*d*(
a^4*d^4*(288*p - 137*q) - 60*b^4*c^4*q + 270*a*b^3*c^3*d*q - 470*a^2*b^2*c^2*d^2*q + 385*a^3*b*c*d^3*q)*r - 60
*b*c*(b^4*c^4 - 5*a*b^3*c^3*d + 10*a^2*b^2*c^2*d^2 - 10*a^3*b*c*d^3 + 5*a^4*d^4)*q*r*Log[c + d*x] + 60*(b*c -
a*d)^5*q*r*Log[(b*(c + d*x))/(b*c - a*d)] + 60*a^5*d^5*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(150*b*d^5) + (2
*(b*c - a*d)^5*p*q*r^2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)])/(5*b*d^5)

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Maple [F]  time = 0.414, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{4} \left ( \ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^4*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x)

[Out]

int((b*x+a)^4*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x)

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Maxima [A]  time = 1.57889, size = 1918, normalized size = 2.08 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x, algorithm="maxima")

[Out]

1/5*(b^4*x^5 + 5*a*b^3*x^4 + 10*a^2*b^2*x^3 + 10*a^3*b*x^2 + 5*a^4*x)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2 +
1/150*(60*a^5*f*p*log(b*x + a)/b - (12*b^4*d^4*f*(p + q)*x^5 + 15*(a*b^3*d^4*f*(4*p + 5*q) - b^4*c*d^3*f*q)*x
^4 + 20*(2*a^2*b^2*d^4*f*(3*p + 5*q) + b^4*c^2*d^2*f*q - 5*a*b^3*c*d^3*f*q)*x^3 + 30*(2*a^3*b*d^4*f*(2*p + 5*q
) - b^4*c^3*d*f*q + 5*a*b^3*c^2*d^2*f*q - 10*a^2*b^2*c*d^3*f*q)*x^2 + 60*(a^4*d^4*f*(p + 5*q) + b^4*c^4*f*q -
5*a*b^3*c^3*d*f*q + 10*a^2*b^2*c^2*d^2*f*q - 10*a^3*b*c*d^3*f*q)*x)/d^4 + 60*(b^4*c^5*f*q - 5*a*b^3*c^4*d*f*q
+ 10*a^2*b^2*c^3*d^2*f*q - 10*a^3*b*c^2*d^3*f*q + 5*a^4*c*d^4*f*q)*log(d*x + c)/d^5)*r*log(((b*x + a)^p*(d*x +
c)^q*f)^r*e)/f - 1/9000*r^2*(60*((12*p*q + 137*q^2)*b^4*c^5*f^2 - 5*(12*p*q + 125*q^2)*a*b^3*c^4*d*f^2 + 20*(
6*p*q + 55*q^2)*a^2*b^2*c^3*d^2*f^2 - 60*(2*p*q + 15*q^2)*a^3*b*c^2*d^3*f^2 + 60*(p*q + 5*q^2)*a^4*c*d^4*f^2)*
log(d*x + c)/d^5 - 3600*(b^5*c^5*f^2*p*q - 5*a*b^4*c^4*d*f^2*p*q + 10*a^2*b^3*c^3*d^2*f^2*p*q - 10*a^3*b^2*c^2
*d^3*f^2*p*q + 5*a^4*b*c*d^4*f^2*p*q - a^5*d^5*f^2*p*q)*(log(b*x + a)*log((b*d*x + a*d)/(b*c - a*d) + 1) + dil
og(-(b*d*x + a*d)/(b*c - a*d)))/(b*d^5) - (144*(p^2 + 2*p*q + q^2)*b^5*d^5*f^2*x^5 - 1800*a^5*d^5*f^2*p^2*log(
b*x + a)^2 - 45*(9*(p*q + q^2)*b^5*c*d^4*f^2 - (16*p^2 + 41*p*q + 25*q^2)*a*b^4*d^5*f^2)*x^4 + 20*((32*p*q + 4
7*q^2)*b^5*c^2*d^3*f^2 - 5*(29*p*q + 35*q^2)*a*b^4*c*d^4*f^2 + (72*p^2 + 257*p*q + 200*q^2)*a^2*b^3*d^5*f^2)*x
^3 - 30*(7*(6*p*q + 11*q^2)*b^5*c^3*d^2*f^2 - 5*(38*p*q + 65*q^2)*a*b^4*c^2*d^3*f^2 + 5*(67*p*q + 100*q^2)*a^2
*b^3*c*d^4*f^2 - (48*p^2 + 283*p*q + 300*q^2)*a^3*b^2*d^5*f^2)*x^2 - 3600*(b^5*c^5*f^2*p*q - 5*a*b^4*c^4*d*f^2
*p*q + 10*a^2*b^3*c^3*d^2*f^2*p*q - 10*a^3*b^2*c^2*d^3*f^2*p*q + 5*a^4*b*c*d^4*f^2*p*q)*log(b*x + a)*log(d*x +
c) - 1800*(b^5*c^5*f^2*q^2 - 5*a*b^4*c^4*d*f^2*q^2 + 10*a^2*b^3*c^3*d^2*f^2*q^2 - 10*a^3*b^2*c^2*d^3*f^2*q^2
+ 5*a^4*b*c*d^4*f^2*q^2)*log(d*x + c)^2 + 60*((72*p*q + 137*q^2)*b^5*c^4*d*f^2 - 5*(66*p*q + 125*q^2)*a*b^4*c^
3*d^2*f^2 + 10*(59*p*q + 110*q^2)*a^2*b^3*c^2*d^3*f^2 - 5*(101*p*q + 180*q^2)*a^3*b^2*c*d^4*f^2 + (12*p^2 + 19
7*p*q + 300*q^2)*a^4*b*d^5*f^2)*x - 60*(60*a*b^4*c^4*d*f^2*p*q - 270*a^2*b^3*c^3*d^2*f^2*p*q + 470*a^3*b^2*c^2
*d^3*f^2*p*q - 385*a^4*b*c*d^4*f^2*p*q + (12*p^2 + 137*p*q)*a^5*d^5*f^2)*log(b*x + a))/(b*d^5))/f^2

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x, algorithm="fricas")

[Out]

integral((b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x + a^4)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^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((b*x+a)**4*ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{4} \log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x, algorithm="giac")

[Out]

integrate((b*x + a)^4*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2, x)