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

Optimal. Leaf size=686 \[ -\frac{2 p q r^2 (b c-a d)^3 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{3 b d^3}+\frac{2 q r (b c-a d)^3 \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d^3}-\frac{2 q r (a+b x) (b c-a d)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d^2}+\frac{2 p q r^2 x (b c-a d)^2}{9 d^2}+\frac{2 q r^2 x (p+q) (b c-a d)^2}{3 d^2}-\frac{2 p q r^2 (b c-a d)^3 \log (c+d x)}{9 b d^3}-\frac{2 p q r^2 (b c-a d)^3 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{3 b d^3}+\frac{5 q^2 r^2 x (b c-a d)^2}{9 d^2}-\frac{q^2 r^2 (b c-a d)^3 \log ^2(c+d x)}{3 b d^3}-\frac{11 q^2 r^2 (b c-a d)^3 \log (c+d x)}{9 b d^3}+\frac{(a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}+\frac{q r (a+b x)^2 (b c-a d) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d}-\frac{2 p r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b}-\frac{2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b}-\frac{b p q r^2 x^2 (b c-a d)}{6 d}-\frac{p q r^2 (a+b x)^2 (b c-a d)}{9 b d}-\frac{a p q r^2 x (b c-a d)}{3 d}-\frac{5 q^2 r^2 (a+b x)^2 (b c-a d)}{18 b d}+\frac{2 p^2 r^2 (a+b x)^3}{27 b}+\frac{4 p q r^2 (a+b x)^3}{27 b}+\frac{2 q^2 r^2 (a+b x)^3}{27 b} \]

[Out]

-(a*(b*c - a*d)*p*q*r^2*x)/(3*d) + (2*(b*c - a*d)^2*p*q*r^2*x)/(9*d^2) + (5*(b*c - a*d)^2*q^2*r^2*x)/(9*d^2) +
(2*(b*c - a*d)^2*q*(p + q)*r^2*x)/(3*d^2) - (b*(b*c - a*d)*p*q*r^2*x^2)/(6*d) - ((b*c - a*d)*p*q*r^2*(a + b*x
)^2)/(9*b*d) - (5*(b*c - a*d)*q^2*r^2*(a + b*x)^2)/(18*b*d) + (2*p^2*r^2*(a + b*x)^3)/(27*b) + (4*p*q*r^2*(a +
b*x)^3)/(27*b) + (2*q^2*r^2*(a + b*x)^3)/(27*b) - (2*(b*c - a*d)^3*p*q*r^2*Log[c + d*x])/(9*b*d^3) - (11*(b*c
- a*d)^3*q^2*r^2*Log[c + d*x])/(9*b*d^3) - (2*(b*c - a*d)^3*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c +
d*x])/(3*b*d^3) - ((b*c - a*d)^3*q^2*r^2*Log[c + d*x]^2)/(3*b*d^3) - (2*(b*c - a*d)^2*q*r*(a + b*x)*Log[e*(f*
(a + b*x)^p*(c + d*x)^q)^r])/(3*b*d^2) + ((b*c - a*d)*q*r*(a + b*x)^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3
*b*d) - (2*p*r*(a + b*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(9*b) - (2*q*r*(a + b*x)^3*Log[e*(f*(a + b*x)
^p*(c + d*x)^q)^r])/(9*b) + (2*(b*c - a*d)^3*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3*b*d^3)
+ ((a + b*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)/(3*b) - (2*(b*c - a*d)^3*p*q*r^2*PolyLog[2, (b*(c + d*x
))/(b*c - a*d)])/(3*b*d^3)

________________________________________________________________________________________

Rubi [A]  time = 0.532864, antiderivative size = 686, normalized size of antiderivative = 1., number of steps used = 24, 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} \[ -\frac{2 p q r^2 (b c-a d)^3 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{3 b d^3}+\frac{2 q r (b c-a d)^3 \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d^3}-\frac{2 q r (a+b x) (b c-a d)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d^2}+\frac{2 p q r^2 x (b c-a d)^2}{9 d^2}+\frac{2 q r^2 x (p+q) (b c-a d)^2}{3 d^2}-\frac{2 p q r^2 (b c-a d)^3 \log (c+d x)}{9 b d^3}-\frac{2 p q r^2 (b c-a d)^3 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{3 b d^3}+\frac{5 q^2 r^2 x (b c-a d)^2}{9 d^2}-\frac{q^2 r^2 (b c-a d)^3 \log ^2(c+d x)}{3 b d^3}-\frac{11 q^2 r^2 (b c-a d)^3 \log (c+d x)}{9 b d^3}+\frac{(a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}+\frac{q r (a+b x)^2 (b c-a d) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d}-\frac{2 p r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b}-\frac{2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b}-\frac{b p q r^2 x^2 (b c-a d)}{6 d}-\frac{p q r^2 (a+b x)^2 (b c-a d)}{9 b d}-\frac{a p q r^2 x (b c-a d)}{3 d}-\frac{5 q^2 r^2 (a+b x)^2 (b c-a d)}{18 b d}+\frac{2 p^2 r^2 (a+b x)^3}{27 b}+\frac{4 p q r^2 (a+b x)^3}{27 b}+\frac{2 q^2 r^2 (a+b x)^3}{27 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*(b*c - a*d)*p*q*r^2*x)/(3*d) + (2*(b*c - a*d)^2*p*q*r^2*x)/(9*d^2) + (5*(b*c - a*d)^2*q^2*r^2*x)/(9*d^2) +
(2*(b*c - a*d)^2*q*(p + q)*r^2*x)/(3*d^2) - (b*(b*c - a*d)*p*q*r^2*x^2)/(6*d) - ((b*c - a*d)*p*q*r^2*(a + b*x
)^2)/(9*b*d) - (5*(b*c - a*d)*q^2*r^2*(a + b*x)^2)/(18*b*d) + (2*p^2*r^2*(a + b*x)^3)/(27*b) + (4*p*q*r^2*(a +
b*x)^3)/(27*b) + (2*q^2*r^2*(a + b*x)^3)/(27*b) - (2*(b*c - a*d)^3*p*q*r^2*Log[c + d*x])/(9*b*d^3) - (11*(b*c
- a*d)^3*q^2*r^2*Log[c + d*x])/(9*b*d^3) - (2*(b*c - a*d)^3*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c +
d*x])/(3*b*d^3) - ((b*c - a*d)^3*q^2*r^2*Log[c + d*x]^2)/(3*b*d^3) - (2*(b*c - a*d)^2*q*r*(a + b*x)*Log[e*(f*
(a + b*x)^p*(c + d*x)^q)^r])/(3*b*d^2) + ((b*c - a*d)*q*r*(a + b*x)^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3
*b*d) - (2*p*r*(a + b*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(9*b) - (2*q*r*(a + b*x)^3*Log[e*(f*(a + b*x)
^p*(c + d*x)^q)^r])/(9*b) + (2*(b*c - a*d)^3*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3*b*d^3)
+ ((a + b*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)/(3*b) - (2*(b*c - a*d)^3*p*q*r^2*PolyLog[2, (b*(c + d*x
))/(b*c - a*d)])/(3*b*d^3)

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)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx &=\frac{(a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}-\frac{1}{3} (2 p r) \int (a+b x)^2 \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)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x} \, dx}{3 b}\\ &=-\frac{2 p r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b}+\frac{(a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}-\frac{(2 d q r) \int \left (\frac{b (b c-a d)^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) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^2}+\frac{b (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d}+\frac{(-b c+a d)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^3 (c+d x)}\right ) \, dx}{3 b}+\frac{1}{9} \left (2 p^2 r^2\right ) \int (a+b x)^2 \, dx+\frac{\left (2 d p q r^2\right ) \int \frac{(a+b x)^3}{c+d x} \, dx}{9 b}\\ &=\frac{2 p^2 r^2 (a+b x)^3}{27 b}-\frac{2 p r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b}+\frac{(a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}-\frac{1}{3} (2 q r) \int (a+b x)^2 \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) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{3 d}-\frac{\left (2 (b c-a d)^2 q r\right ) \int \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{3 d^2}+\frac{\left (2 (b c-a 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}{3 b d^2}+\frac{\left (2 d p q 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}{9 b}\\ &=\frac{2 (b c-a d)^2 p q r^2 x}{9 d^2}-\frac{(b c-a d) p q r^2 (a+b x)^2}{9 b d}+\frac{2 p^2 r^2 (a+b x)^3}{27 b}+\frac{2 p q r^2 (a+b x)^3}{27 b}-\frac{2 (b c-a d)^3 p q r^2 \log (c+d x)}{9 b d^3}-\frac{2 (b c-a d)^2 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d^2}+\frac{(b c-a d) q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d}-\frac{2 p r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b}-\frac{2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b}+\frac{2 (b c-a d)^3 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d^3}+\frac{(a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}+\frac{1}{9} \left (2 p q r^2\right ) \int (a+b x)^2 \, dx-\frac{\left ((b c-a d) p q r^2\right ) \int (a+b x) \, dx}{3 d}-\frac{\left (2 (b c-a d)^3 p q r^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{3 d^3}+\frac{\left (2 d q^2 r^2\right ) \int \frac{(a+b x)^3}{c+d x} \, dx}{9 b}-\frac{\left ((b c-a d) q^2 r^2\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{3 b}-\frac{\left (2 (b c-a d)^3 q^2 r^2\right ) \int \frac{1}{c+d x} \, dx}{3 b d^2}-\frac{\left (2 (b c-a d)^3 q^2 r^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{3 b d^2}+\frac{\left (2 (b c-a d)^2 q (p+q) r^2\right ) \int 1 \, dx}{3 d^2}\\ &=-\frac{a (b c-a d) p q r^2 x}{3 d}+\frac{2 (b c-a d)^2 p q r^2 x}{9 d^2}+\frac{2 (b c-a d)^2 q (p+q) r^2 x}{3 d^2}-\frac{b (b c-a d) p q r^2 x^2}{6 d}-\frac{(b c-a d) p q r^2 (a+b x)^2}{9 b d}+\frac{2 p^2 r^2 (a+b x)^3}{27 b}+\frac{4 p q r^2 (a+b x)^3}{27 b}-\frac{2 (b c-a d)^3 p q r^2 \log (c+d x)}{9 b d^3}-\frac{2 (b c-a d)^3 q^2 r^2 \log (c+d x)}{3 b d^3}-\frac{2 (b c-a d)^3 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b d^3}-\frac{2 (b c-a d)^2 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d^2}+\frac{(b c-a d) q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d}-\frac{2 p r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b}-\frac{2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b}+\frac{2 (b c-a d)^3 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d^3}+\frac{(a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}+\frac{\left (2 (b c-a 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}{3 b d^2}+\frac{\left (2 d 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}{9 b}-\frac{\left ((b c-a d) 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}{3 b}-\frac{\left (2 (b c-a d)^3 q^2 r^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{3 b d^3}\\ &=-\frac{a (b c-a d) p q r^2 x}{3 d}+\frac{2 (b c-a d)^2 p q r^2 x}{9 d^2}+\frac{5 (b c-a d)^2 q^2 r^2 x}{9 d^2}+\frac{2 (b c-a d)^2 q (p+q) r^2 x}{3 d^2}-\frac{b (b c-a d) p q r^2 x^2}{6 d}-\frac{(b c-a d) p q r^2 (a+b x)^2}{9 b d}-\frac{5 (b c-a d) q^2 r^2 (a+b x)^2}{18 b d}+\frac{2 p^2 r^2 (a+b x)^3}{27 b}+\frac{4 p q r^2 (a+b x)^3}{27 b}+\frac{2 q^2 r^2 (a+b x)^3}{27 b}-\frac{2 (b c-a d)^3 p q r^2 \log (c+d x)}{9 b d^3}-\frac{11 (b c-a d)^3 q^2 r^2 \log (c+d x)}{9 b d^3}-\frac{2 (b c-a d)^3 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b d^3}-\frac{(b c-a d)^3 q^2 r^2 \log ^2(c+d x)}{3 b d^3}-\frac{2 (b c-a d)^2 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d^2}+\frac{(b c-a d) q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d}-\frac{2 p r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b}-\frac{2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b}+\frac{2 (b c-a d)^3 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d^3}+\frac{(a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}+\frac{\left (2 (b c-a d)^3 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 )}{3 b d^3}\\ &=-\frac{a (b c-a d) p q r^2 x}{3 d}+\frac{2 (b c-a d)^2 p q r^2 x}{9 d^2}+\frac{5 (b c-a d)^2 q^2 r^2 x}{9 d^2}+\frac{2 (b c-a d)^2 q (p+q) r^2 x}{3 d^2}-\frac{b (b c-a d) p q r^2 x^2}{6 d}-\frac{(b c-a d) p q r^2 (a+b x)^2}{9 b d}-\frac{5 (b c-a d) q^2 r^2 (a+b x)^2}{18 b d}+\frac{2 p^2 r^2 (a+b x)^3}{27 b}+\frac{4 p q r^2 (a+b x)^3}{27 b}+\frac{2 q^2 r^2 (a+b x)^3}{27 b}-\frac{2 (b c-a d)^3 p q r^2 \log (c+d x)}{9 b d^3}-\frac{11 (b c-a d)^3 q^2 r^2 \log (c+d x)}{9 b d^3}-\frac{2 (b c-a d)^3 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b d^3}-\frac{(b c-a d)^3 q^2 r^2 \log ^2(c+d x)}{3 b d^3}-\frac{2 (b c-a d)^2 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d^2}+\frac{(b c-a d) q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d}-\frac{2 p r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b}-\frac{2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b}+\frac{2 (b c-a d)^3 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d^3}+\frac{(a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}-\frac{2 (b c-a d)^3 p q r^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{3 b d^3}\\ \end{align*}

Mathematica [A]  time = 1.12677, size = 1211, normalized size = 1.77 \[ \frac{1}{54} \left (\frac{108 p q r^2 a^3}{b}-\frac{18 p^2 r^2 \log ^2(a+b x) a^3}{b}+\frac{108 p q r^2 \log (c+d x) a^3}{b}-\frac{108 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) a^3}{b}-\frac{108 c p q r^2 a^2}{d}-\frac{54 c q^2 r^2 \log ^2(c+d x) a^2}{d}+54 x \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) a^2+12 p^2 r^2 x a^2+108 q^2 r^2 x a^2+102 p q r^2 x a^2-\frac{108 c q^2 r^2 \log (c+d x) a^2}{d}-\frac{36 c p q r^2 \log (c+d x) a^2}{d}-36 p r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) a^2-108 q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) a^2+\frac{108 c q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) a^2}{d}+\frac{36 b c^2 p q r^2 a}{d^2}+12 b p^2 r^2 x^2 a+27 b q^2 r^2 x^2 a+39 b p q r^2 x^2 a+\frac{54 b c^2 q^2 r^2 \log ^2(c+d x) a}{d^2}+54 b x^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) a-\frac{162 b c q^2 r^2 x a}{d}-\frac{126 b c p q r^2 x a}{d}+\frac{162 b c^2 q^2 r^2 \log (c+d x) a}{d^2}+\frac{36 b c^2 p q r^2 \log (c+d x) a}{d^2}-36 b p r x^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) a-54 b q r x^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) a+\frac{108 b c q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) a}{d}-\frac{108 b c^2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) a}{d^2}+4 b^2 p^2 r^2 x^3+4 b^2 q^2 r^2 x^3+8 b^2 p q r^2 x^3-\frac{15 b^2 c q^2 r^2 x^2}{d}-\frac{15 b^2 c p q r^2 x^2}{d}-\frac{18 b^2 c^3 q^2 r^2 \log ^2(c+d x)}{d^3}+18 b^2 x^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\frac{66 b^2 c^2 q^2 r^2 x}{d^2}+\frac{48 b^2 c^2 p q r^2 x}{d^2}-\frac{66 b^2 c^3 q^2 r^2 \log (c+d x)}{d^3}-\frac{12 b^2 c^3 p q r^2 \log (c+d x)}{d^3}-12 b^2 p r x^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-12 b^2 q r x^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\frac{18 b^2 c q r x^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d}-\frac{36 b^2 c^2 q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^2}+\frac{36 b^2 c^3 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^3}+\frac{6 p r \log (a+b x) \left (6 a^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) d^3+a \left (-6 b^2 q c^2+15 a b d q c+a^2 d^2 (16 p-11 q)\right ) r d-6 b c \left (b^2 c^2-3 a b d c+3 a^2 d^2\right ) q r \log (c+d x)+6 (b c-a d)^3 q r \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )}{b d^3}+\frac{36 (b c-a d)^3 p q r^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )}{b d^3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((108*a^3*p*q*r^2)/b + (36*a*b*c^2*p*q*r^2)/d^2 - (108*a^2*c*p*q*r^2)/d + 12*a^2*p^2*r^2*x + 102*a^2*p*q*r^2*x
+ (48*b^2*c^2*p*q*r^2*x)/d^2 - (126*a*b*c*p*q*r^2*x)/d + 108*a^2*q^2*r^2*x + (66*b^2*c^2*q^2*r^2*x)/d^2 - (16
2*a*b*c*q^2*r^2*x)/d + 12*a*b*p^2*r^2*x^2 + 39*a*b*p*q*r^2*x^2 - (15*b^2*c*p*q*r^2*x^2)/d + 27*a*b*q^2*r^2*x^2
- (15*b^2*c*q^2*r^2*x^2)/d + 4*b^2*p^2*r^2*x^3 + 8*b^2*p*q*r^2*x^3 + 4*b^2*q^2*r^2*x^3 - (18*a^3*p^2*r^2*Log[
a + b*x]^2)/b + (108*a^3*p*q*r^2*Log[c + d*x])/b - (12*b^2*c^3*p*q*r^2*Log[c + d*x])/d^3 + (36*a*b*c^2*p*q*r^2
*Log[c + d*x])/d^2 - (36*a^2*c*p*q*r^2*Log[c + d*x])/d - (66*b^2*c^3*q^2*r^2*Log[c + d*x])/d^3 + (162*a*b*c^2*
q^2*r^2*Log[c + d*x])/d^2 - (108*a^2*c*q^2*r^2*Log[c + d*x])/d - (18*b^2*c^3*q^2*r^2*Log[c + d*x]^2)/d^3 + (54
*a*b*c^2*q^2*r^2*Log[c + d*x]^2)/d^2 - (54*a^2*c*q^2*r^2*Log[c + d*x]^2)/d - (108*a^3*p*r*Log[e*(f*(a + b*x)^p
*(c + d*x)^q)^r])/b - 36*a^2*p*r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - 108*a^2*q*r*x*Log[e*(f*(a + b*x)^p*(
c + d*x)^q)^r] - (36*b^2*c^2*q*r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d^2 + (108*a*b*c*q*r*x*Log[e*(f*(a +
b*x)^p*(c + d*x)^q)^r])/d - 36*a*b*p*r*x^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - 54*a*b*q*r*x^2*Log[e*(f*(a +
b*x)^p*(c + d*x)^q)^r] + (18*b^2*c*q*r*x^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d - 12*b^2*p*r*x^3*Log[e*(f*
(a + b*x)^p*(c + d*x)^q)^r] - 12*b^2*q*r*x^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + (36*b^2*c^3*q*r*Log[c + d*
x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d^3 - (108*a*b*c^2*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)
^r])/d^2 + (108*a^2*c*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d + 54*a^2*x*Log[e*(f*(a + b*x)^p
*(c + d*x)^q)^r]^2 + 54*a*b*x^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2 + 18*b^2*x^3*Log[e*(f*(a + b*x)^p*(c +
d*x)^q)^r]^2 + (6*p*r*Log[a + b*x]*(a*d*(a^2*d^2*(16*p - 11*q) - 6*b^2*c^2*q + 15*a*b*c*d*q)*r - 6*b*c*(b^2*c^
2 - 3*a*b*c*d + 3*a^2*d^2)*q*r*Log[c + d*x] + 6*(b*c - a*d)^3*q*r*Log[(b*(c + d*x))/(b*c - a*d)] + 6*a^3*d^3*L
og[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(b*d^3) + (36*(b*c - a*d)^3*p*q*r^2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a
*d)])/(b*d^3))/54

________________________________________________________________________________________

Maple [F]  time = 0.404, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{2} \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)^2*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.42788, size = 1038, normalized size = 1.51 \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)^2*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x, algorithm="maxima")

[Out]

1/3*(b^2*x^3 + 3*a*b*x^2 + 3*a^2*x)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2 + 1/9*(6*a^3*f*p*log(b*x + a)/b - (
2*b^2*d^2*f*(p + q)*x^3 + 3*(a*b*d^2*f*(2*p + 3*q) - b^2*c*d*f*q)*x^2 + 6*(a^2*d^2*f*(p + 3*q) + b^2*c^2*f*q -
3*a*b*c*d*f*q)*x)/d^2 + 6*(b^2*c^3*f*q - 3*a*b*c^2*d*f*q + 3*a^2*c*d^2*f*q)*log(d*x + c)/d^3)*r*log(((b*x + a
)^p*(d*x + c)^q*f)^r*e)/f - 1/54*r^2*(6*((2*p*q + 11*q^2)*b^2*c^3*f^2 - 3*(2*p*q + 9*q^2)*a*b*c^2*d*f^2 + 6*(p
*q + 3*q^2)*a^2*c*d^2*f^2)*log(d*x + c)/d^3 - 36*(b^3*c^3*f^2*p*q - 3*a*b^2*c^2*d*f^2*p*q + 3*a^2*b*c*d^2*f^2*
p*q - a^3*d^3*f^2*p*q)*(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)))/(
b*d^3) - (4*(p^2 + 2*p*q + q^2)*b^3*d^3*f^2*x^3 - 18*a^3*d^3*f^2*p^2*log(b*x + a)^2 - 3*(5*(p*q + q^2)*b^3*c*d
^2*f^2 - (4*p^2 + 13*p*q + 9*q^2)*a*b^2*d^3*f^2)*x^2 - 36*(b^3*c^3*f^2*p*q - 3*a*b^2*c^2*d*f^2*p*q + 3*a^2*b*c
*d^2*f^2*p*q)*log(b*x + a)*log(d*x + c) - 18*(b^3*c^3*f^2*q^2 - 3*a*b^2*c^2*d*f^2*q^2 + 3*a^2*b*c*d^2*f^2*q^2)
*log(d*x + c)^2 + 6*((8*p*q + 11*q^2)*b^3*c^2*d*f^2 - 3*(7*p*q + 9*q^2)*a*b^2*c*d^2*f^2 + (2*p^2 + 17*p*q + 18
*q^2)*a^2*b*d^3*f^2)*x - 6*(6*a*b^2*c^2*d*f^2*p*q - 15*a^2*b*c*d^2*f^2*p*q + (2*p^2 + 11*p*q)*a^3*d^3*f^2)*log
(b*x + a))/(b*d^3))/f^2

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

[Out]

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

[Out]

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