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

Optimal. Leaf size=540 \[ \frac{p q r^2 (b c-a d)^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b d^2}-\frac{q r (b c-a d)^2 \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d^2}+\frac{p q r^2 (b c-a d)^2 \log (c+d x)}{2 b d^2}+\frac{p q r^2 (b c-a d)^2 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b d^2}+\frac{q^2 r^2 (b c-a d)^2 \log ^2(c+d x)}{2 b d^2}+\frac{3 q^2 r^2 (b c-a d)^2 \log (c+d x)}{2 b d^2}+\frac{(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}+\frac{q r (a+b x) (b c-a d) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}-\frac{p r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac{q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac{p q r^2 x (b c-a d)}{2 d}-\frac{q r^2 x (p+q) (b c-a d)}{d}-\frac{q^2 r^2 x (b c-a d)}{2 d}+\frac{p q r^2 (a+b x)^2}{4 b}+\frac{q^2 r^2 (a+b x)^2}{4 b}+\frac{1}{2} a p^2 r^2 x+\frac{1}{2} a p q r^2 x+\frac{1}{4} b p^2 r^2 x^2+\frac{1}{4} b p q r^2 x^2 \]

[Out]

(a*p^2*r^2*x)/2 + (a*p*q*r^2*x)/2 - ((b*c - a*d)*p*q*r^2*x)/(2*d) - ((b*c - a*d)*q^2*r^2*x)/(2*d) - ((b*c - a*
d)*q*(p + q)*r^2*x)/d + (b*p^2*r^2*x^2)/4 + (b*p*q*r^2*x^2)/4 + (p*q*r^2*(a + b*x)^2)/(4*b) + (q^2*r^2*(a + b*
x)^2)/(4*b) + ((b*c - a*d)^2*p*q*r^2*Log[c + d*x])/(2*b*d^2) + (3*(b*c - a*d)^2*q^2*r^2*Log[c + d*x])/(2*b*d^2
) + ((b*c - a*d)^2*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(b*d^2) + ((b*c - a*d)^2*q^2*r^2*Lo
g[c + d*x]^2)/(2*b*d^2) + ((b*c - a*d)*q*r*(a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(b*d) - (p*r*(a + b
*x)^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*b) - (q*r*(a + b*x)^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*
b) - ((b*c - a*d)^2*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(b*d^2) + ((a + b*x)^2*Log[e*(f*(a
+ b*x)^p*(c + d*x)^q)^r]^2)/(2*b) + ((b*c - a*d)^2*p*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(b*d^2)

________________________________________________________________________________________

Rubi [A]  time = 0.390924, antiderivative size = 540, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 13, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.448, Rules used = {2498, 2495, 43, 2514, 2487, 31, 8, 2494, 2394, 2393, 2391, 2390, 2301} \[ \frac{p q r^2 (b c-a d)^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b d^2}-\frac{q r (b c-a d)^2 \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d^2}+\frac{p q r^2 (b c-a d)^2 \log (c+d x)}{2 b d^2}+\frac{p q r^2 (b c-a d)^2 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b d^2}+\frac{q^2 r^2 (b c-a d)^2 \log ^2(c+d x)}{2 b d^2}+\frac{3 q^2 r^2 (b c-a d)^2 \log (c+d x)}{2 b d^2}+\frac{(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}+\frac{q r (a+b x) (b c-a d) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}-\frac{p r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac{q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac{p q r^2 x (b c-a d)}{2 d}-\frac{q r^2 x (p+q) (b c-a d)}{d}-\frac{q^2 r^2 x (b c-a d)}{2 d}+\frac{p q r^2 (a+b x)^2}{4 b}+\frac{q^2 r^2 (a+b x)^2}{4 b}+\frac{1}{2} a p^2 r^2 x+\frac{1}{2} a p q r^2 x+\frac{1}{4} b p^2 r^2 x^2+\frac{1}{4} b p q r^2 x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*p^2*r^2*x)/2 + (a*p*q*r^2*x)/2 - ((b*c - a*d)*p*q*r^2*x)/(2*d) - ((b*c - a*d)*q^2*r^2*x)/(2*d) - ((b*c - a*
d)*q*(p + q)*r^2*x)/d + (b*p^2*r^2*x^2)/4 + (b*p*q*r^2*x^2)/4 + (p*q*r^2*(a + b*x)^2)/(4*b) + (q^2*r^2*(a + b*
x)^2)/(4*b) + ((b*c - a*d)^2*p*q*r^2*Log[c + d*x])/(2*b*d^2) + (3*(b*c - a*d)^2*q^2*r^2*Log[c + d*x])/(2*b*d^2
) + ((b*c - a*d)^2*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(b*d^2) + ((b*c - a*d)^2*q^2*r^2*Lo
g[c + d*x]^2)/(2*b*d^2) + ((b*c - a*d)*q*r*(a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(b*d) - (p*r*(a + b
*x)^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*b) - (q*r*(a + b*x)^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*
b) - ((b*c - a*d)^2*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(b*d^2) + ((a + b*x)^2*Log[e*(f*(a
+ b*x)^p*(c + d*x)^q)^r]^2)/(2*b) + ((b*c - a*d)^2*p*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(b*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 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) \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx &=\frac{(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-(p r) \int (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx-\frac{(d q r) \int \frac{(a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x} \, dx}{b}\\ &=-\frac{p r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}+\frac{(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac{(d q r) \int \left (-\frac{b (b c-a d) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^2}+\frac{b (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d}+\frac{(-b c+a d)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^2 (c+d x)}\right ) \, dx}{b}+\frac{1}{2} \left (p^2 r^2\right ) \int (a+b x) \, dx+\frac{\left (d p q r^2\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{2 b}\\ &=\frac{1}{2} a p^2 r^2 x+\frac{1}{4} b p^2 r^2 x^2-\frac{p r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}+\frac{(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-(q r) \int (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx+\frac{((b c-a d) q r) \int \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{d}-\frac{\left ((b c-a d)^2 q r\right ) \int \frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x} \, dx}{b d}+\frac{\left (d p q 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}{2 b}\\ &=\frac{1}{2} a p^2 r^2 x-\frac{(b c-a d) p q r^2 x}{2 d}+\frac{1}{4} b p^2 r^2 x^2+\frac{p q r^2 (a+b x)^2}{4 b}+\frac{(b c-a d)^2 p q r^2 \log (c+d x)}{2 b d^2}+\frac{(b c-a d) q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}-\frac{p r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac{q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac{(b c-a 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 d^2}+\frac{(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}+\frac{1}{2} \left (p q r^2\right ) \int (a+b x) \, dx+\frac{\left ((b c-a d)^2 p q r^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{d^2}+\frac{\left (d q^2 r^2\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{2 b}+\frac{\left ((b c-a d)^2 q^2 r^2\right ) \int \frac{1}{c+d x} \, dx}{b d}+\frac{\left ((b c-a d)^2 q^2 r^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b d}-\frac{\left ((b c-a d) q (p+q) r^2\right ) \int 1 \, dx}{d}\\ &=\frac{1}{2} a p^2 r^2 x+\frac{1}{2} a p q r^2 x-\frac{(b c-a d) p q r^2 x}{2 d}-\frac{(b c-a d) q (p+q) r^2 x}{d}+\frac{1}{4} b p^2 r^2 x^2+\frac{1}{4} b p q r^2 x^2+\frac{p q r^2 (a+b x)^2}{4 b}+\frac{(b c-a d)^2 p q r^2 \log (c+d x)}{2 b d^2}+\frac{(b c-a d)^2 q^2 r^2 \log (c+d x)}{b d^2}+\frac{(b c-a d)^2 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d^2}+\frac{(b c-a d) q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}-\frac{p r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac{q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac{(b c-a 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 d^2}+\frac{(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac{\left ((b c-a d)^2 p q r^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b d}+\frac{\left (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}{2 b}+\frac{\left ((b c-a d)^2 q^2 r^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b d^2}\\ &=\frac{1}{2} a p^2 r^2 x+\frac{1}{2} a p q r^2 x-\frac{(b c-a d) p q r^2 x}{2 d}-\frac{(b c-a d) q^2 r^2 x}{2 d}-\frac{(b c-a d) q (p+q) r^2 x}{d}+\frac{1}{4} b p^2 r^2 x^2+\frac{1}{4} b p q r^2 x^2+\frac{p q r^2 (a+b x)^2}{4 b}+\frac{q^2 r^2 (a+b x)^2}{4 b}+\frac{(b c-a d)^2 p q r^2 \log (c+d x)}{2 b d^2}+\frac{3 (b c-a d)^2 q^2 r^2 \log (c+d x)}{2 b d^2}+\frac{(b c-a d)^2 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d^2}+\frac{(b c-a d)^2 q^2 r^2 \log ^2(c+d x)}{2 b d^2}+\frac{(b c-a d) q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}-\frac{p r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac{q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac{(b c-a 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 d^2}+\frac{(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac{\left ((b c-a 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 d^2}\\ &=\frac{1}{2} a p^2 r^2 x+\frac{1}{2} a p q r^2 x-\frac{(b c-a d) p q r^2 x}{2 d}-\frac{(b c-a d) q^2 r^2 x}{2 d}-\frac{(b c-a d) q (p+q) r^2 x}{d}+\frac{1}{4} b p^2 r^2 x^2+\frac{1}{4} b p q r^2 x^2+\frac{p q r^2 (a+b x)^2}{4 b}+\frac{q^2 r^2 (a+b x)^2}{4 b}+\frac{(b c-a d)^2 p q r^2 \log (c+d x)}{2 b d^2}+\frac{3 (b c-a d)^2 q^2 r^2 \log (c+d x)}{2 b d^2}+\frac{(b c-a d)^2 p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d^2}+\frac{(b c-a d)^2 q^2 r^2 \log ^2(c+d x)}{2 b d^2}+\frac{(b c-a d) q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}-\frac{p r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac{q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac{(b c-a 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 d^2}+\frac{(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}+\frac{(b c-a d)^2 p q r^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b d^2}\\ \end{align*}

Mathematica [A]  time = 0.570108, size = 781, normalized size = 1.45 \[ \frac{-4 p q r^2 (b c-a d)^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )-8 a^2 d^2 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 a^2 d^2 p^2 r^2 \log ^2(a+b x)+8 a^2 d^2 p q r^2 \log (c+d x)+8 a^2 d^2 p q r^2-4 b^2 c^2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 b^2 d^2 x^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 b^2 d^2 p r x^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 b^2 d^2 q r x^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+4 b^2 c d q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+4 a b d^2 x \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-4 a b d^2 p r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-8 a b d^2 q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+8 a b c d q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 p r \log (a+b x) \left (a d \left (2 a d \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+3 a d r (p-q)+2 b c q r\right )-2 q r (b c-a d)^2 \log \left (\frac{b (c+d x)}{b c-a d}\right )+2 b c q r (b c-2 a d) \log (c+d x)\right )-4 a b c d p q r^2 \log (c+d x)-4 a b c d p q r^2-4 a b c d q^2 r^2 \log ^2(c+d x)-8 a b c d q^2 r^2 \log (c+d x)+2 a b d^2 p^2 r^2 x+10 a b d^2 p q r^2 x+8 a b d^2 q^2 r^2 x+2 b^2 c^2 p q r^2 \log (c+d x)+2 b^2 c^2 q^2 r^2 \log ^2(c+d x)+6 b^2 c^2 q^2 r^2 \log (c+d x)-6 b^2 c d p q r^2 x-6 b^2 c d q^2 r^2 x+b^2 d^2 p^2 r^2 x^2+2 b^2 d^2 p q r^2 x^2+b^2 d^2 q^2 r^2 x^2}{4 b d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a*b*c*d*p*q*r^2 + 8*a^2*d^2*p*q*r^2 + 2*a*b*d^2*p^2*r^2*x - 6*b^2*c*d*p*q*r^2*x + 10*a*b*d^2*p*q*r^2*x - 6
*b^2*c*d*q^2*r^2*x + 8*a*b*d^2*q^2*r^2*x + b^2*d^2*p^2*r^2*x^2 + 2*b^2*d^2*p*q*r^2*x^2 + b^2*d^2*q^2*r^2*x^2 -
2*a^2*d^2*p^2*r^2*Log[a + b*x]^2 + 2*b^2*c^2*p*q*r^2*Log[c + d*x] - 4*a*b*c*d*p*q*r^2*Log[c + d*x] + 8*a^2*d^
2*p*q*r^2*Log[c + d*x] + 6*b^2*c^2*q^2*r^2*Log[c + d*x] - 8*a*b*c*d*q^2*r^2*Log[c + d*x] + 2*b^2*c^2*q^2*r^2*L
og[c + d*x]^2 - 4*a*b*c*d*q^2*r^2*Log[c + d*x]^2 - 8*a^2*d^2*p*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - 4*a*b*
d^2*p*r*x*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] - 8*a*b*
d^2*q*r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - 2*b^2*d^2*p*r*x^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - 2*b^
2*d^2*q*r*x^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - 4*b^2*c^2*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)
^q)^r] + 8*a*b*c*d*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + 4*a*b*d^2*x*Log[e*(f*(a + b*x)^p*(c
+ d*x)^q)^r]^2 + 2*b^2*d^2*x^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2 + 2*p*r*Log[a + b*x]*(2*b*c*(b*c - 2*a*
d)*q*r*Log[c + d*x] - 2*(b*c - a*d)^2*q*r*Log[(b*(c + d*x))/(b*c - a*d)] + a*d*(3*a*d*(p - q)*r + 2*b*c*q*r +
2*a*d*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])) - 4*(b*c - a*d)^2*p*q*r^2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)
])/(4*b*d^2)

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Maple [F]  time = 0.166, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) \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)*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x)

[Out]

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

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Maxima [A]  time = 1.5037, size = 680, normalized size = 1.26 \begin{align*} \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} \log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} + \frac{{\left (\frac{2 \, a^{2} f p \log \left (b x + a\right )}{b} - \frac{b d f{\left (p + q\right )} x^{2} + 2 \,{\left (a d f{\left (p + 2 \, q\right )} - b c f q\right )} x}{d} - \frac{2 \,{\left (b c^{2} f q - 2 \, a c d f q\right )} \log \left (d x + c\right )}{d^{2}}\right )} r \log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )}{2 \, f} + \frac{r^{2}{\left (\frac{2 \,{\left ({\left (p q + 3 \, q^{2}\right )} b c^{2} f^{2} - 2 \,{\left (p q + 2 \, q^{2}\right )} a c d f^{2}\right )} \log \left (d x + c\right )}{d^{2}} - \frac{4 \,{\left (b^{2} c^{2} f^{2} p q - 2 \, a b c d f^{2} p q + a^{2} d^{2} f^{2} p q\right )}{\left (\log \left (b x + a\right ) \log \left (\frac{b d x + a d}{b c - a d} + 1\right ) +{\rm Li}_2\left (-\frac{b d x + a d}{b c - a d}\right )\right )}}{b d^{2}} - \frac{2 \, a^{2} d^{2} f^{2} p^{2} \log \left (b x + a\right )^{2} -{\left (p^{2} + 2 \, p q + q^{2}\right )} b^{2} d^{2} f^{2} x^{2} - 4 \,{\left (b^{2} c^{2} f^{2} p q - 2 \, a b c d f^{2} p q\right )} \log \left (b x + a\right ) \log \left (d x + c\right ) - 2 \,{\left (b^{2} c^{2} f^{2} q^{2} - 2 \, a b c d f^{2} q^{2}\right )} \log \left (d x + c\right )^{2} + 2 \,{\left (3 \,{\left (p q + q^{2}\right )} b^{2} c d f^{2} -{\left (p^{2} + 5 \, p q + 4 \, q^{2}\right )} a b d^{2} f^{2}\right )} x - 2 \,{\left (2 \, a b c d f^{2} p q -{\left (p^{2} + 3 \, p q\right )} a^{2} d^{2} f^{2}\right )} \log \left (b x + a\right )}{b d^{2}}\right )}}{4 \, f^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*x^2 + 2*a*x)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2 + 1/2*(2*a^2*f*p*log(b*x + a)/b - (b*d*f*(p + q)*x^
2 + 2*(a*d*f*(p + 2*q) - b*c*f*q)*x)/d - 2*(b*c^2*f*q - 2*a*c*d*f*q)*log(d*x + c)/d^2)*r*log(((b*x + a)^p*(d*x
+ c)^q*f)^r*e)/f + 1/4*r^2*(2*((p*q + 3*q^2)*b*c^2*f^2 - 2*(p*q + 2*q^2)*a*c*d*f^2)*log(d*x + c)/d^2 - 4*(b^2
*c^2*f^2*p*q - 2*a*b*c*d*f^2*p*q + a^2*d^2*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^2) - (2*a^2*d^2*f^2*p^2*log(b*x + a)^2 - (p^2 + 2*p*q + q^2)*b^2*d^2*f^2*x^2
- 4*(b^2*c^2*f^2*p*q - 2*a*b*c*d*f^2*p*q)*log(b*x + a)*log(d*x + c) - 2*(b^2*c^2*f^2*q^2 - 2*a*b*c*d*f^2*q^2)*
log(d*x + c)^2 + 2*(3*(p*q + q^2)*b^2*c*d*f^2 - (p^2 + 5*p*q + 4*q^2)*a*b*d^2*f^2)*x - 2*(2*a*b*c*d*f^2*p*q -
(p^2 + 3*p*q)*a^2*d^2*f^2)*log(b*x + a))/(b*d^2))/f^2

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

[Out]

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

[Out]

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