3.1.19 \(\int (a+b x) \log ^2(e (f (a+b x)^p (c+d x)^q)^r) \, dx\) [19]

Optimal. Leaf size=540 \[ \frac {1}{2} a p^2 r^2 x+\frac {1}{2} a p q r^2 x-\frac {(b c-a d) p q r^2 x}{2 d}-\frac {(b c-a d) q^2 r^2 x}{2 d}-\frac {(b c-a d) q (p+q) r^2 x}{d}+\frac {1}{4} b p^2 r^2 x^2+\frac {1}{4} b p q r^2 x^2+\frac {p q r^2 (a+b x)^2}{4 b}+\frac {q^2 r^2 (a+b x)^2}{4 b}+\frac {(b c-a d)^2 p q r^2 \log (c+d x)}{2 b d^2}+\frac {3 (b c-a d)^2 q^2 r^2 \log (c+d x)}{2 b d^2}+\frac {(b c-a d)^2 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d^2}+\frac {(b c-a d)^2 q^2 r^2 \log ^2(c+d x)}{2 b d^2}+\frac {(b c-a d) q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d}-\frac {p r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {(b c-a d)^2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b d^2}+\frac {(a+b x)^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}+\frac {(b c-a d)^2 p q r^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b d^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.27, antiderivative size = 540, normalized size of antiderivative = 1.00, 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 = {2584, 2581, 45, 2594, 2579, 31, 8, 2580, 2441, 2440, 2438, 2437, 2338} \begin {gather*} \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 \end {gather*}

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 8

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

Rule 31

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

Rule 45

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 2338

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 2437

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 2438

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 2440

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 2441

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 2579

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 2580

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 2581

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 2584

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 2594

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]

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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.41, size = 781, normalized size = 1.45 \begin {gather*} \frac {-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 ^2(a+b x)+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 \log ^2(c+d x)-4 a b c d q^2 r^2 \log ^2(c+d x)-8 a^2 d^2 p r \log \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 )+4 b^2 c d q 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 )-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^2 q r \log (c+d 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 )+4 a b d^2 x \log ^2\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 p r \log (a+b x) \left (2 b c (b c-2 a d) q r \log (c+d x)-2 (b c-a d)^2 q r \log \left (\frac {b (c+d x)}{b c-a d}\right )+a d \left (3 a d (p-q) r+2 b c q r+2 a d \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right )-4 (b c-a d)^2 p q r^2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )}{4 b d^2} \end {gather*}

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 = 0.07, size = 0, normalized size = 0.00 \[\int \left (b x +a \right ) \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )^{2}\, dx\]

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 = 0.34, size = 506, normalized size = 0.94 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b x\right ) \log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}^{2}\, dx \end {gather*}

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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2\,\left (a+b\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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