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 \[ \frac {2 q r (b c-a d)^3 \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d^3}-\frac {2 p q r^2 (b c-a d)^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3 b d^3}-\frac {2 p q r^2 (b c-a d)^3 \log (c+d x)}{9 b d^3}-\frac {2 p q r^2 (b c-a d)^3 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{3 b d^3}-\frac {q^2 r^2 (b c-a d)^3 \log ^2(c+d x)}{3 b d^3}-\frac {11 q^2 r^2 (b c-a d)^3 \log (c+d x)}{9 b d^3}-\frac {2 q r (a+b x) (b c-a d)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d^2}+\frac {2 p q r^2 x (b c-a d)^2}{9 d^2}+\frac {2 q r^2 x (p+q) (b c-a d)^2}{3 d^2}+\frac {5 q^2 r^2 x (b c-a d)^2}{9 d^2}+\frac {(a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b}+\frac {q r (a+b x)^2 (b c-a d) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b d}-\frac {2 p r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b}-\frac {2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{9 b}-\frac {b p q r^2 x^2 (b c-a d)}{6 d}-\frac {p q r^2 (a+b x)^2 (b c-a d)}{9 b d}-\frac {a p q r^2 x (b c-a d)}{3 d}-\frac {5 q^2 r^2 (a+b x)^2 (b c-a d)}{18 b d}+\frac {2 p^2 r^2 (a+b x)^3}{27 b}+\frac {4 p q r^2 (a+b x)^3}{27 b}+\frac {2 q^2 r^2 (a+b x)^3}{27 b} \]

[Out]

-1/3*a*(-a*d+b*c)*p*q*r^2*x/d+2/9*(-a*d+b*c)^2*p*q*r^2*x/d^2+5/9*(-a*d+b*c)^2*q^2*r^2*x/d^2+2/3*(-a*d+b*c)^2*q
*(p+q)*r^2*x/d^2-1/6*b*(-a*d+b*c)*p*q*r^2*x^2/d-1/9*(-a*d+b*c)*p*q*r^2*(b*x+a)^2/b/d-5/18*(-a*d+b*c)*q^2*r^2*(
b*x+a)^2/b/d+2/27*p^2*r^2*(b*x+a)^3/b+4/27*p*q*r^2*(b*x+a)^3/b+2/27*q^2*r^2*(b*x+a)^3/b-2/9*(-a*d+b*c)^3*p*q*r
^2*ln(d*x+c)/b/d^3-11/9*(-a*d+b*c)^3*q^2*r^2*ln(d*x+c)/b/d^3-2/3*(-a*d+b*c)^3*p*q*r^2*ln(-d*(b*x+a)/(-a*d+b*c)
)*ln(d*x+c)/b/d^3-1/3*(-a*d+b*c)^3*q^2*r^2*ln(d*x+c)^2/b/d^3-2/3*(-a*d+b*c)^2*q*r*(b*x+a)*ln(e*(f*(b*x+a)^p*(d
*x+c)^q)^r)/b/d^2+1/3*(-a*d+b*c)*q*r*(b*x+a)^2*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/d-2/9*p*r*(b*x+a)^3*ln(e*(f*(
b*x+a)^p*(d*x+c)^q)^r)/b-2/9*q*r*(b*x+a)^3*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b+2/3*(-a*d+b*c)^3*q*r*ln(d*x+c)*ln
(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/d^3+1/3*(b*x+a)^3*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/b-2/3*(-a*d+b*c)^3*p*q*r^2
*polylog(2,b*(d*x+c)/(-a*d+b*c))/b/d^3

________________________________________________________________________________________

Rubi [A]  time = 0.53, antiderivative size = 686, normalized size of antiderivative = 1.00, 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 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 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 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 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 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 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 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 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 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 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 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 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]

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*}

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Mathematica [A]  time = 1.29, 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 {Li}_2\left (\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

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fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}, x\right ) \]

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

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)

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maple [F]  time = 0.30, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right )^{2} \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^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)

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maxima [A]  time = 0.88, size = 769, normalized size = 1.12 \[ \frac {1}{3} \, {\left (b^{2} x^{3} + 3 \, a b x^{2} + 3 \, a^{2} 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 {6 \, a^{3} f p \log \left (b x + a\right )}{b} - \frac {2 \, b^{2} d^{2} f {\left (p + q\right )} x^{3} + 3 \, {\left (a b d^{2} f {\left (2 \, p + 3 \, q\right )} - b^{2} c d f q\right )} x^{2} + 6 \, {\left (a^{2} d^{2} f {\left (p + 3 \, q\right )} + b^{2} c^{2} f q - 3 \, a b c d f q\right )} x}{d^{2}} + \frac {6 \, {\left (b^{2} c^{3} f q - 3 \, a b c^{2} d f q + 3 \, a^{2} c d^{2} f q\right )} \log \left (d x + c\right )}{d^{3}}\right )} r \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{9 \, f} - \frac {r^{2} {\left (\frac {6 \, {\left ({\left (2 \, p q + 11 \, q^{2}\right )} b^{2} c^{3} f^{2} - 3 \, {\left (2 \, p q + 9 \, q^{2}\right )} a b c^{2} d f^{2} + 6 \, {\left (p q + 3 \, q^{2}\right )} a^{2} c d^{2} f^{2}\right )} \log \left (d x + c\right )}{d^{3}} - \frac {36 \, {\left (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\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^{3}} - \frac {4 \, {\left (p^{2} + 2 \, p q + q^{2}\right )} b^{3} d^{3} f^{2} x^{3} - 18 \, a^{3} d^{3} f^{2} p^{2} \log \left (b x + a\right )^{2} - 3 \, {\left (5 \, {\left (p q + q^{2}\right )} b^{3} c d^{2} f^{2} - {\left (4 \, p^{2} + 13 \, p q + 9 \, q^{2}\right )} a b^{2} d^{3} f^{2}\right )} x^{2} - 36 \, {\left (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\right )} \log \left (b x + a\right ) \log \left (d x + c\right ) - 18 \, {\left (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}\right )} \log \left (d x + c\right )^{2} + 6 \, {\left ({\left (8 \, p q + 11 \, q^{2}\right )} b^{3} c^{2} d f^{2} - 3 \, {\left (7 \, p q + 9 \, q^{2}\right )} a b^{2} c d^{2} f^{2} + {\left (2 \, p^{2} + 17 \, p q + 18 \, q^{2}\right )} a^{2} b d^{3} f^{2}\right )} x - 6 \, {\left (6 \, a b^{2} c^{2} d f^{2} p q - 15 \, a^{2} b c d^{2} f^{2} p q + {\left (2 \, p^{2} + 11 \, p q\right )} a^{3} d^{3} f^{2}\right )} \log \left (b x + a\right )}{b d^{3}}\right )}}{54 \, f^{2}} \]

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

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)^2,x)

[Out]

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

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

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]

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

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