\(\int (a+b x)^3 \log ^2(e (f (a+b x)^p (c+d x)^q)^r) \, dx\) [17]

Optimal result

Integrand size = 31, antiderivative size = 805 \[ \int (a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {a (b c-a d)^2 p q r^2 x}{4 d^2}-\frac {(b c-a d)^3 p q r^2 x}{8 d^3}-\frac {13 (b c-a d)^3 q^2 r^2 x}{24 d^3}-\frac {(b c-a d)^3 q (p+q) r^2 x}{2 d^3}+\frac {b (b c-a d)^2 p q r^2 x^2}{8 d^2}+\frac {(b c-a d)^2 p q r^2 (a+b x)^2}{16 b d^2}+\frac {13 (b c-a d)^2 q^2 r^2 (a+b x)^2}{48 b d^2}-\frac {7 (b c-a d) p q r^2 (a+b x)^3}{72 b d}-\frac {7 (b c-a d) q^2 r^2 (a+b x)^3}{72 b d}+\frac {p^2 r^2 (a+b x)^4}{32 b}+\frac {p q r^2 (a+b x)^4}{16 b}+\frac {q^2 r^2 (a+b x)^4}{32 b}+\frac {(b c-a d)^4 p q r^2 \log (c+d x)}{8 b d^4}+\frac {25 (b c-a d)^4 q^2 r^2 \log (c+d x)}{24 b d^4}+\frac {(b c-a d)^4 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b d^4}+\frac {(b c-a d)^4 q^2 r^2 \log ^2(c+d x)}{4 b d^4}+\frac {(b c-a d)^3 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b d^3}-\frac {(b c-a d)^2 q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b d^2}+\frac {(b c-a d) q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{6 b d}-\frac {p r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac {q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac {(b c-a d)^4 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b d^4}+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}+\frac {(b c-a d)^4 p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{2 b d^4} \]

[Out]

1/4*a*(-a*d+b*c)^2*p*q*r^2*x/d^2-1/8*(-a*d+b*c)^3*p*q*r^2*x/d^3-13/24*(-a*d+b*c)^3*q^2*r^2*x/d^3-1/2*(-a*d+b*c
)^3*q*(p+q)*r^2*x/d^3+1/8*b*(-a*d+b*c)^2*p*q*r^2*x^2/d^2+1/16*(-a*d+b*c)^2*p*q*r^2*(b*x+a)^2/b/d^2+13/48*(-a*d
+b*c)^2*q^2*r^2*(b*x+a)^2/b/d^2-7/72*(-a*d+b*c)*p*q*r^2*(b*x+a)^3/b/d-7/72*(-a*d+b*c)*q^2*r^2*(b*x+a)^3/b/d+1/
32*p^2*r^2*(b*x+a)^4/b+1/16*p*q*r^2*(b*x+a)^4/b+1/32*q^2*r^2*(b*x+a)^4/b+1/8*(-a*d+b*c)^4*p*q*r^2*ln(d*x+c)/b/
d^4+25/24*(-a*d+b*c)^4*q^2*r^2*ln(d*x+c)/b/d^4+1/2*(-a*d+b*c)^4*p*q*r^2*ln(-d*(b*x+a)/(-a*d+b*c))*ln(d*x+c)/b/
d^4+1/4*(-a*d+b*c)^4*q^2*r^2*ln(d*x+c)^2/b/d^4+1/2*(-a*d+b*c)^3*q*r*(b*x+a)*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/
d^3-1/4*(-a*d+b*c)^2*q*r*(b*x+a)^2*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/d^2+1/6*(-a*d+b*c)*q*r*(b*x+a)^3*ln(e*(f*
(b*x+a)^p*(d*x+c)^q)^r)/b/d-1/8*p*r*(b*x+a)^4*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b-1/8*q*r*(b*x+a)^4*ln(e*(f*(b*x
+a)^p*(d*x+c)^q)^r)/b-1/2*(-a*d+b*c)^4*q*r*ln(d*x+c)*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/d^4+1/4*(b*x+a)^4*ln(e*
(f*(b*x+a)^p*(d*x+c)^q)^r)^2/b+1/2*(-a*d+b*c)^4*p*q*r^2*polylog(2,b*(d*x+c)/(-a*d+b*c))/b/d^4

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 805, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules used = {2584, 2581, 32, 45, 2594, 2579, 31, 8, 2580, 2441, 2440, 2438, 2437, 2338} \[ \int (a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {q^2 r^2 \log ^2(c+d x) (b c-a d)^4}{4 b d^4}+\frac {25 q^2 r^2 \log (c+d x) (b c-a d)^4}{24 b d^4}+\frac {p q r^2 \log (c+d x) (b c-a d)^4}{8 b d^4}+\frac {p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) (b c-a d)^4}{2 b d^4}-\frac {q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^4}{2 b d^4}+\frac {p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) (b c-a d)^4}{2 b d^4}-\frac {13 q^2 r^2 x (b c-a d)^3}{24 d^3}-\frac {p q r^2 x (b c-a d)^3}{8 d^3}-\frac {q (p+q) r^2 x (b c-a d)^3}{2 d^3}+\frac {q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^3}{2 b d^3}+\frac {b p q r^2 x^2 (b c-a d)^2}{8 d^2}+\frac {13 q^2 r^2 (a+b x)^2 (b c-a d)^2}{48 b d^2}+\frac {p q r^2 (a+b x)^2 (b c-a d)^2}{16 b d^2}+\frac {a p q r^2 x (b c-a d)^2}{4 d^2}-\frac {q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^2}{4 b d^2}-\frac {7 q^2 r^2 (a+b x)^3 (b c-a d)}{72 b d}-\frac {7 p q r^2 (a+b x)^3 (b c-a d)}{72 b d}+\frac {q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)}{6 b d}+\frac {p^2 r^2 (a+b x)^4}{32 b}+\frac {q^2 r^2 (a+b x)^4}{32 b}+\frac {p q r^2 (a+b x)^4}{16 b}+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}-\frac {p r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac {q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b} \]

[In]

Int[(a + b*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2,x]

[Out]

(a*(b*c - a*d)^2*p*q*r^2*x)/(4*d^2) - ((b*c - a*d)^3*p*q*r^2*x)/(8*d^3) - (13*(b*c - a*d)^3*q^2*r^2*x)/(24*d^3
) - ((b*c - a*d)^3*q*(p + q)*r^2*x)/(2*d^3) + (b*(b*c - a*d)^2*p*q*r^2*x^2)/(8*d^2) + ((b*c - a*d)^2*p*q*r^2*(
a + b*x)^2)/(16*b*d^2) + (13*(b*c - a*d)^2*q^2*r^2*(a + b*x)^2)/(48*b*d^2) - (7*(b*c - a*d)*p*q*r^2*(a + b*x)^
3)/(72*b*d) - (7*(b*c - a*d)*q^2*r^2*(a + b*x)^3)/(72*b*d) + (p^2*r^2*(a + b*x)^4)/(32*b) + (p*q*r^2*(a + b*x)
^4)/(16*b) + (q^2*r^2*(a + b*x)^4)/(32*b) + ((b*c - a*d)^4*p*q*r^2*Log[c + d*x])/(8*b*d^4) + (25*(b*c - a*d)^4
*q^2*r^2*Log[c + d*x])/(24*b*d^4) + ((b*c - a*d)^4*p*q*r^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(2*
b*d^4) + ((b*c - a*d)^4*q^2*r^2*Log[c + d*x]^2)/(4*b*d^4) + ((b*c - a*d)^3*q*r*(a + b*x)*Log[e*(f*(a + b*x)^p*
(c + d*x)^q)^r])/(2*b*d^3) - ((b*c - a*d)^2*q*r*(a + b*x)^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(4*b*d^2) +
((b*c - a*d)*q*r*(a + b*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(6*b*d) - (p*r*(a + b*x)^4*Log[e*(f*(a + b*
x)^p*(c + d*x)^q)^r])/(8*b) - (q*r*(a + b*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(8*b) - ((b*c - a*d)^4*q*
r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*b*d^4) + ((a + b*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q
)^r]^2)/(4*b) + ((b*c - a*d)^4*p*q*r^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(2*b*d^4)

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 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*} \text {integral}& = \frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}-\frac {1}{2} (p r) \int (a+b x)^3 \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)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x} \, dx}{2 b} \\ & = -\frac {p r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}-\frac {(d q r) \int \left (-\frac {b (b c-a d)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^4}+\frac {b (b c-a d)^2 (a+b x) \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)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^2}+\frac {b (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d}+\frac {(-b c+a d)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^4 (c+d x)}\right ) \, dx}{2 b}+\frac {1}{8} \left (p^2 r^2\right ) \int (a+b x)^3 \, dx+\frac {\left (d p q r^2\right ) \int \frac {(a+b x)^4}{c+d x} \, dx}{8 b} \\ & = \frac {p^2 r^2 (a+b x)^4}{32 b}-\frac {p r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}-\frac {1}{2} (q r) \int (a+b x)^3 \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 (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{2 d}-\frac {\left ((b c-a d)^2 q r\right ) \int (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{2 d^2}+\frac {\left ((b c-a d)^3 q r\right ) \int \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{2 d^3}-\frac {\left ((b c-a d)^4 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}{2 b d^3}+\frac {\left (d p q r^2\right ) \int \left (-\frac {b (b c-a d)^3}{d^4}+\frac {b (b c-a d)^2 (a+b x)}{d^3}-\frac {b (b c-a d) (a+b x)^2}{d^2}+\frac {b (a+b x)^3}{d}+\frac {(-b c+a d)^4}{d^4 (c+d x)}\right ) \, dx}{8 b} \\ & = -\frac {(b c-a d)^3 p q r^2 x}{8 d^3}+\frac {(b c-a d)^2 p q r^2 (a+b x)^2}{16 b d^2}-\frac {(b c-a d) p q r^2 (a+b x)^3}{24 b d}+\frac {p^2 r^2 (a+b x)^4}{32 b}+\frac {p q r^2 (a+b x)^4}{32 b}+\frac {(b c-a d)^4 p q r^2 \log (c+d x)}{8 b d^4}+\frac {(b c-a d)^3 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b d^3}-\frac {(b c-a d)^2 q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b d^2}+\frac {(b c-a d) q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{6 b d}-\frac {p r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac {q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac {(b c-a d)^4 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b d^4}+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}+\frac {1}{8} \left (p q r^2\right ) \int (a+b x)^3 \, dx-\frac {\left ((b c-a d) p q r^2\right ) \int (a+b x)^2 \, dx}{6 d}+\frac {\left ((b c-a d)^2 p q r^2\right ) \int (a+b x) \, dx}{4 d^2}+\frac {\left ((b c-a d)^4 p q r^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{2 d^4}+\frac {\left (d q^2 r^2\right ) \int \frac {(a+b x)^4}{c+d x} \, dx}{8 b}-\frac {\left ((b c-a d) q^2 r^2\right ) \int \frac {(a+b x)^3}{c+d x} \, dx}{6 b}+\frac {\left ((b c-a d)^2 q^2 r^2\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{4 b d}+\frac {\left ((b c-a d)^4 q^2 r^2\right ) \int \frac {1}{c+d x} \, dx}{2 b d^3}+\frac {\left ((b c-a d)^4 q^2 r^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{2 b d^3}-\frac {\left ((b c-a d)^3 q (p+q) r^2\right ) \int 1 \, dx}{2 d^3} \\ & = \frac {a (b c-a d)^2 p q r^2 x}{4 d^2}-\frac {(b c-a d)^3 p q r^2 x}{8 d^3}-\frac {(b c-a d)^3 q (p+q) r^2 x}{2 d^3}+\frac {b (b c-a d)^2 p q r^2 x^2}{8 d^2}+\frac {(b c-a d)^2 p q r^2 (a+b x)^2}{16 b d^2}-\frac {7 (b c-a d) p q r^2 (a+b x)^3}{72 b d}+\frac {p^2 r^2 (a+b x)^4}{32 b}+\frac {p q r^2 (a+b x)^4}{16 b}+\frac {(b c-a d)^4 p q r^2 \log (c+d x)}{8 b d^4}+\frac {(b c-a d)^4 q^2 r^2 \log (c+d x)}{2 b d^4}+\frac {(b c-a d)^4 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b d^4}+\frac {(b c-a d)^3 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b d^3}-\frac {(b c-a d)^2 q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b d^2}+\frac {(b c-a d) q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{6 b d}-\frac {p r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac {q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac {(b c-a d)^4 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b d^4}+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}-\frac {\left ((b c-a d)^4 p q r^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{2 b d^3}+\frac {\left (d q^2 r^2\right ) \int \left (-\frac {b (b c-a d)^3}{d^4}+\frac {b (b c-a d)^2 (a+b x)}{d^3}-\frac {b (b c-a d) (a+b x)^2}{d^2}+\frac {b (a+b x)^3}{d}+\frac {(-b c+a d)^4}{d^4 (c+d x)}\right ) \, dx}{8 b}-\frac {\left ((b c-a 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}{6 b}+\frac {\left ((b c-a d)^2 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}{4 b d}+\frac {\left ((b c-a d)^4 q^2 r^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{2 b d^4} \\ & = \frac {a (b c-a d)^2 p q r^2 x}{4 d^2}-\frac {(b c-a d)^3 p q r^2 x}{8 d^3}-\frac {13 (b c-a d)^3 q^2 r^2 x}{24 d^3}-\frac {(b c-a d)^3 q (p+q) r^2 x}{2 d^3}+\frac {b (b c-a d)^2 p q r^2 x^2}{8 d^2}+\frac {(b c-a d)^2 p q r^2 (a+b x)^2}{16 b d^2}+\frac {13 (b c-a d)^2 q^2 r^2 (a+b x)^2}{48 b d^2}-\frac {7 (b c-a d) p q r^2 (a+b x)^3}{72 b d}-\frac {7 (b c-a d) q^2 r^2 (a+b x)^3}{72 b d}+\frac {p^2 r^2 (a+b x)^4}{32 b}+\frac {p q r^2 (a+b x)^4}{16 b}+\frac {q^2 r^2 (a+b x)^4}{32 b}+\frac {(b c-a d)^4 p q r^2 \log (c+d x)}{8 b d^4}+\frac {25 (b c-a d)^4 q^2 r^2 \log (c+d x)}{24 b d^4}+\frac {(b c-a d)^4 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b d^4}+\frac {(b c-a d)^4 q^2 r^2 \log ^2(c+d x)}{4 b d^4}+\frac {(b c-a d)^3 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b d^3}-\frac {(b c-a d)^2 q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b d^2}+\frac {(b c-a d) q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{6 b d}-\frac {p r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac {q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac {(b c-a d)^4 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b d^4}+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}-\frac {\left ((b c-a d)^4 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 )}{2 b d^4} \\ & = \frac {a (b c-a d)^2 p q r^2 x}{4 d^2}-\frac {(b c-a d)^3 p q r^2 x}{8 d^3}-\frac {13 (b c-a d)^3 q^2 r^2 x}{24 d^3}-\frac {(b c-a d)^3 q (p+q) r^2 x}{2 d^3}+\frac {b (b c-a d)^2 p q r^2 x^2}{8 d^2}+\frac {(b c-a d)^2 p q r^2 (a+b x)^2}{16 b d^2}+\frac {13 (b c-a d)^2 q^2 r^2 (a+b x)^2}{48 b d^2}-\frac {7 (b c-a d) p q r^2 (a+b x)^3}{72 b d}-\frac {7 (b c-a d) q^2 r^2 (a+b x)^3}{72 b d}+\frac {p^2 r^2 (a+b x)^4}{32 b}+\frac {p q r^2 (a+b x)^4}{16 b}+\frac {q^2 r^2 (a+b x)^4}{32 b}+\frac {(b c-a d)^4 p q r^2 \log (c+d x)}{8 b d^4}+\frac {25 (b c-a d)^4 q^2 r^2 \log (c+d x)}{24 b d^4}+\frac {(b c-a d)^4 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b d^4}+\frac {(b c-a d)^4 q^2 r^2 \log ^2(c+d x)}{4 b d^4}+\frac {(b c-a d)^3 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b d^3}-\frac {(b c-a d)^2 q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b d^2}+\frac {(b c-a d) q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{6 b d}-\frac {p r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac {q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac {(b c-a d)^4 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b d^4}+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}+\frac {(b c-a d)^4 p q r^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2 b d^4} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1853\) vs. \(2(805)=1610\).

Time = 0.95 (sec) , antiderivative size = 1853, normalized size of antiderivative = 2.30 \[ \int (a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {2 a^4 p q r^2}{b}-\frac {a b^2 c^3 p q r^2}{2 d^3}+\frac {2 a^2 b c^2 p q r^2}{d^2}-\frac {3 a^3 c p q r^2}{d}+\frac {1}{8} a^3 p^2 r^2 x+\frac {37}{24} a^3 p q r^2 x-\frac {5 b^3 c^3 p q r^2 x}{8 d^3}+\frac {9 a b^2 c^2 p q r^2 x}{4 d^2}-\frac {35 a^2 b c p q r^2 x}{12 d}+2 a^3 q^2 r^2 x-\frac {25 b^3 c^3 q^2 r^2 x}{24 d^3}+\frac {11 a b^2 c^2 q^2 r^2 x}{3 d^2}-\frac {9 a^2 b c q^2 r^2 x}{2 d}+\frac {3}{16} a^2 b p^2 r^2 x^2+\frac {41}{48} a^2 b p q r^2 x^2+\frac {3 b^3 c^2 p q r^2 x^2}{16 d^2}-\frac {2 a b^2 c p q r^2 x^2}{3 d}+\frac {3}{4} a^2 b q^2 r^2 x^2+\frac {13 b^3 c^2 q^2 r^2 x^2}{48 d^2}-\frac {5 a b^2 c q^2 r^2 x^2}{6 d}+\frac {1}{8} a b^2 p^2 r^2 x^3+\frac {25}{72} a b^2 p q r^2 x^3-\frac {7 b^3 c p q r^2 x^3}{72 d}+\frac {2}{9} a b^2 q^2 r^2 x^3-\frac {7 b^3 c q^2 r^2 x^3}{72 d}+\frac {1}{32} b^3 p^2 r^2 x^4+\frac {1}{16} b^3 p q r^2 x^4+\frac {1}{32} b^3 q^2 r^2 x^4-\frac {a^4 p^2 r^2 \log ^2(a+b x)}{4 b}+\frac {2 a^4 p q r^2 \log (c+d x)}{b}+\frac {b^3 c^4 p q r^2 \log (c+d x)}{8 d^4}-\frac {a b^2 c^3 p q r^2 \log (c+d x)}{2 d^3}+\frac {3 a^2 b c^2 p q r^2 \log (c+d x)}{4 d^2}-\frac {a^3 c p q r^2 \log (c+d x)}{2 d}+\frac {25 b^3 c^4 q^2 r^2 \log (c+d x)}{24 d^4}-\frac {11 a b^2 c^3 q^2 r^2 \log (c+d x)}{3 d^3}+\frac {9 a^2 b c^2 q^2 r^2 \log (c+d x)}{2 d^2}-\frac {2 a^3 c q^2 r^2 \log (c+d x)}{d}+\frac {b^3 c^4 q^2 r^2 \log ^2(c+d x)}{4 d^4}-\frac {a b^2 c^3 q^2 r^2 \log ^2(c+d x)}{d^3}+\frac {3 a^2 b c^2 q^2 r^2 \log ^2(c+d x)}{2 d^2}-\frac {a^3 c q^2 r^2 \log ^2(c+d x)}{d}-\frac {2 a^4 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b}-\frac {1}{2} a^3 p r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 a^3 q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\frac {b^3 c^3 q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 d^3}-\frac {2 a 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 {3 a^2 b c q r x \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d}-\frac {3}{4} a^2 b p r x^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-\frac {3}{2} a^2 b q r x^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-\frac {b^3 c^2 q r x^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 d^2}+\frac {a 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 {1}{2} a b^2 p r x^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-\frac {2}{3} a b^2 q r x^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\frac {b^3 c q r x^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{6 d}-\frac {1}{8} b^3 p r x^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-\frac {1}{8} b^3 q r x^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-\frac {b^3 c^4 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 d^4}+\frac {2 a 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 {3 a^2 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 )}{d^2}+\frac {2 a^3 c q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d}+a^3 x \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\frac {3}{2} a^2 b x^2 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+a b^2 x^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\frac {1}{4} b^3 x^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+\frac {p r \log (a+b x) \left (a d \left (5 a^3 d^3 (9 p-5 q)+12 b^3 c^3 q-42 a b^2 c^2 d q+52 a^2 b c d^2 q\right ) r+12 b c \left (b^3 c^3-4 a b^2 c^2 d+6 a^2 b c d^2-4 a^3 d^3\right ) q r \log (c+d x)-12 (b c-a d)^4 q r \log \left (\frac {b (c+d x)}{b c-a d}\right )+12 a^4 d^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{24 b d^4}-\frac {(b c-a d)^4 p q r^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )}{2 b d^4} \]

[In]

Integrate[(a + b*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2,x]

[Out]

(2*a^4*p*q*r^2)/b - (a*b^2*c^3*p*q*r^2)/(2*d^3) + (2*a^2*b*c^2*p*q*r^2)/d^2 - (3*a^3*c*p*q*r^2)/d + (a^3*p^2*r
^2*x)/8 + (37*a^3*p*q*r^2*x)/24 - (5*b^3*c^3*p*q*r^2*x)/(8*d^3) + (9*a*b^2*c^2*p*q*r^2*x)/(4*d^2) - (35*a^2*b*
c*p*q*r^2*x)/(12*d) + 2*a^3*q^2*r^2*x - (25*b^3*c^3*q^2*r^2*x)/(24*d^3) + (11*a*b^2*c^2*q^2*r^2*x)/(3*d^2) - (
9*a^2*b*c*q^2*r^2*x)/(2*d) + (3*a^2*b*p^2*r^2*x^2)/16 + (41*a^2*b*p*q*r^2*x^2)/48 + (3*b^3*c^2*p*q*r^2*x^2)/(1
6*d^2) - (2*a*b^2*c*p*q*r^2*x^2)/(3*d) + (3*a^2*b*q^2*r^2*x^2)/4 + (13*b^3*c^2*q^2*r^2*x^2)/(48*d^2) - (5*a*b^
2*c*q^2*r^2*x^2)/(6*d) + (a*b^2*p^2*r^2*x^3)/8 + (25*a*b^2*p*q*r^2*x^3)/72 - (7*b^3*c*p*q*r^2*x^3)/(72*d) + (2
*a*b^2*q^2*r^2*x^3)/9 - (7*b^3*c*q^2*r^2*x^3)/(72*d) + (b^3*p^2*r^2*x^4)/32 + (b^3*p*q*r^2*x^4)/16 + (b^3*q^2*
r^2*x^4)/32 - (a^4*p^2*r^2*Log[a + b*x]^2)/(4*b) + (2*a^4*p*q*r^2*Log[c + d*x])/b + (b^3*c^4*p*q*r^2*Log[c + d
*x])/(8*d^4) - (a*b^2*c^3*p*q*r^2*Log[c + d*x])/(2*d^3) + (3*a^2*b*c^2*p*q*r^2*Log[c + d*x])/(4*d^2) - (a^3*c*
p*q*r^2*Log[c + d*x])/(2*d) + (25*b^3*c^4*q^2*r^2*Log[c + d*x])/(24*d^4) - (11*a*b^2*c^3*q^2*r^2*Log[c + d*x])
/(3*d^3) + (9*a^2*b*c^2*q^2*r^2*Log[c + d*x])/(2*d^2) - (2*a^3*c*q^2*r^2*Log[c + d*x])/d + (b^3*c^4*q^2*r^2*Lo
g[c + d*x]^2)/(4*d^4) - (a*b^2*c^3*q^2*r^2*Log[c + d*x]^2)/d^3 + (3*a^2*b*c^2*q^2*r^2*Log[c + d*x]^2)/(2*d^2)
- (a^3*c*q^2*r^2*Log[c + d*x]^2)/d - (2*a^4*p*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/b - (a^3*p*r*x*Log[e*(f*
(a + b*x)^p*(c + d*x)^q)^r])/2 - 2*a^3*q*r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + (b^3*c^3*q*r*x*Log[e*(f*(a
+ b*x)^p*(c + d*x)^q)^r])/(2*d^3) - (2*a*b^2*c^2*q*r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d^2 + (3*a^2*b*c
*q*r*x*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d - (3*a^2*b*p*r*x^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/4 - (3
*a^2*b*q*r*x^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/2 - (b^3*c^2*q*r*x^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]
)/(4*d^2) + (a*b^2*c*q*r*x^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d - (a*b^2*p*r*x^3*Log[e*(f*(a + b*x)^p*(c
+ d*x)^q)^r])/2 - (2*a*b^2*q*r*x^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/3 + (b^3*c*q*r*x^3*Log[e*(f*(a + b*x)
^p*(c + d*x)^q)^r])/(6*d) - (b^3*p*r*x^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/8 - (b^3*q*r*x^4*Log[e*(f*(a +
b*x)^p*(c + d*x)^q)^r])/8 - (b^3*c^4*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(2*d^4) + (2*a*b^2
*c^3*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d^3 - (3*a^2*b*c^2*q*r*Log[c + d*x]*Log[e*(f*(a +
b*x)^p*(c + d*x)^q)^r])/d^2 + (2*a^3*c*q*r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/d + a^3*x*Log[e*
(f*(a + b*x)^p*(c + d*x)^q)^r]^2 + (3*a^2*b*x^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)/2 + a*b^2*x^3*Log[e*(f
*(a + b*x)^p*(c + d*x)^q)^r]^2 + (b^3*x^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2)/4 + (p*r*Log[a + b*x]*(a*d*(
5*a^3*d^3*(9*p - 5*q) + 12*b^3*c^3*q - 42*a*b^2*c^2*d*q + 52*a^2*b*c*d^2*q)*r + 12*b*c*(b^3*c^3 - 4*a*b^2*c^2*
d + 6*a^2*b*c*d^2 - 4*a^3*d^3)*q*r*Log[c + d*x] - 12*(b*c - a*d)^4*q*r*Log[(b*(c + d*x))/(b*c - a*d)] + 12*a^4
*d^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]))/(24*b*d^4) - ((b*c - a*d)^4*p*q*r^2*PolyLog[2, (d*(a + b*x))/(-(b*
c) + a*d)])/(2*b*d^4)

Maple [F]

\[\int \left (b x +a \right )^{3} {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}^{2}d x\]

[In]

int((b*x+a)^3*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x)

[Out]

int((b*x+a)^3*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x)

Fricas [F]

\[ \int (a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int { {\left (b x + a\right )}^{3} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} \,d x } \]

[In]

integrate((b*x+a)^3*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int (a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int \left (a + b x\right )^{3} \log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}^{2}\, dx \]

[In]

integrate((b*x+a)**3*ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)**2,x)

[Out]

Integral((a + b*x)**3*log(e*(f*(a + b*x)**p*(c + d*x)**q)**r)**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 1071, normalized size of antiderivative = 1.33 \[ \int (a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^3*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x, algorithm="maxima")

[Out]

1/4*(b^3*x^4 + 4*a*b^2*x^3 + 6*a^2*b*x^2 + 4*a^3*x)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2 + 1/24*(12*a^4*f*p*
log(b*x + a)/b - (3*b^3*d^3*f*(p + q)*x^4 + 4*(a*b^2*d^3*f*(3*p + 4*q) - b^3*c*d^2*f*q)*x^3 + 6*(3*a^2*b*d^3*f
*(p + 2*q) + b^3*c^2*d*f*q - 4*a*b^2*c*d^2*f*q)*x^2 + 12*(a^3*d^3*f*(p + 4*q) - b^3*c^3*f*q + 4*a*b^2*c^2*d*f*
q - 6*a^2*b*c*d^2*f*q)*x)/d^3 - 12*(b^3*c^4*f*q - 4*a*b^2*c^3*d*f*q + 6*a^2*b*c^2*d^2*f*q - 4*a^3*c*d^3*f*q)*l
og(d*x + c)/d^4)*r*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/f + 1/288*r^2*(12*((3*p*q + 25*q^2)*b^3*c^4*f^2 - 4*(3
*p*q + 22*q^2)*a*b^2*c^3*d*f^2 + 18*(p*q + 6*q^2)*a^2*b*c^2*d^2*f^2 - 12*(p*q + 4*q^2)*a^3*c*d^3*f^2)*log(d*x
+ c)/d^4 - 144*(b^4*c^4*f^2*p*q - 4*a*b^3*c^3*d*f^2*p*q + 6*a^2*b^2*c^2*d^2*f^2*p*q - 4*a^3*b*c*d^3*f^2*p*q +
a^4*d^4*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^4)
+ (9*(p^2 + 2*p*q + q^2)*b^4*d^4*f^2*x^4 - 72*a^4*d^4*f^2*p^2*log(b*x + a)^2 - 4*(7*(p*q + q^2)*b^4*c*d^3*f^2
- (9*p^2 + 25*p*q + 16*q^2)*a*b^3*d^4*f^2)*x^3 + 6*((9*p*q + 13*q^2)*b^4*c^2*d^2*f^2 - 8*(4*p*q + 5*q^2)*a*b^
3*c*d^3*f^2 + (9*p^2 + 41*p*q + 36*q^2)*a^2*b^2*d^4*f^2)*x^2 + 144*(b^4*c^4*f^2*p*q - 4*a*b^3*c^3*d*f^2*p*q +
6*a^2*b^2*c^2*d^2*f^2*p*q - 4*a^3*b*c*d^3*f^2*p*q)*log(b*x + a)*log(d*x + c) + 72*(b^4*c^4*f^2*q^2 - 4*a*b^3*c
^3*d*f^2*q^2 + 6*a^2*b^2*c^2*d^2*f^2*q^2 - 4*a^3*b*c*d^3*f^2*q^2)*log(d*x + c)^2 - 12*(5*(3*p*q + 5*q^2)*b^4*c
^3*d*f^2 - 2*(27*p*q + 44*q^2)*a*b^3*c^2*d^2*f^2 + 2*(35*p*q + 54*q^2)*a^2*b^2*c*d^3*f^2 - (3*p^2 + 37*p*q + 4
8*q^2)*a^3*b*d^4*f^2)*x + 12*(12*a*b^3*c^3*d*f^2*p*q - 42*a^2*b^2*c^2*d^2*f^2*p*q + 52*a^3*b*c*d^3*f^2*p*q - (
3*p^2 + 25*p*q)*a^4*d^4*f^2)*log(b*x + a))/(b*d^4))/f^2

Giac [F]

\[ \int (a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int { {\left (b x + a\right )}^{3} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} \,d x } \]

[In]

integrate((b*x+a)^3*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2,x, algorithm="giac")

[Out]

integrate((b*x + a)^3*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)^2, x)

Mupad [F(-1)]

Timed out. \[ \int (a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\int {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2\,{\left (a+b\,x\right )}^3 \,d x \]

[In]

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^2*(a + b*x)^3,x)

[Out]

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^2*(a + b*x)^3, x)