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

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.66471, antiderivative size = 805, normalized size of antiderivative = 1., number of steps used = 28, 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} \[ \frac{q^2 r^2 \log ^2(c+d x) (b c-a d)^4}{4 b d^4}+\frac{25 q^2 r^2 \log (c+d x) (b c-a d)^4}{24 b d^4}+\frac{p q r^2 \log (c+d x) (b c-a d)^4}{8 b d^4}+\frac{p q r^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) (b c-a d)^4}{2 b d^4}-\frac{q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^4}{2 b d^4}+\frac{p q r^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) (b c-a d)^4}{2 b d^4}-\frac{13 q^2 r^2 x (b c-a d)^3}{24 d^3}-\frac{p q r^2 x (b c-a d)^3}{8 d^3}-\frac{q (p+q) r^2 x (b c-a d)^3}{2 d^3}+\frac{q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^3}{2 b d^3}+\frac{b p q r^2 x^2 (b c-a d)^2}{8 d^2}+\frac{13 q^2 r^2 (a+b x)^2 (b c-a d)^2}{48 b d^2}+\frac{p q r^2 (a+b x)^2 (b c-a d)^2}{16 b d^2}+\frac{a p q r^2 x (b c-a d)^2}{4 d^2}-\frac{q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^2}{4 b d^2}-\frac{7 q^2 r^2 (a+b x)^3 (b c-a d)}{72 b d}-\frac{7 p q r^2 (a+b x)^3 (b c-a d)}{72 b d}+\frac{q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)}{6 b d}+\frac{p^2 r^2 (a+b x)^4}{32 b}+\frac{q^2 r^2 (a+b x)^4}{32 b}+\frac{p q r^2 (a+b x)^4}{16 b}+\frac{(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}-\frac{p r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac{q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b} \]

Antiderivative was successfully verified.

[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 2498

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_)*((g_.) + (h_.)*(x_))^(
m_.), x_Symbol] :> Simp[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(h*(m + 1)), x] + (-Dist[(b
*p*r*s)/(h*(m + 1)), Int[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/(a + b*x), x], x] -
Dist[(d*q*r*s)/(h*(m + 1)), Int[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/(c + d*x), x]
, x]) /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && NeQ[m, -1]

Rule 2495

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*((g_.) + (h_.)*(x_))^(m_.),
x_Symbol] :> Simp[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(h*(m + 1)), x] + (-Dist[(b*p*r)/(
h*(m + 1)), Int[(g + h*x)^(m + 1)/(a + b*x), x], x] - Dist[(d*q*r)/(h*(m + 1)), Int[(g + h*x)^(m + 1)/(c + d*x
), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2514

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(RFx_), x_Symbol] :>
With[{u = ExpandIntegrand[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a,
b, c, d, e, f, p, q, r, s}, x] && RationalFunctionQ[RFx, x] && IGtQ[s, 0]

Rule 2487

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[((
a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + (Dist[(q*r*s*(b*c - a*d))/b, Int[Log[e*(f*(a + b*x)^p
*(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] - Dist[r*s*(p + q), Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1
), x], x]) /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && NeQ[p + q, 0] && IGtQ[s, 0] &&
LtQ[s, 4]

Rule 31

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

Rule 8

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

Rule 2494

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]/((g_.) + (h_.)*(x_)), x_Sym
bol] :> Simp[(Log[g + h*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/h, x] + (-Dist[(b*p*r)/h, Int[Log[g + h*x]/(a
+ b*x), x], x] - Dist[(d*q*r)/h, Int[Log[g + h*x]/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, p, q,
r}, x] && NeQ[b*c - a*d, 0]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2301

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

Rubi steps

\begin{align*} \int (a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx &=\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 ) \operatorname{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 ) \operatorname{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]  time = 1.85919, size = 1853, normalized size = 2.3 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[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]  time = 0.402, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{3} \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)^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)

________________________________________________________________________________________

Maxima [A]  time = 1.47114, size = 1446, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\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)^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(-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)**3*ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)