3.59 \(\int \frac{(c i+d i x) (A+B \log (\frac{e (a+b x)}{c+d x}))^2}{a g+b g x} \, dx\)

Optimal. Leaf size=286 \[ \frac{2 B i (b c-a d) \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^2 g}+\frac{2 B^2 i (b c-a d) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{b^2 g}+\frac{2 B^2 i (b c-a d) \text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )}{b^2 g}+\frac{2 B i (b c-a d) \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^2 g}+\frac{d i (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{b^2 g}-\frac{i (b c-a d) \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{b^2 g} \]

[Out]

(2*B*(b*c - a*d)*i*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^2*g) + (d*i*(a + b*
x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(b^2*g) - ((b*c - a*d)*i*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2*Log
[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^2*g) + (2*B^2*(b*c - a*d)*i*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^
2*g) + (2*B*(b*c - a*d)*i*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^2*g
) + (2*B^2*(b*c - a*d)*i*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b^2*g)

________________________________________________________________________________________

Rubi [B]  time = 2.93939, antiderivative size = 644, normalized size of antiderivative = 2.25, number of steps used = 39, number of rules used = 19, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.475, Rules used = {2528, 2523, 12, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 6742, 2411, 2344, 2317, 2507, 2488, 2506, 6610} \[ \frac{2 A B i (b c-a d) \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g}+\frac{2 B^2 i (b c-a d) \text{PolyLog}\left (2,\frac{b c-a d}{d (a+b x)}+1\right ) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g}+\frac{2 a B^2 d i \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g}+\frac{2 B^2 i (b c-a d) \text{PolyLog}\left (3,\frac{b c-a d}{d (a+b x)}+1\right )}{b^2 g}+\frac{2 B^2 c i \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{2 a B d i \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^2 g}+\frac{i (b c-a d) \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{b^2 g}-\frac{A B i (b c-a d) \log ^2(a+b x)}{b^2 g}+\frac{2 A B i (b c-a d) \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac{2 B c i \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b g}+\frac{d i x \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{b g}-\frac{B^2 i (b c-a d) \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g}-\frac{B^2 i (b c-a d) \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g}+\frac{2 a B^2 d i \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac{a B^2 d i \log ^2(a+b x)}{b^2 g}+\frac{2 B^2 c i \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b g}-\frac{B^2 c i \log ^2(c+d x)}{b g} \]

Antiderivative was successfully verified.

[In]

Int[((c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(a*g + b*g*x),x]

[Out]

-((a*B^2*d*i*Log[a + b*x]^2)/(b^2*g)) - (A*B*(b*c - a*d)*i*Log[a + b*x]^2)/(b^2*g) - (B^2*(b*c - a*d)*i*Log[-(
(b*c - a*d)/(d*(a + b*x)))]*Log[(e*(a + b*x))/(c + d*x)]^2)/(b^2*g) - (B^2*(b*c - a*d)*i*Log[a + b*x]*Log[(e*(
a + b*x))/(c + d*x)]^2)/(b^2*g) + (2*a*B*d*i*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^2*g) + (d*i
*x*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(b*g) + ((b*c - a*d)*i*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d
*x)])^2)/(b^2*g) + (2*B^2*c*i*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(b*g) - (2*B*c*i*(A + B*Log[(e*(
a + b*x))/(c + d*x)])*Log[c + d*x])/(b*g) - (B^2*c*i*Log[c + d*x]^2)/(b*g) + (2*a*B^2*d*i*Log[a + b*x]*Log[(b*
(c + d*x))/(b*c - a*d)])/(b^2*g) + (2*A*B*(b*c - a*d)*i*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(b^2*g) +
(2*a*B^2*d*i*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(b^2*g) + (2*A*B*(b*c - a*d)*i*PolyLog[2, -((d*(a + b*
x))/(b*c - a*d))])/(b^2*g) + (2*B^2*c*i*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(b*g) + (2*B^2*(b*c - a*d)*i*Lo
g[(e*(a + b*x))/(c + d*x)]*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/(b^2*g) + (2*B^2*(b*c - a*d)*i*PolyLog[3
, 1 + (b*c - a*d)/(d*(a + b*x))])/(b^2*g)

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 2523

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[(x*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2524

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*(a + b
*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

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]

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 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2344

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

Rule 2317

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

Rule 2507

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*Log[(i_.)*((j_.)*((g_
.) + (h_.)*(x_))^(t_.))^(u_.)]*(v_), x_Symbol] :> With[{k = Simplify[v*(a + b*x)*(c + d*x)]}, Simp[(k*Log[i*(j
*(g + h*x)^t)^u]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1))/(p*r*(s + 1)*(b*c - a*d)), x] - Dist[(k*h*t*u)/
(p*r*(s + 1)*(b*c - a*d)), Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1)/(g + h*x), x], x] /; FreeQ[k, x]]
/; FreeQ[{a, b, c, d, e, f, g, h, i, j, p, q, r, s, t, u}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && NeQ[s,
-1]

Rule 2488

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

Rule 2506

Int[Log[v_]*Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_), x_Symbo
l] :> With[{g = Simplify[((v - 1)*(c + d*x))/(a + b*x)], h = Simplify[u*(a + b*x)*(c + d*x)]}, -Simp[(h*PolyLo
g[2, 1 - v]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(b*c - a*d), x] + Dist[h*p*r*s, Int[(PolyLog[2, 1 - v]*Log
[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{g, h}, x]] /; FreeQ[{a, b,
c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0]

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps

\begin{align*} \int \frac{(59 c+59 d x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{a g+b g x} \, dx &=\int \left (\frac{59 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac{59 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b g (a+b x)}\right ) \, dx\\ &=\frac{(59 d) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \, dx}{b g}+\frac{(59 (b c-a d)) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{a+b x} \, dx}{b g}\\ &=\frac{59 d x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac{59 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}-\frac{(118 B d) \int \frac{(b c-a d) x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{b g}-\frac{(118 B (b c-a d)) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{e (a+b x)} \, dx}{b^2 g}\\ &=\frac{59 d x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac{59 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}-\frac{(118 B d (b c-a d)) \int \frac{x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{b g}-\frac{(118 B (b c-a d)) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{b^2 e g}\\ &=\frac{59 d x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac{59 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}-\frac{(118 B d (b c-a d)) \int \left (-\frac{a \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)}+\frac{c \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d x)}\right ) \, dx}{b g}-\frac{(118 B (b c-a d)) \int \frac{(b c-a d) e \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{b^2 e g}\\ &=\frac{59 d x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac{59 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}+\frac{(118 a B d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b g}-\frac{(118 B c d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b g}-\frac{\left (118 B (b c-a d)^2\right ) \int \frac{\log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{b^2 g}\\ &=\frac{118 a B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g}+\frac{59 d x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac{59 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}-\frac{118 B c \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b g}+\frac{\left (118 B^2 c\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{b g}-\frac{\left (118 a B^2 d\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{b^2 g}-\frac{\left (118 B (b c-a d)^2\right ) \int \left (\frac{A \log (a+b x)}{(a+b x) (c+d x)}+\frac{B \log (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)}\right ) \, dx}{b^2 g}\\ &=\frac{118 a B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g}+\frac{59 d x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac{59 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}-\frac{118 B c \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b g}-\frac{\left (118 A B (b c-a d)^2\right ) \int \frac{\log (a+b x)}{(a+b x) (c+d x)} \, dx}{b^2 g}-\frac{\left (118 B^2 (b c-a d)^2\right ) \int \frac{\log (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)} \, dx}{b^2 g}+\frac{\left (118 B^2 c\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b e g}-\frac{\left (118 a B^2 d\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^2 e g}\\ &=-\frac{59 B^2 (b c-a d) \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g}+\frac{118 a B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g}+\frac{59 d x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac{59 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}-\frac{118 B c \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b g}+\frac{\left (59 B^2 (b c-a d)\right ) \int \frac{\log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b g}-\frac{\left (118 A B (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x \left (\frac{b c-a d}{b}+\frac{d x}{b}\right )} \, dx,x,a+b x\right )}{b^3 g}+\frac{\left (118 B^2 c\right ) \int \left (\frac{b e \log (c+d x)}{a+b x}-\frac{d e \log (c+d x)}{c+d x}\right ) \, dx}{b e g}-\frac{\left (118 a B^2 d\right ) \int \left (\frac{b e \log (a+b x)}{a+b x}-\frac{d e \log (a+b x)}{c+d x}\right ) \, dx}{b^2 e g}\\ &=-\frac{59 B^2 (b c-a d) \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g}-\frac{59 B^2 (b c-a d) \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g}+\frac{118 a B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g}+\frac{59 d x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac{59 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}-\frac{118 B c \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b g}+\frac{\left (118 B^2 c\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{g}-\frac{\left (118 a B^2 d\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b g}-\frac{\left (118 B^2 c d\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b g}+\frac{\left (118 a B^2 d^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^2 g}-\frac{(118 A B (b c-a d)) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^2 g}+\frac{(118 A B d (b c-a d)) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{b c-a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{b^3 g}+\frac{\left (118 B^2 (b c-a d)^2\right ) \int \frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)} \, dx}{b^2 g}\\ &=-\frac{59 A B (b c-a d) \log ^2(a+b x)}{b^2 g}-\frac{59 B^2 (b c-a d) \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g}-\frac{59 B^2 (b c-a d) \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g}+\frac{118 a B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g}+\frac{59 d x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac{59 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}+\frac{118 B^2 c \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b g}-\frac{118 B c \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b g}+\frac{118 a B^2 d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac{118 A B (b c-a d) \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac{118 B^2 (b c-a d) \log \left (\frac{e (a+b x)}{c+d x}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^2 g}-\frac{\left (118 B^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b g}-\frac{\left (118 a B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^2 g}-\frac{\left (118 a B^2 d\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b g}-\frac{\left (118 B^2 c d\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b g}-\frac{(118 A B (b c-a d)) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 g}-\frac{\left (118 B^2 (b c-a d)^2\right ) \int \frac{\text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b^2 g}\\ &=-\frac{59 a B^2 d \log ^2(a+b x)}{b^2 g}-\frac{59 A B (b c-a d) \log ^2(a+b x)}{b^2 g}-\frac{59 B^2 (b c-a d) \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g}-\frac{59 B^2 (b c-a d) \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g}+\frac{118 a B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g}+\frac{59 d x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac{59 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}+\frac{118 B^2 c \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b g}-\frac{118 B c \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b g}-\frac{59 B^2 c \log ^2(c+d x)}{b g}+\frac{118 a B^2 d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac{118 A B (b c-a d) \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac{118 A B (b c-a d) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g}+\frac{118 B^2 (b c-a d) \log \left (\frac{e (a+b x)}{c+d x}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^2 g}+\frac{118 B^2 (b c-a d) \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^2 g}-\frac{\left (118 B^2 c\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b g}-\frac{\left (118 a B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 g}\\ &=-\frac{59 a B^2 d \log ^2(a+b x)}{b^2 g}-\frac{59 A B (b c-a d) \log ^2(a+b x)}{b^2 g}-\frac{59 B^2 (b c-a d) \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g}-\frac{59 B^2 (b c-a d) \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{b^2 g}+\frac{118 a B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g}+\frac{59 d x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac{59 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}+\frac{118 B^2 c \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b g}-\frac{118 B c \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b g}-\frac{59 B^2 c \log ^2(c+d x)}{b g}+\frac{118 a B^2 d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac{118 A B (b c-a d) \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac{118 a B^2 d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g}+\frac{118 A B (b c-a d) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g}+\frac{118 B^2 c \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b g}+\frac{118 B^2 (b c-a d) \log \left (\frac{e (a+b x)}{c+d x}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^2 g}+\frac{118 B^2 (b c-a d) \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^2 g}\\ \end{align*}

Mathematica [B]  time = 1.31181, size = 987, normalized size = 3.45 \[ \frac{i \left (3 b d x A^2+3 (b c-a d) \log (a+b x) A^2-3 B \left (a d \log ^2\left (\frac{a}{b}+x\right )-2 a d (\log (a+b x)+1) \log \left (\frac{a}{b}+x\right )+2 \left (-b c+a d+\log \left (\frac{c}{d}+x\right ) \left (b c+a d \log (a+b x)-a d \log \left (\frac{d (a+b x)}{a d-b c}\right )\right )+(a d \log (a+b x)-b d x) \log \left (\frac{e (a+b x)}{c+d x}\right )\right )-2 a d \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )\right ) A+3 b B c \left (\log ^2\left (\frac{a}{b}+x\right )-2 \log (a+b x) \left (\log \left (\frac{a}{b}+x\right )-\log \left (\frac{c}{d}+x\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right )-2 \left (\log \left (\frac{c}{d}+x\right ) \log \left (\frac{d (a+b x)}{a d-b c}\right )+\text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )\right )\right ) A-B^2 \left (a d \log ^3\left (\frac{a}{b}+x\right )-3 d (b x-a \log (a+b x)) \left (-\log \left (\frac{a}{b}+x\right )+\log \left (\frac{c}{d}+x\right )+\log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2-3 d \left ((a+b x) \log ^2\left (\frac{a}{b}+x\right )-2 (a+b x) \log \left (\frac{a}{b}+x\right )+2 b x\right )-3 b \left ((c+d x) \log ^2\left (\frac{c}{d}+x\right )-2 (c+d x) \log \left (\frac{c}{d}+x\right )+2 d x\right )+6 \left (a d+2 b x d-b x \log \left (\frac{c}{d}+x\right ) d-b c \log (c+d x)+\log \left (\frac{a}{b}+x\right ) \left (-d (a+b x)+d \log \left (\frac{c}{d}+x\right ) (a+b x)+(b c-a d) \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )+(b c-a d) \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )-3 \left (\log \left (\frac{a}{b}+x\right )-\log \left (\frac{c}{d}+x\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \left (a d \log ^2\left (\frac{a}{b}+x\right )-2 d (a+b x) \log \left (\frac{a}{b}+x\right )-2 b c+2 a d+2 \log \left (\frac{c}{d}+x\right ) \left (b (c+d x)-a d \log \left (\frac{d (a+b x)}{a d-b c}\right )\right )-2 a d \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )\right )-3 a d \left (\left (\log \left (\frac{c}{d}+x\right )-\log \left (\frac{b (c+d x)}{b c-a d}\right )\right ) \log ^2\left (\frac{a}{b}+x\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right ) \log \left (\frac{a}{b}+x\right )+2 \text{PolyLog}\left (3,\frac{d (a+b x)}{a d-b c}\right )\right )+3 a d \left (\log \left (\frac{d (a+b x)}{a d-b c}\right ) \log ^2\left (\frac{c}{d}+x\right )+2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) \log \left (\frac{c}{d}+x\right )-2 \text{PolyLog}\left (3,\frac{b (c+d x)}{b c-a d}\right )\right )\right )-3 b B^2 c \left (\log \left (\frac{a d-b c}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )-2 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right ) \log \left (\frac{e (a+b x)}{c+d x}\right )-2 \text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )\right )\right )}{3 b^2 g} \]

Antiderivative was successfully verified.

[In]

Integrate[((c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(a*g + b*g*x),x]

[Out]

(i*(3*A^2*b*d*x + 3*A^2*(b*c - a*d)*Log[a + b*x] - 3*A*B*(a*d*Log[a/b + x]^2 - 2*a*d*Log[a/b + x]*(1 + Log[a +
b*x]) + 2*(-(b*c) + a*d + Log[c/d + x]*(b*c + a*d*Log[a + b*x] - a*d*Log[(d*(a + b*x))/(-(b*c) + a*d)]) + (-(
b*d*x) + a*d*Log[a + b*x])*Log[(e*(a + b*x))/(c + d*x)]) - 2*a*d*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]) + 3*A*
b*B*c*(Log[a/b + x]^2 - 2*Log[a + b*x]*(Log[a/b + x] - Log[c/d + x] - Log[(e*(a + b*x))/(c + d*x)]) - 2*(Log[c
/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] + PolyLog[2, (b*(c + d*x))/(b*c - a*d)])) - B^2*(a*d*Log[a/b + x]^3
- 3*d*(2*b*x - 2*(a + b*x)*Log[a/b + x] + (a + b*x)*Log[a/b + x]^2) - 3*b*(2*d*x - 2*(c + d*x)*Log[c/d + x] +
(c + d*x)*Log[c/d + x]^2) - 3*d*(b*x - a*Log[a + b*x])*(-Log[a/b + x] + Log[c/d + x] + Log[(e*(a + b*x))/(c +
d*x)])^2 + 6*(a*d + 2*b*d*x - b*d*x*Log[c/d + x] - b*c*Log[c + d*x] + Log[a/b + x]*(-(d*(a + b*x)) + d*(a + b*
x)*Log[c/d + x] + (b*c - a*d)*Log[(b*(c + d*x))/(b*c - a*d)]) + (b*c - a*d)*PolyLog[2, (d*(a + b*x))/(-(b*c) +
a*d)]) - 3*(Log[a/b + x] - Log[c/d + x] - Log[(e*(a + b*x))/(c + d*x)])*(-2*b*c + 2*a*d - 2*d*(a + b*x)*Log[a
/b + x] + a*d*Log[a/b + x]^2 + 2*Log[c/d + x]*(b*(c + d*x) - a*d*Log[(d*(a + b*x))/(-(b*c) + a*d)]) - 2*a*d*Po
lyLog[2, (b*(c + d*x))/(b*c - a*d)]) - 3*a*d*(Log[a/b + x]^2*(Log[c/d + x] - Log[(b*(c + d*x))/(b*c - a*d)]) -
2*Log[a/b + x]*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] + 2*PolyLog[3, (d*(a + b*x))/(-(b*c) + a*d)]) + 3*a*d
*(Log[c/d + x]^2*Log[(d*(a + b*x))/(-(b*c) + a*d)] + 2*Log[c/d + x]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)] - 2*
PolyLog[3, (b*(c + d*x))/(b*c - a*d)])) - 3*b*B^2*c*(Log[(-(b*c) + a*d)/(d*(a + b*x))]*Log[(e*(a + b*x))/(c +
d*x)]^2 - 2*Log[(e*(a + b*x))/(c + d*x)]*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))] - 2*PolyLog[3, (b*(c + d*x))/
(d*(a + b*x))])))/(3*b^2*g)

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Maple [F]  time = 2.77, size = 0, normalized size = 0. \begin{align*} \int{\frac{dix+ci}{bgx+ag} \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) }{dx+c}} \right ) \right ) ^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g),x)

[Out]

int((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} A^{2} d i{\left (\frac{x}{b g} - \frac{a \log \left (b x + a\right )}{b^{2} g}\right )} + \frac{A^{2} c i \log \left (b g x + a g\right )}{b g} + \frac{{\left (B^{2} b d i x +{\left (b c i - a d i\right )} B^{2} \log \left (b x + a\right )\right )} \log \left (d x + c\right )^{2}}{b^{2} g} - \int -\frac{B^{2} b^{2} c^{2} i \log \left (e\right )^{2} + 2 \, A B b^{2} c^{2} i \log \left (e\right ) +{\left (B^{2} b^{2} d^{2} i \log \left (e\right )^{2} + 2 \, A B b^{2} d^{2} i \log \left (e\right )\right )} x^{2} +{\left (B^{2} b^{2} d^{2} i x^{2} + 2 \, B^{2} b^{2} c d i x + B^{2} b^{2} c^{2} i\right )} \log \left (b x + a\right )^{2} + 2 \,{\left (B^{2} b^{2} c d i \log \left (e\right )^{2} + 2 \, A B b^{2} c d i \log \left (e\right )\right )} x + 2 \,{\left (B^{2} b^{2} c^{2} i \log \left (e\right ) + A B b^{2} c^{2} i +{\left (B^{2} b^{2} d^{2} i \log \left (e\right ) + A B b^{2} d^{2} i\right )} x^{2} + 2 \,{\left (B^{2} b^{2} c d i \log \left (e\right ) + A B b^{2} c d i\right )} x\right )} \log \left (b x + a\right ) - 2 \,{\left (B^{2} b^{2} c^{2} i \log \left (e\right ) + A B b^{2} c^{2} i +{\left ({\left (i \log \left (e\right ) + i\right )} B^{2} b^{2} d^{2} + A B b^{2} d^{2} i\right )} x^{2} +{\left (2 \, A B b^{2} c d i +{\left (2 \, b^{2} c d i \log \left (e\right ) + a b d^{2} i\right )} B^{2}\right )} x +{\left (B^{2} b^{2} d^{2} i x^{2} +{\left (3 \, b^{2} c d i - a b d^{2} i\right )} B^{2} x +{\left (b^{2} c^{2} i + a b c d i - a^{2} d^{2} i\right )} B^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{b^{3} d g x^{2} + a b^{2} c g +{\left (b^{3} c g + a b^{2} d g\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g),x, algorithm="maxima")

[Out]

A^2*d*i*(x/(b*g) - a*log(b*x + a)/(b^2*g)) + A^2*c*i*log(b*g*x + a*g)/(b*g) + (B^2*b*d*i*x + (b*c*i - a*d*i)*B
^2*log(b*x + a))*log(d*x + c)^2/(b^2*g) - integrate(-(B^2*b^2*c^2*i*log(e)^2 + 2*A*B*b^2*c^2*i*log(e) + (B^2*b
^2*d^2*i*log(e)^2 + 2*A*B*b^2*d^2*i*log(e))*x^2 + (B^2*b^2*d^2*i*x^2 + 2*B^2*b^2*c*d*i*x + B^2*b^2*c^2*i)*log(
b*x + a)^2 + 2*(B^2*b^2*c*d*i*log(e)^2 + 2*A*B*b^2*c*d*i*log(e))*x + 2*(B^2*b^2*c^2*i*log(e) + A*B*b^2*c^2*i +
(B^2*b^2*d^2*i*log(e) + A*B*b^2*d^2*i)*x^2 + 2*(B^2*b^2*c*d*i*log(e) + A*B*b^2*c*d*i)*x)*log(b*x + a) - 2*(B^
2*b^2*c^2*i*log(e) + A*B*b^2*c^2*i + ((i*log(e) + i)*B^2*b^2*d^2 + A*B*b^2*d^2*i)*x^2 + (2*A*B*b^2*c*d*i + (2*
b^2*c*d*i*log(e) + a*b*d^2*i)*B^2)*x + (B^2*b^2*d^2*i*x^2 + (3*b^2*c*d*i - a*b*d^2*i)*B^2*x + (b^2*c^2*i + a*b
*c*d*i - a^2*d^2*i)*B^2)*log(b*x + a))*log(d*x + c))/(b^3*d*g*x^2 + a*b^2*c*g + (b^3*c*g + a*b^2*d*g)*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{A^{2} d i x + A^{2} c i +{\left (B^{2} d i x + B^{2} c i\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} + 2 \,{\left (A B d i x + A B c i\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{b g x + a g}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g),x, algorithm="fricas")

[Out]

integral((A^2*d*i*x + A^2*c*i + (B^2*d*i*x + B^2*c*i)*log((b*e*x + a*e)/(d*x + c))^2 + 2*(A*B*d*i*x + A*B*c*i)
*log((b*e*x + a*e)/(d*x + c)))/(b*g*x + a*g), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))**2/(b*g*x+a*g),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d i x + c i\right )}{\left (B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{b g x + a g}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g),x, algorithm="giac")

[Out]

integrate((d*i*x + c*i)*(B*log((b*x + a)*e/(d*x + c)) + A)^2/(b*g*x + a*g), x)