\(\int (a g+b g x) (c i+d i x)^3 (A+B \log (\frac {e (a+b x)}{c+d x}))^2 \, dx\) [76]

Optimal result

Integrand size = 40, antiderivative size = 730 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\frac {B^2 (b c-a d)^4 g i^3 x}{60 b^3 d}+\frac {B^2 (b c-a d)^3 g i^3 (c+d x)^2}{30 b^2 d^2}+\frac {B^2 (b c-a d)^2 g i^3 (c+d x)^3}{30 b d^2}-\frac {B^2 (b c-a d)^5 g i^3 \log \left (\frac {a+b x}{c+d x}\right )}{12 b^4 d^2}-\frac {B (b c-a d)^4 g i^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^4 d}-\frac {B (b c-a d)^3 g i^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^4}+\frac {3 B (b c-a d)^3 g i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{20 b^2 d^2}+\frac {B (b c-a d)^2 g i^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^2}-\frac {B (b c-a d) g i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 d^2}+\frac {(b c-a d)^3 g i^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{20 b^4}+\frac {(b c-a d)^2 g i^3 (a+b x)^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{10 b^3}+\frac {3 (b c-a d) g i^3 (a+b x)^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{20 b^2}+\frac {g i^3 (a+b x)^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b}-\frac {B (b c-a d)^5 g i^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^4 d^2}-\frac {11 B^2 (b c-a d)^5 g i^3 \log (c+d x)}{60 b^4 d^2}-\frac {B^2 (b c-a d)^5 g i^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{10 b^4 d^2} \]

[Out]

1/60*B^2*(-a*d+b*c)^4*g*i^3*x/b^3/d+1/30*B^2*(-a*d+b*c)^3*g*i^3*(d*x+c)^2/b^2/d^2+1/30*B^2*(-a*d+b*c)^2*g*i^3*
(d*x+c)^3/b/d^2-1/12*B^2*(-a*d+b*c)^5*g*i^3*ln((b*x+a)/(d*x+c))/b^4/d^2-1/10*B*(-a*d+b*c)^4*g*i^3*(b*x+a)*(A+B
*ln(e*(b*x+a)/(d*x+c)))/b^4/d-1/10*B*(-a*d+b*c)^3*g*i^3*(b*x+a)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^4+3/20*B*(-a*d
+b*c)^3*g*i^3*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^2/d^2+1/30*B*(-a*d+b*c)^2*g*i^3*(d*x+c)^3*(A+B*ln(e*(b*x
+a)/(d*x+c)))/b/d^2-1/10*B*(-a*d+b*c)*g*i^3*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^2+1/20*(-a*d+b*c)^3*g*i^3*
(b*x+a)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/b^4+1/10*(-a*d+b*c)^2*g*i^3*(b*x+a)^2*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c
)))^2/b^3+3/20*(-a*d+b*c)*g*i^3*(b*x+a)^2*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/b^2+1/5*g*i^3*(b*x+a)^2*(d*x
+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/b-1/10*B*(-a*d+b*c)^5*g*i^3*ln((-a*d+b*c)/b/(d*x+c))*(A+B+B*ln(e*(b*x+a)/(
d*x+c)))/b^4/d^2-11/60*B^2*(-a*d+b*c)^5*g*i^3*ln(d*x+c)/b^4/d^2-1/10*B^2*(-a*d+b*c)^5*g*i^3*polylog(2,d*(b*x+a
)/b/(d*x+c))/b^4/d^2

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 730, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {2562, 2383, 2381, 2384, 2354, 2438, 2373, 45, 2382, 12, 78} \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=-\frac {B g i^3 (b c-a d)^5 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A+B\right )}{10 b^4 d^2}-\frac {B g i^3 (a+b x) (b c-a d)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{10 b^4 d}+\frac {g i^3 (a+b x)^2 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{20 b^4}-\frac {B g i^3 (a+b x)^2 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{10 b^4}+\frac {g i^3 (a+b x)^2 (c+d x) (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{10 b^3}+\frac {3 B g i^3 (c+d x)^2 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{20 b^2 d^2}+\frac {3 g i^3 (a+b x)^2 (c+d x)^2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{20 b^2}+\frac {B g i^3 (c+d x)^3 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{30 b d^2}-\frac {B g i^3 (c+d x)^4 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{10 d^2}+\frac {g i^3 (a+b x)^2 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{5 b}-\frac {B^2 g i^3 (b c-a d)^5 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{10 b^4 d^2}-\frac {B^2 g i^3 (b c-a d)^5 \log \left (\frac {a+b x}{c+d x}\right )}{12 b^4 d^2}-\frac {11 B^2 g i^3 (b c-a d)^5 \log (c+d x)}{60 b^4 d^2}+\frac {B^2 g i^3 x (b c-a d)^4}{60 b^3 d}+\frac {B^2 g i^3 (c+d x)^2 (b c-a d)^3}{30 b^2 d^2}+\frac {B^2 g i^3 (c+d x)^3 (b c-a d)^2}{30 b d^2} \]

[In]

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

[Out]

(B^2*(b*c - a*d)^4*g*i^3*x)/(60*b^3*d) + (B^2*(b*c - a*d)^3*g*i^3*(c + d*x)^2)/(30*b^2*d^2) + (B^2*(b*c - a*d)
^2*g*i^3*(c + d*x)^3)/(30*b*d^2) - (B^2*(b*c - a*d)^5*g*i^3*Log[(a + b*x)/(c + d*x)])/(12*b^4*d^2) - (B*(b*c -
a*d)^4*g*i^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(10*b^4*d) - (B*(b*c - a*d)^3*g*i^3*(a + b*x)^2*
(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(10*b^4) + (3*B*(b*c - a*d)^3*g*i^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))
/(c + d*x)]))/(20*b^2*d^2) + (B*(b*c - a*d)^2*g*i^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(30*b*d^
2) - (B*(b*c - a*d)*g*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(10*d^2) + ((b*c - a*d)^3*g*i^3*(a
+ b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(20*b^4) + ((b*c - a*d)^2*g*i^3*(a + b*x)^2*(c + d*x)*(A + B
*Log[(e*(a + b*x))/(c + d*x)])^2)/(10*b^3) + (3*(b*c - a*d)*g*i^3*(a + b*x)^2*(c + d*x)^2*(A + B*Log[(e*(a + b
*x))/(c + d*x)])^2)/(20*b^2) + (g*i^3*(a + b*x)^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(5*b) -
(B*(b*c - a*d)^5*g*i^3*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B + B*Log[(e*(a + b*x))/(c + d*x)]))/(10*b^4*d^2) -
(11*B^2*(b*c - a*d)^5*g*i^3*Log[c + d*x])/(60*b^4*d^2) - (B^2*(b*c - a*d)^5*g*i^3*PolyLog[2, (d*(a + b*x))/(b
*(c + d*x))])/(10*b^4*d^2)

Rule 12

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

Rule 45

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 2354

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 2373

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

Rule 2381

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> Simp
[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(d*f*(q + 1))), x] + Dist[b*n*(p/(d*(q + 1))), Int[(
f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[m
+ q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 2382

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> With[{u = IntHide[
x^m*(d + e*x)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ
[{a, b, c, d, e, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[m, 0]

Rule 2383

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> Simp
[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(d*f*(q + 1))), x] + (Dist[(m + q + 2)/(d*(q + 1)),
Int[(f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p, x], x] + Dist[b*n*(p/(d*(q + 1))), Int[(f*x)^m*(d + e*x)^(
q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[p,
0] && LtQ[q, -1] && GtQ[m, 0]

Rule 2384

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(f*x
)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])/(e*(q + 1))), x] - Dist[f/(e*(q + 1)), Int[(f*x)^(m - 1)*(d + e*x)^(
q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 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 2562

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_)
)^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q, Subst[Int[x^m*(
(A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h,
i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i
, 0] && IntegersQ[m, q]

Rubi steps \begin{align*} \text {integral}& = \left ((b c-a d)^5 g i^3\right ) \text {Subst}\left (\int \frac {x (A+B \log (e x))^2}{(b-d x)^6} \, dx,x,\frac {a+b x}{c+d x}\right ) \\ & = \frac {g i^3 (a+b x)^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b}+\frac {\left (3 (b c-a d)^5 g i^3\right ) \text {Subst}\left (\int \frac {x (A+B \log (e x))^2}{(b-d x)^5} \, dx,x,\frac {a+b x}{c+d x}\right )}{5 b}-\frac {\left (2 B (b c-a d)^5 g i^3\right ) \text {Subst}\left (\int \frac {x (A+B \log (e x))}{(b-d x)^5} \, dx,x,\frac {a+b x}{c+d x}\right )}{5 b} \\ & = \frac {2 B (b c-a d)^2 g i^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b d^2}-\frac {B (b c-a d) g i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 d^2}+\frac {3 (b c-a d) g i^3 (a+b x)^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{20 b^2}+\frac {g i^3 (a+b x)^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b}+\frac {\left (3 (b c-a d)^5 g i^3\right ) \text {Subst}\left (\int \frac {x (A+B \log (e x))^2}{(b-d x)^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{10 b^2}-\frac {\left (3 B (b c-a d)^5 g i^3\right ) \text {Subst}\left (\int \frac {x (A+B \log (e x))}{(b-d x)^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{10 b^2}+\frac {\left (2 B^2 (b c-a d)^5 g i^3\right ) \text {Subst}\left (\int \frac {-b+4 d x}{12 d^2 x (b-d x)^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{5 b} \\ & = \frac {3 B (b c-a d)^3 g i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{20 b^2 d^2}+\frac {B (b c-a d)^2 g i^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^2}-\frac {B (b c-a d) g i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 d^2}+\frac {(b c-a d)^2 g i^3 (a+b x)^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{10 b^3}+\frac {3 (b c-a d) g i^3 (a+b x)^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{20 b^2}+\frac {g i^3 (a+b x)^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b}+\frac {\left ((b c-a d)^5 g i^3\right ) \text {Subst}\left (\int \frac {x (A+B \log (e x))^2}{(b-d x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{10 b^3}-\frac {\left (B (b c-a d)^5 g i^3\right ) \text {Subst}\left (\int \frac {x (A+B \log (e x))}{(b-d x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{5 b^3}+\frac {\left (3 B^2 (b c-a d)^5 g i^3\right ) \text {Subst}\left (\int \frac {-b+3 d x}{6 d^2 x (b-d x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{10 b^2}+\frac {\left (B^2 (b c-a d)^5 g i^3\right ) \text {Subst}\left (\int \frac {-b+4 d x}{x (b-d x)^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{30 b d^2} \\ & = -\frac {B (b c-a d)^3 g i^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^4}+\frac {3 B (b c-a d)^3 g i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{20 b^2 d^2}+\frac {B (b c-a d)^2 g i^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^2}-\frac {B (b c-a d) g i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 d^2}+\frac {(b c-a d)^3 g i^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{20 b^4}+\frac {(b c-a d)^2 g i^3 (a+b x)^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{10 b^3}+\frac {3 (b c-a d) g i^3 (a+b x)^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{20 b^2}+\frac {g i^3 (a+b x)^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b}-\frac {\left (B (b c-a d)^5 g i^3\right ) \text {Subst}\left (\int \frac {x (A+B \log (e x))}{(b-d x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{10 b^4}+\frac {\left (B^2 (b c-a d)^5 g i^3\right ) \text {Subst}\left (\int \frac {x}{(b-d x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{10 b^4}+\frac {\left (B^2 (b c-a d)^5 g i^3\right ) \text {Subst}\left (\int \frac {-b+3 d x}{x (b-d x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{20 b^2 d^2}+\frac {\left (B^2 (b c-a d)^5 g i^3\right ) \text {Subst}\left (\int \left (-\frac {1}{b^3 x}+\frac {3 d}{(b-d x)^4}-\frac {d}{b (b-d x)^3}-\frac {d}{b^2 (b-d x)^2}-\frac {d}{b^3 (b-d x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{30 b d^2} \\ & = -\frac {B^2 (b c-a d)^4 g i^3 x}{30 b^3 d}-\frac {B^2 (b c-a d)^3 g i^3 (c+d x)^2}{60 b^2 d^2}+\frac {B^2 (b c-a d)^2 g i^3 (c+d x)^3}{30 b d^2}-\frac {B^2 (b c-a d)^5 g i^3 \log \left (\frac {a+b x}{c+d x}\right )}{30 b^4 d^2}-\frac {B (b c-a d)^4 g i^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^4 d}-\frac {B (b c-a d)^3 g i^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^4}+\frac {3 B (b c-a d)^3 g i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{20 b^2 d^2}+\frac {B (b c-a d)^2 g i^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^2}-\frac {B (b c-a d) g i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 d^2}+\frac {(b c-a d)^3 g i^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{20 b^4}+\frac {(b c-a d)^2 g i^3 (a+b x)^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{10 b^3}+\frac {3 (b c-a d) g i^3 (a+b x)^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{20 b^2}+\frac {g i^3 (a+b x)^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b}-\frac {B^2 (b c-a d)^5 g i^3 \log (c+d x)}{30 b^4 d^2}+\frac {\left (B^2 (b c-a d)^5 g i^3\right ) \text {Subst}\left (\int \left (\frac {b}{d (-b+d x)^2}+\frac {1}{d (-b+d x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{10 b^4}+\frac {\left (B^2 (b c-a d)^5 g i^3\right ) \text {Subst}\left (\int \left (-\frac {1}{b^2 x}+\frac {2 d}{(b-d x)^3}-\frac {d}{b (b-d x)^2}-\frac {d}{b^2 (b-d x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{20 b^2 d^2}+\frac {\left (B (b c-a d)^5 g i^3\right ) \text {Subst}\left (\int \frac {A+B+B \log (e x)}{b-d x} \, dx,x,\frac {a+b x}{c+d x}\right )}{10 b^4 d} \\ & = \frac {B^2 (b c-a d)^4 g i^3 x}{60 b^3 d}+\frac {B^2 (b c-a d)^3 g i^3 (c+d x)^2}{30 b^2 d^2}+\frac {B^2 (b c-a d)^2 g i^3 (c+d x)^3}{30 b d^2}-\frac {B^2 (b c-a d)^5 g i^3 \log \left (\frac {a+b x}{c+d x}\right )}{12 b^4 d^2}-\frac {B (b c-a d)^4 g i^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^4 d}-\frac {B (b c-a d)^3 g i^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^4}+\frac {3 B (b c-a d)^3 g i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{20 b^2 d^2}+\frac {B (b c-a d)^2 g i^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^2}-\frac {B (b c-a d) g i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 d^2}+\frac {(b c-a d)^3 g i^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{20 b^4}+\frac {(b c-a d)^2 g i^3 (a+b x)^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{10 b^3}+\frac {3 (b c-a d) g i^3 (a+b x)^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{20 b^2}+\frac {g i^3 (a+b x)^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b}-\frac {B (b c-a d)^5 g i^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^4 d^2}-\frac {11 B^2 (b c-a d)^5 g i^3 \log (c+d x)}{60 b^4 d^2}+\frac {\left (B^2 (b c-a d)^5 g i^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d x}{b}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{10 b^4 d^2} \\ & = \frac {B^2 (b c-a d)^4 g i^3 x}{60 b^3 d}+\frac {B^2 (b c-a d)^3 g i^3 (c+d x)^2}{30 b^2 d^2}+\frac {B^2 (b c-a d)^2 g i^3 (c+d x)^3}{30 b d^2}-\frac {B^2 (b c-a d)^5 g i^3 \log \left (\frac {a+b x}{c+d x}\right )}{12 b^4 d^2}-\frac {B (b c-a d)^4 g i^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^4 d}-\frac {B (b c-a d)^3 g i^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^4}+\frac {3 B (b c-a d)^3 g i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{20 b^2 d^2}+\frac {B (b c-a d)^2 g i^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^2}-\frac {B (b c-a d) g i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 d^2}+\frac {(b c-a d)^3 g i^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{20 b^4}+\frac {(b c-a d)^2 g i^3 (a+b x)^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{10 b^3}+\frac {3 (b c-a d) g i^3 (a+b x)^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{20 b^2}+\frac {g i^3 (a+b x)^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b}-\frac {B (b c-a d)^5 g i^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b^4 d^2}-\frac {11 B^2 (b c-a d)^5 g i^3 \log (c+d x)}{60 b^4 d^2}-\frac {B^2 (b c-a d)^5 g i^3 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{10 b^4 d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 901, normalized size of antiderivative = 1.23 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\frac {g i^3 \left (-5 (b c-a d) (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+4 b (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+\frac {5 B (b c-a d)^2 \left (6 A b d (b c-a d)^2 x-3 B (b c-a d)^2 (b d x+(b c-a d) \log (a+b x))-B (b c-a d) \left (2 b d (b c-a d) x+b^2 (c+d x)^2+2 (b c-a d)^2 \log (a+b x)\right )+6 B d (b c-a d)^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+3 b^2 (b c-a d) (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 b^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+6 (b c-a d)^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-6 B (b c-a d)^3 \log (c+d x)-3 B (b c-a d)^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{3 b^4}-\frac {B (b c-a d) \left (24 A b d (b c-a d)^3 x-12 B (b c-a d)^3 (b d x+(b c-a d) \log (a+b x))-4 B (b c-a d)^2 \left (2 b d (b c-a d) x+b^2 (c+d x)^2+2 (b c-a d)^2 \log (a+b x)\right )-B (b c-a d) \left (6 b d (b c-a d)^2 x+3 b^2 (b c-a d) (c+d x)^2+2 b^3 (c+d x)^3+6 (b c-a d)^3 \log (a+b x)\right )+24 B d (b c-a d)^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+12 b^2 (b c-a d)^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+8 b^3 (b c-a d) (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+6 b^4 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+24 (b c-a d)^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-24 B (b c-a d)^4 \log (c+d x)-12 B (b c-a d)^4 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{3 b^4}\right )}{20 d^2} \]

[In]

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

[Out]

(g*i^3*(-5*(b*c - a*d)*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 4*b*(c + d*x)^5*(A + B*Log[(e*(a +
b*x))/(c + d*x)])^2 + (5*B*(b*c - a*d)^2*(6*A*b*d*(b*c - a*d)^2*x - 3*B*(b*c - a*d)^2*(b*d*x + (b*c - a*d)*Lo
g[a + b*x]) - B*(b*c - a*d)*(2*b*d*(b*c - a*d)*x + b^2*(c + d*x)^2 + 2*(b*c - a*d)^2*Log[a + b*x]) + 6*B*d*(b*
c - a*d)^2*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] + 3*b^2*(b*c - a*d)*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c
+ d*x)]) + 2*b^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 6*(b*c - a*d)^3*Log[a + b*x]*(A + B*Log[(e
*(a + b*x))/(c + d*x)]) - 6*B*(b*c - a*d)^3*Log[c + d*x] - 3*B*(b*c - a*d)^3*(Log[a + b*x]*(Log[a + b*x] - 2*L
og[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)])))/(3*b^4) - (B*(b*c - a*d)*(24*A*
b*d*(b*c - a*d)^3*x - 12*B*(b*c - a*d)^3*(b*d*x + (b*c - a*d)*Log[a + b*x]) - 4*B*(b*c - a*d)^2*(2*b*d*(b*c -
a*d)*x + b^2*(c + d*x)^2 + 2*(b*c - a*d)^2*Log[a + b*x]) - B*(b*c - a*d)*(6*b*d*(b*c - a*d)^2*x + 3*b^2*(b*c -
a*d)*(c + d*x)^2 + 2*b^3*(c + d*x)^3 + 6*(b*c - a*d)^3*Log[a + b*x]) + 24*B*d*(b*c - a*d)^3*(a + b*x)*Log[(e*
(a + b*x))/(c + d*x)] + 12*b^2*(b*c - a*d)^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 8*b^3*(b*c - a
*d)*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 6*b^4*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])
+ 24*(b*c - a*d)^4*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 24*B*(b*c - a*d)^4*Log[c + d*x] - 12*B*
(b*c - a*d)^4*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(
b*c) + a*d)])))/(3*b^4)))/(20*d^2)

Maple [F]

\[\int \left (b g x +a g \right ) \left (d i x +c i \right )^{3} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}d x\]

[In]

int((b*g*x+a*g)*(d*i*x+c*i)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2,x)

[Out]

int((b*g*x+a*g)*(d*i*x+c*i)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2,x)

Fricas [F]

\[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )} {\left (d i x + c i\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2} \,d x } \]

[In]

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

[Out]

integral(A^2*b*d^3*g*i^3*x^4 + A^2*a*c^3*g*i^3 + (3*A^2*b*c*d^2 + A^2*a*d^3)*g*i^3*x^3 + 3*(A^2*b*c^2*d + A^2*
a*c*d^2)*g*i^3*x^2 + (A^2*b*c^3 + 3*A^2*a*c^2*d)*g*i^3*x + (B^2*b*d^3*g*i^3*x^4 + B^2*a*c^3*g*i^3 + (3*B^2*b*c
*d^2 + B^2*a*d^3)*g*i^3*x^3 + 3*(B^2*b*c^2*d + B^2*a*c*d^2)*g*i^3*x^2 + (B^2*b*c^3 + 3*B^2*a*c^2*d)*g*i^3*x)*l
og((b*e*x + a*e)/(d*x + c))^2 + 2*(A*B*b*d^3*g*i^3*x^4 + A*B*a*c^3*g*i^3 + (3*A*B*b*c*d^2 + A*B*a*d^3)*g*i^3*x
^3 + 3*(A*B*b*c^2*d + A*B*a*c*d^2)*g*i^3*x^2 + (A*B*b*c^3 + 3*A*B*a*c^2*d)*g*i^3*x)*log((b*e*x + a*e)/(d*x + c
)), x)

Sympy [F(-1)]

Timed out. \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3218 vs. \(2 (697) = 1394\).

Time = 0.32 (sec) , antiderivative size = 3218, normalized size of antiderivative = 4.41 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/5*A^2*b*d^3*g*i^3*x^5 + 3/4*A^2*b*c*d^2*g*i^3*x^4 + 1/4*A^2*a*d^3*g*i^3*x^4 + A^2*b*c^2*d*g*i^3*x^3 + A^2*a*
c*d^2*g*i^3*x^3 + 1/2*A^2*b*c^3*g*i^3*x^2 + 3/2*A^2*a*c^2*d*g*i^3*x^2 + 2*(x*log(b*e*x/(d*x + c) + a*e/(d*x +
c)) + a*log(b*x + a)/b - c*log(d*x + c)/d)*A*B*a*c^3*g*i^3 + (x^2*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - a^2*l
og(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*A*B*b*c^3*g*i^3 + 3*(x^2*log(b*e*x/(d*x + c) + a
*e/(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*A*B*a*c^2*d*g*i^3 + (2*x^3*
log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*
x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*A*B*b*c^2*d*g*i^3 + (2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 2*
a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))
*A*B*a*c*d^2*g*i^3 + 1/4*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x
+ c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3)
)*A*B*b*c*d^2*g*i^3 + 1/12*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*
x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^
3))*A*B*a*d^3*g*i^3 + 1/30*(12*x^5*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 12*a^5*log(b*x + a)/b^5 - 12*c^5*log
(d*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 - a^2*b^2*d^4)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)
*x^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4*d^4))*A*B*b*d^3*g*i^3 + A^2*a*c^3*g*i^3*x - 1/60*(47*a^2*b^2*c^3*d^2*g*i
^3 - 27*a^3*b*c^2*d^3*g*i^3 + 6*a^4*c*d^4*g*i^3 - (6*g*i^3*log(e) - 5*g*i^3)*b^4*c^5 + (30*g*i^3*log(e) - 31*g
*i^3)*a*b^3*c^4*d)*B^2*log(d*x + c)/(b^3*d^2) + 1/10*(b^5*c^5*g*i^3 - 5*a*b^4*c^4*d*g*i^3 + 10*a^2*b^3*c^3*d^2
*g*i^3 - 10*a^3*b^2*c^2*d^3*g*i^3 + 5*a^4*b*c*d^4*g*i^3 - a^5*d^5*g*i^3)*(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^2/(b^4*d^2) + 1/60*(12*B^2*b^5*d^5*g*i^3*x^5*log(e)^2 + 3*(
(15*g*i^3*log(e)^2 - 2*g*i^3*log(e))*b^5*c*d^4 + (5*g*i^3*log(e)^2 + 2*g*i^3*log(e))*a*b^4*d^5)*B^2*x^4 + 2*((
30*g*i^3*log(e)^2 - 11*g*i^3*log(e) + g*i^3)*b^5*c^2*d^3 + 2*(15*g*i^3*log(e)^2 + 5*g*i^3*log(e) - g*i^3)*a*b^
4*c*d^4 + (g*i^3*log(e) + g*i^3)*a^2*b^3*d^5)*B^2*x^3 + ((30*g*i^3*log(e)^2 - 27*g*i^3*log(e) + 8*g*i^3)*b^5*c
^3*d^2 + 3*(30*g*i^3*log(e)^2 + 5*g*i^3*log(e) - 6*g*i^3)*a*b^4*c^2*d^3 + 3*(5*g*i^3*log(e) + 4*g*i^3)*a^2*b^3
*c*d^4 - (3*g*i^3*log(e) + 2*g*i^3)*a^3*b^2*d^5)*B^2*x^2 - ((6*g*i^3*log(e) - 11*g*i^3)*b^5*c^4*d - 2*(30*g*i^
3*log(e)^2 - 15*g*i^3*log(e) - 14*g*i^3)*a*b^4*c^3*d^2 - 12*(5*g*i^3*log(e) + 2*g*i^3)*a^2*b^3*c^2*d^3 + 2*(15
*g*i^3*log(e) + 4*g*i^3)*a^3*b^2*c*d^4 - (6*g*i^3*log(e) + g*i^3)*a^4*b*d^5)*B^2*x + 3*(4*B^2*b^5*d^5*g*i^3*x^
5 + 20*B^2*a*b^4*c^3*d^2*g*i^3*x + 5*(3*b^5*c*d^4*g*i^3 + a*b^4*d^5*g*i^3)*B^2*x^4 + 20*(b^5*c^2*d^3*g*i^3 + a
*b^4*c*d^4*g*i^3)*B^2*x^3 + 10*(b^5*c^3*d^2*g*i^3 + 3*a*b^4*c^2*d^3*g*i^3)*B^2*x^2 + (10*a^2*b^3*c^3*d^2*g*i^3
- 10*a^3*b^2*c^2*d^3*g*i^3 + 5*a^4*b*c*d^4*g*i^3 - a^5*d^5*g*i^3)*B^2)*log(b*x + a)^2 + 3*(4*B^2*b^5*d^5*g*i^
3*x^5 + 20*B^2*a*b^4*c^3*d^2*g*i^3*x + 5*(3*b^5*c*d^4*g*i^3 + a*b^4*d^5*g*i^3)*B^2*x^4 + 20*(b^5*c^2*d^3*g*i^3
+ a*b^4*c*d^4*g*i^3)*B^2*x^3 + 10*(b^5*c^3*d^2*g*i^3 + 3*a*b^4*c^2*d^3*g*i^3)*B^2*x^2 - (b^5*c^5*g*i^3 - 5*a*
b^4*c^4*d*g*i^3)*B^2)*log(d*x + c)^2 + (24*B^2*b^5*d^5*g*i^3*x^5*log(e) + 6*((15*g*i^3*log(e) - g*i^3)*b^5*c*d
^4 + (5*g*i^3*log(e) + g*i^3)*a*b^4*d^5)*B^2*x^4 + 2*(a^2*b^3*d^5*g*i^3 + (60*g*i^3*log(e) - 11*g*i^3)*b^5*c^2
*d^3 + 10*(6*g*i^3*log(e) + g*i^3)*a*b^4*c*d^4)*B^2*x^3 + 3*(5*a^2*b^3*c*d^4*g*i^3 - a^3*b^2*d^5*g*i^3 + (20*g
*i^3*log(e) - 9*g*i^3)*b^5*c^3*d^2 + 5*(12*g*i^3*log(e) + g*i^3)*a*b^4*c^2*d^3)*B^2*x^2 - 6*(b^5*c^4*d*g*i^3 -
10*a^2*b^3*c^2*d^3*g*i^3 + 5*a^3*b^2*c*d^4*g*i^3 - a^4*b*d^5*g*i^3 - 5*(4*g*i^3*log(e) - g*i^3)*a*b^4*c^3*d^2
)*B^2*x - (6*a*b^4*c^4*d*g*i^3 - 3*(20*g*i^3*log(e) - g*i^3)*a^2*b^3*c^3*d^2 + (60*g*i^3*log(e) - 23*g*i^3)*a^
3*b^2*c^2*d^3 - (30*g*i^3*log(e) - 19*g*i^3)*a^4*b*c*d^4 + (6*g*i^3*log(e) - 5*g*i^3)*a^5*d^5)*B^2)*log(b*x +
a) - (24*B^2*b^5*d^5*g*i^3*x^5*log(e) + 6*((15*g*i^3*log(e) - g*i^3)*b^5*c*d^4 + (5*g*i^3*log(e) + g*i^3)*a*b^
4*d^5)*B^2*x^4 + 2*(a^2*b^3*d^5*g*i^3 + (60*g*i^3*log(e) - 11*g*i^3)*b^5*c^2*d^3 + 10*(6*g*i^3*log(e) + g*i^3)
*a*b^4*c*d^4)*B^2*x^3 + 3*(5*a^2*b^3*c*d^4*g*i^3 - a^3*b^2*d^5*g*i^3 + (20*g*i^3*log(e) - 9*g*i^3)*b^5*c^3*d^2
+ 5*(12*g*i^3*log(e) + g*i^3)*a*b^4*c^2*d^3)*B^2*x^2 - 6*(b^5*c^4*d*g*i^3 - 10*a^2*b^3*c^2*d^3*g*i^3 + 5*a^3*
b^2*c*d^4*g*i^3 - a^4*b*d^5*g*i^3 - 5*(4*g*i^3*log(e) - g*i^3)*a*b^4*c^3*d^2)*B^2*x + 6*(4*B^2*b^5*d^5*g*i^3*x
^5 + 20*B^2*a*b^4*c^3*d^2*g*i^3*x + 5*(3*b^5*c*d^4*g*i^3 + a*b^4*d^5*g*i^3)*B^2*x^4 + 20*(b^5*c^2*d^3*g*i^3 +
a*b^4*c*d^4*g*i^3)*B^2*x^3 + 10*(b^5*c^3*d^2*g*i^3 + 3*a*b^4*c^2*d^3*g*i^3)*B^2*x^2 + (10*a^2*b^3*c^3*d^2*g*i^
3 - 10*a^3*b^2*c^2*d^3*g*i^3 + 5*a^4*b*c*d^4*g*i^3 - a^5*d^5*g*i^3)*B^2)*log(b*x + a))*log(d*x + c))/(b^4*d^2)

Giac [F]

\[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )} {\left (d i x + c i\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\int \left (a\,g+b\,g\,x\right )\,{\left (c\,i+d\,i\,x\right )}^3\,{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2 \,d x \]

[In]

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

[Out]

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