\(\int (a g+b g x)^2 (c i+d i x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))^2 \, dx\) [65]

Optimal result

Integrand size = 42, antiderivative size = 761 \[ \int (a g+b g x)^2 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=-\frac {B^2 (b c-a d)^4 g^2 i^2 x}{10 b^2 d^2}-\frac {B^2 (b c-a d)^3 g^2 i^2 (c+d x)^2}{20 b d^3}+\frac {B^2 (b c-a d)^2 g^2 i^2 (c+d x)^3}{30 d^3}+\frac {B^2 (b c-a d)^5 g^2 i^2 \log \left (\frac {a+b x}{c+d x}\right )}{30 b^3 d^3}-\frac {B (b c-a d)^3 g^2 i^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b^3 d}-\frac {B (b c-a d)^2 g^2 i^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3}-\frac {B (b c-a d)^3 g^2 i^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b d^3}+\frac {4 B (b c-a d)^2 g^2 i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 d^3}-\frac {b B (b c-a d) g^2 i^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 d^3}+\frac {(b c-a d)^2 g^2 i^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{30 b^3}+\frac {(b c-a d) g^2 i^2 (a+b x)^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{10 b^2}+\frac {g^2 i^2 (a+b x)^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b}+\frac {B (b c-a d)^4 g^2 i^2 (a+b x) \left (2 A+B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b^3 d^2}+\frac {B (b c-a d)^5 g^2 i^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (2 A+3 B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b^3 d^3}+\frac {B^2 (b c-a d)^5 g^2 i^2 \log (c+d x)}{10 b^3 d^3}+\frac {B^2 (b c-a d)^5 g^2 i^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{15 b^3 d^3} \]

[Out]

-1/10*B^2*(-a*d+b*c)^4*g^2*i^2*x/b^2/d^2-1/20*B^2*(-a*d+b*c)^3*g^2*i^2*(d*x+c)^2/b/d^3+1/30*B^2*(-a*d+b*c)^2*g
^2*i^2*(d*x+c)^3/d^3+1/30*B^2*(-a*d+b*c)^5*g^2*i^2*ln((b*x+a)/(d*x+c))/b^3/d^3-1/30*B*(-a*d+b*c)^3*g^2*i^2*(b*
x+a)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^3/d-1/15*B*(-a*d+b*c)^2*g^2*i^2*(b*x+a)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^3
-1/5*B*(-a*d+b*c)^3*g^2*i^2*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/b/d^3+4/15*B*(-a*d+b*c)^2*g^2*i^2*(d*x+c)^3*
(A+B*ln(e*(b*x+a)/(d*x+c)))/d^3-1/10*b*B*(-a*d+b*c)*g^2*i^2*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^3+1/30*(-a
*d+b*c)^2*g^2*i^2*(b*x+a)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/b^3+1/10*(-a*d+b*c)*g^2*i^2*(b*x+a)^3*(d*x+c)*(A+B*l
n(e*(b*x+a)/(d*x+c)))^2/b^2+1/5*g^2*i^2*(b*x+a)^3*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/b+1/30*B*(-a*d+b*c)^
4*g^2*i^2*(b*x+a)*(2*A+B+2*B*ln(e*(b*x+a)/(d*x+c)))/b^3/d^2+1/30*B*(-a*d+b*c)^5*g^2*i^2*ln((-a*d+b*c)/b/(d*x+c
))*(2*A+3*B+2*B*ln(e*(b*x+a)/(d*x+c)))/b^3/d^3+1/10*B^2*(-a*d+b*c)^5*g^2*i^2*ln(d*x+c)/b^3/d^3+1/15*B^2*(-a*d+
b*c)^5*g^2*i^2*polylog(2,d*(b*x+a)/b/(d*x+c))/b^3/d^3

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 761, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.262, Rules used = {2562, 2383, 2381, 2384, 2354, 2438, 2373, 45, 2382, 12, 907} \[ \int (a g+b g x)^2 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\frac {B g^2 i^2 (b c-a d)^5 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (2 B \log \left (\frac {e (a+b x)}{c+d x}\right )+2 A+3 B\right )}{30 b^3 d^3}+\frac {B g^2 i^2 (a+b x) (b c-a d)^4 \left (2 B \log \left (\frac {e (a+b x)}{c+d x}\right )+2 A+B\right )}{30 b^3 d^2}-\frac {B g^2 i^2 (a+b x)^2 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{30 b^3 d}+\frac {g^2 i^2 (a+b x)^3 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{30 b^3}-\frac {B g^2 i^2 (a+b x)^3 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{15 b^3}+\frac {g^2 i^2 (a+b x)^3 (c+d x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{10 b^2}-\frac {B g^2 i^2 (c+d x)^2 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 b d^3}+\frac {4 B g^2 i^2 (c+d x)^3 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{15 d^3}-\frac {b B g^2 i^2 (c+d x)^4 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{10 d^3}+\frac {g^2 i^2 (a+b x)^3 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{5 b}+\frac {B^2 g^2 i^2 (b c-a d)^5 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{15 b^3 d^3}+\frac {B^2 g^2 i^2 (b c-a d)^5 \log \left (\frac {a+b x}{c+d x}\right )}{30 b^3 d^3}+\frac {B^2 g^2 i^2 (b c-a d)^5 \log (c+d x)}{10 b^3 d^3}-\frac {B^2 g^2 i^2 x (b c-a d)^4}{10 b^2 d^2}-\frac {B^2 g^2 i^2 (c+d x)^2 (b c-a d)^3}{20 b d^3}+\frac {B^2 g^2 i^2 (c+d x)^3 (b c-a d)^2}{30 d^3} \]

[In]

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

[Out]

-1/10*(B^2*(b*c - a*d)^4*g^2*i^2*x)/(b^2*d^2) - (B^2*(b*c - a*d)^3*g^2*i^2*(c + d*x)^2)/(20*b*d^3) + (B^2*(b*c
- a*d)^2*g^2*i^2*(c + d*x)^3)/(30*d^3) + (B^2*(b*c - a*d)^5*g^2*i^2*Log[(a + b*x)/(c + d*x)])/(30*b^3*d^3) -
(B*(b*c - a*d)^3*g^2*i^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(30*b^3*d) - (B*(b*c - a*d)^2*g^2*i
^2*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(15*b^3) - (B*(b*c - a*d)^3*g^2*i^2*(c + d*x)^2*(A + B*Lo
g[(e*(a + b*x))/(c + d*x)]))/(5*b*d^3) + (4*B*(b*c - a*d)^2*g^2*i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c +
d*x)]))/(15*d^3) - (b*B*(b*c - a*d)*g^2*i^2*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(10*d^3) + ((b*c
- a*d)^2*g^2*i^2*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(30*b^3) + ((b*c - a*d)*g^2*i^2*(a + b*x
)^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(10*b^2) + (g^2*i^2*(a + b*x)^3*(c + d*x)^2*(A + B*Log[(
e*(a + b*x))/(c + d*x)])^2)/(5*b) + (B*(b*c - a*d)^4*g^2*i^2*(a + b*x)*(2*A + B + 2*B*Log[(e*(a + b*x))/(c + d
*x)]))/(30*b^3*d^2) + (B*(b*c - a*d)^5*g^2*i^2*Log[(b*c - a*d)/(b*(c + d*x))]*(2*A + 3*B + 2*B*Log[(e*(a + b*x
))/(c + d*x)]))/(30*b^3*d^3) + (B^2*(b*c - a*d)^5*g^2*i^2*Log[c + d*x])/(10*b^3*d^3) + (B^2*(b*c - a*d)^5*g^2*
i^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(15*b^3*d^3)

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 907

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

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

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 1194, normalized size of antiderivative = 1.57 \[ \int (a g+b g x)^2 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\frac {g^2 i^2 \left (20 d^3 (b c-a d)^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+30 d^4 (b c-a d) (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+12 d^5 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+20 B (b c-a d)^3 \left (2 A b d (b c-a d) x+2 B d (b c-a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2 B (b c-a d)^2 \log (c+d x)-2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+B (b c-a d) (b d x+(-b c+a d) \log (c+d x))+B (b c-a d)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )-10 B (b c-a d)^2 \left (6 A b d (b c-a d)^2 x+6 B d (b c-a d)^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+3 d^2 (-b c+a d) (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 d^3 (a+b x)^3 \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)-6 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+B (b c-a d) \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )+3 B (b c-a d)^2 (b d x+(-b c+a d) \log (c+d x))+3 B (b c-a d)^3 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )+B (b c-a d) \left (24 A b d (b c-a d)^3 x+24 B d (b c-a d)^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-12 d^2 (b c-a d)^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+8 d^3 (b c-a d) (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-6 d^4 (a+b x)^4 \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)-24 (b c-a d)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+4 B (b c-a d)^2 \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )+B (b c-a d) \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )+12 B (b c-a d)^3 (b d x+(-b c+a d) \log (c+d x))+12 B (b c-a d)^4 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )\right )}{60 b^3 d^3} \]

[In]

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

[Out]

(g^2*i^2*(20*d^3*(b*c - a*d)^2*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 30*d^4*(b*c - a*d)*(a + b*
x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 12*d^5*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 20*B
*(b*c - a*d)^3*(2*A*b*d*(b*c - a*d)*x + 2*B*d*(b*c - a*d)*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] - d^2*(a + b*
x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 2*B*(b*c - a*d)^2*Log[c + d*x] - 2*(b*c - a*d)^2*(A + B*Log[(e*(a
+ b*x))/(c + d*x)])*Log[c + d*x] + B*(b*c - a*d)*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + B*(b*c - a*d)^2*((2*L
og[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])) - 10
*B*(b*c - a*d)^2*(6*A*b*d*(b*c - a*d)^2*x + 6*B*d*(b*c - a*d)^2*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] + 3*d^2
*(-(b*c) + a*d)*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 2*d^3*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/
(c + d*x)]) - 6*B*(b*c - a*d)^3*Log[c + d*x] - 6*(b*c - a*d)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*
x] + B*(b*c - a*d)*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d*x]) + 3*B*(b*c - a*d)^2*
(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + 3*B*(b*c - a*d)^3*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x]
)*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])) + B*(b*c - a*d)*(24*A*b*d*(b*c - a*d)^3*x + 24*B*d*
(b*c - a*d)^3*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] - 12*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x
))/(c + d*x)]) + 8*d^3*(b*c - a*d)*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 6*d^4*(a + b*x)^4*(A + B
*Log[(e*(a + b*x))/(c + d*x)]) - 24*B*(b*c - a*d)^4*Log[c + d*x] - 24*(b*c - a*d)^4*(A + B*Log[(e*(a + b*x))/(
c + d*x)])*Log[c + d*x] + 4*B*(b*c - a*d)^2*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d
*x]) + B*(b*c - a*d)*(6*b*d*(b*c - a*d)^2*x + 3*d^2*(-(b*c) + a*d)*(a + b*x)^2 + 2*d^3*(a + b*x)^3 - 6*(b*c -
a*d)^3*Log[c + d*x]) + 12*B*(b*c - a*d)^3*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + 12*B*(b*c - a*d)^4*((2*Log[(
d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]))))/(60*b^
3*d^3)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

integrate((b*g*x+a*g)**2*(d*i*x+c*i)**2*(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. 3656 vs. \(2 (728) = 1456\).

Time = 0.34 (sec) , antiderivative size = 3656, normalized size of antiderivative = 4.80 \[ \int (a g+b g x)^2 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/5*A^2*b^2*d^2*g^2*i^2*x^5 + 1/2*A^2*b^2*c*d*g^2*i^2*x^4 + 1/2*A^2*a*b*d^2*g^2*i^2*x^4 + 1/3*A^2*b^2*c^2*g^2*
i^2*x^3 + 4/3*A^2*a*b*c*d*g^2*i^2*x^3 + 1/3*A^2*a^2*d^2*g^2*i^2*x^3 + A^2*a*b*c^2*g^2*i^2*x^2 + A^2*a^2*c*d*g^
2*i^2*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^2*c^2*g^2*i
^2 + 2*(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*b*c^2*g^2*i^2 + 1/3*(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^2*c^2*g^2*i^2 + 2*(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^2*c*d*g^2*i^2 + 4/3*(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*b*c*d*g^2*i^2 + 1/6*(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^2*c*d*g^2*i^2 + 1/3*(2*x^3*l
og(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^2*d^2*g^2*i^2 + 1/6*(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*b*d^2*g^2*i^2 + 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^2*d^2*g^2*i^2 +
A^2*a^2*c^2*g^2*i^2*x - 1/30*(2*b^4*c^5*g^2*i^2*log(e) + 9*a^3*b*c^2*d^3*g^2*i^2 - 2*a^4*c*d^4*g^2*i^2 - 2*(5*
g^2*i^2*log(e) - g^2*i^2)*a*b^3*c^4*d + (20*g^2*i^2*log(e) - 9*g^2*i^2)*a^2*b^2*c^3*d^2)*B^2*log(d*x + c)/(b^2
*d^3) - 1/15*(b^5*c^5*g^2*i^2 - 5*a*b^4*c^4*d*g^2*i^2 + 10*a^2*b^3*c^3*d^2*g^2*i^2 - 10*a^3*b^2*c^2*d^3*g^2*i^
2 + 5*a^4*b*c*d^4*g^2*i^2 - a^5*d^5*g^2*i^2)*(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^3*d^3) + 1/60*(12*B^2*b^5*d^5*g^2*i^2*x^5*log(e)^2 + 6*((5*g^2*i^2*log(e)^2 - g^2*
i^2*log(e))*b^5*c*d^4 + (5*g^2*i^2*log(e)^2 + g^2*i^2*log(e))*a*b^4*d^5)*B^2*x^4 + 2*((10*g^2*i^2*log(e)^2 - 6
*g^2*i^2*log(e) + g^2*i^2)*b^5*c^2*d^3 + 2*(20*g^2*i^2*log(e)^2 - g^2*i^2)*a*b^4*c*d^4 + (10*g^2*i^2*log(e)^2
+ 6*g^2*i^2*log(e) + g^2*i^2)*a^2*b^3*d^5)*B^2*x^3 - ((2*g^2*i^2*log(e) - 3*g^2*i^2)*b^5*c^3*d^2 - 3*(20*g^2*i
^2*log(e)^2 - 10*g^2*i^2*log(e) - g^2*i^2)*a*b^4*c^2*d^3 - 3*(20*g^2*i^2*log(e)^2 + 10*g^2*i^2*log(e) - g^2*i^
2)*a^2*b^3*c*d^4 - (2*g^2*i^2*log(e) + 3*g^2*i^2)*a^3*b^2*d^5)*B^2*x^2 + 2*(2*(g^2*i^2*log(e) - g^2*i^2)*b^5*c
^4*d - (10*g^2*i^2*log(e) - 11*g^2*i^2)*a*b^4*c^3*d^2 + 6*(5*g^2*i^2*log(e)^2 - 3*g^2*i^2)*a^2*b^3*c^2*d^3 + (
10*g^2*i^2*log(e) + 11*g^2*i^2)*a^3*b^2*c*d^4 - 2*(g^2*i^2*log(e) + g^2*i^2)*a^4*b*d^5)*B^2*x + 2*(6*B^2*b^5*d
^5*g^2*i^2*x^5 + 30*B^2*a^2*b^3*c^2*d^3*g^2*i^2*x + 15*(b^5*c*d^4*g^2*i^2 + a*b^4*d^5*g^2*i^2)*B^2*x^4 + 10*(b
^5*c^2*d^3*g^2*i^2 + 4*a*b^4*c*d^4*g^2*i^2 + a^2*b^3*d^5*g^2*i^2)*B^2*x^3 + 30*(a*b^4*c^2*d^3*g^2*i^2 + a^2*b^
3*c*d^4*g^2*i^2)*B^2*x^2 + (10*a^3*b^2*c^2*d^3*g^2*i^2 - 5*a^4*b*c*d^4*g^2*i^2 + a^5*d^5*g^2*i^2)*B^2)*log(b*x
+ a)^2 + 2*(6*B^2*b^5*d^5*g^2*i^2*x^5 + 30*B^2*a^2*b^3*c^2*d^3*g^2*i^2*x + 15*(b^5*c*d^4*g^2*i^2 + a*b^4*d^5*
g^2*i^2)*B^2*x^4 + 10*(b^5*c^2*d^3*g^2*i^2 + 4*a*b^4*c*d^4*g^2*i^2 + a^2*b^3*d^5*g^2*i^2)*B^2*x^3 + 30*(a*b^4*
c^2*d^3*g^2*i^2 + a^2*b^3*c*d^4*g^2*i^2)*B^2*x^2 + (b^5*c^5*g^2*i^2 - 5*a*b^4*c^4*d*g^2*i^2 + 10*a^2*b^3*c^3*d
^2*g^2*i^2)*B^2)*log(d*x + c)^2 + 2*(12*B^2*b^5*d^5*g^2*i^2*x^5*log(e) + 3*((10*g^2*i^2*log(e) - g^2*i^2)*b^5*
c*d^4 + (10*g^2*i^2*log(e) + g^2*i^2)*a*b^4*d^5)*B^2*x^4 + 2*(40*a*b^4*c*d^4*g^2*i^2*log(e) + (10*g^2*i^2*log(
e) - 3*g^2*i^2)*b^5*c^2*d^3 + (10*g^2*i^2*log(e) + 3*g^2*i^2)*a^2*b^3*d^5)*B^2*x^3 - (b^5*c^3*d^2*g^2*i^2 - a^
3*b^2*d^5*g^2*i^2 - 15*(4*g^2*i^2*log(e) - g^2*i^2)*a*b^4*c^2*d^3 - 15*(4*g^2*i^2*log(e) + g^2*i^2)*a^2*b^3*c*
d^4)*B^2*x^2 + 2*(30*a^2*b^3*c^2*d^3*g^2*i^2*log(e) + b^5*c^4*d*g^2*i^2 - 5*a*b^4*c^3*d^2*g^2*i^2 + 5*a^3*b^2*
c*d^4*g^2*i^2 - a^4*b*d^5*g^2*i^2)*B^2*x + (2*a^5*d^5*g^2*i^2*log(e) + 2*a*b^4*c^4*d*g^2*i^2 - 9*a^2*b^3*c^3*d
^2*g^2*i^2 + (20*g^2*i^2*log(e) + 9*g^2*i^2)*a^3*b^2*c^2*d^3 - 2*(5*g^2*i^2*log(e) + g^2*i^2)*a^4*b*c*d^4)*B^2
)*log(b*x + a) - 2*(12*B^2*b^5*d^5*g^2*i^2*x^5*log(e) + 3*((10*g^2*i^2*log(e) - g^2*i^2)*b^5*c*d^4 + (10*g^2*i
^2*log(e) + g^2*i^2)*a*b^4*d^5)*B^2*x^4 + 2*(40*a*b^4*c*d^4*g^2*i^2*log(e) + (10*g^2*i^2*log(e) - 3*g^2*i^2)*b
^5*c^2*d^3 + (10*g^2*i^2*log(e) + 3*g^2*i^2)*a^2*b^3*d^5)*B^2*x^3 - (b^5*c^3*d^2*g^2*i^2 - a^3*b^2*d^5*g^2*i^2
- 15*(4*g^2*i^2*log(e) - g^2*i^2)*a*b^4*c^2*d^3 - 15*(4*g^2*i^2*log(e) + g^2*i^2)*a^2*b^3*c*d^4)*B^2*x^2 + 2*
(30*a^2*b^3*c^2*d^3*g^2*i^2*log(e) + b^5*c^4*d*g^2*i^2 - 5*a*b^4*c^3*d^2*g^2*i^2 + 5*a^3*b^2*c*d^4*g^2*i^2 - a
^4*b*d^5*g^2*i^2)*B^2*x + 2*(6*B^2*b^5*d^5*g^2*i^2*x^5 + 30*B^2*a^2*b^3*c^2*d^3*g^2*i^2*x + 15*(b^5*c*d^4*g^2*
i^2 + a*b^4*d^5*g^2*i^2)*B^2*x^4 + 10*(b^5*c^2*d^3*g^2*i^2 + 4*a*b^4*c*d^4*g^2*i^2 + a^2*b^3*d^5*g^2*i^2)*B^2*
x^3 + 30*(a*b^4*c^2*d^3*g^2*i^2 + a^2*b^3*c*d^4*g^2*i^2)*B^2*x^2 + (10*a^3*b^2*c^2*d^3*g^2*i^2 - 5*a^4*b*c*d^4
*g^2*i^2 + a^5*d^5*g^2*i^2)*B^2)*log(b*x + a))*log(d*x + c))/(b^3*d^3)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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