Integrand size = 32, antiderivative size = 365 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=-\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b}+\frac {B (b c-a d)^2 g^4 (a+b x)^3 \left (4 A+B+4 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^2}-\frac {B (b c-a d)^3 g^4 (a+b x)^2 \left (12 A+7 B+12 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 b d^3}+\frac {B (b c-a d)^4 g^4 (a+b x) \left (12 A+13 B+12 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^4}+\frac {B (b c-a d)^5 g^4 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (12 A+25 B+12 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^5}+\frac {2 B^2 (b c-a d)^5 g^4 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5} \]
-1/10*B*(-a*d+b*c)*g^4*(b*x+a)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/b/d+1/5*g^4*( b*x+a)^5*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/b+1/30*B*(-a*d+b*c)^2*g^4*(b*x+a)^3 *(4*A+B+4*B*ln(e*(b*x+a)/(d*x+c)))/b/d^2-1/60*B*(-a*d+b*c)^3*g^4*(b*x+a)^2 *(12*A+7*B+12*B*ln(e*(b*x+a)/(d*x+c)))/b/d^3+1/30*B*(-a*d+b*c)^4*g^4*(b*x+ a)*(12*A+13*B+12*B*ln(e*(b*x+a)/(d*x+c)))/b/d^4+1/30*B*(-a*d+b*c)^5*g^4*ln ((-a*d+b*c)/b/(d*x+c))*(12*A+25*B+12*B*ln(e*(b*x+a)/(d*x+c)))/b/d^5+2/5*B^ 2*(-a*d+b*c)^5*g^4*polylog(2,d*(b*x+a)/b/(d*x+c))/b/d^5
Time = 0.31 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.40 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\frac {g^4 \left ((a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+\frac {B (b c-a d) \left (24 A b d (b c-a d)^3 x+24 B d (b c-a d)^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-12 d^2 (b c-a d)^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+8 d^3 (b c-a d) (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-6 d^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-24 B (b c-a d)^4 \log (c+d x)-24 (b c-a d)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+4 B (b c-a d)^2 \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )+B (b c-a d) \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )+12 B (b c-a d)^3 (b d x+(-b c+a d) \log (c+d x))+12 B (b c-a d)^4 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{12 d^5}\right )}{5 b} \]
(g^4*((a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + (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)])))/(12*d^5)))/(5*b)
Time = 0.92 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.19, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2950, 2781, 2784, 2784, 2784, 27, 2784, 2754, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (a g+b g x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2 \, dx\) |
| \(\Big \downarrow \) 2950 |
| \(\displaystyle g^4 (b c-a d)^5 \int \frac {(a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}\) |
| \(\Big \downarrow \) 2781 |
| \(\displaystyle g^4 (b c-a d)^5 \left (\frac {(a+b x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{5 b (c+d x)^5 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B \int \frac {(a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{5 b}\right )\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle g^4 (b c-a d)^5 \left (\frac {(a+b x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{5 b (c+d x)^5 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B \left (\frac {(a+b x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 d (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {\int \frac {(a+b x)^3 \left (4 A+B+4 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{4 d}\right )}{5 b}\right )\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle g^4 (b c-a d)^5 \left (\frac {(a+b x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{5 b (c+d x)^5 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B \left (\frac {(a+b x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 d (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {\frac {(a+b x)^3 \left (4 B \log \left (\frac {e (a+b x)}{c+d x}\right )+4 A+B\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\int \frac {(a+b x)^2 \left (12 A+7 B+12 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 d}}{4 d}\right )}{5 b}\right )\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle g^4 (b c-a d)^5 \left (\frac {(a+b x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{5 b (c+d x)^5 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B \left (\frac {(a+b x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 d (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {\frac {(a+b x)^3 \left (4 B \log \left (\frac {e (a+b x)}{c+d x}\right )+4 A+B\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (12 B \log \left (\frac {e (a+b x)}{c+d x}\right )+12 A+7 B\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\int \frac {2 (a+b x) \left (12 A+13 B+12 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{2 d}}{3 d}}{4 d}\right )}{5 b}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle g^4 (b c-a d)^5 \left (\frac {(a+b x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{5 b (c+d x)^5 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B \left (\frac {(a+b x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 d (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {\frac {(a+b x)^3 \left (4 B \log \left (\frac {e (a+b x)}{c+d x}\right )+4 A+B\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (12 B \log \left (\frac {e (a+b x)}{c+d x}\right )+12 A+7 B\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\int \frac {(a+b x) \left (12 A+13 B+12 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{d}}{3 d}}{4 d}\right )}{5 b}\right )\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle g^4 (b c-a d)^5 \left (\frac {(a+b x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{5 b (c+d x)^5 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B \left (\frac {(a+b x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 d (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {\frac {(a+b x)^3 \left (4 B \log \left (\frac {e (a+b x)}{c+d x}\right )+4 A+B\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (12 B \log \left (\frac {e (a+b x)}{c+d x}\right )+12 A+7 B\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (12 B \log \left (\frac {e (a+b x)}{c+d x}\right )+12 A+13 B\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\int \frac {12 A+25 B+12 B \log \left (\frac {e (a+b x)}{c+d x}\right )}{b-\frac {d (a+b x)}{c+d x}}d\frac {a+b x}{c+d x}}{d}}{d}}{3 d}}{4 d}\right )}{5 b}\right )\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle g^4 (b c-a d)^5 \left (\frac {(a+b x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{5 b (c+d x)^5 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B \left (\frac {(a+b x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 d (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {\frac {(a+b x)^3 \left (4 B \log \left (\frac {e (a+b x)}{c+d x}\right )+4 A+B\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (12 B \log \left (\frac {e (a+b x)}{c+d x}\right )+12 A+7 B\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (12 B \log \left (\frac {e (a+b x)}{c+d x}\right )+12 A+13 B\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\frac {12 B \int \frac {(c+d x) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{d}-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (12 B \log \left (\frac {e (a+b x)}{c+d x}\right )+12 A+25 B\right )}{d}}{d}}{d}}{3 d}}{4 d}\right )}{5 b}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle g^4 (b c-a d)^5 \left (\frac {(a+b x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{5 b (c+d x)^5 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B \left (\frac {(a+b x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 d (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {\frac {(a+b x)^3 \left (4 B \log \left (\frac {e (a+b x)}{c+d x}\right )+4 A+B\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (12 B \log \left (\frac {e (a+b x)}{c+d x}\right )+12 A+7 B\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (12 B \log \left (\frac {e (a+b x)}{c+d x}\right )+12 A+13 B\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (12 B \log \left (\frac {e (a+b x)}{c+d x}\right )+12 A+25 B\right )}{d}-\frac {12 B \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d}}{d}}{d}}{3 d}}{4 d}\right )}{5 b}\right )\) |
(b*c - a*d)^5*g^4*(((a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(5 *b*(c + d*x)^5*(b - (d*(a + b*x))/(c + d*x))^5) - (2*B*(((a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*d*(c + d*x)^4*(b - (d*(a + b*x))/(c + d*x))^4) - (((a + b*x)^3*(4*A + B + 4*B*Log[(e*(a + b*x))/(c + d*x)]))/(3* d*(c + d*x)^3*(b - (d*(a + b*x))/(c + d*x))^3) - (((a + b*x)^2*(12*A + 7*B + 12*B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d*(c + d*x)^2*(b - (d*(a + b*x)) /(c + d*x))^2) - (((a + b*x)*(12*A + 13*B + 12*B*Log[(e*(a + b*x))/(c + d* x)]))/(d*(c + d*x)*(b - (d*(a + b*x))/(c + d*x))) - (-(((12*A + 25*B + 12* B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/d) - (12*B*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/d)/d)/d)/(3*d))/(4*d)))/(5 *b))
3.1.97.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[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]
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] + Simp[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]
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] - Simp[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]
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]
Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)^( m + 1)*(g/b)^m Subst[Int[x^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + 2)), x] , x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] & & EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] && E qQ[b*f - a*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
\[\int \left (b g x +a g \right )^{4} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}d x\]
\[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )}^{4} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2} \,d x } \]
integral(A^2*b^4*g^4*x^4 + 4*A^2*a*b^3*g^4*x^3 + 6*A^2*a^2*b^2*g^4*x^2 + 4 *A^2*a^3*b*g^4*x + A^2*a^4*g^4 + (B^2*b^4*g^4*x^4 + 4*B^2*a*b^3*g^4*x^3 + 6*B^2*a^2*b^2*g^4*x^2 + 4*B^2*a^3*b*g^4*x + B^2*a^4*g^4)*log((b*e*x + a*e) /(d*x + c))^2 + 2*(A*B*b^4*g^4*x^4 + 4*A*B*a*b^3*g^4*x^3 + 6*A*B*a^2*b^2*g ^4*x^2 + 4*A*B*a^3*b*g^4*x + A*B*a^4*g^4)*log((b*e*x + a*e)/(d*x + c)), x)
Timed out. \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 2389 vs. \(2 (350) = 700\).
Time = 0.32 (sec) , antiderivative size = 2389, normalized size of antiderivative = 6.55 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\text {Too large to display} \]
1/5*A^2*b^4*g^4*x^5 + A^2*a*b^3*g^4*x^4 + 2*A^2*a^2*b^2*g^4*x^3 + 2*A^2*a^ 3*b*g^4*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^4*g^4 + 4*(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^3*b*g^4 + 2*(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^2*b^2*g^4 + 1/3*(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^3*g^4 + 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^4*g^4 + A^2*a^4*g^4*x - 1/30*((12*g^4*log(e) + 25*g^4)*b^4*c^5 - (60*g^4*log(e) + 113*g^4)*a*b^3*c^4*d + 4*(30*g^4*log(e) + 49*g^4)*a^2*b^2*c^3*d^2 - 12*( 10*g^4*log(e) + 13*g^4)*a^3*b*c^2*d^3 + 12*(5*g^4*log(e) + 4*g^4)*a^4*c*d^ 4)*B^2*log(d*x + c)/d^5 - 2/5*(b^5*c^5*g^4 - 5*a*b^4*c^4*d*g^4 + 10*a^2*b^ 3*c^3*d^2*g^4 - 10*a^3*b^2*c^2*d^3*g^4 + 5*a^4*b*c*d^4*g^4 - a^5*d^5*g^4)* (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*d^5) + 1/60*(12*B^2*b^5*d^5*g^4*x^5*log(e)^2 - 6*(b^...
\[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )}^{4} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2} \,d x } \]
Timed out. \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\int {\left (a\,g+b\,g\,x\right )}^4\,{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2 \,d x \]