Integrand size = 40, antiderivative size = 198 \[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d i}-\frac {(b c-a d) g^2 (a+b x) \left (2 A+B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d^2 i}-\frac {(b c-a d)^2 g^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (2 A+3 B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d^3 i}-\frac {B (b c-a d)^2 g^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i} \]
1/2*g^2*(b*x+a)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/d/i-1/2*(-a*d+b*c)*g^2*(b*x+ a)*(2*A+B+2*B*ln(e*(b*x+a)/(d*x+c)))/d^2/i-1/2*(-a*d+b*c)^2*g^2*ln((-a*d+b *c)/b/(d*x+c))*(2*A+3*B+2*B*ln(e*(b*x+a)/(d*x+c)))/d^3/i-B*(-a*d+b*c)^2*g^ 2*polylog(2,d*(b*x+a)/b/(d*x+c))/d^3/i
Time = 0.12 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.28 \[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\frac {g^2 \left (-2 A b d (b c-a d) x+2 B d (-b c+a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 B (b c-a d)^2 \log (c+d x)-B (b c-a d) (b d x+(-b c+a d) \log (c+d x))+2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (i (c+d x))-B (b c-a d)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (i (c+d x))\right ) \log (i (c+d x))+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{2 d^3 i} \]
(g^2*(-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] - B*(b*c - a*d)*(b*d*x + (-(b*c) + a*d)*Log [c + d*x]) + 2*(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[i*(c + d*x)] - B*(b*c - a*d)^2*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[i*( c + d*x)])*Log[i*(c + d*x)] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])))/( 2*d^3*i)
Time = 0.58 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2962, 2784, 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.
| \(\displaystyle \int \frac {(a g+b g x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c i+d i x} \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle \frac {g^2 (b c-a d)^2 \int \frac {(a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{i}\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {g^2 (b c-a d)^2 \left (\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\int \frac {(a+b x) \left (2 A+B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{2 d}\right )}{i}\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {g^2 (b c-a d)^2 \left (\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (2 B \log \left (\frac {e (a+b x)}{c+d x}\right )+2 A+B\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\int \frac {2 A+3 B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )}{b-\frac {d (a+b x)}{c+d x}}d\frac {a+b x}{c+d x}}{d}}{2 d}\right )}{i}\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle \frac {g^2 (b c-a d)^2 \left (\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (2 B \log \left (\frac {e (a+b x)}{c+d x}\right )+2 A+B\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\frac {2 B \int \frac {(c+d x) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{d}-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (2 B \log \left (\frac {e (a+b x)}{c+d x}\right )+2 A+3 B\right )}{d}}{d}}{2 d}\right )}{i}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {g^2 (b c-a d)^2 \left (\frac {(a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (2 B \log \left (\frac {e (a+b x)}{c+d x}\right )+2 A+B\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (2 B \log \left (\frac {e (a+b x)}{c+d x}\right )+2 A+3 B\right )}{d}-\frac {2 B \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d}}{d}}{2 d}\right )}{i}\) |
((b*c - a*d)^2*g^2*(((a + b*x)^2*(A + 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)*(2*A + B + 2* B*Log[(e*(a + b*x))/(c + d*x)]))/(d*(c + d*x)*(b - (d*(a + b*x))/(c + d*x) )) - (-(((2*A + 3*B + 2*B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (d*(a + b* x))/(b*(c + d*x))])/d) - (2*B*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/d)/ d)/(2*d)))/i
3.1.32.3.1 Defintions of rubi rules used
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_.))*((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_.)*((h_.) + (i_.)*(x_))^(q_.), x_Sy mbol] :> Simp[(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] && IGt Q[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && I ntegersQ[m, q]
Leaf count of result is larger than twice the leaf count of optimal. \(656\) vs. \(2(192)=384\).
Time = 1.77 (sec) , antiderivative size = 657, normalized size of antiderivative = 3.32
| method | result | size |
| parts | \(\frac {g^{2} A \left (\frac {b \left (\frac {1}{2} b d \,x^{2}+2 x a d -b c x \right )}{d^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{3}}\right )}{i}-\frac {g^{2} B \left (\frac {e^{2} b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (-\frac {\ln \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}{2 e^{2} b^{2} d}-\frac {1}{2 e b d \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -2 b e \right )}{2 e^{2} b^{2} \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )^{2}}\right )}{d}+\frac {2 b e \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\ln \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}{b e d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}\right )}{d}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{d}\right )}{i d}\) | \(657\) |
| derivativedivides | \(-\frac {e \left (a d -c b \right ) \left (-\frac {A \,d^{2} g^{2} \left (a d -c b \right ) \left (-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{d^{3}}-\frac {2 b e}{d^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {b^{2} e^{2}}{2 d^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}\right )}{e i}-\frac {B \,d^{2} g^{2} \left (a d -c b \right ) \left (-\frac {\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}}{d^{2}}-\frac {2 b e \left (\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b e d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right )}{d^{2}}-\frac {b^{2} e^{2} \left (-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{2 b^{2} e^{2} d}+\frac {1}{2 b e d \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (2 b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{2 b^{2} e^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}\right )}{d^{2}}\right )}{e i}\right )}{d^{2}}\) | \(712\) |
| default | \(-\frac {e \left (a d -c b \right ) \left (-\frac {A \,d^{2} g^{2} \left (a d -c b \right ) \left (-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{d^{3}}-\frac {2 b e}{d^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {b^{2} e^{2}}{2 d^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}\right )}{e i}-\frac {B \,d^{2} g^{2} \left (a d -c b \right ) \left (-\frac {\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}}{d^{2}}-\frac {2 b e \left (\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b e d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right )}{d^{2}}-\frac {b^{2} e^{2} \left (-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{2 b^{2} e^{2} d}+\frac {1}{2 b e d \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (2 b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{2 b^{2} e^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}\right )}{d^{2}}\right )}{e i}\right )}{d^{2}}\) | \(712\) |
| risch | \(\text {Expression too large to display}\) | \(1998\) |
g^2*A/i*(b/d^2*(1/2*b*d*x^2+2*x*a*d-b*c*x)+(a^2*d^2-2*a*b*c*d+b^2*c^2)/d^3 *ln(d*x+c))-g^2*B/i/d*(e^2*b^2*(a^2*d^2-2*a*b*c*d+b^2*c^2)/d*(-1/2/e^2/b^2 /d*ln((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-b*e)-1/2/e/b/d/((b*e/d+(a*d-b*c)*e/d /(d*x+c))*d-b*e)+1/2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/ (d*x+c))*((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-2*b*e)/e^2/b^2/((b*e/d+(a*d-b*c) *e/d/(d*x+c))*d-b*e)^2)+2*b*e*(a^2*d^2-2*a*b*c*d+b^2*c^2)/d*(1/b/e/d*ln((b *e/d+(a*d-b*c)*e/d/(d*x+c))*d-b*e)-ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+ (a*d-b*c)*e/d/(d*x+c))/b/e/((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-b*e))+1/d*(a^2 *d^2-2*a*b*c*d+b^2*c^2)*(dilog(-((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-b*e)/b/e) /d+ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(-((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-b* e)/b/e)/d))
\[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\int { \frac {{\left (b g x + a g\right )}^{2} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}}{d i x + c i} \,d x } \]
integral((A*b^2*g^2*x^2 + 2*A*a*b*g^2*x + A*a^2*g^2 + (B*b^2*g^2*x^2 + 2*B *a*b*g^2*x + B*a^2*g^2)*log((b*e*x + a*e)/(d*x + c)))/(d*i*x + c*i), x)
\[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\frac {g^{2} \left (\int \frac {A a^{2}}{c + d x}\, dx + \int \frac {A b^{2} x^{2}}{c + d x}\, dx + \int \frac {B a^{2} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{c + d x}\, dx + \int \frac {2 A a b x}{c + d x}\, dx + \int \frac {B b^{2} x^{2} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{c + d x}\, dx + \int \frac {2 B a b x \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{c + d x}\, dx\right )}{i} \]
g**2*(Integral(A*a**2/(c + d*x), x) + Integral(A*b**2*x**2/(c + d*x), x) + Integral(B*a**2*log(a*e/(c + d*x) + b*e*x/(c + d*x))/(c + d*x), x) + Inte gral(2*A*a*b*x/(c + d*x), x) + Integral(B*b**2*x**2*log(a*e/(c + d*x) + b* e*x/(c + d*x))/(c + d*x), x) + Integral(2*B*a*b*x*log(a*e/(c + d*x) + b*e* x/(c + d*x))/(c + d*x), x))/i
Leaf count of result is larger than twice the leaf count of optimal. 477 vs. \(2 (191) = 382\).
Time = 0.26 (sec) , antiderivative size = 477, normalized size of antiderivative = 2.41 \[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=2 \, A a b g^{2} {\left (\frac {x}{d i} - \frac {c \log \left (d x + c\right )}{d^{2} i}\right )} + \frac {1}{2} \, A b^{2} g^{2} {\left (\frac {2 \, c^{2} \log \left (d x + c\right )}{d^{3} i} + \frac {d x^{2} - 2 \, c x}{d^{2} i}\right )} + \frac {A a^{2} g^{2} \log \left (d i x + c i\right )}{d i} + \frac {{\left (b^{2} c^{2} g^{2} - 2 \, a b c d g^{2} + a^{2} d^{2} g^{2}\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B}{d^{3} i} + \frac {{\left (2 \, a^{2} d^{2} g^{2} \log \left (e\right ) + {\left (2 \, g^{2} \log \left (e\right ) + 3 \, g^{2}\right )} b^{2} c^{2} - 4 \, {\left (g^{2} \log \left (e\right ) + g^{2}\right )} a b c d\right )} B \log \left (d x + c\right )}{2 \, d^{3} i} + \frac {B b^{2} d^{2} g^{2} x^{2} \log \left (e\right ) - {\left (b^{2} c^{2} g^{2} - 2 \, a b c d g^{2} + a^{2} d^{2} g^{2}\right )} B \log \left (d x + c\right )^{2} - {\left ({\left (2 \, g^{2} \log \left (e\right ) + g^{2}\right )} b^{2} c d - {\left (4 \, g^{2} \log \left (e\right ) + g^{2}\right )} a b d^{2}\right )} B x + {\left (B b^{2} d^{2} g^{2} x^{2} - 2 \, {\left (b^{2} c d g^{2} - 2 \, a b d^{2} g^{2}\right )} B x - {\left (2 \, a b c d g^{2} - 3 \, a^{2} d^{2} g^{2}\right )} B\right )} \log \left (b x + a\right ) - {\left (B b^{2} d^{2} g^{2} x^{2} - 2 \, {\left (b^{2} c d g^{2} - 2 \, a b d^{2} g^{2}\right )} B x\right )} \log \left (d x + c\right )}{2 \, d^{3} i} \]
2*A*a*b*g^2*(x/(d*i) - c*log(d*x + c)/(d^2*i)) + 1/2*A*b^2*g^2*(2*c^2*log( d*x + c)/(d^3*i) + (d*x^2 - 2*c*x)/(d^2*i)) + A*a^2*g^2*log(d*i*x + c*i)/( d*i) + (b^2*c^2*g^2 - 2*a*b*c*d*g^2 + a^2*d^2*g^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/(d^3*i) + 1/2*(2*a^2*d^2*g^2*log(e) + (2*g^2*log(e) + 3*g^2)*b^2*c^2 - 4*(g^2*log(e ) + g^2)*a*b*c*d)*B*log(d*x + c)/(d^3*i) + 1/2*(B*b^2*d^2*g^2*x^2*log(e) - (b^2*c^2*g^2 - 2*a*b*c*d*g^2 + a^2*d^2*g^2)*B*log(d*x + c)^2 - ((2*g^2*lo g(e) + g^2)*b^2*c*d - (4*g^2*log(e) + g^2)*a*b*d^2)*B*x + (B*b^2*d^2*g^2*x ^2 - 2*(b^2*c*d*g^2 - 2*a*b*d^2*g^2)*B*x - (2*a*b*c*d*g^2 - 3*a^2*d^2*g^2) *B)*log(b*x + a) - (B*b^2*d^2*g^2*x^2 - 2*(b^2*c*d*g^2 - 2*a*b*d^2*g^2)*B* x)*log(d*x + c))/(d^3*i)
Leaf count of result is larger than twice the leaf count of optimal. 2364 vs. \(2 (191) = 382\).
Time = 55.79 (sec) , antiderivative size = 2364, normalized size of antiderivative = 11.94 \[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\text {Too large to display} \]
1/24*(2*(B*b^7*c^5*e^5*g^2 - 5*B*a*b^6*c^4*d*e^5*g^2 + 10*B*a^2*b^5*c^3*d^ 2*e^5*g^2 - 10*B*a^3*b^4*c^2*d^3*e^5*g^2 + 5*B*a^4*b^3*c*d^4*e^5*g^2 - B*a ^5*b^2*d^5*e^5*g^2 - 4*(b*e*x + a*e)*B*b^6*c^5*d*e^4*g^2/(d*x + c) + 20*(b *e*x + a*e)*B*a*b^5*c^4*d^2*e^4*g^2/(d*x + c) - 40*(b*e*x + a*e)*B*a^2*b^4 *c^3*d^3*e^4*g^2/(d*x + c) + 40*(b*e*x + a*e)*B*a^3*b^3*c^2*d^4*e^4*g^2/(d *x + c) - 20*(b*e*x + a*e)*B*a^4*b^2*c*d^5*e^4*g^2/(d*x + c) + 4*(b*e*x + a*e)*B*a^5*b*d^6*e^4*g^2/(d*x + c) + 6*(b*e*x + a*e)^2*B*b^5*c^5*d^2*e^3*g ^2/(d*x + c)^2 - 30*(b*e*x + a*e)^2*B*a*b^4*c^4*d^3*e^3*g^2/(d*x + c)^2 + 60*(b*e*x + a*e)^2*B*a^2*b^3*c^3*d^4*e^3*g^2/(d*x + c)^2 - 60*(b*e*x + a*e )^2*B*a^3*b^2*c^2*d^5*e^3*g^2/(d*x + c)^2 + 30*(b*e*x + a*e)^2*B*a^4*b*c*d ^6*e^3*g^2/(d*x + c)^2 - 6*(b*e*x + a*e)^2*B*a^5*d^7*e^3*g^2/(d*x + c)^2)* log((b*e*x + a*e)/(d*x + c))/(b^4*d^3*e^4*i - 4*(b*e*x + a*e)*b^3*d^4*e^3* i/(d*x + c) + 6*(b*e*x + a*e)^2*b^2*d^5*e^2*i/(d*x + c)^2 - 4*(b*e*x + a*e )^3*b*d^6*e*i/(d*x + c)^3 + (b*e*x + a*e)^4*d^7*i/(d*x + c)^4) + (2*A*b^8* c^5*e^5*g^2 + B*b^8*c^5*e^5*g^2 - 10*A*a*b^7*c^4*d*e^5*g^2 - 5*B*a*b^7*c^4 *d*e^5*g^2 + 20*A*a^2*b^6*c^3*d^2*e^5*g^2 + 10*B*a^2*b^6*c^3*d^2*e^5*g^2 - 20*A*a^3*b^5*c^2*d^3*e^5*g^2 - 10*B*a^3*b^5*c^2*d^3*e^5*g^2 + 10*A*a^4*b^ 4*c*d^4*e^5*g^2 + 5*B*a^4*b^4*c*d^4*e^5*g^2 - 2*A*a^5*b^3*d^5*e^5*g^2 - B* a^5*b^3*d^5*e^5*g^2 - 8*(b*e*x + a*e)*A*b^7*c^5*d*e^4*g^2/(d*x + c) - 2*(b *e*x + a*e)*B*b^7*c^5*d*e^4*g^2/(d*x + c) + 40*(b*e*x + a*e)*A*a*b^6*c^...
Timed out. \[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\int \frac {{\left (a\,g+b\,g\,x\right )}^2\,\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}{c\,i+d\,i\,x} \,d x \]