Integrand size = 32, antiderivative size = 126 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2} \, dx=-\frac {2 B^2 (c+d x)}{(b c-a d) g^2 (a+b x)}-\frac {2 B (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) g^2 (a+b x)}-\frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d) g^2 (a+b x)} \]
-2*B^2*(d*x+c)/(-a*d+b*c)/g^2/(b*x+a)-2*B*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c )))/(-a*d+b*c)/g^2/(b*x+a)-(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c )/g^2/(b*x+a)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.29 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.49 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2} \, dx=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+\frac {B \left (2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 d (a+b x) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2 d (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+2 B (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-B d (a+b x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+B d (a+b x) \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{b c-a d}}{b g^2 (a+b x)} \]
-(((A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + (B*(2*(b*c - a*d)*(A + B*Log[( e*(a + b*x))/(c + d*x)]) + 2*d*(a + b*x)*Log[a + b*x]*(A + B*Log[(e*(a + b *x))/(c + d*x)]) - 2*d*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[ c + d*x] + 2*B*(b*c - a*d + d*(a + b*x)*Log[a + b*x] - d*(a + b*x)*Log[c + d*x]) - B*d*(a + b*x)*(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)]) + B*d*(a + b*x) *((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*Po lyLog[2, (b*(c + d*x))/(b*c - a*d)])))/(b*c - a*d))/(b*g^2*(a + b*x)))
Time = 0.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.81, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2950, 2742, 2741}
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.
| \(\displaystyle \int \frac {\left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{(a g+b g x)^2} \, dx\) |
| \(\Big \downarrow \) 2950 |
| \(\displaystyle \frac {\int \frac {(c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^2}d\frac {a+b x}{c+d x}}{g^2 (b c-a d)}\) |
| \(\Big \downarrow \) 2742 |
| \(\displaystyle \frac {2 B \int \frac {(c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2}d\frac {a+b x}{c+d x}-\frac {(c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{a+b x}}{g^2 (b c-a d)}\) |
| \(\Big \downarrow \) 2741 |
| \(\displaystyle \frac {2 B \left (-\frac {(c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-\frac {B (c+d x)}{a+b x}\right )-\frac {(c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{a+b x}}{g^2 (b c-a d)}\) |
(-(((c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(a + b*x)) + 2*B*(-( (B*(c + d*x))/(a + b*x)) - ((c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]) )/(a + b*x)))/((b*c - a*d)*g^2)
3.2.2.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^( m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbo l] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])^p/(d*(m + 1))), x] - Simp[b*n* (p/(m + 1)) Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b , c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]
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])
Time = 0.68 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.40
| method | result | size |
| norman | \(\frac {\frac {\left (A^{2}+2 B A +2 B^{2}\right ) x}{g a}+\frac {B^{2} c \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g \left (a d -c b \right )}+\frac {B^{2} d x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g \left (a d -c b \right )}+\frac {2 c B \left (A +B \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g \left (a d -c b \right )}+\frac {2 B d \left (A +B \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g \left (a d -c b \right )}}{g \left (b x +a \right )}\) | \(176\) |
| parallelrisch | \(-\frac {-2 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} c d -2 A B \,b^{3} c d +A^{2} a \,b^{2} d^{2}-A^{2} b^{3} c d +2 B^{2} a \,b^{2} d^{2}-2 B^{2} b^{3} c d +2 A B a \,b^{2} d^{2}-2 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} d^{2}-B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{3} d^{2}-2 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} d^{2}-B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{3} c d -2 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} c d}{g^{2} \left (b x +a \right ) b^{3} d \left (a d -c b \right )}\) | \(251\) |
| parts | \(-\frac {A^{2}}{g^{2} \left (b x +a \right ) b}-\frac {B^{2} e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{g^{2} \left (a d -c b \right )}-\frac {2 B A e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{g^{2} \left (a d -c b \right )}\) | \(300\) |
| derivativedivides | \(-\frac {e \left (a d -c b \right ) \left (-\frac {d^{2} A^{2}}{\left (a d -c b \right )^{2} g^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {2 d^{2} A B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{\left (a d -c b \right )^{2} g^{2}}+\frac {d^{2} B^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{\left (a d -c b \right )^{2} g^{2}}\right )}{d^{2}}\) | \(349\) |
| default | \(-\frac {e \left (a d -c b \right ) \left (-\frac {d^{2} A^{2}}{\left (a d -c b \right )^{2} g^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {2 d^{2} A B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{\left (a d -c b \right )^{2} g^{2}}+\frac {d^{2} B^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {2}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{\left (a d -c b \right )^{2} g^{2}}\right )}{d^{2}}\) | \(349\) |
| risch | \(-\frac {A^{2}}{g^{2} \left (b x +a \right ) b}+\frac {B^{2} e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{g^{2} \left (a d -c b \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}+\frac {2 B^{2} e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{g^{2} \left (a d -c b \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}+\frac {2 B^{2} e}{g^{2} \left (a d -c b \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}+\frac {2 B A e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{g^{2} \left (a d -c b \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}+\frac {2 B A e}{g^{2} \left (a d -c b \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}\) | \(368\) |
((A^2+2*A*B+2*B^2)/g/a*x+B^2*c/g/(a*d-b*c)*ln(e*(b*x+a)/(d*x+c))^2+B^2*d/g /(a*d-b*c)*x*ln(e*(b*x+a)/(d*x+c))^2+2*c*B*(A+B)/g/(a*d-b*c)*ln(e*(b*x+a)/ (d*x+c))+2*B*d*(A+B)/g/(a*d-b*c)*x*ln(e*(b*x+a)/(d*x+c)))/g/(b*x+a)
Time = 0.26 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.19 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2} \, dx=-\frac {{\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} b c - {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a d + {\left (B^{2} b d x + B^{2} b c\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 2 \, {\left ({\left (A B + B^{2}\right )} b d x + {\left (A B + B^{2}\right )} b c\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b^{3} c - a b^{2} d\right )} g^{2} x + {\left (a b^{2} c - a^{2} b d\right )} g^{2}} \]
-((A^2 + 2*A*B + 2*B^2)*b*c - (A^2 + 2*A*B + 2*B^2)*a*d + (B^2*b*d*x + B^2 *b*c)*log((b*e*x + a*e)/(d*x + c))^2 + 2*((A*B + B^2)*b*d*x + (A*B + B^2)* b*c)*log((b*e*x + a*e)/(d*x + c)))/((b^3*c - a*b^2*d)*g^2*x + (a*b^2*c - a ^2*b*d)*g^2)
Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (105) = 210\).
Time = 1.09 (sec) , antiderivative size = 434, normalized size of antiderivative = 3.44 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2} \, dx=- \frac {2 B d \left (A + B\right ) \log {\left (x + \frac {2 A B a d^{2} + 2 A B b c d + 2 B^{2} a d^{2} + 2 B^{2} b c d - \frac {2 B a^{2} d^{3} \left (A + B\right )}{a d - b c} + \frac {4 B a b c d^{2} \left (A + B\right )}{a d - b c} - \frac {2 B b^{2} c^{2} d \left (A + B\right )}{a d - b c}}{4 A B b d^{2} + 4 B^{2} b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac {2 B d \left (A + B\right ) \log {\left (x + \frac {2 A B a d^{2} + 2 A B b c d + 2 B^{2} a d^{2} + 2 B^{2} b c d + \frac {2 B a^{2} d^{3} \left (A + B\right )}{a d - b c} - \frac {4 B a b c d^{2} \left (A + B\right )}{a d - b c} + \frac {2 B b^{2} c^{2} d \left (A + B\right )}{a d - b c}}{4 A B b d^{2} + 4 B^{2} b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac {\left (- 2 A B - 2 B^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{a b g^{2} + b^{2} g^{2} x} + \frac {\left (B^{2} c + B^{2} d x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{a^{2} d g^{2} - a b c g^{2} + a b d g^{2} x - b^{2} c g^{2} x} + \frac {- A^{2} - 2 A B - 2 B^{2}}{a b g^{2} + b^{2} g^{2} x} \]
-2*B*d*(A + B)*log(x + (2*A*B*a*d**2 + 2*A*B*b*c*d + 2*B**2*a*d**2 + 2*B** 2*b*c*d - 2*B*a**2*d**3*(A + B)/(a*d - b*c) + 4*B*a*b*c*d**2*(A + B)/(a*d - b*c) - 2*B*b**2*c**2*d*(A + B)/(a*d - b*c))/(4*A*B*b*d**2 + 4*B**2*b*d** 2))/(b*g**2*(a*d - b*c)) + 2*B*d*(A + B)*log(x + (2*A*B*a*d**2 + 2*A*B*b*c *d + 2*B**2*a*d**2 + 2*B**2*b*c*d + 2*B*a**2*d**3*(A + B)/(a*d - b*c) - 4* B*a*b*c*d**2*(A + B)/(a*d - b*c) + 2*B*b**2*c**2*d*(A + B)/(a*d - b*c))/(4 *A*B*b*d**2 + 4*B**2*b*d**2))/(b*g**2*(a*d - b*c)) + (-2*A*B - 2*B**2)*log (e*(a + b*x)/(c + d*x))/(a*b*g**2 + b**2*g**2*x) + (B**2*c + B**2*d*x)*log (e*(a + b*x)/(c + d*x))**2/(a**2*d*g**2 - a*b*c*g**2 + a*b*d*g**2*x - b**2 *c*g**2*x) + (-A**2 - 2*A*B - 2*B**2)/(a*b*g**2 + b**2*g**2*x)
Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (126) = 252\).
Time = 0.21 (sec) , antiderivative size = 416, normalized size of antiderivative = 3.30 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2} \, dx=-{\left (2 \, {\left (\frac {1}{b^{2} g^{2} x + a b g^{2}} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {{\left (b d x + a d\right )} \log \left (b x + a\right )^{2} + {\left (b d x + a d\right )} \log \left (d x + c\right )^{2} - 2 \, b c + 2 \, a d - 2 \, {\left (b d x + a d\right )} \log \left (b x + a\right ) + 2 \, {\left (b d x + a d - {\left (b d x + a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{a b^{2} c g^{2} - a^{2} b d g^{2} + {\left (b^{3} c g^{2} - a b^{2} d g^{2}\right )} x}\right )} B^{2} - 2 \, A B {\left (\frac {\log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b^{2} g^{2} x + a b g^{2}} + \frac {1}{b^{2} g^{2} x + a b g^{2}} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} - \frac {B^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )^{2}}{b^{2} g^{2} x + a b g^{2}} - \frac {A^{2}}{b^{2} g^{2} x + a b g^{2}} \]
-(2*(1/(b^2*g^2*x + a*b*g^2) + d*log(b*x + a)/((b^2*c - a*b*d)*g^2) - d*lo g(d*x + c)/((b^2*c - a*b*d)*g^2))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - ( (b*d*x + a*d)*log(b*x + a)^2 + (b*d*x + a*d)*log(d*x + c)^2 - 2*b*c + 2*a* d - 2*(b*d*x + a*d)*log(b*x + a) + 2*(b*d*x + a*d - (b*d*x + a*d)*log(b*x + a))*log(d*x + c))/(a*b^2*c*g^2 - a^2*b*d*g^2 + (b^3*c*g^2 - a*b^2*d*g^2) *x))*B^2 - 2*A*B*(log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^2*g^2*x + a*b*g^ 2) + 1/(b^2*g^2*x + a*b*g^2) + d*log(b*x + a)/((b^2*c - a*b*d)*g^2) - d*lo g(d*x + c)/((b^2*c - a*b*d)*g^2)) - B^2*log(b*e*x/(d*x + c) + a*e/(d*x + c ))^2/(b^2*g^2*x + a*b*g^2) - A^2/(b^2*g^2*x + a*b*g^2)
Time = 0.40 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.52 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2} \, dx=-{\left (\frac {{\left (d x + c\right )} B^{2} e^{2} \log \left (\frac {b e x + a e}{d x + c}\right )^{2}}{{\left (b e x + a e\right )} g^{2}} + \frac {2 \, {\left (A B e^{2} + B^{2} e^{2}\right )} {\left (d x + c\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b e x + a e\right )} g^{2}} + \frac {{\left (A^{2} e^{2} + 2 \, A B e^{2} + 2 \, B^{2} e^{2}\right )} {\left (d x + c\right )}}{{\left (b e x + a e\right )} g^{2}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \]
-((d*x + c)*B^2*e^2*log((b*e*x + a*e)/(d*x + c))^2/((b*e*x + a*e)*g^2) + 2 *(A*B*e^2 + B^2*e^2)*(d*x + c)*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e) *g^2) + (A^2*e^2 + 2*A*B*e^2 + 2*B^2*e^2)*(d*x + c)/((b*e*x + a*e)*g^2))*( b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))
Time = 2.21 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.76 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2} \, dx=-\frac {A^2+2\,A\,B+2\,B^2}{x\,b^2\,g^2+a\,b\,g^2}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {B^2}{b^2\,g^2\,\left (x+\frac {a}{b}\right )}-\frac {B^2\,d}{b\,g^2\,\left (a\,d-b\,c\right )}\right )-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {2\,B^2}{b^2\,d\,g^2}+\frac {2\,A\,B}{b^2\,d\,g^2}\right )}{\frac {x}{d}+\frac {a}{b\,d}}-\frac {B\,d\,\mathrm {atan}\left (\frac {\left (2\,b\,d\,x+\frac {c\,b^2\,g^2+a\,d\,b\,g^2}{b\,g^2}\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (A+B\right )\,4{}\mathrm {i}}{b\,g^2\,\left (a\,d-b\,c\right )} \]
- (A^2 + 2*B^2 + 2*A*B)/(b^2*g^2*x + a*b*g^2) - log((e*(a + b*x))/(c + d*x ))^2*(B^2/(b^2*g^2*(x + a/b)) - (B^2*d)/(b*g^2*(a*d - b*c))) - (log((e*(a + b*x))/(c + d*x))*((2*B^2)/(b^2*d*g^2) + (2*A*B)/(b^2*d*g^2)))/(x/d + a/( b*d)) - (B*d*atan(((2*b*d*x + (b^2*c*g^2 + a*b*d*g^2)/(b*g^2))*1i)/(a*d - b*c))*(A + B)*4i)/(b*g^2*(a*d - b*c))