Integrand size = 30, antiderivative size = 118 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \, dx=-\frac {B (b c-a d)^2 g^2 x}{3 d^2}+\frac {B (b c-a d) g^2 (a+b x)^2}{6 b d}+\frac {B (b c-a d)^3 g^2 \log (c+d x)}{3 b d^3}+\frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b} \]
-1/3*B*(-a*d+b*c)^2*g^2*x/d^2+1/6*B*(-a*d+b*c)*g^2*(b*x+a)^2/b/d+1/3*B*(-a *d+b*c)^3*g^2*ln(d*x+c)/b/d^3+1/3*g^2*(b*x+a)^3*(A+B*ln(e*(d*x+c)/(b*x+a)) )/b
Time = 0.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.84 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \, dx=\frac {g^2 \left (\frac {B (b c-a d) \left (d \left (a^2 d+4 a b d x+b^2 x (-2 c+d x)\right )+2 (b c-a d)^2 \log (c+d x)\right )}{2 d^3}+(a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )\right )}{3 b} \]
(g^2*((B*(b*c - a*d)*(d*(a^2*d + 4*a*b*d*x + b^2*x*(-2*c + d*x)) + 2*(b*c - a*d)^2*Log[c + d*x]))/(2*d^3) + (a + b*x)^3*(A + B*Log[(e*(c + d*x))/(a + b*x)])))/(3*b)
Time = 0.26 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2948, 27, 49, 2009}
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.
| \(\displaystyle \int (a g+b g x)^2 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right ) \, dx\) |
| \(\Big \downarrow \) 2948 |
| \(\displaystyle \frac {B (b c-a d) \int \frac {g^3 (a+b x)^2}{c+d x}dx}{3 b g}+\frac {g^2 (a+b x)^3 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{3 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B g^2 (b c-a d) \int \frac {(a+b x)^2}{c+d x}dx}{3 b}+\frac {g^2 (a+b x)^3 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{3 b}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {B g^2 (b c-a d) \int \left (\frac {(a d-b c)^2}{d^2 (c+d x)}-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}\right )dx}{3 b}+\frac {g^2 (a+b x)^3 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{3 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {g^2 (a+b x)^3 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{3 b}+\frac {B g^2 (b c-a d) \left (\frac {(b c-a d)^2 \log (c+d x)}{d^3}-\frac {b x (b c-a d)}{d^2}+\frac {(a+b x)^2}{2 d}\right )}{3 b}\) |
(B*(b*c - a*d)*g^2*(-((b*(b*c - a*d)*x)/d^2) + (a + b*x)^2/(2*d) + ((b*c - a*d)^2*Log[c + d*x])/d^3))/(3*b) + (g^2*(a + b*x)^3*(A + B*Log[(e*(c + d* x))/(a + b*x)]))/(3*b)
3.2.75.3.1 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_.) + (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}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(f + g*x)^(m + 1)*( (A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])/(g*(m + 1))), x] - Simp[B*n*((b*c - a*d)/(g*(m + 1))) Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] / ; FreeQ[{a, b, c, d, e, f, g, A, B, m, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && !(EqQ[m, -2] && IntegerQ[n])
Time = 0.79 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.75
| method | result | size |
| risch | \(\frac {\left (b x +a \right )^{3} g^{2} B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )}{3 b}+\frac {g^{2} b^{2} A \,x^{3}}{3}+g^{2} b A a \,x^{2}-\frac {g^{2} b B a \,x^{2}}{6}+\frac {g^{2} b^{2} B c \,x^{2}}{6 d}+g^{2} A \,a^{2} x -\frac {g^{2} B \ln \left (d x +c \right ) a^{3}}{3 b}+\frac {g^{2} B \ln \left (d x +c \right ) a^{2} c}{d}-\frac {g^{2} b B \ln \left (d x +c \right ) a \,c^{2}}{d^{2}}+\frac {g^{2} b^{2} B \ln \left (d x +c \right ) c^{3}}{3 d^{3}}-\frac {2 g^{2} B \,a^{2} x}{3}+\frac {g^{2} b B a c x}{d}-\frac {g^{2} b^{2} B \,c^{2} x}{3 d^{2}}\) | \(206\) |
| parts | \(\frac {A \,g^{2} \left (b x +a \right )^{3}}{3 b}+B \,g^{2} e^{3} \left (a d -c b \right )^{3} \left (-\frac {1}{6 d e b \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )^{2}}+\frac {\ln \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )}{3 d^{3} e^{3} b}+\frac {1}{3 d^{2} e^{2} b \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )}-\frac {\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) \left (b^{2} \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{2}-3 \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b d e +3 d^{2} e^{2}\right )}{3 d^{3} e^{3} \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )^{3}}\right )\) | \(358\) |
| derivativedivides | \(\frac {e \left (a d -c b \right ) \left (-\frac {A b \,e^{2} g^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{3 \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )^{3}}+B \,b^{2} e^{2} g^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (-\frac {1}{6 d e b \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )^{2}}+\frac {\ln \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )}{3 d^{3} e^{3} b}+\frac {1}{3 d^{2} e^{2} b \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )}-\frac {\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) \left (b^{2} \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{2}-3 \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b d e +3 d^{2} e^{2}\right )}{3 d^{3} e^{3} \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )^{3}}\right )\right )}{b^{2}}\) | \(437\) |
| default | \(\frac {e \left (a d -c b \right ) \left (-\frac {A b \,e^{2} g^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{3 \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )^{3}}+B \,b^{2} e^{2} g^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (-\frac {1}{6 d e b \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )^{2}}+\frac {\ln \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )}{3 d^{3} e^{3} b}+\frac {1}{3 d^{2} e^{2} b \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )}-\frac {\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) \left (b^{2} \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{2}-3 \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b d e +3 d^{2} e^{2}\right )}{3 d^{3} e^{3} \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )^{3}}\right )\right )}{b^{2}}\) | \(437\) |
| parallelrisch | \(\frac {2 B \,x^{3} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) b^{3} d^{3} g^{2}+4 B \,a^{3} d^{3} g^{2}+6 A \,x^{2} a \,b^{2} d^{3} g^{2}-B \,x^{2} a \,b^{2} d^{3} g^{2}+B \,x^{2} b^{3} c \,d^{2} g^{2}+6 A x \,a^{2} b \,d^{3} g^{2}-4 B x \,a^{2} b \,d^{3} g^{2}-2 B x \,b^{3} c^{2} d \,g^{2}+6 B \,x^{2} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a \,b^{2} d^{3} g^{2}+6 B x \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a^{2} b \,d^{3} g^{2}+6 B x a \,b^{2} c \,d^{2} g^{2}+6 B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a^{2} b c \,d^{2} g^{2}-6 B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a \,b^{2} c^{2} d \,g^{2}+6 B \ln \left (b x +a \right ) a^{2} b c \,d^{2} g^{2}-6 B \ln \left (b x +a \right ) a \,b^{2} c^{2} d \,g^{2}-6 A \,a^{3} d^{3} g^{2}-B \,a^{2} b c \,d^{2} g^{2}-5 B a \,b^{2} c^{2} d \,g^{2}-12 A \,a^{2} b c \,d^{2} g^{2}+2 B \,c^{3} g^{2} b^{3}+2 B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) b^{3} c^{3} g^{2}-2 B \ln \left (b x +a \right ) a^{3} d^{3} g^{2}+2 B \ln \left (b x +a \right ) b^{3} c^{3} g^{2}+2 A \,x^{3} b^{3} d^{3} g^{2}}{6 b \,d^{3}}\) | \(458\) |
1/3*(b*x+a)^3*g^2*B/b*ln(e*(d*x+c)/(b*x+a))+1/3*g^2*b^2*A*x^3+g^2*b*A*a*x^ 2-1/6*g^2*b*B*a*x^2+1/6*g^2*b^2/d*B*c*x^2+g^2*A*a^2*x-1/3*g^2/b*B*ln(d*x+c )*a^3+g^2/d*B*ln(d*x+c)*a^2*c-g^2*b/d^2*B*ln(d*x+c)*a*c^2+1/3*g^2*b^2/d^3* B*ln(d*x+c)*c^3-2/3*g^2*B*a^2*x+g^2*b/d*B*a*c*x-1/3*g^2*b^2/d^2*B*c^2*x
Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (110) = 220\).
Time = 0.28 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.89 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \, dx=\frac {2 \, A b^{3} d^{3} g^{2} x^{3} - 2 \, B a^{3} d^{3} g^{2} \log \left (b x + a\right ) + {\left (B b^{3} c d^{2} + {\left (6 \, A - B\right )} a b^{2} d^{3}\right )} g^{2} x^{2} - 2 \, {\left (B b^{3} c^{2} d - 3 \, B a b^{2} c d^{2} - {\left (3 \, A - 2 \, B\right )} a^{2} b d^{3}\right )} g^{2} x + 2 \, {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} g^{2} \log \left (d x + c\right ) + 2 \, {\left (B b^{3} d^{3} g^{2} x^{3} + 3 \, B a b^{2} d^{3} g^{2} x^{2} + 3 \, B a^{2} b d^{3} g^{2} x\right )} \log \left (\frac {d e x + c e}{b x + a}\right )}{6 \, b d^{3}} \]
1/6*(2*A*b^3*d^3*g^2*x^3 - 2*B*a^3*d^3*g^2*log(b*x + a) + (B*b^3*c*d^2 + ( 6*A - B)*a*b^2*d^3)*g^2*x^2 - 2*(B*b^3*c^2*d - 3*B*a*b^2*c*d^2 - (3*A - 2* B)*a^2*b*d^3)*g^2*x + 2*(B*b^3*c^3 - 3*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2)*g^ 2*log(d*x + c) + 2*(B*b^3*d^3*g^2*x^3 + 3*B*a*b^2*d^3*g^2*x^2 + 3*B*a^2*b* d^3*g^2*x)*log((d*e*x + c*e)/(b*x + a)))/(b*d^3)
Leaf count of result is larger than twice the leaf count of optimal. 491 vs. \(2 (100) = 200\).
Time = 1.48 (sec) , antiderivative size = 491, normalized size of antiderivative = 4.16 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \, dx=\frac {A b^{2} g^{2} x^{3}}{3} - \frac {B a^{3} g^{2} \log {\left (x + \frac {\frac {B a^{4} d^{3} g^{2}}{b} + 3 B a^{3} c d^{2} g^{2} - 3 B a^{2} b c^{2} d g^{2} + B a b^{2} c^{3} g^{2}}{B a^{3} d^{3} g^{2} + 3 B a^{2} b c d^{2} g^{2} - 3 B a b^{2} c^{2} d g^{2} + B b^{3} c^{3} g^{2}} \right )}}{3 b} + \frac {B c g^{2} \cdot \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {4 B a^{3} c d^{2} g^{2} - 3 B a^{2} b c^{2} d g^{2} + B a b^{2} c^{3} g^{2} - B a c g^{2} \cdot \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) + \frac {B b c^{2} g^{2} \cdot \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{d}}{B a^{3} d^{3} g^{2} + 3 B a^{2} b c d^{2} g^{2} - 3 B a b^{2} c^{2} d g^{2} + B b^{3} c^{3} g^{2}} \right )}}{3 d^{3}} + x^{2} \left (A a b g^{2} - \frac {B a b g^{2}}{6} + \frac {B b^{2} c g^{2}}{6 d}\right ) + x \left (A a^{2} g^{2} - \frac {2 B a^{2} g^{2}}{3} + \frac {B a b c g^{2}}{d} - \frac {B b^{2} c^{2} g^{2}}{3 d^{2}}\right ) + \left (B a^{2} g^{2} x + B a b g^{2} x^{2} + \frac {B b^{2} g^{2} x^{3}}{3}\right ) \log {\left (\frac {e \left (c + d x\right )}{a + b x} \right )} \]
A*b**2*g**2*x**3/3 - B*a**3*g**2*log(x + (B*a**4*d**3*g**2/b + 3*B*a**3*c* d**2*g**2 - 3*B*a**2*b*c**2*d*g**2 + B*a*b**2*c**3*g**2)/(B*a**3*d**3*g**2 + 3*B*a**2*b*c*d**2*g**2 - 3*B*a*b**2*c**2*d*g**2 + B*b**3*c**3*g**2))/(3 *b) + B*c*g**2*(3*a**2*d**2 - 3*a*b*c*d + b**2*c**2)*log(x + (4*B*a**3*c*d **2*g**2 - 3*B*a**2*b*c**2*d*g**2 + B*a*b**2*c**3*g**2 - B*a*c*g**2*(3*a** 2*d**2 - 3*a*b*c*d + b**2*c**2) + B*b*c**2*g**2*(3*a**2*d**2 - 3*a*b*c*d + b**2*c**2)/d)/(B*a**3*d**3*g**2 + 3*B*a**2*b*c*d**2*g**2 - 3*B*a*b**2*c** 2*d*g**2 + B*b**3*c**3*g**2))/(3*d**3) + x**2*(A*a*b*g**2 - B*a*b*g**2/6 + B*b**2*c*g**2/(6*d)) + x*(A*a**2*g**2 - 2*B*a**2*g**2/3 + B*a*b*c*g**2/d - B*b**2*c**2*g**2/(3*d**2)) + (B*a**2*g**2*x + B*a*b*g**2*x**2 + B*b**2*g **2*x**3/3)*log(e*(c + d*x)/(a + b*x))
Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (110) = 220\).
Time = 0.22 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.36 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \, dx=\frac {1}{3} \, A b^{2} g^{2} x^{3} + A a b g^{2} x^{2} + {\left (x \log \left (\frac {d e x}{b x + a} + \frac {c e}{b x + a}\right ) - \frac {a \log \left (b x + a\right )}{b} + \frac {c \log \left (d x + c\right )}{d}\right )} B a^{2} g^{2} + {\left (x^{2} \log \left (\frac {d e x}{b x + a} + \frac {c e}{b x + a}\right ) + \frac {a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c - a d\right )} x}{b d}\right )} B a b g^{2} + \frac {1}{6} \, {\left (2 \, x^{3} \log \left (\frac {d e x}{b x + a} + \frac {c e}{b x + a}\right ) - \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} + \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} + \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B b^{2} g^{2} + A a^{2} g^{2} x \]
1/3*A*b^2*g^2*x^3 + A*a*b*g^2*x^2 + (x*log(d*e*x/(b*x + a) + c*e/(b*x + a) ) - a*log(b*x + a)/b + c*log(d*x + c)/d)*B*a^2*g^2 + (x^2*log(d*e*x/(b*x + a) + c*e/(b*x + a)) + a^2*log(b*x + a)/b^2 - c^2*log(d*x + c)/d^2 + (b*c - a*d)*x/(b*d))*B*a*b*g^2 + 1/6*(2*x^3*log(d*e*x/(b*x + a) + c*e/(b*x + a) ) - 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))*B*b^2*g^2 + A*a^2*g^2*x
Leaf count of result is larger than twice the leaf count of optimal. 1056 vs. \(2 (110) = 220\).
Time = 0.40 (sec) , antiderivative size = 1056, normalized size of antiderivative = 8.95 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \, dx=-\frac {1}{6} \, {\left (\frac {2 \, {\left (B b^{4} c^{4} e^{4} g^{2} - 4 \, B a b^{3} c^{3} d e^{4} g^{2} + 6 \, B a^{2} b^{2} c^{2} d^{2} e^{4} g^{2} - 4 \, B a^{3} b c d^{3} e^{4} g^{2} + B a^{4} d^{4} e^{4} g^{2}\right )} \log \left (\frac {d e x + c e}{b x + a}\right )}{b d^{3} e^{3} - \frac {3 \, {\left (d e x + c e\right )} b^{2} d^{2} e^{2}}{b x + a} + \frac {3 \, {\left (d e x + c e\right )}^{2} b^{3} d e}{{\left (b x + a\right )}^{2}} - \frac {{\left (d e x + c e\right )}^{3} b^{4}}{{\left (b x + a\right )}^{3}}} + \frac {2 \, A b^{4} c^{4} d^{2} e^{4} g^{2} - 3 \, B b^{4} c^{4} d^{2} e^{4} g^{2} - 8 \, A a b^{3} c^{3} d^{3} e^{4} g^{2} + 12 \, B a b^{3} c^{3} d^{3} e^{4} g^{2} + 12 \, A a^{2} b^{2} c^{2} d^{4} e^{4} g^{2} - 18 \, B a^{2} b^{2} c^{2} d^{4} e^{4} g^{2} - 8 \, A a^{3} b c d^{5} e^{4} g^{2} + 12 \, B a^{3} b c d^{5} e^{4} g^{2} + 2 \, A a^{4} d^{6} e^{4} g^{2} - 3 \, B a^{4} d^{6} e^{4} g^{2} + \frac {5 \, {\left (d e x + c e\right )} B b^{5} c^{4} d e^{3} g^{2}}{b x + a} - \frac {20 \, {\left (d e x + c e\right )} B a b^{4} c^{3} d^{2} e^{3} g^{2}}{b x + a} + \frac {30 \, {\left (d e x + c e\right )} B a^{2} b^{3} c^{2} d^{3} e^{3} g^{2}}{b x + a} - \frac {20 \, {\left (d e x + c e\right )} B a^{3} b^{2} c d^{4} e^{3} g^{2}}{b x + a} + \frac {5 \, {\left (d e x + c e\right )} B a^{4} b d^{5} e^{3} g^{2}}{b x + a} - \frac {2 \, {\left (d e x + c e\right )}^{2} B b^{6} c^{4} e^{2} g^{2}}{{\left (b x + a\right )}^{2}} + \frac {8 \, {\left (d e x + c e\right )}^{2} B a b^{5} c^{3} d e^{2} g^{2}}{{\left (b x + a\right )}^{2}} - \frac {12 \, {\left (d e x + c e\right )}^{2} B a^{2} b^{4} c^{2} d^{2} e^{2} g^{2}}{{\left (b x + a\right )}^{2}} + \frac {8 \, {\left (d e x + c e\right )}^{2} B a^{3} b^{3} c d^{3} e^{2} g^{2}}{{\left (b x + a\right )}^{2}} - \frac {2 \, {\left (d e x + c e\right )}^{2} B a^{4} b^{2} d^{4} e^{2} g^{2}}{{\left (b x + a\right )}^{2}}}{b d^{5} e^{3} - \frac {3 \, {\left (d e x + c e\right )} b^{2} d^{4} e^{2}}{b x + a} + \frac {3 \, {\left (d e x + c e\right )}^{2} b^{3} d^{3} e}{{\left (b x + a\right )}^{2}} - \frac {{\left (d e x + c e\right )}^{3} b^{4} d^{2}}{{\left (b x + a\right )}^{3}}} + \frac {2 \, {\left (B b^{4} c^{4} e g^{2} - 4 \, B a b^{3} c^{3} d e g^{2} + 6 \, B a^{2} b^{2} c^{2} d^{2} e g^{2} - 4 \, B a^{3} b c d^{3} e g^{2} + B a^{4} d^{4} e g^{2}\right )} \log \left (-d e + \frac {{\left (d e x + c e\right )} b}{b x + a}\right )}{b d^{3}} - \frac {2 \, {\left (B b^{4} c^{4} e g^{2} - 4 \, B a b^{3} c^{3} d e g^{2} + 6 \, B a^{2} b^{2} c^{2} d^{2} e g^{2} - 4 \, B a^{3} b c d^{3} e g^{2} + B a^{4} d^{4} e g^{2}\right )} \log \left (\frac {d e x + c e}{b x + a}\right )}{b d^{3}}\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 )} \]
-1/6*(2*(B*b^4*c^4*e^4*g^2 - 4*B*a*b^3*c^3*d*e^4*g^2 + 6*B*a^2*b^2*c^2*d^2 *e^4*g^2 - 4*B*a^3*b*c*d^3*e^4*g^2 + B*a^4*d^4*e^4*g^2)*log((d*e*x + c*e)/ (b*x + a))/(b*d^3*e^3 - 3*(d*e*x + c*e)*b^2*d^2*e^2/(b*x + a) + 3*(d*e*x + c*e)^2*b^3*d*e/(b*x + a)^2 - (d*e*x + c*e)^3*b^4/(b*x + a)^3) + (2*A*b^4* c^4*d^2*e^4*g^2 - 3*B*b^4*c^4*d^2*e^4*g^2 - 8*A*a*b^3*c^3*d^3*e^4*g^2 + 12 *B*a*b^3*c^3*d^3*e^4*g^2 + 12*A*a^2*b^2*c^2*d^4*e^4*g^2 - 18*B*a^2*b^2*c^2 *d^4*e^4*g^2 - 8*A*a^3*b*c*d^5*e^4*g^2 + 12*B*a^3*b*c*d^5*e^4*g^2 + 2*A*a^ 4*d^6*e^4*g^2 - 3*B*a^4*d^6*e^4*g^2 + 5*(d*e*x + c*e)*B*b^5*c^4*d*e^3*g^2/ (b*x + a) - 20*(d*e*x + c*e)*B*a*b^4*c^3*d^2*e^3*g^2/(b*x + a) + 30*(d*e*x + c*e)*B*a^2*b^3*c^2*d^3*e^3*g^2/(b*x + a) - 20*(d*e*x + c*e)*B*a^3*b^2*c *d^4*e^3*g^2/(b*x + a) + 5*(d*e*x + c*e)*B*a^4*b*d^5*e^3*g^2/(b*x + a) - 2 *(d*e*x + c*e)^2*B*b^6*c^4*e^2*g^2/(b*x + a)^2 + 8*(d*e*x + c*e)^2*B*a*b^5 *c^3*d*e^2*g^2/(b*x + a)^2 - 12*(d*e*x + c*e)^2*B*a^2*b^4*c^2*d^2*e^2*g^2/ (b*x + a)^2 + 8*(d*e*x + c*e)^2*B*a^3*b^3*c*d^3*e^2*g^2/(b*x + a)^2 - 2*(d *e*x + c*e)^2*B*a^4*b^2*d^4*e^2*g^2/(b*x + a)^2)/(b*d^5*e^3 - 3*(d*e*x + c *e)*b^2*d^4*e^2/(b*x + a) + 3*(d*e*x + c*e)^2*b^3*d^3*e/(b*x + a)^2 - (d*e *x + c*e)^3*b^4*d^2/(b*x + a)^3) + 2*(B*b^4*c^4*e*g^2 - 4*B*a*b^3*c^3*d*e* g^2 + 6*B*a^2*b^2*c^2*d^2*e*g^2 - 4*B*a^3*b*c*d^3*e*g^2 + B*a^4*d^4*e*g^2) *log(-d*e + (d*e*x + c*e)*b/(b*x + a))/(b*d^3) - 2*(B*b^4*c^4*e*g^2 - 4*B* a*b^3*c^3*d*e*g^2 + 6*B*a^2*b^2*c^2*d^2*e*g^2 - 4*B*a^3*b*c*d^3*e*g^2 +...
Time = 1.23 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.46 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \, dx=x^2\,\left (\frac {b\,g^2\,\left (9\,A\,a\,d+3\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{6\,d}-\frac {A\,b\,g^2\,\left (3\,a\,d+3\,b\,c\right )}{6\,d}\right )-x\,\left (\frac {\left (3\,a\,d+3\,b\,c\right )\,\left (\frac {b\,g^2\,\left (9\,A\,a\,d+3\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{3\,d}-\frac {A\,b\,g^2\,\left (3\,a\,d+3\,b\,c\right )}{3\,d}\right )}{3\,b\,d}-\frac {a\,g^2\,\left (3\,A\,a\,d+3\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{d}+\frac {A\,a\,b\,c\,g^2}{d}\right )+\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )\,\left (B\,a^2\,g^2\,x+B\,a\,b\,g^2\,x^2+\frac {B\,b^2\,g^2\,x^3}{3}\right )+\frac {\ln \left (c+d\,x\right )\,\left (3\,B\,a^2\,c\,d^2\,g^2-3\,B\,a\,b\,c^2\,d\,g^2+B\,b^2\,c^3\,g^2\right )}{3\,d^3}+\frac {A\,b^2\,g^2\,x^3}{3}-\frac {B\,a^3\,g^2\,\ln \left (a+b\,x\right )}{3\,b} \]
x^2*((b*g^2*(9*A*a*d + 3*A*b*c - B*a*d + B*b*c))/(6*d) - (A*b*g^2*(3*a*d + 3*b*c))/(6*d)) - x*(((3*a*d + 3*b*c)*((b*g^2*(9*A*a*d + 3*A*b*c - B*a*d + B*b*c))/(3*d) - (A*b*g^2*(3*a*d + 3*b*c))/(3*d)))/(3*b*d) - (a*g^2*(3*A*a *d + 3*A*b*c - B*a*d + B*b*c))/d + (A*a*b*c*g^2)/d) + log((e*(c + d*x))/(a + b*x))*((B*b^2*g^2*x^3)/3 + B*a^2*g^2*x + B*a*b*g^2*x^2) + (log(c + d*x) *(B*b^2*c^3*g^2 + 3*B*a^2*c*d^2*g^2 - 3*B*a*b*c^2*d*g^2))/(3*d^3) + (A*b^2 *g^2*x^3)/3 - (B*a^3*g^2*log(a + b*x))/(3*b)