Integrand size = 30, antiderivative size = 175 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4} \, dx=-\frac {B}{9 b g^4 (a+b x)^3}+\frac {B d}{6 b (b c-a d) g^4 (a+b x)^2}-\frac {B d^2}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac {B d^3 \log (a+b x)}{3 b (b c-a d)^3 g^4}-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{3 b g^4 (a+b x)^3}+\frac {B d^3 \log (c+d x)}{3 b (b c-a d)^3 g^4} \] Output:
-1/9*B/b/g^4/(b*x+a)^3+1/6*B*d/b/(-a*d+b*c)/g^4/(b*x+a)^2-1/3*B*d^2/b/(-a* d+b*c)^2/g^4/(b*x+a)-1/3*B*d^3*ln(b*x+a)/b/(-a*d+b*c)^3/g^4-1/3*(A+B*ln(e* (b*x+a)/(d*x+c)))/b/g^4/(b*x+a)^3+1/3*B*d^3*ln(d*x+c)/b/(-a*d+b*c)^3/g^4
Time = 0.16 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.81 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4} \, dx=-\frac {6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+\frac {B \left ((b c-a d) \left (11 a^2 d^2+a b d (-7 c+15 d x)+b^2 \left (2 c^2-3 c d x+6 d^2 x^2\right )\right )+6 d^3 (a+b x)^3 \log (a+b x)-6 d^3 (a+b x)^3 \log (c+d x)\right )}{(b c-a d)^3}}{18 b g^4 (a+b x)^3} \] Input:
Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])/(a*g + b*g*x)^4,x]
Output:
-1/18*(6*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + (B*((b*c - a*d)*(11*a^2*d^ 2 + a*b*d*(-7*c + 15*d*x) + b^2*(2*c^2 - 3*c*d*x + 6*d^2*x^2)) + 6*d^3*(a + b*x)^3*Log[a + b*x] - 6*d^3*(a + b*x)^3*Log[c + d*x]))/(b*c - a*d)^3)/(b *g^4*(a + b*x)^3)
Time = 0.37 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2948, 27, 54, 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.
| \(\displaystyle \int \frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{(a g+b g x)^4} \, dx\) |
| \(\Big \downarrow \) 2948 |
| \(\displaystyle \frac {B (b c-a d) \int \frac {1}{g^3 (a+b x)^4 (c+d x)}dx}{3 b g}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 b g^4 (a+b x)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B (b c-a d) \int \frac {1}{(a+b x)^4 (c+d x)}dx}{3 b g^4}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 b g^4 (a+b x)^3}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {B (b c-a d) \int \left (\frac {d^4}{(b c-a d)^4 (c+d x)}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b}{(b c-a d) (a+b x)^4}\right )dx}{3 b g^4}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 b g^4 (a+b x)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {B (b c-a d) \left (-\frac {d^3 \log (a+b x)}{(b c-a d)^4}+\frac {d^3 \log (c+d x)}{(b c-a d)^4}-\frac {d^2}{(a+b x) (b c-a d)^3}+\frac {d}{2 (a+b x)^2 (b c-a d)^2}-\frac {1}{3 (a+b x)^3 (b c-a d)}\right )}{3 b g^4}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 b g^4 (a+b x)^3}\) |
Input:
Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])/(a*g + b*g*x)^4,x]
Output:
-1/3*(A + B*Log[(e*(a + b*x))/(c + d*x)])/(b*g^4*(a + b*x)^3) + (B*(b*c - a*d)*(-1/3*1/((b*c - a*d)*(a + b*x)^3) + d/(2*(b*c - a*d)^2*(a + b*x)^2) - d^2/((b*c - a*d)^3*(a + b*x)) - (d^3*Log[a + b*x])/(b*c - a*d)^4 + (d^3*L og[c + d*x])/(b*c - a*d)^4))/(3*b*g^4)
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[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[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])
Leaf count of result is larger than twice the leaf count of optimal. \(355\) vs. \(2(163)=326\).
Time = 1.26 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.03
| method | result | size |
| orering | \(\frac {\left (b x +a \right ) \left (15 b^{2} d^{3} x^{3}+39 a b \,d^{3} x^{2}+6 b^{2} c \,d^{2} x^{2}+31 a^{2} d^{3} x +16 a b c \,d^{2} x -2 b^{2} c^{2} d x +31 a^{2} d^{2} c -23 a b \,c^{2} d +7 b^{2} c^{3}\right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{9 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (b g x +a g \right )^{4}}+\frac {\left (6 b^{2} d^{2} x^{2}+15 a b \,d^{2} x -3 b^{2} c d x +11 a^{2} d^{2}-7 a c d b +2 c^{2} b^{2}\right ) \left (b x +a \right )^{2} \left (d x +c \right ) \left (\frac {B \left (\frac {e b}{d x +c}-\frac {e \left (b x +a \right ) d}{\left (d x +c \right )^{2}}\right ) \left (d x +c \right )}{e \left (b x +a \right ) \left (b g x +a g \right )^{4}}-\frac {4 \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) b g}{\left (b g x +a g \right )^{5}}\right )}{18 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(356\) |
| parts | \(-\frac {A}{3 g^{4} \left (b x +a \right )^{3} b}-\frac {B \left (d a -b c \right ) e \left (\frac {d^{4} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{\left (d a -b c \right )^{4}}-\frac {2 d^{3} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{4}}+\frac {d^{2} b^{2} e^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (d a -b c \right )^{4}}\right )}{g^{4} d^{2}}\) | \(361\) |
| risch | \(-\frac {B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{3 b \,g^{4} \left (b x +a \right )^{3}}-\frac {6 B \ln \left (d x +c \right ) b^{3} d^{3} x^{3}-6 B \ln \left (-b x -a \right ) b^{3} d^{3} x^{3}+18 B \ln \left (d x +c \right ) a \,b^{2} d^{3} x^{2}-18 B \ln \left (-b x -a \right ) a \,b^{2} d^{3} x^{2}+18 B \ln \left (d x +c \right ) a^{2} b \,d^{3} x -18 B \ln \left (-b x -a \right ) a^{2} b \,d^{3} x +6 B a \,b^{2} d^{3} x^{2}-6 B \,b^{3} c \,d^{2} x^{2}+6 B \ln \left (d x +c \right ) a^{3} d^{3}-6 B \ln \left (-b x -a \right ) a^{3} d^{3}+15 B \,a^{2} b \,d^{3} x -18 B a \,b^{2} c \,d^{2} x +3 B \,b^{3} c^{2} d x +6 A \,a^{3} d^{3}-18 A \,a^{2} b c \,d^{2}+18 A a \,b^{2} c^{2} d -6 A \,b^{3} c^{3}+11 B \,a^{3} d^{3}-18 B \,a^{2} b c \,d^{2}+9 B a \,b^{2} c^{2} d -2 B \,c^{3} b^{3}}{18 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (b x +a \right )^{3} g^{4} b}\) | \(377\) |
| parallelrisch | \(-\frac {-18 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{6} d^{4}-18 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{5} d^{4}-18 B x a \,b^{6} c \,d^{3}-18 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{5} c \,d^{3}+18 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{6} c^{2} d^{2}-18 A \,a^{2} b^{5} c \,d^{3}+18 A a \,b^{6} c^{2} d^{2}+6 A \,a^{3} b^{4} d^{4}-6 A \,b^{7} c^{3} d +11 B \,a^{3} b^{4} d^{4}-2 B \,b^{7} c^{3} d -18 B \,a^{2} b^{5} c \,d^{3}+9 B a \,b^{6} c^{2} d^{2}-6 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{7} d^{4}+6 B \,x^{2} a \,b^{6} d^{4}-6 B \,x^{2} b^{7} c \,d^{3}+15 B x \,a^{2} b^{5} d^{4}+3 B x \,b^{7} c^{2} d^{2}-6 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{7} c^{3} d}{18 g^{4} \left (b x +a \right )^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{5} d}\) | \(382\) |
| norman | \(\frac {\frac {B \,a^{2} d^{3} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) g}+\frac {B a b \,d^{3} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) g}-\frac {6 A \,a^{2} b \,d^{2}-12 A a \,b^{2} c d +6 A \,b^{3} c^{2}+9 B \,a^{2} b \,d^{2}-7 B a \,b^{2} c d +2 B \,b^{3} c^{2}}{18 g \,b^{2} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}-\frac {\left (3 B a b \,d^{2}-B \,b^{2} c d \right ) x}{6 g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) b}+\frac {B c \left (3 a^{2} d^{2}-3 a c d b +c^{2} b^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{3 g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {B \,b^{2} d^{2} x^{3}}{9 a g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {B \,b^{2} d^{3} x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{3 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) g}}{\left (b x +a \right )^{3} g^{3}}\) | \(472\) |
| derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (-\frac {d^{2} A \,b^{2} e^{2}}{3 \left (d a -b c \right )^{4} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}+\frac {d^{3} A b e}{\left (d a -b c \right )^{4} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {d^{4} A}{\left (d a -b c \right )^{4} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{2} B \,b^{2} e^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (d a -b c \right )^{4} g^{4}}-\frac {2 d^{3} B b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{4} g^{4}}+\frac {d^{4} B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{\left (d a -b c \right )^{4} g^{4}}\right )}{d^{2}}\) | \(503\) |
| default | \(-\frac {e \left (d a -b c \right ) \left (-\frac {d^{2} A \,b^{2} e^{2}}{3 \left (d a -b c \right )^{4} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}+\frac {d^{3} A b e}{\left (d a -b c \right )^{4} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {d^{4} A}{\left (d a -b c \right )^{4} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{2} B \,b^{2} e^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (d a -b c \right )^{4} g^{4}}-\frac {2 d^{3} B b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{4} g^{4}}+\frac {d^{4} B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{\left (d a -b c \right )^{4} g^{4}}\right )}{d^{2}}\) | \(503\) |
Input:
int((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x,method=_RETURNVERBOSE)
Output:
1/9*(b*x+a)*(15*b^2*d^3*x^3+39*a*b*d^3*x^2+6*b^2*c*d^2*x^2+31*a^2*d^3*x+16 *a*b*c*d^2*x-2*b^2*c^2*d*x+31*a^2*c*d^2-23*a*b*c^2*d+7*b^2*c^3)/(a^3*d^3-3 *a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g )^4+1/18/b*(6*b^2*d^2*x^2+15*a*b*d^2*x-3*b^2*c*d*x+11*a^2*d^2-7*a*b*c*d+2* b^2*c^2)/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*(b*x+a)^2*(d*x+c)*( B*(e*b/(d*x+c)-e*(b*x+a)/(d*x+c)^2*d)/e/(b*x+a)*(d*x+c)/(b*g*x+a*g)^4-4*(A +B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5*b*g)
Leaf count of result is larger than twice the leaf count of optimal. 406 vs. \(2 (163) = 326\).
Time = 0.09 (sec) , antiderivative size = 406, normalized size of antiderivative = 2.32 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4} \, dx=-\frac {2 \, {\left (3 \, A + B\right )} b^{3} c^{3} - 9 \, {\left (2 \, A + B\right )} a b^{2} c^{2} d + 18 \, {\left (A + B\right )} a^{2} b c d^{2} - {\left (6 \, A + 11 \, B\right )} a^{3} d^{3} + 6 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} x^{2} - 3 \, {\left (B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} + 5 \, B a^{2} b d^{3}\right )} x + 6 \, {\left (B b^{3} d^{3} x^{3} + 3 \, B a b^{2} d^{3} x^{2} + 3 \, B a^{2} b d^{3} x + B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{18 \, {\left ({\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} g^{4} x + {\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}\right )} g^{4}\right )}} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x, algorithm="fricas" )
Output:
-1/18*(2*(3*A + B)*b^3*c^3 - 9*(2*A + B)*a*b^2*c^2*d + 18*(A + B)*a^2*b*c* d^2 - (6*A + 11*B)*a^3*d^3 + 6*(B*b^3*c*d^2 - B*a*b^2*d^3)*x^2 - 3*(B*b^3* c^2*d - 6*B*a*b^2*c*d^2 + 5*B*a^2*b*d^3)*x + 6*(B*b^3*d^3*x^3 + 3*B*a*b^2* d^3*x^2 + 3*B*a^2*b*d^3*x + B*b^3*c^3 - 3*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2) *log((b*e*x + a*e)/(d*x + c)))/((b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*g^4*x^3 + 3*(a*b^6*c^3 - 3*a^2*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*g^4*x^2 + 3*(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d ^2 - a^5*b^2*d^3)*g^4*x + (a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3)*g^4)
Leaf count of result is larger than twice the leaf count of optimal. 656 vs. \(2 (150) = 300\).
Time = 1.78 (sec) , antiderivative size = 656, normalized size of antiderivative = 3.75 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4} \, dx=- \frac {B \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{3 a^{3} b g^{4} + 9 a^{2} b^{2} g^{4} x + 9 a b^{3} g^{4} x^{2} + 3 b^{4} g^{4} x^{3}} - \frac {B d^{3} \log {\left (x + \frac {- \frac {B a^{4} d^{7}}{\left (a d - b c\right )^{3}} + \frac {4 B a^{3} b c d^{6}}{\left (a d - b c\right )^{3}} - \frac {6 B a^{2} b^{2} c^{2} d^{5}}{\left (a d - b c\right )^{3}} + \frac {4 B a b^{3} c^{3} d^{4}}{\left (a d - b c\right )^{3}} + B a d^{4} - \frac {B b^{4} c^{4} d^{3}}{\left (a d - b c\right )^{3}} + B b c d^{3}}{2 B b d^{4}} \right )}}{3 b g^{4} \left (a d - b c\right )^{3}} + \frac {B d^{3} \log {\left (x + \frac {\frac {B a^{4} d^{7}}{\left (a d - b c\right )^{3}} - \frac {4 B a^{3} b c d^{6}}{\left (a d - b c\right )^{3}} + \frac {6 B a^{2} b^{2} c^{2} d^{5}}{\left (a d - b c\right )^{3}} - \frac {4 B a b^{3} c^{3} d^{4}}{\left (a d - b c\right )^{3}} + B a d^{4} + \frac {B b^{4} c^{4} d^{3}}{\left (a d - b c\right )^{3}} + B b c d^{3}}{2 B b d^{4}} \right )}}{3 b g^{4} \left (a d - b c\right )^{3}} + \frac {- 6 A a^{2} d^{2} + 12 A a b c d - 6 A b^{2} c^{2} - 11 B a^{2} d^{2} + 7 B a b c d - 2 B b^{2} c^{2} - 6 B b^{2} d^{2} x^{2} + x \left (- 15 B a b d^{2} + 3 B b^{2} c d\right )}{18 a^{5} b d^{2} g^{4} - 36 a^{4} b^{2} c d g^{4} + 18 a^{3} b^{3} c^{2} g^{4} + x^{3} \cdot \left (18 a^{2} b^{4} d^{2} g^{4} - 36 a b^{5} c d g^{4} + 18 b^{6} c^{2} g^{4}\right ) + x^{2} \cdot \left (54 a^{3} b^{3} d^{2} g^{4} - 108 a^{2} b^{4} c d g^{4} + 54 a b^{5} c^{2} g^{4}\right ) + x \left (54 a^{4} b^{2} d^{2} g^{4} - 108 a^{3} b^{3} c d g^{4} + 54 a^{2} b^{4} c^{2} g^{4}\right )} \] Input:
integrate((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**4,x)
Output:
-B*log(e*(a + b*x)/(c + d*x))/(3*a**3*b*g**4 + 9*a**2*b**2*g**4*x + 9*a*b* *3*g**4*x**2 + 3*b**4*g**4*x**3) - B*d**3*log(x + (-B*a**4*d**7/(a*d - b*c )**3 + 4*B*a**3*b*c*d**6/(a*d - b*c)**3 - 6*B*a**2*b**2*c**2*d**5/(a*d - b *c)**3 + 4*B*a*b**3*c**3*d**4/(a*d - b*c)**3 + B*a*d**4 - B*b**4*c**4*d**3 /(a*d - b*c)**3 + B*b*c*d**3)/(2*B*b*d**4))/(3*b*g**4*(a*d - b*c)**3) + B* d**3*log(x + (B*a**4*d**7/(a*d - b*c)**3 - 4*B*a**3*b*c*d**6/(a*d - b*c)** 3 + 6*B*a**2*b**2*c**2*d**5/(a*d - b*c)**3 - 4*B*a*b**3*c**3*d**4/(a*d - b *c)**3 + B*a*d**4 + B*b**4*c**4*d**3/(a*d - b*c)**3 + B*b*c*d**3)/(2*B*b*d **4))/(3*b*g**4*(a*d - b*c)**3) + (-6*A*a**2*d**2 + 12*A*a*b*c*d - 6*A*b** 2*c**2 - 11*B*a**2*d**2 + 7*B*a*b*c*d - 2*B*b**2*c**2 - 6*B*b**2*d**2*x**2 + x*(-15*B*a*b*d**2 + 3*B*b**2*c*d))/(18*a**5*b*d**2*g**4 - 36*a**4*b**2* c*d*g**4 + 18*a**3*b**3*c**2*g**4 + x**3*(18*a**2*b**4*d**2*g**4 - 36*a*b* *5*c*d*g**4 + 18*b**6*c**2*g**4) + x**2*(54*a**3*b**3*d**2*g**4 - 108*a**2 *b**4*c*d*g**4 + 54*a*b**5*c**2*g**4) + x*(54*a**4*b**2*d**2*g**4 - 108*a* *3*b**3*c*d*g**4 + 54*a**2*b**4*c**2*g**4))
Leaf count of result is larger than twice the leaf count of optimal. 428 vs. \(2 (163) = 326\).
Time = 0.05 (sec) , antiderivative size = 428, normalized size of antiderivative = 2.45 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4} \, dx=-\frac {1}{18} \, B {\left (\frac {6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \, {\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} g^{4} x + {\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} g^{4}} + \frac {6 \, \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}} + \frac {6 \, d^{3} \log \left (b x + a\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}} - \frac {6 \, d^{3} \log \left (d x + c\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}}\right )} - \frac {A}{3 \, {\left (b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}\right )}} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x, algorithm="maxima" )
Output:
-1/18*B*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5* c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c* d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*g^4) + 6*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3* a^2*b^2*g^4*x + a^3*b*g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4) - 6*d^3*log(d*x + c)/((b^4*c^3 - 3*a*b ^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4)) - 1/3*A/(b^4*g^4*x^3 + 3*a*b ^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4)
Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (163) = 326\).
Time = 0.26 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.38 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4} \, dx=-\frac {1}{18} \, {\left (\frac {6 \, {\left (B b^{2} e^{4} - \frac {3 \, {\left (b e x + a e\right )} B b d e^{3}}{d x + c} + \frac {3 \, {\left (b e x + a e\right )}^{2} B d^{2} e^{2}}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{3} b^{2} c^{2} g^{4}}{{\left (d x + c\right )}^{3}} - \frac {2 \, {\left (b e x + a e\right )}^{3} a b c d g^{4}}{{\left (d x + c\right )}^{3}} + \frac {{\left (b e x + a e\right )}^{3} a^{2} d^{2} g^{4}}{{\left (d x + c\right )}^{3}}} + \frac {6 \, A b^{2} e^{4} + 2 \, B b^{2} e^{4} - \frac {18 \, {\left (b e x + a e\right )} A b d e^{3}}{d x + c} - \frac {9 \, {\left (b e x + a e\right )} B b d e^{3}}{d x + c} + \frac {18 \, {\left (b e x + a e\right )}^{2} A d^{2} e^{2}}{{\left (d x + c\right )}^{2}} + \frac {18 \, {\left (b e x + a e\right )}^{2} B d^{2} e^{2}}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b e x + a e\right )}^{3} b^{2} c^{2} g^{4}}{{\left (d x + c\right )}^{3}} - \frac {2 \, {\left (b e x + a e\right )}^{3} a b c d g^{4}}{{\left (d x + c\right )}^{3}} + \frac {{\left (b e x + a e\right )}^{3} a^{2} d^{2} g^{4}}{{\left (d x + c\right )}^{3}}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \] Input:
integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x, algorithm="giac")
Output:
-1/18*(6*(B*b^2*e^4 - 3*(b*e*x + a*e)*B*b*d*e^3/(d*x + c) + 3*(b*e*x + a*e )^2*B*d^2*e^2/(d*x + c)^2)*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)^3*b ^2*c^2*g^4/(d*x + c)^3 - 2*(b*e*x + a*e)^3*a*b*c*d*g^4/(d*x + c)^3 + (b*e* x + a*e)^3*a^2*d^2*g^4/(d*x + c)^3) + (6*A*b^2*e^4 + 2*B*b^2*e^4 - 18*(b*e *x + a*e)*A*b*d*e^3/(d*x + c) - 9*(b*e*x + a*e)*B*b*d*e^3/(d*x + c) + 18*( b*e*x + a*e)^2*A*d^2*e^2/(d*x + c)^2 + 18*(b*e*x + a*e)^2*B*d^2*e^2/(d*x + c)^2)/((b*e*x + a*e)^3*b^2*c^2*g^4/(d*x + c)^3 - 2*(b*e*x + a*e)^3*a*b*c* d*g^4/(d*x + c)^3 + (b*e*x + a*e)^3*a^2*d^2*g^4/(d*x + c)^3))*(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 = 26.55 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.94 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4} \, dx=\frac {2\,A\,a\,c\,d}{3\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {A\,b\,c^2}{3\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,b\,c^2}{9\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {A\,a^2\,d^2}{3\,b\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {11\,B\,a^2\,d^2}{18\,b\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {5\,B\,a\,d^2\,x}{6\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,b\,d^2\,x^2}{3\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{3\,b\,g^4\,{\left (a+b\,x\right )}^3}+\frac {7\,B\,a\,c\,d}{18\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}+\frac {B\,b\,c\,d\,x}{6\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,d^3\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,2{}\mathrm {i}}{3\,b\,g^4\,{\left (a\,d-b\,c\right )}^3} \] Input:
int((A + B*log((e*(a + b*x))/(c + d*x)))/(a*g + b*g*x)^4,x)
Output:
(2*A*a*c*d)/(3*g^4*(a*d - b*c)^2*(a + b*x)^3) - (B*d^3*atan((a*d*1i + b*c* 1i + b*d*x*2i)/(a*d - b*c))*2i)/(3*b*g^4*(a*d - b*c)^3) - (A*b*c^2)/(3*g^4 *(a*d - b*c)^2*(a + b*x)^3) - (B*b*c^2)/(9*g^4*(a*d - b*c)^2*(a + b*x)^3) - (A*a^2*d^2)/(3*b*g^4*(a*d - b*c)^2*(a + b*x)^3) - (11*B*a^2*d^2)/(18*b*g ^4*(a*d - b*c)^2*(a + b*x)^3) - (5*B*a*d^2*x)/(6*g^4*(a*d - b*c)^2*(a + b* x)^3) - (B*b*d^2*x^2)/(3*g^4*(a*d - b*c)^2*(a + b*x)^3) - (B*log((e*(a + b *x))/(c + d*x)))/(3*b*g^4*(a + b*x)^3) + (7*B*a*c*d)/(18*g^4*(a*d - b*c)^2 *(a + b*x)^3) + (B*b*c*d*x)/(6*g^4*(a*d - b*c)^2*(a + b*x)^3)
Time = 0.16 (sec) , antiderivative size = 606, normalized size of antiderivative = 3.46 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4} \, dx=\frac {16 a^{3} b^{2} c \,d^{2}-9 a^{3} b^{2} d^{3} x -9 a^{2} b^{3} c^{2} d +2 a \,b^{4} d^{3} x^{3}-2 b^{5} c \,d^{2} x^{3}+6 \,\mathrm {log}\left (b x +a \right ) a^{4} b \,d^{3}-6 \,\mathrm {log}\left (d x +c \right ) a^{4} b \,d^{3}-6 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{4} b \,d^{3}+6 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{4} c^{3}-9 a^{4} b \,d^{3}+2 a \,b^{4} c^{3}+18 a^{4} b c \,d^{2}-18 a^{3} b^{2} c^{2} d +18 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{2} d^{3} x +18 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{3} d^{3} x^{2}+6 \,\mathrm {log}\left (b x +a \right ) a \,b^{4} d^{3} x^{3}-18 \,\mathrm {log}\left (d x +c \right ) a^{3} b^{2} d^{3} x -18 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{3} d^{3} x^{2}-6 \,\mathrm {log}\left (d x +c \right ) a \,b^{4} d^{3} x^{3}+18 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{3} b^{2} c \,d^{2}-18 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{2} b^{3} c^{2} d +12 a^{2} b^{3} c \,d^{2} x -3 a \,b^{4} c^{2} d x -6 a^{5} d^{3}+6 a^{2} b^{3} c^{3}}{18 a b \,g^{4} \left (a^{3} b^{3} d^{3} x^{3}-3 a^{2} b^{4} c \,d^{2} x^{3}+3 a \,b^{5} c^{2} d \,x^{3}-b^{6} c^{3} x^{3}+3 a^{4} b^{2} d^{3} x^{2}-9 a^{3} b^{3} c \,d^{2} x^{2}+9 a^{2} b^{4} c^{2} d \,x^{2}-3 a \,b^{5} c^{3} x^{2}+3 a^{5} b \,d^{3} x -9 a^{4} b^{2} c \,d^{2} x +9 a^{3} b^{3} c^{2} d x -3 a^{2} b^{4} c^{3} x +a^{6} d^{3}-3 a^{5} b c \,d^{2}+3 a^{4} b^{2} c^{2} d -a^{3} b^{3} c^{3}\right )} \] Input:
int((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x)
Output:
(6*log(a + b*x)*a**4*b*d**3 + 18*log(a + b*x)*a**3*b**2*d**3*x + 18*log(a + b*x)*a**2*b**3*d**3*x**2 + 6*log(a + b*x)*a*b**4*d**3*x**3 - 6*log(c + d *x)*a**4*b*d**3 - 18*log(c + d*x)*a**3*b**2*d**3*x - 18*log(c + d*x)*a**2* b**3*d**3*x**2 - 6*log(c + d*x)*a*b**4*d**3*x**3 - 6*log((a*e + b*e*x)/(c + d*x))*a**4*b*d**3 + 18*log((a*e + b*e*x)/(c + d*x))*a**3*b**2*c*d**2 - 1 8*log((a*e + b*e*x)/(c + d*x))*a**2*b**3*c**2*d + 6*log((a*e + b*e*x)/(c + d*x))*a*b**4*c**3 - 6*a**5*d**3 + 18*a**4*b*c*d**2 - 9*a**4*b*d**3 - 18*a **3*b**2*c**2*d + 16*a**3*b**2*c*d**2 - 9*a**3*b**2*d**3*x + 6*a**2*b**3*c **3 - 9*a**2*b**3*c**2*d + 12*a**2*b**3*c*d**2*x + 2*a*b**4*c**3 - 3*a*b** 4*c**2*d*x + 2*a*b**4*d**3*x**3 - 2*b**5*c*d**2*x**3)/(18*a*b*g**4*(a**6*d **3 - 3*a**5*b*c*d**2 + 3*a**5*b*d**3*x + 3*a**4*b**2*c**2*d - 9*a**4*b**2 *c*d**2*x + 3*a**4*b**2*d**3*x**2 - a**3*b**3*c**3 + 9*a**3*b**3*c**2*d*x - 9*a**3*b**3*c*d**2*x**2 + a**3*b**3*d**3*x**3 - 3*a**2*b**4*c**3*x + 9*a **2*b**4*c**2*d*x**2 - 3*a**2*b**4*c*d**2*x**3 - 3*a*b**5*c**3*x**2 + 3*a* b**5*c**2*d*x**3 - b**6*c**3*x**3))