Integrand size = 40, antiderivative size = 365 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\frac {B d^3 (a+b x)^2}{4 (b c-a d)^4 g^2 i^3 (c+d x)^2}-\frac {3 b B d^2 (a+b x)}{(b c-a d)^4 g^2 i^3 (c+d x)}-\frac {b^3 B (c+d x)}{(b c-a d)^4 g^2 i^3 (a+b x)}+\frac {3 b^2 B d \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^4 g^2 i^3}-\frac {d^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^4 g^2 i^3 (c+d x)^2}+\frac {3 b d^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^2 i^3 (c+d x)}-\frac {b^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^2 i^3 (a+b x)}-\frac {3 b^2 d \log \left (\frac {a+b x}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^2 i^3} \]
[Out]
Time = 0.17 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2562, 45, 2372, 2338} \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=-\frac {b^3 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^3 (a+b x) (b c-a d)^4}-\frac {3 b^2 d \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^3 (b c-a d)^4}-\frac {d^3 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^2 i^3 (c+d x)^2 (b c-a d)^4}+\frac {3 b d^2 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^3 (c+d x) (b c-a d)^4}-\frac {b^3 B (c+d x)}{g^2 i^3 (a+b x) (b c-a d)^4}+\frac {3 b^2 B d \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 g^2 i^3 (b c-a d)^4}+\frac {B d^3 (a+b x)^2}{4 g^2 i^3 (c+d x)^2 (b c-a d)^4}-\frac {3 b B d^2 (a+b x)}{g^2 i^3 (c+d x) (b c-a d)^4} \]
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Rule 45
Rule 2338
Rule 2372
Rule 2562
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(b-d x)^3 (A+B \log (e x))}{x^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^4 g^2 i^3} \\ & = -\frac {d^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^4 g^2 i^3 (c+d x)^2}+\frac {3 b d^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^2 i^3 (c+d x)}-\frac {b^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^2 i^3 (a+b x)}-\frac {3 b^2 d \log \left (\frac {a+b x}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^2 i^3}-\frac {B \text {Subst}\left (\int \left (3 b d^2-\frac {b^3}{x^2}-\frac {d^3 x}{2}-\frac {3 b^2 d \log (x)}{x}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^4 g^2 i^3} \\ & = \frac {B d^3 (a+b x)^2}{4 (b c-a d)^4 g^2 i^3 (c+d x)^2}-\frac {3 b B d^2 (a+b x)}{(b c-a d)^4 g^2 i^3 (c+d x)}-\frac {b^3 B (c+d x)}{(b c-a d)^4 g^2 i^3 (a+b x)}-\frac {d^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^4 g^2 i^3 (c+d x)^2}+\frac {3 b d^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^2 i^3 (c+d x)}-\frac {b^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^2 i^3 (a+b x)}-\frac {3 b^2 d \log \left (\frac {a+b x}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^2 i^3}+\frac {\left (3 b^2 B d\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^4 g^2 i^3} \\ & = \frac {B d^3 (a+b x)^2}{4 (b c-a d)^4 g^2 i^3 (c+d x)^2}-\frac {3 b B d^2 (a+b x)}{(b c-a d)^4 g^2 i^3 (c+d x)}-\frac {b^3 B (c+d x)}{(b c-a d)^4 g^2 i^3 (a+b x)}+\frac {3 b^2 B d \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^4 g^2 i^3}-\frac {d^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^4 g^2 i^3 (c+d x)^2}+\frac {3 b d^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^2 i^3 (c+d x)}-\frac {b^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^2 i^3 (a+b x)}-\frac {3 b^2 d \log \left (\frac {a+b x}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^2 i^3} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.38 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.24 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\frac {-\frac {4 b^3 B c}{a+b x}+\frac {4 a b^2 B d}{a+b x}+\frac {B d (b c-a d)^2}{(c+d x)^2}+\frac {8 b^2 B c d}{c+d x}-\frac {8 a b B d^2}{c+d x}+\frac {2 b B d (b c-a d)}{c+d x}+6 b^2 B d \log (a+b x)-\frac {4 b^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x}-\frac {2 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x)^2}-\frac {8 b d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x}-12 b^2 d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-6 b^2 B d \log (c+d x)+12 b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+6 b^2 B d \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )-6 b^2 B d \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{4 (b c-a d)^4 g^2 i^3} \]
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Time = 2.83 (sec) , antiderivative size = 506, normalized size of antiderivative = 1.39
method | result | size |
parts | \(\frac {A \left (-\frac {d}{2 \left (a d -c b \right )^{2} \left (d x +c \right )^{2}}+\frac {3 d \,b^{2} \ln \left (d x +c \right )}{\left (a d -c b \right )^{4}}+\frac {2 d b}{\left (a d -c b \right )^{3} \left (d x +c \right )}+\frac {b^{2}}{\left (a d -c b \right )^{3} \left (b x +a \right )}-\frac {3 d \,b^{2} \ln \left (b x +a \right )}{\left (a d -c b \right )^{4}}\right )}{g^{2} i^{3}}-\frac {B d \left (\frac {d^{2} \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{\left (a d -c b \right )^{2}}-\frac {3 d b e \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{\left (a d -c b \right )^{2}}+\frac {3 b^{2} e^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (a d -c b \right )^{2}}-\frac {b^{3} e^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{d \left (a d -c b \right )^{2}}\right )}{g^{2} i^{3} \left (a d -c b \right )^{2} e^{2}}\) | \(506\) |
derivativedivides | \(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A \,b^{3}}{i^{3} \left (a d -c b \right )^{5} g^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {3 d^{3} A \,b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{3} \left (a d -c b \right )^{5} g^{2}}-\frac {3 d^{4} A b \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{3} \left (a d -c b \right )^{5} g^{2}}+\frac {d^{5} A \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e^{3} i^{3} \left (a d -c b \right )^{5} g^{2}}-\frac {d^{2} B \,b^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{i^{3} \left (a d -c b \right )^{5} g^{2}}+\frac {3 d^{3} B \,b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e \,i^{3} \left (a d -c b \right )^{5} g^{2}}-\frac {3 d^{4} B b \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{e^{2} i^{3} \left (a d -c b \right )^{5} g^{2}}+\frac {d^{5} B \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{e^{3} i^{3} \left (a d -c b \right )^{5} g^{2}}\right )}{d^{2}}\) | \(632\) |
default | \(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A \,b^{3}}{i^{3} \left (a d -c b \right )^{5} g^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {3 d^{3} A \,b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{3} \left (a d -c b \right )^{5} g^{2}}-\frac {3 d^{4} A b \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{3} \left (a d -c b \right )^{5} g^{2}}+\frac {d^{5} A \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e^{3} i^{3} \left (a d -c b \right )^{5} g^{2}}-\frac {d^{2} B \,b^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{i^{3} \left (a d -c b \right )^{5} g^{2}}+\frac {3 d^{3} B \,b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e \,i^{3} \left (a d -c b \right )^{5} g^{2}}-\frac {3 d^{4} B b \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{e^{2} i^{3} \left (a d -c b \right )^{5} g^{2}}+\frac {d^{5} B \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{e^{3} i^{3} \left (a d -c b \right )^{5} g^{2}}\right )}{d^{2}}\) | \(632\) |
risch | \(-\frac {A d}{2 g^{2} i^{3} \left (a d -c b \right )^{2} \left (d x +c \right )^{2}}+\frac {3 A d \,b^{2} \ln \left (d x +c \right )}{g^{2} i^{3} \left (a d -c b \right )^{4}}+\frac {2 A d b}{g^{2} i^{3} \left (a d -c b \right )^{3} \left (d x +c \right )}+\frac {A \,b^{2}}{g^{2} i^{3} \left (a d -c b \right )^{3} \left (b x +a \right )}-\frac {3 A d \,b^{2} \ln \left (b x +a \right )}{g^{2} i^{3} \left (a d -c b \right )^{4}}+\frac {5 B d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) b^{2}}{2 g^{2} i^{3} \left (a d -c b \right )^{4}}+\frac {2 B \,d^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) b a}{g^{2} i^{3} \left (a d -c b \right )^{4} \left (d x +c \right )}-\frac {2 B d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) b^{2} c}{g^{2} i^{3} \left (a d -c b \right )^{4} \left (d x +c \right )}-\frac {B \,d^{3} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) a^{2}}{2 g^{2} i^{3} \left (a d -c b \right )^{4} \left (d x +c \right )^{2}}+\frac {B \,d^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) a c b}{g^{2} i^{3} \left (a d -c b \right )^{4} \left (d x +c \right )^{2}}-\frac {B d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) c^{2} b^{2}}{2 g^{2} i^{3} \left (a d -c b \right )^{4} \left (d x +c \right )^{2}}-\frac {11 B d \,b^{2}}{4 g^{2} i^{3} \left (a d -c b \right )^{4}}-\frac {5 B \,d^{2} b a}{2 g^{2} i^{3} \left (a d -c b \right )^{4} \left (d x +c \right )}+\frac {5 B d \,b^{2} c}{2 g^{2} i^{3} \left (a d -c b \right )^{4} \left (d x +c \right )}+\frac {B \,d^{3} a^{2}}{4 g^{2} i^{3} \left (a d -c b \right )^{4} \left (d x +c \right )^{2}}-\frac {B \,d^{2} a c b}{2 g^{2} i^{3} \left (a d -c b \right )^{4} \left (d x +c \right )^{2}}+\frac {B d \,c^{2} b^{2}}{4 g^{2} i^{3} \left (a d -c b \right )^{4} \left (d x +c \right )^{2}}-\frac {3 B d \,b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 g^{2} i^{3} \left (a d -c b \right )^{4}}-\frac {B e \,b^{3} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{g^{2} i^{3} \left (a d -c b \right )^{4} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}-\frac {B e \,b^{3}}{g^{2} i^{3} \left (a d -c b \right )^{4} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}\) | \(869\) |
parallelrisch | \(-\frac {6 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{5} d^{7}+12 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{6} c \,d^{6}+12 A \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} d^{7}+24 A \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} c \,d^{6}-18 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} d^{7}+6 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{6} c^{2} d^{5}+12 A x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} c^{2} d^{5}-6 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{4} d^{7}+12 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} c^{2} d^{5}+2 A \,a^{3} b^{3} d^{7}+4 A \,b^{6} c^{3} d^{4}-B \,a^{3} b^{3} d^{7}+4 B \,b^{6} c^{3} d^{4}+6 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{6} d^{7}+12 A \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} d^{7}-6 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} d^{7}-12 A \,x^{2} a \,b^{5} d^{7}+12 A \,x^{2} b^{6} c \,d^{6}+6 B \,x^{2} a \,b^{5} d^{7}-6 B \,x^{2} b^{6} c \,d^{6}-6 A x \,a^{2} b^{4} d^{7}+18 A x \,b^{6} c^{2} d^{5}+9 B x \,a^{2} b^{4} d^{7}-3 B x \,b^{6} c^{2} d^{5}+2 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b^{3} d^{7}+4 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} c^{3} d^{4}-12 a^{2} A \,b^{4} c \,d^{6}+6 a A \,b^{5} c^{2} d^{5}+12 a^{2} B \,b^{4} c \,d^{6}-15 a B \,b^{5} c^{2} d^{5}+6 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{5} c^{2} d^{5}-12 A x a \,b^{5} c \,d^{6}+12 A \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} c^{2} d^{5}-6 B x a \,b^{5} c \,d^{6}-12 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{4} c \,d^{6}+12 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{5} c \,d^{6}+24 A x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} c \,d^{6}-24 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} c \,d^{6}}{4 i^{3} g^{2} \left (d x +c \right )^{2} \left (b x +a \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 ) b^{3} d^{4} \left (a d -c b \right )}\) | \(870\) |
norman | \(\text {Expression too large to display}\) | \(1178\) |
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Time = 0.31 (sec) , antiderivative size = 672, normalized size of antiderivative = 1.84 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=-\frac {4 \, {\left (A + B\right )} b^{3} c^{3} + 3 \, {\left (2 \, A - 5 \, B\right )} a b^{2} c^{2} d - 12 \, {\left (A - B\right )} a^{2} b c d^{2} + {\left (2 \, A - B\right )} a^{3} d^{3} + 6 \, {\left ({\left (2 \, A - B\right )} b^{3} c d^{2} - {\left (2 \, A - B\right )} a b^{2} d^{3}\right )} x^{2} + 6 \, {\left (B b^{3} d^{3} x^{3} + B a b^{2} c^{2} d + {\left (2 \, B b^{3} c d^{2} + B a b^{2} d^{3}\right )} x^{2} + {\left (B b^{3} c^{2} d + 2 \, B a b^{2} c d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 3 \, {\left ({\left (6 \, A - B\right )} b^{3} c^{2} d - 2 \, {\left (2 \, A + B\right )} a b^{2} c d^{2} - {\left (2 \, A - 3 \, B\right )} a^{2} b d^{3}\right )} x + 2 \, {\left (3 \, {\left (2 \, A - B\right )} b^{3} d^{3} x^{3} + 2 \, B b^{3} c^{3} + 6 \, A a b^{2} c^{2} d - 6 \, B a^{2} b c d^{2} + B a^{3} d^{3} + 3 \, {\left (4 \, A b^{3} c d^{2} + {\left (2 \, A - 3 \, B\right )} a b^{2} d^{3}\right )} x^{2} + 3 \, {\left (2 \, {\left (A + B\right )} b^{3} c^{2} d + 4 \, {\left (A - B\right )} a b^{2} c d^{2} - B a^{2} b d^{3}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} g^{2} i^{3} x^{3} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} g^{2} i^{3} x^{2} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} g^{2} i^{3} x + {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4}\right )} g^{2} i^{3}\right )}} \]
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Timed out. \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1721 vs. \(2 (359) = 718\).
Time = 0.33 (sec) , antiderivative size = 1721, normalized size of antiderivative = 4.72 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\text {Too large to display} \]
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\[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\int { \frac {B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A}{{\left (b g x + a g\right )}^{2} {\left (d i x + c i\right )}^{3}} \,d x } \]
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Time = 5.71 (sec) , antiderivative size = 983, normalized size of antiderivative = 2.69 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\frac {A\,b^2\,c^2}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {A\,a^2\,d^2}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {3\,B\,b^2\,d\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^4}+\frac {B\,a^2\,d^2}{4\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {B\,b^2\,c^2}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {B\,a\,d\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {B\,b\,c\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {3\,A\,b^2\,d^2\,x^2}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {3\,B\,b^2\,d^2\,x^2}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {5\,A\,a\,b\,c\,d}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {11\,B\,a\,b\,c\,d}{4\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {3\,B\,b\,d\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {3\,B\,b^2\,d^2\,x^2\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {3\,A\,a\,b\,d^2\,x}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {9\,B\,a\,b\,d^2\,x}{4\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {9\,A\,b^2\,c\,d\,x}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}-\frac {3\,B\,b^2\,c\,d\,x}{4\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {3\,B\,a\,b\,c\,d\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {3\,B\,a\,b\,d^2\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {3\,B\,b^2\,c\,d\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2}+\frac {A\,b^2\,d\,\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 )\,6{}\mathrm {i}}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^4}-\frac {B\,b^2\,d\,\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 )\,3{}\mathrm {i}}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^4} \]
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