Integrand size = 29, antiderivative size = 144 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{f+g x} \, dx=-\frac {2 B \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{g}+\frac {2 B \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}-\frac {2 B \operatorname {PolyLog}\left (2,\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {2 B \operatorname {PolyLog}\left (2,\frac {d (f+g x)}{d f-c g}\right )}{g} \]
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Time = 0.08 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2546, 2441, 2440, 2438} \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{f+g x} \, dx=\frac {\log (f+g x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{g}-\frac {2 B \operatorname {PolyLog}\left (2,\frac {b (f+g x)}{b f-a g}\right )}{g}-\frac {2 B \log (f+g x) \log \left (-\frac {g (a+b x)}{b f-a g}\right )}{g}+\frac {2 B \operatorname {PolyLog}\left (2,\frac {d (f+g x)}{d f-c g}\right )}{g}+\frac {2 B \log (f+g x) \log \left (-\frac {g (c+d x)}{d f-c g}\right )}{g} \]
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Rule 2438
Rule 2440
Rule 2441
Rule 2546
Rubi steps \begin{align*} \text {integral}& = \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{g}-\frac {(2 b B) \int \frac {\log (f+g x)}{a+b x} \, dx}{g}+\frac {(2 B d) \int \frac {\log (f+g x)}{c+d x} \, dx}{g} \\ & = -\frac {2 B \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{g}+\frac {2 B \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}+(2 B) \int \frac {\log \left (\frac {g (a+b x)}{-b f+a g}\right )}{f+g x} \, dx-(2 B) \int \frac {\log \left (\frac {g (c+d x)}{-d f+c g}\right )}{f+g x} \, dx \\ & = -\frac {2 B \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{g}+\frac {2 B \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}+\frac {(2 B) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b f+a g}\right )}{x} \, dx,x,f+g x\right )}{g}-\frac {(2 B) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{-d f+c g}\right )}{x} \, dx,x,f+g x\right )}{g} \\ & = -\frac {2 B \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (f+g x)}{g}+\frac {2 B \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}-\frac {2 B \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {2 B \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.83 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{f+g x} \, dx=\frac {\left (A-2 B \log \left (\frac {g (a+b x)}{-b f+a g}\right )+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+2 B \log \left (\frac {g (c+d x)}{-d f+c g}\right )\right ) \log (f+g x)-2 B \operatorname {PolyLog}\left (2,\frac {b (f+g x)}{b f-a g}\right )+2 B \operatorname {PolyLog}\left (2,\frac {d (f+g x)}{d f-c g}\right )}{g} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(395\) vs. \(2(144)=288\).
Time = 3.37 (sec) , antiderivative size = 396, normalized size of antiderivative = 2.75
method | result | size |
parts | \(\frac {A \ln \left (g x +f \right )}{g}+B \left (-\frac {\ln \left (\frac {1}{d x +c}\right ) \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )-\left (2 a d -2 c b \right ) \left (\frac {\operatorname {dilog}\left (\frac {\frac {a d -c b}{d x +c}+b}{b}\right )}{a d -c b}+\frac {\ln \left (\frac {1}{d x +c}\right ) \ln \left (\frac {\frac {a d -c b}{d x +c}+b}{b}\right )}{a d -c b}\right )}{g}+\frac {\left (\frac {\ln \left (\frac {c g -d f}{d x +c}-g \right ) \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{c g -d f}-\frac {2 \left (a d -c b \right ) \left (\frac {\operatorname {dilog}\left (\frac {\left (\frac {c g -d f}{d x +c}-g \right ) \left (a d -c b \right )+a d g -b d f}{a d g -b d f}\right )}{a d -c b}+\frac {\ln \left (\frac {c g -d f}{d x +c}-g \right ) \ln \left (\frac {\left (\frac {c g -d f}{d x +c}-g \right ) \left (a d -c b \right )+a d g -b d f}{a d g -b d f}\right )}{a d -c b}\right )}{c g -d f}\right ) \left (c g -d f \right )}{g}\right )\) | \(396\) |
derivativedivides | \(-\frac {-d A \left (-\frac {\ln \left (\frac {1}{d x +c}\right )}{g}+\frac {\ln \left (\frac {c g}{d x +c}-\frac {f d}{d x +c}-g \right )}{g}\right )-d B \left (-\frac {\ln \left (\frac {1}{d x +c}\right ) \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )-\left (2 a d -2 c b \right ) \left (\frac {\operatorname {dilog}\left (\frac {\frac {a d -c b}{d x +c}+b}{b}\right )}{a d -c b}+\frac {\ln \left (\frac {1}{d x +c}\right ) \ln \left (\frac {\frac {a d -c b}{d x +c}+b}{b}\right )}{a d -c b}\right )}{g}+\frac {\left (\frac {\ln \left (\frac {c g -d f}{d x +c}-g \right ) \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{c g -d f}-\frac {2 \left (a d -c b \right ) \left (\frac {\operatorname {dilog}\left (\frac {\left (\frac {c g -d f}{d x +c}-g \right ) \left (a d -c b \right )+a d g -b d f}{a d g -b d f}\right )}{a d -c b}+\frac {\ln \left (\frac {c g -d f}{d x +c}-g \right ) \ln \left (\frac {\left (\frac {c g -d f}{d x +c}-g \right ) \left (a d -c b \right )+a d g -b d f}{a d g -b d f}\right )}{a d -c b}\right )}{c g -d f}\right ) \left (c g -d f \right )}{g}\right )}{d}\) | \(440\) |
default | \(-\frac {-d A \left (-\frac {\ln \left (\frac {1}{d x +c}\right )}{g}+\frac {\ln \left (\frac {c g}{d x +c}-\frac {f d}{d x +c}-g \right )}{g}\right )-d B \left (-\frac {\ln \left (\frac {1}{d x +c}\right ) \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )-\left (2 a d -2 c b \right ) \left (\frac {\operatorname {dilog}\left (\frac {\frac {a d -c b}{d x +c}+b}{b}\right )}{a d -c b}+\frac {\ln \left (\frac {1}{d x +c}\right ) \ln \left (\frac {\frac {a d -c b}{d x +c}+b}{b}\right )}{a d -c b}\right )}{g}+\frac {\left (\frac {\ln \left (\frac {c g -d f}{d x +c}-g \right ) \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{c g -d f}-\frac {2 \left (a d -c b \right ) \left (\frac {\operatorname {dilog}\left (\frac {\left (\frac {c g -d f}{d x +c}-g \right ) \left (a d -c b \right )+a d g -b d f}{a d g -b d f}\right )}{a d -c b}+\frac {\ln \left (\frac {c g -d f}{d x +c}-g \right ) \ln \left (\frac {\left (\frac {c g -d f}{d x +c}-g \right ) \left (a d -c b \right )+a d g -b d f}{a d g -b d f}\right )}{a d -c b}\right )}{c g -d f}\right ) \left (c g -d f \right )}{g}\right )}{d}\) | \(440\) |
risch | \(\text {Expression too large to display}\) | \(1109\) |
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\[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{f+g x} \, dx=\int { \frac {B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A}{g x + f} \,d x } \]
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Timed out. \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{f+g x} \, dx=\text {Timed out} \]
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\[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{f+g x} \, dx=\int { \frac {B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A}{g x + f} \,d x } \]
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\[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{f+g x} \, dx=\int { \frac {B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A}{g x + f} \,d x } \]
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Timed out. \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{f+g x} \, dx=\int \frac {A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )}{f+g\,x} \,d x \]
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