Integrand size = 38, antiderivative size = 125 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\frac {g (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d i}+\frac {(b c-a d) g \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2 i}+\frac {B (b c-a d) g \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i} \]
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Time = 0.10 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2562, 2384, 2354, 2438} \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\frac {g (b c-a d) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A+B\right )}{d^2 i}+\frac {g (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d i}+\frac {B g (b c-a d) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i} \]
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Rule 2354
Rule 2384
Rule 2438
Rule 2562
Rubi steps \begin{align*} \text {integral}& = \frac {((b c-a d) g) \text {Subst}\left (\int \frac {x (A+B \log (e x))}{(b-d x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{i} \\ & = \frac {g (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d i}-\frac {((b c-a d) g) \text {Subst}\left (\int \frac {A+B+B \log (e x)}{b-d x} \, dx,x,\frac {a+b x}{c+d x}\right )}{d i} \\ & = \frac {g (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d i}+\frac {(b c-a d) g \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2 i}-\frac {(B (b c-a d) g) \text {Subst}\left (\int \frac {\log \left (1-\frac {d x}{b}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{d^2 i} \\ & = \frac {g (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d i}+\frac {(b c-a d) g \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2 i}+\frac {B (b c-a d) g \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.30 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\frac {g \left (2 A b d x+2 B d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-2 B (b c-a d) \log (c+d x)-2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+B (b c-a d) \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{2 d^2 i} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(331\) vs. \(2(125)=250\).
Time = 1.34 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.66
method | result | size |
parts | \(\frac {g A \left (\frac {b x}{d}+\frac {\left (a d -c b \right ) \ln \left (d x +c \right )}{d^{2}}\right )}{i}-\frac {g B \left (b e \left (a d -c b \right ) \left (\frac {\ln \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}{b e d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}\right )+\left (a d -c b \right ) \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )\right )}{i d}\) | \(332\) |
derivativedivides | \(-\frac {e \left (a d -c b \right ) \left (\frac {g A b}{i \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {g A \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e i}+\frac {g B \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e i}+\frac {g d B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e i \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {g B \operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{e i}+\frac {g B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{e i}\right )}{d^{2}}\) | \(380\) |
default | \(-\frac {e \left (a d -c b \right ) \left (\frac {g A b}{i \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {g A \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e i}+\frac {g B \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e i}+\frac {g d B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e i \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {g B \operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{e i}+\frac {g B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{e i}\right )}{d^{2}}\) | \(380\) |
risch | \(\frac {g A b x}{i d}+\frac {g A \ln \left (d x +c \right ) a}{i d}-\frac {g A \ln \left (d x +c \right ) c b}{i \,d^{2}}-\frac {g B \ln \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right ) a}{i d}+\frac {g B b \ln \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right ) c}{i \,d^{2}}+\frac {g B b e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) a}{i d \left (\frac {e d a}{d x +c}-\frac {e c b}{d x +c}\right )}-\frac {g B \,b^{2} e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) c}{i \,d^{2} \left (\frac {e d a}{d x +c}-\frac {e c b}{d x +c}\right )}+\frac {g B e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) a^{2}}{i \left (\frac {e d a}{d x +c}-\frac {e c b}{d x +c}\right ) \left (d x +c \right )}-\frac {2 g B b e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) a c}{i d \left (\frac {e d a}{d x +c}-\frac {e c b}{d x +c}\right ) \left (d x +c \right )}+\frac {g B \,b^{2} e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) c^{2}}{i \,d^{2} \left (\frac {e d a}{d x +c}-\frac {e c b}{d x +c}\right ) \left (d x +c \right )}-\frac {g B \operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right ) a}{i d}+\frac {g B \operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right ) c b}{i \,d^{2}}-\frac {g B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right ) a}{i d}+\frac {g B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right ) c b}{i \,d^{2}}\) | \(769\) |
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\[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\int { \frac {{\left (b g x + a g\right )} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}}{d i x + c i} \,d x } \]
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\[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\frac {g \left (\int \frac {A a}{c + d x}\, dx + \int \frac {A b x}{c + d x}\, dx + \int \frac {B a \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{c + d x}\, dx + \int \frac {B b x \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{c + d x}\, dx\right )}{i} \]
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Time = 0.25 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.77 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=A b g {\left (\frac {x}{d i} - \frac {c \log \left (d x + c\right )}{d^{2} i}\right )} + \frac {A a g \log \left (d i x + c i\right )}{d i} - \frac {{\left (b c g - a d g\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B}{d^{2} i} + \frac {{\left (a d g \log \left (e\right ) - {\left (g \log \left (e\right ) + g\right )} b c\right )} B \log \left (d x + c\right )}{d^{2} i} - \frac {2 \, B b d g x \log \left (d x + c\right ) - 2 \, B b d g x \log \left (e\right ) - {\left (b c g - a d g\right )} B \log \left (d x + c\right )^{2} - 2 \, {\left (B b d g x + B a d g\right )} \log \left (b x + a\right )}{2 \, d^{2} i} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1230 vs. \(2 (124) = 248\).
Time = 42.82 (sec) , antiderivative size = 1230, normalized size of antiderivative = 9.84 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\int \frac {\left (a\,g+b\,g\,x\right )\,\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}{c\,i+d\,i\,x} \,d x \]
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