Integrand size = 31, antiderivative size = 148 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x} \, dx=-\frac {B n \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log (g+h x)}{h}+\frac {B n \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \log (g+h x)}{h}-\frac {B n \operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{h}+\frac {B n \operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{h} \]
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Time = 0.09 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2546, 2441, 2440, 2438} \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x} \, dx=\frac {\log (g+h x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{h}-\frac {B n \operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{h}-\frac {B n \log (g+h x) \log \left (-\frac {h (a+b x)}{b g-a h}\right )}{h}+\frac {B n \operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{h}+\frac {B n \log (g+h x) \log \left (-\frac {h (c+d x)}{d g-c h}\right )}{h} \]
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Rule 2438
Rule 2440
Rule 2441
Rule 2546
Rubi steps \begin{align*} \text {integral}& = \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \log (g+h x)}{h}-\frac {(b B n) \int \frac {\log (g+h x)}{a+b x} \, dx}{h}+\frac {(B d n) \int \frac {\log (g+h x)}{c+d x} \, dx}{h} \\ & = -\frac {B n \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log (g+h x)}{h}+\frac {B n \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \log (g+h x)}{h}+(B n) \int \frac {\log \left (\frac {h (a+b x)}{-b g+a h}\right )}{g+h x} \, dx-(B n) \int \frac {\log \left (\frac {h (c+d x)}{-d g+c h}\right )}{g+h x} \, dx \\ & = -\frac {B n \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log (g+h x)}{h}+\frac {B n \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \log (g+h x)}{h}+\frac {(B n) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b g+a h}\right )}{x} \, dx,x,g+h x\right )}{h}-\frac {(B n) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{-d g+c h}\right )}{x} \, dx,x,g+h x\right )}{h} \\ & = -\frac {B n \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log (g+h x)}{h}+\frac {B n \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \log (g+h x)}{h}-\frac {B n \text {Li}_2\left (\frac {b (g+h x)}{b g-a h}\right )}{h}+\frac {B n \text {Li}_2\left (\frac {d (g+h x)}{d g-c h}\right )}{h} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.01 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x} \, dx=\frac {\left (A+B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right ) \log (g+h x)+B n \left (\log (a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right )+\operatorname {PolyLog}\left (2,\frac {h (a+b x)}{-b g+a h}\right )\right )-B n \left (\log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right )+\operatorname {PolyLog}\left (2,\frac {h (c+d x)}{-d g+c h}\right )\right )}{h} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.52 (sec) , antiderivative size = 521, normalized size of antiderivative = 3.52
method | result | size |
risch | \(\frac {\left (-i B \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )+i B \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{2}-i B \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+i B \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}+i B \pi \,\operatorname {csgn}\left (i \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}-i B \pi \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3}+i B \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{2}-i B \pi \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{3}+2 B \ln \left (e \right )+2 A \right ) \ln \left (h x +g \right )}{2 h}+\frac {B \ln \left (\left (b x +a \right )^{n}\right ) \ln \left (h x +g \right )}{h}-\frac {B n \operatorname {dilog}\left (\frac {\left (h x +g \right ) b +a h -b g}{a h -b g}\right )}{h}-\frac {B n \ln \left (h x +g \right ) \ln \left (\frac {\left (h x +g \right ) b +a h -b g}{a h -b g}\right )}{h}-\frac {B \ln \left (\left (d x +c \right )^{n}\right ) \ln \left (h x +g \right )}{h}+\frac {B n \operatorname {dilog}\left (\frac {d \left (h x +g \right )+c h -d g}{c h -d g}\right )}{h}+\frac {B n \ln \left (h x +g \right ) \ln \left (\frac {d \left (h x +g \right )+c h -d g}{c h -d g}\right )}{h}\) | \(521\) |
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\[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x} \, dx=\int { \frac {B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A}{h x + g} \,d x } \]
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Exception generated. \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x} \, dx=\int { \frac {B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A}{h x + g} \,d x } \]
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\[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x} \, dx=\int { \frac {B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A}{h x + g} \,d x } \]
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Timed out. \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x} \, dx=\int \frac {A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}{g+h\,x} \,d x \]
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