Integrand size = 23, antiderivative size = 138 \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x)^2} \, dx=-\frac {b e f n \log (d+e x)}{g^2 (e f-d g)}+\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 (f+g x)}+\frac {b e f n \log (f+g x)}{g^2 (e f-d g)}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^2} \]
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Time = 0.12 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {45, 2463, 2442, 36, 31, 2441, 2440, 2438} \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x)^2} \, dx=\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 (f+g x)}+\frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^2}-\frac {b e f n \log (d+e x)}{g^2 (e f-d g)}+\frac {b e f n \log (f+g x)}{g^2 (e f-d g)} \]
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Rule 31
Rule 36
Rule 45
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2463
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (f+g x)^2}+\frac {a+b \log \left (c (d+e x)^n\right )}{g (f+g x)}\right ) \, dx \\ & = \frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx}{g}-\frac {f \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^2} \, dx}{g} \\ & = \frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 (f+g x)}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}-\frac {(b e n) \int \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{g^2}-\frac {(b e f n) \int \frac {1}{(d+e x) (f+g x)} \, dx}{g^2} \\ & = \frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 (f+g x)}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}-\frac {(b n) \text {Subst}\left (\int \frac {\log \left (1+\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g^2}-\frac {\left (b e^2 f n\right ) \int \frac {1}{d+e x} \, dx}{g^2 (e f-d g)}+\frac {(b e f n) \int \frac {1}{f+g x} \, dx}{g (e f-d g)} \\ & = -\frac {b e f n \log (d+e x)}{g^2 (e f-d g)}+\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 (f+g x)}+\frac {b e f n \log (f+g x)}{g^2 (e f-d g)}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}+\frac {b n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^2} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.83 \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x)^2} \, dx=\frac {\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x}-\frac {b e f n (\log (d+e x)-\log (f+g x))}{e f-d g}+\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )+b n \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )}{g^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.80 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.25
method | result | size |
risch | \(\frac {b \ln \left (\left (e x +d \right )^{n}\right ) f}{g^{2} \left (g x +f \right )}+\frac {b \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{g^{2}}-\frac {b n \operatorname {dilog}\left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{g^{2}}-\frac {b n \ln \left (g x +f \right ) \ln \left (\frac {\left (g x +f \right ) e +d g -e f}{d g -e f}\right )}{g^{2}}-\frac {b e n f \ln \left (g x +f \right )}{g^{2} \left (d g -e f \right )}+\frac {b e n f \ln \left (\left (g x +f \right ) e +d g -e f \right )}{g^{2} \left (d g -e f \right )}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}-\frac {i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b}{2}+b \ln \left (c \right )+a \right ) \left (\frac {f}{g^{2} \left (g x +f \right )}+\frac {\ln \left (g x +f \right )}{g^{2}}\right )\) | \(311\) |
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\[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x)^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x}{{\left (g x + f\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x)^2} \, dx=\text {Timed out} \]
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\[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x)^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x}{{\left (g x + f\right )}^{2}} \,d x } \]
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\[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x)^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x}{{\left (g x + f\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x)^2} \, dx=\int \frac {x\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{{\left (f+g\,x\right )}^2} \,d x \]
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