Integrand size = 20, antiderivative size = 77 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d e}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d e} \]
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Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2355, 2354, 2438} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=-\frac {2 b n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d e}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d e} \]
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Rule 2354
Rule 2355
Rule 2438
Rubi steps \begin{align*} \text {integral}& = \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d} \\ & = \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d e}+\frac {\left (2 b^2 n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d e} \\ & = \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d e}-\frac {2 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d e} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right ) \left (a e x+b e x \log \left (c x^n\right )-2 b n (d+e x) \log \left (1+\frac {e x}{d}\right )\right )-2 b^2 n^2 (d+e x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d e (d+e x)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.40 (sec) , antiderivative size = 369, normalized size of antiderivative = 4.79
method | result | size |
risch | \(-\frac {b^{2} \ln \left (x^{n}\right )^{2}}{e \left (e x +d \right )}-\frac {2 b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{e d}+\frac {2 b^{2} n \ln \left (x^{n}\right ) \ln \left (x \right )}{e d}-\frac {b^{2} n^{2} \ln \left (x \right )^{2}}{e d}+\frac {2 b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e d}+\frac {2 b^{2} n^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e d}+\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right ) b \left (-\frac {\ln \left (x^{n}\right )}{e \left (e x +d \right )}+\frac {n \left (-\frac {\ln \left (e x +d \right )}{d}+\frac {\ln \left (x \right )}{d}\right )}{e}\right )-\frac {{\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right )}^{2}}{4 \left (e x +d \right ) e}\) | \(369\) |
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{\left (d + e x\right )^{2}}\, dx \]
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x\right )}^2} \,d x \]
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