Integrand size = 20, antiderivative size = 126 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)}-\frac {b n \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}+\frac {b^2 n^2 \log (d+e x)}{d^2 e}+\frac {b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^2 e} \]
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Time = 0.10 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2356, 2389, 2379, 2438, 2351, 31} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=-\frac {b n \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 e}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}+\frac {b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^2 e}+\frac {b^2 n^2 \log (d+e x)}{d^2 e} \]
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Rule 31
Rule 2351
Rule 2356
Rule 2379
Rule 2389
Rule 2438
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}+\frac {(b n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{e} \\ & = -\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}-\frac {(b n) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d}+\frac {(b n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{d e} \\ & = -\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)}-\frac {b n \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}+\frac {\left (b^2 n^2\right ) \int \frac {1}{d+e x} \, dx}{d^2}+\frac {\left (b^2 n^2\right ) \int \frac {\log \left (1+\frac {d}{e x}\right )}{x} \, dx}{d^2 e} \\ & = -\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)}-\frac {b n \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}+\frac {b^2 n^2 \log (d+e x)}{d^2 e}+\frac {b^2 n^2 \text {Li}_2\left (-\frac {d}{e x}\right )}{d^2 e} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.16 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}+\frac {b n \left (\frac {a+b \log \left (c x^n\right )}{d (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b d^2 n}-\frac {b n \left (\frac {\log (x)}{d}-\frac {\log (d+e x)}{d}\right )}{d}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {d+e x}{d}\right )}{d^2}-\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^2}\right )}{e} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.48 (sec) , antiderivative size = 435, normalized size of antiderivative = 3.45
method | result | size |
risch | \(-\frac {b^{2} \ln \left (x^{n}\right )^{2}}{2 e \left (e x +d \right )^{2}}-\frac {b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{e \,d^{2}}+\frac {b^{2} n \ln \left (x^{n}\right )}{e d \left (e x +d \right )}+\frac {b^{2} n \ln \left (x^{n}\right ) \ln \left (x \right )}{e \,d^{2}}-\frac {b^{2} n^{2} \ln \left (x \right )^{2}}{2 e \,d^{2}}+\frac {b^{2} n^{2} \ln \left (e x +d \right )}{d^{2} e}-\frac {b^{2} n^{2} \ln \left (x \right )}{e \,d^{2}}+\frac {b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e \,d^{2}}+\frac {b^{2} n^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e \,d^{2}}+\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right ) b \left (-\frac {\ln \left (x^{n}\right )}{2 e \left (e x +d \right )^{2}}+\frac {n \left (-\frac {\ln \left (e x +d \right )}{d^{2}}+\frac {1}{d \left (e x +d \right )}+\frac {\ln \left (x \right )}{d^{2}}\right )}{2 e}\right )-\frac {{\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right )}^{2}}{8 \left (e x +d \right )^{2} e}\) | \(435\) |
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{\left (d + e x\right )^{3}}\, dx \]
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \]
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