Integrand size = 21, antiderivative size = 112 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{d e (d+e x)}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 d (d+e x)^2}-\frac {b n \left (a+b n+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d e^2}-\frac {b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d e^2} \]
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Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2381, 2384, 2354, 2438} \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=-\frac {b n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )+b n\right )}{d e^2}+\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{d e (d+e x)}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 d (d+e x)^2}-\frac {b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d e^2} \]
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Rule 2354
Rule 2381
Rule 2384
Rule 2438
Rubi steps \begin{align*} \text {integral}& = \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 d (d+e x)^2}-\frac {(b n) \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx}{d} \\ & = \frac {b n x \left (a+b \log \left (c x^n\right )\right )}{d e (d+e x)}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 d (d+e x)^2}-\frac {(b n) \int \frac {a+b n+b \log \left (c x^n\right )}{d+e x} \, dx}{d e} \\ & = \frac {b n x \left (a+b \log \left (c x^n\right )\right )}{d e (d+e x)}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 d (d+e x)^2}-\frac {b n \left (a+b n+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d e^2}+\frac {\left (b^2 n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d e^2} \\ & = \frac {b n x \left (a+b \log \left (c x^n\right )\right )}{d e (d+e x)}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 d (d+e x)^2}-\frac {b n \left (a+b n+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d e^2}-\frac {b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d e^2} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.38 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\frac {-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{d+e x}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+\frac {2 b^2 n^2 (\log (x)-\log (d+e x))}{d}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d}}{2 e^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.52 (sec) , antiderivative size = 484, normalized size of antiderivative = 4.32
method | result | size |
risch | \(-\frac {b^{2} \ln \left (x^{n}\right )^{2}}{e^{2} \left (e x +d \right )}+\frac {b^{2} \ln \left (x^{n}\right )^{2} d}{2 e^{2} \left (e x +d \right )^{2}}-\frac {b^{2} n \ln \left (x^{n}\right )}{e^{2} \left (e x +d \right )}-\frac {b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{e^{2} d}+\frac {b^{2} n \ln \left (x^{n}\right ) \ln \left (x \right )}{e^{2} d}-\frac {b^{2} n^{2} \ln \left (x \right )^{2}}{2 e^{2} d}-\frac {b^{2} n^{2} \ln \left (e x +d \right )}{e^{2} d}+\frac {b^{2} n^{2} \ln \left (x \right )}{e^{2} d}+\frac {b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{2} d}+\frac {b^{2} n^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{2} 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^{2} \left (e x +d \right )}+\frac {\ln \left (x^{n}\right ) d}{2 e^{2} \left (e x +d \right )^{2}}-\frac {n \left (\frac {\ln \left (e x +d \right )}{d}+\frac {1}{e x +d}-\frac {\ln \left (x \right )}{d}\right )}{2 e^{2}}\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} \left (-\frac {1}{e^{2} \left (e x +d \right )}+\frac {d}{2 e^{2} \left (e x +d \right )^{2}}\right )}{4}\) | \(484\) |
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\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x}{{\left (e x + d\right )}^{3}} \,d x } \]
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\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int \frac {x \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{\left (d + e x\right )^{3}}\, dx \]
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\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x}{{\left (e x + d\right )}^{3}} \,d x } \]
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\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x}{{\left (e x + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int \frac {x\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \]
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