Integrand size = 23, antiderivative size = 273 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^4 (d+e x)} \, dx=-\frac {2 b^2 n^2}{27 d x^3}+\frac {b^2 e n^2}{4 d^2 x^2}-\frac {2 b^2 e^2 n^2}{d^3 x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{9 d x^3}+\frac {b e n \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}-\frac {2 b e^2 n \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 d x^3}+\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 x^2}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}+\frac {e^3 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^4}-\frac {2 b e^3 n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^4}-\frac {2 b^2 e^3 n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^4} \]
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Time = 0.33 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2380, 2342, 2341, 2379, 2421, 6724} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^4 (d+e x)} \, dx=-\frac {2 b e^3 n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}+\frac {e^3 \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^4}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}-\frac {2 b e^2 n \left (a+b \log \left (c x^n\right )\right )}{d^3 x}+\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 x^2}+\frac {b e n \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 d x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{9 d x^3}-\frac {2 b^2 e^3 n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^4}-\frac {2 b^2 e^2 n^2}{d^3 x}+\frac {b^2 e n^2}{4 d^2 x^2}-\frac {2 b^2 n^2}{27 d x^3} \]
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Rule 2341
Rule 2342
Rule 2379
Rule 2380
Rule 2421
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)} \, dx}{d} \\ & = -\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 d x^3}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx}{d^2}+\frac {e^2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)} \, dx}{d^2}+\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{x^4} \, dx}{3 d} \\ & = -\frac {2 b^2 n^2}{27 d x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{9 d x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 d x^3}+\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 x^2}+\frac {e^2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx}{d^3}-\frac {e^3 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)} \, dx}{d^3}-\frac {(b e n) \int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx}{d^2} \\ & = -\frac {2 b^2 n^2}{27 d x^3}+\frac {b^2 e n^2}{4 d^2 x^2}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{9 d x^3}+\frac {b e n \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 d x^3}+\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 x^2}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}+\frac {e^3 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^4}+\frac {\left (2 b e^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{d^3}-\frac {\left (2 b e^3 n\right ) \int \frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{d^4} \\ & = -\frac {2 b^2 n^2}{27 d x^3}+\frac {b^2 e n^2}{4 d^2 x^2}-\frac {2 b^2 e^2 n^2}{d^3 x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{9 d x^3}+\frac {b e n \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}-\frac {2 b e^2 n \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 d x^3}+\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 x^2}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}+\frac {e^3 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^4}-\frac {2 b e^3 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {d}{e x}\right )}{d^4}+\frac {\left (2 b^2 e^3 n^2\right ) \int \frac {\text {Li}_2\left (-\frac {d}{e x}\right )}{x} \, dx}{d^4} \\ & = -\frac {2 b^2 n^2}{27 d x^3}+\frac {b^2 e n^2}{4 d^2 x^2}-\frac {2 b^2 e^2 n^2}{d^3 x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{9 d x^3}+\frac {b e n \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}-\frac {2 b e^2 n \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 d x^3}+\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 x^2}-\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}+\frac {e^3 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^4}-\frac {2 b e^3 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {d}{e x}\right )}{d^4}-\frac {2 b^2 e^3 n^2 \text {Li}_3\left (-\frac {d}{e x}\right )}{d^4} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^4 (d+e x)} \, dx=\frac {-\frac {36 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{x^3}+\frac {54 d^2 e \left (a+b \log \left (c x^n\right )\right )^2}{x^2}-\frac {108 d e^2 \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {36 e^3 \left (a+b \log \left (c x^n\right )\right )^3}{b n}-\frac {216 b d e^2 n \left (a+b n+b \log \left (c x^n\right )\right )}{x}+\frac {27 b d^2 e n \left (2 a+b n+2 b \log \left (c x^n\right )\right )}{x^2}-\frac {8 b d^3 n \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{x^3}+108 e^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )+216 b e^3 n \left (\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-b n \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )}{108 d^4} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.53 (sec) , antiderivative size = 828, normalized size of antiderivative = 3.03
method | result | size |
risch | \(\frac {b^{2} \ln \left (x^{n}\right )^{2} e^{3} \ln \left (e x +d \right )}{d^{4}}-\frac {b^{2} \ln \left (x^{n}\right )^{2}}{3 d \,x^{3}}-\frac {b^{2} \ln \left (x^{n}\right )^{2} e^{2}}{d^{3} x}+\frac {b^{2} \ln \left (x^{n}\right )^{2} e}{2 d^{2} x^{2}}-\frac {b^{2} \ln \left (x^{n}\right )^{2} e^{3} \ln \left (x \right )}{d^{4}}-\frac {2 b^{2} n \ln \left (x^{n}\right ) e^{2}}{d^{3} x}+\frac {b^{2} n \ln \left (x^{n}\right ) e}{2 d^{2} x^{2}}-\frac {2 b^{2} n \ln \left (x^{n}\right )}{9 d \,x^{3}}-\frac {2 b^{2} e^{2} n^{2}}{d^{3} x}+\frac {b^{2} e \,n^{2}}{4 d^{2} x^{2}}-\frac {2 b^{2} n^{2}}{27 d \,x^{3}}+\frac {b^{2} n \,e^{3} \ln \left (x^{n}\right ) \ln \left (x \right )^{2}}{d^{4}}-\frac {b^{2} e^{3} \ln \left (x \right )^{3} n^{2}}{3 d^{4}}+\frac {2 b^{2} e^{3} \ln \left (x \right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right ) n^{2}}{d^{4}}+\frac {2 b^{2} e^{3} \ln \left (x \right ) \operatorname {dilog}\left (-\frac {e x}{d}\right ) n^{2}}{d^{4}}-\frac {2 b^{2} n \,e^{3} \ln \left (x^{n}\right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{4}}-\frac {2 b^{2} n \,e^{3} \ln \left (x^{n}\right ) \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{4}}-\frac {b^{2} e^{3} n^{2} \ln \left (e x +d \right ) \ln \left (x \right )^{2}}{d^{4}}+\frac {b^{2} e^{3} n^{2} \ln \left (x \right )^{2} \ln \left (1+\frac {e x}{d}\right )}{d^{4}}+\frac {2 b^{2} e^{3} n^{2} \ln \left (x \right ) \operatorname {Li}_{2}\left (-\frac {e x}{d}\right )}{d^{4}}-\frac {2 b^{2} e^{3} n^{2} \operatorname {Li}_{3}\left (-\frac {e x}{d}\right )}{d^{4}}+\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^{3} \ln \left (e x +d \right )}{d^{4}}-\frac {\ln \left (x^{n}\right )}{3 d \,x^{3}}-\frac {\ln \left (x^{n}\right ) e^{2}}{d^{3} x}+\frac {\ln \left (x^{n}\right ) e}{2 d^{2} x^{2}}-\frac {\ln \left (x^{n}\right ) e^{3} \ln \left (x \right )}{d^{4}}-\frac {n \left (-\frac {-\frac {6 e^{2}}{x}+\frac {3 d e}{2 x^{2}}-\frac {2 d^{2}}{3 x^{3}}}{d^{3}}-\frac {3 e^{3} \ln \left (x \right )^{2}}{d^{4}}+\frac {6 e^{3} \left (\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )\right )}{d^{4}}\right )}{6}\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 {e^{3} \ln \left (e x +d \right )}{d^{4}}-\frac {1}{3 d \,x^{3}}-\frac {e^{2}}{d^{3} x}+\frac {e}{2 d^{2} x^{2}}-\frac {e^{3} \ln \left (x \right )}{d^{4}}\right )}{4}\) | \(828\) |
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^4 (d+e x)} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )} x^{4}} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^4 (d+e x)} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{x^{4} \left (d + e x\right )}\, dx \]
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^4 (d+e x)} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )} x^{4}} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^4 (d+e x)} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^4 (d+e x)} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^4\,\left (d+e\,x\right )} \,d x \]
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