Integrand size = 23, antiderivative size = 232 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{e^2 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}-\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac {b^2 n^2 \log (d+e x)}{e^3}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {3 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^3}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^3}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^3} \]
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Time = 0.30 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.13, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {2395, 2356, 2389, 2379, 2438, 2351, 31, 2355, 2354, 2421, 6724} \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}+\frac {2 b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {b n \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {4 b n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{e^2 (d+e x)}-\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac {b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{e^3}+\frac {4 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^3}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^3}+\frac {b^2 n^2 \log (d+e x)}{e^3} \]
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Rule 31
Rule 2351
Rule 2354
Rule 2355
Rule 2356
Rule 2379
Rule 2389
Rule 2395
Rule 2421
Rule 2438
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)^3}-\frac {2 d \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}\right ) \, dx \\ & = \frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{e^2}-\frac {(2 d) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{e^2}+\frac {d^2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx}{e^2} \\ & = -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}-\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^3}-\frac {(2 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^3}+\frac {\left (b d^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{e^3}+\frac {(4 b n) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^2} \\ & = -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}-\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac {4 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^3}+\frac {(b d n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{e^3}-\frac {(b d n) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^2}-\frac {\left (2 b^2 n^2\right ) \int \frac {\text {Li}_2\left (-\frac {e x}{d}\right )}{x} \, dx}{e^3}-\frac {\left (4 b^2 n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^3} \\ & = -\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{e^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 )}{e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}-\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac {4 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {4 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{e^3}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^3}-\frac {2 b^2 n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^3}+\frac {\left (b^2 n^2\right ) \int \frac {\log \left (1+\frac {d}{e x}\right )}{x} \, dx}{e^3}+\frac {\left (b^2 n^2\right ) \int \frac {1}{d+e x} \, dx}{e^2} \\ & = -\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{e^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 )}{e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}-\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac {b^2 n^2 \log (d+e x)}{e^3}+\frac {4 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {b^2 n^2 \text {Li}_2\left (-\frac {d}{e x}\right )}{e^3}+\frac {4 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{e^3}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^3}-\frac {2 b^2 n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^3} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.91 \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\frac {\frac {2 b d n \left (a+b \log \left (c x^n\right )\right )}{d+e x}-3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}+\frac {4 d \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}-2 b^2 n^2 (\log (x)-\log (d+e x))+6 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )+2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )+6 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+4 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-4 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{2 e^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.45 (sec) , antiderivative size = 738, normalized size of antiderivative = 3.18
method | result | size |
risch | \(\frac {b^{2} \ln \left (x^{n}\right )^{2} \ln \left (e x +d \right )}{e^{3}}+\frac {2 b^{2} \ln \left (x^{n}\right )^{2} d}{e^{3} \left (e x +d \right )}-\frac {b^{2} \ln \left (x^{n}\right )^{2} d^{2}}{2 e^{3} \left (e x +d \right )^{2}}+\frac {b^{2} n \ln \left (x^{n}\right ) d}{e^{3} \left (e x +d \right )}+\frac {3 b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{e^{3}}-\frac {3 b^{2} n \ln \left (x^{n}\right ) \ln \left (x \right )}{e^{3}}+\frac {b^{2} n^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {b^{2} n^{2} \ln \left (x \right )}{e^{3}}+\frac {3 b^{2} n^{2} \ln \left (x \right )^{2}}{2 e^{3}}-\frac {3 b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{3}}-\frac {3 b^{2} n^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{3}}+\frac {2 b^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right ) \ln \left (x \right ) n^{2}}{e^{3}}-\frac {2 b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{3}}+\frac {2 b^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right ) \ln \left (x \right ) n^{2}}{e^{3}}-\frac {2 b^{2} n \ln \left (x^{n}\right ) \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{3}}-\frac {b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (x \right )^{2}}{e^{3}}+\frac {b^{2} n^{2} \ln \left (x \right )^{2} \ln \left (1+\frac {e x}{d}\right )}{e^{3}}+\frac {2 b^{2} n^{2} \ln \left (x \right ) \operatorname {Li}_{2}\left (-\frac {e x}{d}\right )}{e^{3}}-\frac {2 b^{2} n^{2} \operatorname {Li}_{3}\left (-\frac {e x}{d}\right )}{e^{3}}+\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 ) \ln \left (e x +d \right )}{e^{3}}+\frac {2 \ln \left (x^{n}\right ) d}{e^{3} \left (e x +d \right )}-\frac {\ln \left (x^{n}\right ) d^{2}}{2 e^{3} \left (e x +d \right )^{2}}-\frac {n \left (-\frac {d}{e^{3} \left (e x +d \right )}-\frac {3 \ln \left (e x +d \right )}{e^{3}}+\frac {3 \ln \left (e x \right )}{e^{3}}+\frac {2 \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{3}}+\frac {2 \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{3}}\right )}{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 {\ln \left (e x +d \right )}{e^{3}}+\frac {2 d}{e^{3} \left (e x +d \right )}-\frac {d^{2}}{2 e^{3} \left (e x +d \right )^{2}}\right )}{4}\) | \(738\) |
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\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]
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\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int \frac {x^{2} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{\left (d + e x\right )^{3}}\, dx \]
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\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]
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\[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{2}}{{\left (e x + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int \frac {x^2\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \]
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