Integrand size = 23, antiderivative size = 296 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=-\frac {2 a b n x}{e^3}+\frac {2 b^2 n^2 x}{e^3}-\frac {2 b^2 n x \log \left (c x^n\right )}{e^3}+\frac {b d n x \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^3}+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4 (d+e x)^2}+\frac {3 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}-\frac {b^2 d n^2 \log (d+e x)}{e^4}-\frac {5 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^4}-\frac {3 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^4}-\frac {5 b^2 d n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^4}-\frac {6 b d n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^4}+\frac {6 b^2 d n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^4} \]
[Out]
Time = 0.29 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.10, number of steps used = 17, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {2395, 2333, 2332, 2356, 2389, 2379, 2438, 2351, 31, 2355, 2354, 2421, 6724} \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4 (d+e x)^2}-\frac {6 b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac {b d n \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac {3 d \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac {6 b d n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac {3 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}+\frac {b d n x \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {2 a b n x}{e^3}-\frac {2 b^2 n x \log \left (c x^n\right )}{e^3}-\frac {b^2 d n^2 \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{e^4}-\frac {6 b^2 d n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^4}+\frac {6 b^2 d n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^4}-\frac {b^2 d n^2 \log (d+e x)}{e^4}+\frac {2 b^2 n^2 x}{e^3} \]
[In]
[Out]
Rule 31
Rule 2332
Rule 2333
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 {\left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)^3}+\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)^2}-\frac {3 d \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}\right ) \, dx \\ & = \frac {\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{e^3}-\frac {(3 d) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{e^3}+\frac {\left (3 d^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{e^3}-\frac {d^3 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx}{e^3} \\ & = \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^3}+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4 (d+e x)^2}+\frac {3 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}-\frac {3 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^4}+\frac {(6 b d n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^4}-\frac {\left (b d^3 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{e^4}-\frac {(2 b n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^3}-\frac {(6 b d n) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^3} \\ & = -\frac {2 a b n x}{e^3}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^3}+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4 (d+e x)^2}+\frac {3 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}-\frac {6 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^4}-\frac {3 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^4}-\frac {6 b d n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}-\frac {\left (b d^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{e^4}-\frac {\left (2 b^2 n\right ) \int \log \left (c x^n\right ) \, dx}{e^3}+\frac {\left (b d^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^3}+\frac {\left (6 b^2 d n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^4}+\frac {\left (6 b^2 d n^2\right ) \int \frac {\text {Li}_2\left (-\frac {e x}{d}\right )}{x} \, dx}{e^4} \\ & = -\frac {2 a b n x}{e^3}+\frac {2 b^2 n^2 x}{e^3}-\frac {2 b^2 n x \log \left (c x^n\right )}{e^3}+\frac {b d n x \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}+\frac {b d n \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^3}+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4 (d+e x)^2}+\frac {3 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}-\frac {6 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^4}-\frac {3 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^4}-\frac {6 b^2 d n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}-\frac {6 b d n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}+\frac {6 b^2 d n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^4}-\frac {\left (b^2 d n^2\right ) \int \frac {\log \left (1+\frac {d}{e x}\right )}{x} \, dx}{e^4}-\frac {\left (b^2 d n^2\right ) \int \frac {1}{d+e x} \, dx}{e^3} \\ & = -\frac {2 a b n x}{e^3}+\frac {2 b^2 n^2 x}{e^3}-\frac {2 b^2 n x \log \left (c x^n\right )}{e^3}+\frac {b d n x \left (a+b \log \left (c x^n\right )\right )}{e^3 (d+e x)}+\frac {b d n \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^3}+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4 (d+e x)^2}+\frac {3 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}-\frac {b^2 d n^2 \log (d+e x)}{e^4}-\frac {6 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^4}-\frac {3 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^4}-\frac {b^2 d n^2 \text {Li}_2\left (-\frac {d}{e x}\right )}{e^4}-\frac {6 b^2 d n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}-\frac {6 b d n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}+\frac {6 b^2 d n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^4} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.87 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\frac {-\frac {2 b d^2 n \left (a+b \log \left (c x^n\right )\right )}{d+e x}+5 d \left (a+b \log \left (c x^n\right )\right )^2+2 e x \left (a+b \log \left (c x^n\right )\right )^2+\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}-\frac {6 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}-4 b e n x \left (a-b n+b \log \left (c x^n\right )\right )+2 b^2 d n^2 (\log (x)-\log (d+e x))-10 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-6 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )-10 b^2 d n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-12 b d n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+12 b^2 d n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{2 e^4} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.52 (sec) , antiderivative size = 827, normalized size of antiderivative = 2.79
[In]
[Out]
\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{3}}{{\left (e x + d\right )}^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int \frac {x^{3} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{\left (d + e x\right )^{3}}\, dx \]
[In]
[Out]
\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{3}}{{\left (e x + d\right )}^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{3}}{{\left (e x + d\right )}^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx=\int \frac {x^3\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \]
[In]
[Out]