Optimal. Leaf size=420 \[ \frac {3 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^5 (d+e x)}-\frac {8 b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{3 d^5 (d+e x)}+\frac {8 b e n \text {Li}_2\left (-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}-\frac {4 e \left (a+b \log \left (c x^n\right )\right )^3}{3 b d^5 n}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^5}+\frac {4 e \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^5}-\frac {26 b e n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^5}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^4 x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{d^4 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}+\frac {b e n \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^2}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^2 (d+e x)^3}-\frac {26 b^2 e n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{3 d^5}-\frac {8 b^2 e n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{d^5}-\frac {b^2 e n^2 \log (x)}{3 d^5}+\frac {3 b^2 e n^2 \log (d+e x)}{d^5}-\frac {b^2 e n^2}{3 d^4 (d+e x)}-\frac {2 b^2 n^2}{d^4 x} \]
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Rubi [A] time = 0.89, antiderivative size = 420, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 17, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.739, Rules used = {2353, 2305, 2304, 2302, 30, 2319, 2347, 2344, 2301, 2317, 2391, 2314, 31, 44, 2318, 2374, 6589} \[ \frac {8 b e n \text {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}-\frac {26 b^2 e n^2 \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{3 d^5}-\frac {8 b^2 e n^2 \text {PolyLog}\left (3,-\frac {e x}{d}\right )}{d^5}+\frac {3 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^5 (d+e x)}-\frac {8 b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{3 d^5 (d+e x)}-\frac {4 e \left (a+b \log \left (c x^n\right )\right )^3}{3 b d^5 n}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^5}+\frac {4 e \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^5}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^2 (d+e x)^3}-\frac {26 b e n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^5}+\frac {b e n \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^4 x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{d^4 x}-\frac {b^2 e n^2}{3 d^4 (d+e x)}-\frac {b^2 e n^2 \log (x)}{3 d^5}+\frac {3 b^2 e n^2 \log (d+e x)}{d^5}-\frac {2 b^2 n^2}{d^4 x} \]
Antiderivative was successfully verified.
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Rule 30
Rule 31
Rule 44
Rule 2301
Rule 2302
Rule 2304
Rule 2305
Rule 2314
Rule 2317
Rule 2318
Rule 2319
Rule 2344
Rule 2347
Rule 2353
Rule 2374
Rule 2391
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)^4} \, dx &=\int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^4 x^2}-\frac {4 e \left (a+b \log \left (c x^n\right )\right )^2}{d^5 x}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^2 (d+e x)^4}+\frac {2 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^3}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)^2}+\frac {4 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^5 (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx}{d^4}-\frac {(4 e) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{d^5}+\frac {\left (4 e^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{d^5}+\frac {\left (3 e^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{d^4}+\frac {\left (2 e^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx}{d^3}+\frac {e^2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx}{d^2}\\ &=-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^4 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^2 (d+e x)^3}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}+\frac {3 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^5 (d+e x)}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d^5}-\frac {(4 e) \operatorname {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{b d^5 n}+\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{d^4}-\frac {(8 b e n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^5}+\frac {(2 b e n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{d^3}+\frac {(2 b e n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^3} \, dx}{3 d^2}-\frac {\left (6 b e^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^5}\\ &=-\frac {2 b^2 n^2}{d^4 x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{d^4 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^4 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^2 (d+e x)^3}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}+\frac {3 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^5 (d+e x)}-\frac {4 e \left (a+b \log \left (c x^n\right )\right )^3}{3 b d^5 n}-\frac {6 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^5}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d^5}+\frac {8 b e n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d^5}+\frac {(2 b e n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{d^4}+\frac {(2 b e n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{3 d^3}-\frac {\left (2 b e^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d^4}-\frac {\left (2 b e^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{3 d^3}+\frac {\left (6 b^2 e n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^5}-\frac {\left (8 b^2 e n^2\right ) \int \frac {\text {Li}_2\left (-\frac {e x}{d}\right )}{x} \, dx}{d^5}\\ &=-\frac {2 b^2 n^2}{d^4 x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{d^4 x}+\frac {b e n \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^2}-\frac {2 b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^4 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^2 (d+e x)^3}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}+\frac {3 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^5 (d+e x)}-\frac {4 e \left (a+b \log \left (c x^n\right )\right )^3}{3 b d^5 n}-\frac {6 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^5}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d^5}-\frac {6 b^2 e n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d^5}+\frac {8 b e n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d^5}-\frac {8 b^2 e n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{d^5}+\frac {(2 b e n) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{d^5}+\frac {(2 b e n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{3 d^4}-\frac {\left (2 b e^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^5}-\frac {\left (2 b e^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{3 d^4}-\frac {\left (b^2 e n^2\right ) \int \frac {1}{x (d+e x)^2} \, dx}{3 d^3}+\frac {\left (2 b^2 e^2 n^2\right ) \int \frac {1}{d+e x} \, dx}{d^5}\\ &=-\frac {2 b^2 n^2}{d^4 x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{d^4 x}+\frac {b e n \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^2}-\frac {8 b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{3 d^5 (d+e x)}+\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{d^5}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^4 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^2 (d+e x)^3}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}+\frac {3 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^5 (d+e x)}-\frac {4 e \left (a+b \log \left (c x^n\right )\right )^3}{3 b d^5 n}+\frac {2 b^2 e n^2 \log (d+e x)}{d^5}-\frac {8 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^5}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d^5}-\frac {6 b^2 e n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d^5}+\frac {8 b e n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d^5}-\frac {8 b^2 e n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{d^5}+\frac {(2 b e n) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{3 d^5}-\frac {\left (2 b e^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{3 d^5}+\frac {\left (2 b^2 e n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^5}-\frac {\left (b^2 e n^2\right ) \int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx}{3 d^3}+\frac {\left (2 b^2 e^2 n^2\right ) \int \frac {1}{d+e x} \, dx}{3 d^5}\\ &=-\frac {2 b^2 n^2}{d^4 x}-\frac {b^2 e n^2}{3 d^4 (d+e x)}-\frac {b^2 e n^2 \log (x)}{3 d^5}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{d^4 x}+\frac {b e n \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^2}-\frac {8 b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{3 d^5 (d+e x)}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^5}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^4 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^2 (d+e x)^3}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}+\frac {3 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^5 (d+e x)}-\frac {4 e \left (a+b \log \left (c x^n\right )\right )^3}{3 b d^5 n}+\frac {3 b^2 e n^2 \log (d+e x)}{d^5}-\frac {26 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{3 d^5}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d^5}-\frac {8 b^2 e n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d^5}+\frac {8 b e n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d^5}-\frac {8 b^2 e n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{d^5}+\frac {\left (2 b^2 e n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{3 d^5}\\ &=-\frac {2 b^2 n^2}{d^4 x}-\frac {b^2 e n^2}{3 d^4 (d+e x)}-\frac {b^2 e n^2 \log (x)}{3 d^5}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{d^4 x}+\frac {b e n \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^2}-\frac {8 b e^2 n x \left (a+b \log \left (c x^n\right )\right )}{3 d^5 (d+e x)}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^5}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^4 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{3 d^2 (d+e x)^3}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}+\frac {3 e^2 x \left (a+b \log \left (c x^n\right )\right )^2}{d^5 (d+e x)}-\frac {4 e \left (a+b \log \left (c x^n\right )\right )^3}{3 b d^5 n}+\frac {3 b^2 e n^2 \log (d+e x)}{d^5}-\frac {26 b e n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{3 d^5}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d^5}-\frac {26 b^2 e n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{3 d^5}+\frac {8 b e n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d^5}-\frac {8 b^2 e n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{d^5}\\ \end {align*}
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Mathematica [A] time = 0.67, size = 378, normalized size = 0.90 \[ -\frac {\frac {d^3 e \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3}+\frac {3 d^2 e \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}-\frac {b d^2 e n \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}-24 b e n \text {Li}_2\left (-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {9 d e \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}-\frac {8 b d e n \left (a+b \log \left (c x^n\right )\right )}{d+e x}-12 e \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+26 b e n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {3 d \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {6 b d n \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )^3}{b n}-13 e \left (a+b \log \left (c x^n\right )\right )^2+26 b^2 e n^2 \text {Li}_2\left (-\frac {e x}{d}\right )+24 b^2 e n^2 \text {Li}_3\left (-\frac {e x}{d}\right )+8 b^2 e n^2 (\log (x)-\log (d+e x))+\frac {b^2 e n^2 (\log (x) (d+e x)-(d+e x) \log (d+e x)+d)}{d+e x}+\frac {6 b^2 d n^2}{x}}{3 d^5} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}}{e^{4} x^{6} + 4 \, d e^{3} x^{5} + 6 \, d^{2} e^{2} x^{4} + 4 \, d^{3} e x^{3} + d^{4} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{4} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.86, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2}}{\left (e x +d \right )^{4} x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{3} \, a^{2} {\left (\frac {12 \, e^{3} x^{3} + 30 \, d e^{2} x^{2} + 22 \, d^{2} e x + 3 \, d^{3}}{d^{4} e^{3} x^{4} + 3 \, d^{5} e^{2} x^{3} + 3 \, d^{6} e x^{2} + d^{7} x} - \frac {12 \, e \log \left (e x + d\right )}{d^{5}} + \frac {12 \, e \log \relax (x)}{d^{5}}\right )} + \int \frac {b^{2} \log \relax (c)^{2} + b^{2} \log \left (x^{n}\right )^{2} + 2 \, a b \log \relax (c) + 2 \, {\left (b^{2} \log \relax (c) + a b\right )} \log \left (x^{n}\right )}{e^{4} x^{6} + 4 \, d e^{3} x^{5} + 6 \, d^{2} e^{2} x^{4} + 4 \, d^{3} e x^{3} + d^{4} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^2\,{\left (d+e\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{x^{2} \left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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