Optimal. Leaf size=285 \[ -\frac {e^3 x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}+\frac {6 b e^2 n \text {Li}_2\left (-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}-\frac {3 e^2 \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^4}+\frac {2 b e^2 n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}+\frac {4 b e n \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}+\frac {2 b^2 e^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d^4}+\frac {6 b^2 e^2 n^2 \text {Li}_3\left (-\frac {d}{e x}\right )}{d^4}+\frac {4 b^2 e n^2}{d^3 x}-\frac {b^2 n^2}{4 d^2 x^2} \]
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Rubi [A] time = 0.38, antiderivative size = 304, normalized size of antiderivative = 1.07, number of steps used = 14, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {2353, 2305, 2304, 2302, 30, 2318, 2317, 2391, 2374, 6589} \[ -\frac {6 b e^2 n \text {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}+\frac {2 b^2 e^2 n^2 \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^4}+\frac {6 b^2 e^2 n^2 \text {PolyLog}\left (3,-\frac {e x}{d}\right )}{d^4}-\frac {e^3 x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^3}{b d^4 n}-\frac {3 e^2 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^4}+\frac {2 b e^2 n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}+\frac {4 b e n \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}+\frac {4 b^2 e n^2}{d^3 x}-\frac {b^2 n^2}{4 d^2 x^2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2302
Rule 2304
Rule 2305
Rule 2317
Rule 2318
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^3 (d+e x)^2} \, dx &=\int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^2 x^3}-\frac {2 e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x^2}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^4 x}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )^2}{d^3 (d+e x)^2}-\frac {3 e^3 \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx}{d^2}-\frac {(2 e) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx}{d^3}+\frac {\left (3 e^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{d^4}-\frac {\left (3 e^3\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{d^4}-\frac {e^3 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{d^3}\\ &=-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 x^2}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}-\frac {e^3 x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}-\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d^4}+\frac {\left (3 e^2\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{b d^4 n}+\frac {(b n) \int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx}{d^2}-\frac {(4 b e n) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{d^3}+\frac {\left (6 b e^2 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^4}+\frac {\left (2 b e^3 n\right ) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^4}\\ &=-\frac {b^2 n^2}{4 d^2 x^2}+\frac {4 b^2 e n^2}{d^3 x}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}+\frac {4 b e 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}{2 d^2 x^2}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}-\frac {e^3 x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^3}{b d^4 n}+\frac {2 b e^2 n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^4}-\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d^4}-\frac {6 b e^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d^4}-\frac {\left (2 b^2 e^2 n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^4}+\frac {\left (6 b^2 e^2 n^2\right ) \int \frac {\text {Li}_2\left (-\frac {e x}{d}\right )}{x} \, dx}{d^4}\\ &=-\frac {b^2 n^2}{4 d^2 x^2}+\frac {4 b^2 e n^2}{d^3 x}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{2 d^2 x^2}+\frac {4 b e 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}{2 d^2 x^2}+\frac {2 e \left (a+b \log \left (c x^n\right )\right )^2}{d^3 x}-\frac {e^3 x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^3}{b d^4 n}+\frac {2 b e^2 n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^4}-\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d^4}+\frac {2 b^2 e^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d^4}-\frac {6 b e^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{d^4}+\frac {6 b^2 e^2 n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{d^4}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 268, normalized size = 0.94 \[ \frac {4 e^2 \left (2 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )-\left (a+b \log \left (c x^n\right )\right ) \left (a+b \log \left (c x^n\right )-2 b n \log \left (\frac {e x}{d}+1\right )\right )\right )-\frac {2 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^2}-\frac {b d^2 n \left (2 a+2 b \log \left (c x^n\right )+b n\right )}{x^2}-24 b e^2 n \left (\text {Li}_2\left (-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \text {Li}_3\left (-\frac {e x}{d}\right )\right )-12 e^2 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {4 d e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+\frac {8 d e \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {16 b d e n \left (a+b \log \left (c x^n\right )+b n\right )}{x}+\frac {4 e^2 \left (a+b \log \left (c x^n\right )\right )^3}{b n}}{4 d^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}}{e^{2} x^{5} + 2 \, d e x^{4} + d^{2} x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.76, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2}}{\left (e x +d \right )^{2} x^{3}}\, dx \]
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 \[ \frac {1}{2} \, a^{2} {\left (\frac {6 \, e^{2} x^{2} + 3 \, d e x - d^{2}}{d^{3} e x^{3} + d^{4} x^{2}} - \frac {6 \, e^{2} \log \left (e x + d\right )}{d^{4}} + \frac {6 \, e^{2} \log \relax (x)}{d^{4}}\right )} + \int \frac {b^{2} \log \relax (c)^{2} + b^{2} \log \left (x^{n}\right )^{2} + 2 \, a b \log \relax (c) + 2 \, {\left (b^{2} \log \relax (c) + a b\right )} \log \left (x^{n}\right )}{e^{2} x^{5} + 2 \, d e x^{4} + d^{2} x^{3}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^3\,{\left (d+e\,x\right )}^2} \,d x \]
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 \[ \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{x^{3} \left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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