Optimal. Leaf size=232 \[ -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}+\frac {2 b n \text {Li}_2\left (-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {3 b n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{e^2 (d+e x)}-\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e^3}+\frac {3 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{e^3}-\frac {2 b^2 n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^3}+\frac {b^2 n^2 \log (d+e x)}{e^3} \]
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Rubi [A] time = 0.45, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {2353, 2319, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2318, 2374, 6589} \[ \frac {2 b n \text {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {3 b^2 n^2 \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^3}-\frac {2 b^2 n^2 \text {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^3}-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 (d+e x)^2}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{e^2 (d+e x)}+\frac {3 b n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}+\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e^3}+\frac {b^2 n^2 \log (d+e x)}{e^3} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2301
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 {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx &=\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 {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 {(b n) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{e^3}-\frac {(b n) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^2}+\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 {\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 {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 {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 {\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 \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*}
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Mathematica [A] time = 0.28, size = 212, normalized size = 0.91 \[ \frac {-\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}+4 b n \text {Li}_2\left (-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2 b d n \left (a+b \log \left (c x^n\right )\right )}{d+e x}+6 b n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {4 d \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+2 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-3 \left (a+b \log \left (c x^n\right )\right )^2+6 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )-4 b^2 n^2 \text {Li}_3\left (-\frac {e x}{d}\right )-2 b^2 n^2 (\log (x)-\log (d+e x))}{2 e^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} x^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b x^{2} \log \left (c x^{n}\right ) + a^{2} x^{2}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{2}}{{\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.77, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} x^{2}}{\left (e x +d \right )^{3}}\, dx \]
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 \[ \frac {1}{2} \, a^{2} {\left (\frac {4 \, d e x + 3 \, d^{2}}{e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}} + \frac {2 \, \log \left (e x + d\right )}{e^{3}}\right )} + \int \frac {b^{2} x^{2} \log \left (x^{n}\right )^{2} + 2 \, {\left (b^{2} \log \relax (c) + a b\right )} x^{2} \log \left (x^{n}\right ) + {\left (b^{2} \log \relax (c)^{2} + 2 \, a b \log \relax (c)\right )} x^{2}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}\,{d x} \]
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 \[ \int \frac {x^2\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \]
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 \[ \int \frac {x^{2} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{\left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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