Optimal. Leaf size=200 \[ \frac {2 b d^2 n \text {Li}_2\left (-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {d^2 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {d x \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {b n x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e}+\frac {2 a b d n x}{e^2}+\frac {2 b^2 d n x \log \left (c x^n\right )}{e^2}-\frac {2 b^2 d^2 n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^3}-\frac {2 b^2 d n^2 x}{e^2}+\frac {b^2 n^2 x^2}{4 e} \]
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Rubi [A] time = 0.22, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2353, 2296, 2295, 2305, 2304, 2317, 2374, 6589} \[ \frac {2 b d^2 n \text {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^3}-\frac {2 b^2 d^2 n^2 \text {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^3}+\frac {d^2 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {d x \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {b n x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e}+\frac {2 a b d n x}{e^2}+\frac {2 b^2 d n x \log \left (c x^n\right )}{e^2}-\frac {2 b^2 d n^2 x}{e^2}+\frac {b^2 n^2 x^2}{4 e} \]
Antiderivative was successfully verified.
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Rule 2295
Rule 2296
Rule 2304
Rule 2305
Rule 2317
Rule 2353
Rule 2374
Rule 6589
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx &=\int \left (-\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{e^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac {d \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{e^2}+\frac {d^2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{e^2}+\frac {\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{e}\\ &=-\frac {d x \left (a+b \log \left (c x^n\right )\right )^2}{e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^3}-\frac {\left (2 b d^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}{e^3}+\frac {(2 b d n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^2}-\frac {(b n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{e}\\ &=\frac {2 a b d n x}{e^2}+\frac {b^2 n^2 x^2}{4 e}-\frac {b n x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac {d x \left (a+b \log \left (c x^n\right )\right )^2}{e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {2 b d^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^3}+\frac {\left (2 b^2 d n\right ) \int \log \left (c x^n\right ) \, dx}{e^2}-\frac {\left (2 b^2 d^2 n^2\right ) \int \frac {\text {Li}_2\left (-\frac {e x}{d}\right )}{x} \, dx}{e^3}\\ &=\frac {2 a b d n x}{e^2}-\frac {2 b^2 d n^2 x}{e^2}+\frac {b^2 n^2 x^2}{4 e}+\frac {2 b^2 d n x \log \left (c x^n\right )}{e^2}-\frac {b n x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac {d x \left (a+b \log \left (c x^n\right )\right )^2}{e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^3}+\frac {2 b d^2 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 d^2 n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 158, normalized size = 0.79 \[ \frac {8 b d^2 n \left (\text {Li}_2\left (-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \text {Li}_3\left (-\frac {e x}{d}\right )\right )+4 d^2 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-4 d e x \left (a+b \log \left (c x^n\right )\right )^2+8 b d e n x \left (a+b \log \left (c x^n\right )-b n\right )+2 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2+b e^2 n x^2 \left (b n-2 \left (a+b \log \left (c x^n\right )\right )\right )}{4 e^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} x^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b x^{2} \log \left (c x^{n}\right ) + a^{2} x^{2}}{e x + d}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{2}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.71, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} x^{2}}{e x +d}\, dx \]
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 \[ \frac {1}{2} \, a^{2} {\left (\frac {2 \, d^{2} \log \left (e x + d\right )}{e^{3}} + \frac {e x^{2} - 2 \, d x}{e^{2}}\right )} + \int \frac {b^{2} x^{2} \log \left (x^{n}\right )^{2} + 2 \, {\left (b^{2} \log \relax (c) + a b\right )} x^{2} \log \left (x^{n}\right ) + {\left (b^{2} \log \relax (c)^{2} + 2 \, a b \log \relax (c)\right )} x^{2}}{e x + d}\,{d x} \]
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 \[ \int \frac {x^2\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{d+e\,x} \,d x \]
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 \[ \int \frac {x^{2} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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