Optimal. Leaf size=271 \[ -\frac {2 b d^3 n \text {Li}_2\left (-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac {d^3 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^4}+\frac {d^2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+\frac {b d n x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e}-\frac {2 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{9 e}-\frac {2 a b d^2 n x}{e^3}-\frac {2 b^2 d^2 n x \log \left (c x^n\right )}{e^3}+\frac {2 b^2 d^3 n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^4}+\frac {2 b^2 d^2 n^2 x}{e^3}-\frac {b^2 d n^2 x^2}{4 e^2}+\frac {2 b^2 n^2 x^3}{27 e} \]
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Rubi [A] time = 0.28, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 12, 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} \[ -\frac {2 b d^3 n \text {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac {2 b^2 d^3 n^2 \text {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^4}-\frac {d^3 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^4}+\frac {d^2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+\frac {b d n x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e}-\frac {2 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{9 e}-\frac {2 a b d^2 n x}{e^3}-\frac {2 b^2 d^2 n x \log \left (c x^n\right )}{e^3}+\frac {2 b^2 d^2 n^2 x}{e^3}-\frac {b^2 d n^2 x^2}{4 e^2}+\frac {2 b^2 n^2 x^3}{27 e} \]
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^3 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx &=\int \left (\frac {d^2 \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 {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^3 (d+e x)}\right ) \, dx\\ &=\frac {d^2 \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{e^3}-\frac {d^3 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{e^3}-\frac {d \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{e^2}+\frac {\int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{e}\\ &=\frac {d^2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^4}+\frac {\left (2 b d^3 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^4}-\frac {\left (2 b d^2 n\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^3}+\frac {(b d n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^2}-\frac {(2 b n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 e}\\ &=-\frac {2 a b d^2 n x}{e^3}-\frac {b^2 d n^2 x^2}{4 e^2}+\frac {2 b^2 n^2 x^3}{27 e}+\frac {b d n x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {2 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{9 e}+\frac {d^2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^4}-\frac {2 b d^3 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}-\frac {\left (2 b^2 d^2 n\right ) \int \log \left (c x^n\right ) \, dx}{e^3}+\frac {\left (2 b^2 d^3 n^2\right ) \int \frac {\text {Li}_2\left (-\frac {e x}{d}\right )}{x} \, dx}{e^4}\\ &=-\frac {2 a b d^2 n x}{e^3}+\frac {2 b^2 d^2 n^2 x}{e^3}-\frac {b^2 d n^2 x^2}{4 e^2}+\frac {2 b^2 n^2 x^3}{27 e}-\frac {2 b^2 d^2 n x \log \left (c x^n\right )}{e^3}+\frac {b d n x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {2 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{9 e}+\frac {d^2 x \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{3 e}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^4}-\frac {2 b d^3 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^4}+\frac {2 b^2 d^3 n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^4}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 211, normalized size = 0.78 \[ -\frac {216 b d^3 n \left (\text {Li}_2\left (-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \text {Li}_3\left (-\frac {e x}{d}\right )\right )+108 d^3 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-108 d^2 e x \left (a+b \log \left (c x^n\right )\right )^2+216 b d^2 e n x \left (a+b \log \left (c x^n\right )-b n\right )+54 d e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2+27 b d e^2 n x^2 \left (b n-2 \left (a+b \log \left (c x^n\right )\right )\right )-36 e^3 x^3 \left (a+b \log \left (c x^n\right )\right )^2-8 b e^3 n x^3 \left (b n-3 \left (a+b \log \left (c x^n\right )\right )\right )}{108 e^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} x^{3} \log \left (c x^{n}\right )^{2} + 2 \, a b x^{3} \log \left (c x^{n}\right ) + a^{2} x^{3}}{e x + d}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{3}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.72, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} x^{3}}{e x +d}\, dx \]
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 \[ -\frac {1}{6} \, a^{2} {\left (\frac {6 \, d^{3} \log \left (e x + d\right )}{e^{4}} - \frac {2 \, e^{2} x^{3} - 3 \, d e x^{2} + 6 \, d^{2} x}{e^{3}}\right )} + \int \frac {b^{2} x^{3} \log \left (x^{n}\right )^{2} + 2 \, {\left (b^{2} \log \relax (c) + a b\right )} x^{3} \log \left (x^{n}\right ) + {\left (b^{2} \log \relax (c)^{2} + 2 \, a b \log \relax (c)\right )} x^{3}}{e x + d}\,{d x} \]
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 \[ \int \frac {x^3\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{d+e\,x} \,d x \]
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 \[ \int \frac {x^{3} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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