Optimal. Leaf size=112 \[ -\frac {b n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )+b n\right )}{d e^2}+\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{d e (d+e x)}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 d (d+e x)^2}-\frac {b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d e^2} \]
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Rubi [A] time = 0.36, antiderivative size = 176, normalized size of antiderivative = 1.57, number of steps used = 13, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {2353, 2319, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2318} \[ -\frac {b^2 n^2 \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{d e^2}-\frac {b n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d e^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}+\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{d e (d+e x)}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d e (d+e x)}-\frac {b^2 n^2 \log (d+e x)}{d e^2} \]
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 2391
Rubi steps
\begin {align*} \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx &=\int \left (-\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{e (d+e x)^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{e (d+e x)^2}\right ) \, dx\\ &=\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{e}-\frac {d \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx}{e}\\ &=\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d e (d+e x)}-\frac {(b d n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{e^2}-\frac {(2 b n) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d e}\\ &=\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d e (d+e x)}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d e^2}-\frac {(b n) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{e^2}+\frac {(b n) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e}+\frac {\left (2 b^2 n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d e^2}\\ &=\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{d e (d+e x)}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d e (d+e x)}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d e^2}-\frac {2 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d e^2}-\frac {(b n) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{d e^2}+\frac {(b n) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d e}-\frac {\left (b^2 n^2\right ) \int \frac {1}{d+e x} \, dx}{d e}\\ &=\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{d e (d+e x)}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d e (d+e x)}-\frac {b^2 n^2 \log (d+e x)}{d e^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d e^2}-\frac {2 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d e^2}-\frac {\left (b^2 n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d e^2}\\ &=\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{d e (d+e x)}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d e (d+e x)}-\frac {b^2 n^2 \log (d+e x)}{d e^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d e^2}-\frac {b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d e^2}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 155, normalized size = 1.38 \[ \frac {-\frac {2 b n \left (a+b \log \left (c x^n\right )\right )}{d+e x}-\frac {2 b n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d}-\frac {2 b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{d}+\frac {2 b^2 n^2 (\log (x)-\log (d+e x))}{d}}{2 e^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} x \log \left (c x^{n}\right )^{2} + 2 \, a b x \log \left (c x^{n}\right ) + a^{2} x}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x}{{\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.32, size = 1199, normalized size = 10.71 \[ \text {result too large to display} \]
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 \[ -a b n {\left (\frac {1}{e^{3} x + d e^{2}} + \frac {\log \left (e x + d\right )}{d e^{2}} - \frac {\log \relax (x)}{d e^{2}}\right )} - \frac {1}{2} \, {\left (\frac {{\left (2 \, e x + d\right )} \log \left (x^{n}\right )^{2}}{e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}} - 2 \, \int \frac {e^{2} x^{2} \log \relax (c)^{2} + {\left (3 \, d e n x + d^{2} n + 2 \, {\left (e^{2} n + e^{2} \log \relax (c)\right )} x^{2}\right )} \log \left (x^{n}\right )}{e^{5} x^{4} + 3 \, d e^{4} x^{3} + 3 \, d^{2} e^{3} x^{2} + d^{3} e^{2} x}\,{d x}\right )} b^{2} - \frac {{\left (2 \, e x + d\right )} a b \log \left (c x^{n}\right )}{e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}} - \frac {{\left (2 \, e x + d\right )} a^{2}}{2 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \]
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 \[ \int \frac {x \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{\left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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