Optimal. Leaf size=398 \[ -\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5 (d+e x)^3}+\frac {2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^5 (d+e x)^2}+\frac {b d^3 n \left (a+b \log \left (c x^n\right )\right )}{3 e^5 (d+e x)^2}-\frac {8 b d n \text {Li}_2\left (-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^5}-\frac {5 d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5}-\frac {4 d \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^5}-\frac {26 b d n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^5}+\frac {6 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}+\frac {10 b d n x \left (a+b \log \left (c x^n\right )\right )}{3 e^4 (d+e x)}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac {2 a b n x}{e^4}-\frac {2 b^2 n x \log \left (c x^n\right )}{e^4}-\frac {b^2 d^2 n^2}{3 e^5 (d+e x)}-\frac {26 b^2 d n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{3 e^5}+\frac {8 b^2 d n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^5}-\frac {b^2 d n^2 \log (x)}{3 e^5}-\frac {3 b^2 d n^2 \log (d+e x)}{e^5}+\frac {2 b^2 n^2 x}{e^4} \]
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Rubi [A] time = 0.83, antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 15, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {2353, 2296, 2295, 2319, 2347, 2344, 2301, 2317, 2391, 2314, 31, 44, 2318, 2374, 6589} \[ -\frac {8 b d n \text {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^5}-\frac {26 b^2 d n^2 \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{3 e^5}+\frac {8 b^2 d n^2 \text {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^5}-\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5 (d+e x)^3}+\frac {2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^5 (d+e x)^2}+\frac {b d^3 n \left (a+b \log \left (c x^n\right )\right )}{3 e^5 (d+e x)^2}+\frac {6 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}-\frac {5 d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5}+\frac {10 b d n x \left (a+b \log \left (c x^n\right )\right )}{3 e^4 (d+e x)}-\frac {4 d \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^5}-\frac {26 b d n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^5}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac {2 a b n x}{e^4}-\frac {2 b^2 n x \log \left (c x^n\right )}{e^4}-\frac {b^2 d^2 n^2}{3 e^5 (d+e x)}-\frac {b^2 d n^2 \log (x)}{3 e^5}-\frac {3 b^2 d n^2 \log (d+e x)}{e^5}+\frac {2 b^2 n^2 x}{e^4} \]
Antiderivative was successfully verified.
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Rule 31
Rule 44
Rule 2295
Rule 2296
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^4 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx &=\int \left (\frac {\left (a+b \log \left (c x^n\right )\right )^2}{e^4}+\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)^4}-\frac {4 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)^3}+\frac {6 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)^2}-\frac {4 d \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}\right ) \, dx\\ &=\frac {\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{e^4}-\frac {(4 d) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{e^4}+\frac {\left (6 d^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{e^4}-\frac {\left (4 d^3\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx}{e^4}+\frac {d^4 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx}{e^4}\\ &=\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5 (d+e x)^3}+\frac {2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^5 (d+e x)^2}+\frac {6 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}-\frac {4 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^5}+\frac {(8 b d n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^5}-\frac {\left (4 b d^3 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{e^5}+\frac {\left (2 b d^4 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^3} \, dx}{3 e^5}-\frac {(2 b n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^4}-\frac {(12 b d n) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^4}\\ &=-\frac {2 a b n x}{e^4}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5 (d+e x)^3}+\frac {2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^5 (d+e x)^2}+\frac {6 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}-\frac {12 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^5}-\frac {4 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^5}-\frac {8 b d n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^5}-\frac {\left (4 b d^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{e^5}+\frac {\left (2 b d^3 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{3 e^5}-\frac {\left (2 b^2 n\right ) \int \log \left (c x^n\right ) \, dx}{e^4}+\frac {\left (4 b d^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^4}-\frac {\left (2 b d^3 n\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{3 e^4}+\frac {\left (8 b^2 d n^2\right ) \int \frac {\text {Li}_2\left (-\frac {e x}{d}\right )}{x} \, dx}{e^5}+\frac {\left (12 b^2 d n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^5}\\ &=-\frac {2 a b n x}{e^4}+\frac {2 b^2 n^2 x}{e^4}-\frac {2 b^2 n x \log \left (c x^n\right )}{e^4}+\frac {b d^3 n \left (a+b \log \left (c x^n\right )\right )}{3 e^5 (d+e x)^2}+\frac {4 b d n x \left (a+b \log \left (c x^n\right )\right )}{e^4 (d+e x)}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5 (d+e x)^3}+\frac {2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^5 (d+e x)^2}+\frac {6 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}-\frac {12 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^5}-\frac {4 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^5}-\frac {12 b^2 d n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{e^5}-\frac {8 b d n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^5}+\frac {8 b^2 d n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^5}-\frac {(4 b d n) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{e^5}+\frac {\left (2 b d^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{3 e^5}+\frac {(4 b d n) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^4}-\frac {\left (2 b d^2 n\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{3 e^4}-\frac {\left (b^2 d^3 n^2\right ) \int \frac {1}{x (d+e x)^2} \, dx}{3 e^5}-\frac {\left (4 b^2 d n^2\right ) \int \frac {1}{d+e x} \, dx}{e^4}\\ &=-\frac {2 a b n x}{e^4}+\frac {2 b^2 n^2 x}{e^4}-\frac {2 b^2 n x \log \left (c x^n\right )}{e^4}+\frac {b d^3 n \left (a+b \log \left (c x^n\right )\right )}{3 e^5 (d+e x)^2}+\frac {10 b d n x \left (a+b \log \left (c x^n\right )\right )}{3 e^4 (d+e x)}-\frac {2 d \left (a+b \log \left (c x^n\right )\right )^2}{e^5}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5 (d+e x)^3}+\frac {2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^5 (d+e x)^2}+\frac {6 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}-\frac {4 b^2 d n^2 \log (d+e x)}{e^5}-\frac {8 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^5}-\frac {4 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^5}-\frac {12 b^2 d n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{e^5}-\frac {8 b d n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^5}+\frac {8 b^2 d n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^5}+\frac {(2 b d n) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{3 e^5}-\frac {(2 b d n) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{3 e^4}-\frac {\left (4 b^2 d n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^5}-\frac {\left (b^2 d^3 n^2\right ) \int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx}{3 e^5}+\frac {\left (2 b^2 d n^2\right ) \int \frac {1}{d+e x} \, dx}{3 e^4}\\ &=-\frac {2 a b n x}{e^4}+\frac {2 b^2 n^2 x}{e^4}-\frac {b^2 d^2 n^2}{3 e^5 (d+e x)}-\frac {b^2 d n^2 \log (x)}{3 e^5}-\frac {2 b^2 n x \log \left (c x^n\right )}{e^4}+\frac {b d^3 n \left (a+b \log \left (c x^n\right )\right )}{3 e^5 (d+e x)^2}+\frac {10 b d n x \left (a+b \log \left (c x^n\right )\right )}{3 e^4 (d+e x)}-\frac {5 d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5 (d+e x)^3}+\frac {2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^5 (d+e x)^2}+\frac {6 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}-\frac {3 b^2 d n^2 \log (d+e x)}{e^5}-\frac {26 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{3 e^5}-\frac {4 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^5}-\frac {8 b^2 d n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{e^5}-\frac {8 b d n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^5}+\frac {8 b^2 d n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^5}+\frac {\left (2 b^2 d n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{3 e^5}\\ &=-\frac {2 a b n x}{e^4}+\frac {2 b^2 n^2 x}{e^4}-\frac {b^2 d^2 n^2}{3 e^5 (d+e x)}-\frac {b^2 d n^2 \log (x)}{3 e^5}-\frac {2 b^2 n x \log \left (c x^n\right )}{e^4}+\frac {b d^3 n \left (a+b \log \left (c x^n\right )\right )}{3 e^5 (d+e x)^2}+\frac {10 b d n x \left (a+b \log \left (c x^n\right )\right )}{3 e^4 (d+e x)}-\frac {5 d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e^4}-\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 e^5 (d+e x)^3}+\frac {2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{e^5 (d+e x)^2}+\frac {6 d x \left (a+b \log \left (c x^n\right )\right )^2}{e^4 (d+e x)}-\frac {3 b^2 d n^2 \log (d+e x)}{e^5}-\frac {26 b d n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{3 e^5}-\frac {4 d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^5}-\frac {26 b^2 d n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{3 e^5}-\frac {8 b d n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )}{e^5}+\frac {8 b^2 d n^2 \text {Li}_3\left (-\frac {e x}{d}\right )}{e^5}\\ \end {align*}
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Mathematica [A] time = 0.65, size = 344, normalized size = 0.86 \[ -\frac {\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3}-\frac {6 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}-\frac {b d^3 n \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}+\frac {18 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+\frac {10 b d^2 n \left (a+b \log \left (c x^n\right )\right )}{d+e x}+24 b d n \text {Li}_2\left (-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )+12 d \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+26 b d n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )-13 d \left (a+b \log \left (c x^n\right )\right )^2-3 e x \left (a+b \log \left (c x^n\right )\right )^2+6 b e n x \left (a+b \log \left (c x^n\right )-b n\right )+26 b^2 d n^2 \text {Li}_2\left (-\frac {e x}{d}\right )-24 b^2 d n^2 \text {Li}_3\left (-\frac {e x}{d}\right )-10 b^2 d n^2 (\log (x)-\log (d+e x))+\frac {b^2 d n^2 (\log (x) (d+e x)-(d+e x) \log (d+e x)+d)}{d+e x}}{3 e^5} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} x^{4} \log \left (c x^{n}\right )^{2} + 2 \, a b x^{4} \log \left (c x^{n}\right ) + a^{2} x^{4}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{4}}{{\left (e x + d\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.89, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} x^{4}}{\left (e x +d \right )^{4}}\, dx \]
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 \[ -\frac {1}{3} \, a^{2} {\left (\frac {18 \, d^{2} e^{2} x^{2} + 30 \, d^{3} e x + 13 \, d^{4}}{e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}} - \frac {3 \, x}{e^{4}} + \frac {12 \, d \log \left (e x + d\right )}{e^{5}}\right )} + \int \frac {b^{2} x^{4} \log \left (x^{n}\right )^{2} + 2 \, {\left (b^{2} \log \relax (c) + a b\right )} x^{4} \log \left (x^{n}\right ) + {\left (b^{2} \log \relax (c)^{2} + 2 \, a b \log \relax (c)\right )} x^{4}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}\,{d x} \]
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 \[ \int \frac {x^4\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x\right )}^4} \,d x \]
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 \[ \int \frac {x^{4} \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{\left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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