Optimal. Leaf size=211 \[ \frac {3 e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)}+\frac {4 e \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}-\frac {a+b \log \left (c x^n\right )}{d^4 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)^2}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 (d+e x)^3}-\frac {4 b e n \text {Li}_2\left (-\frac {d}{e x}\right )}{d^5}+\frac {4 b e n \log (x)}{3 d^5}-\frac {13 b e n \log (d+e x)}{3 d^5}+\frac {4 b e n}{3 d^4 (d+e x)}-\frac {b n}{d^4 x}+\frac {b e n}{6 d^3 (d+e x)^2} \]
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Rubi [A] time = 0.30, antiderivative size = 231, normalized size of antiderivative = 1.09, number of steps used = 14, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {44, 2351, 2304, 2301, 2319, 2314, 31, 2317, 2391} \[ \frac {4 b e n \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^5}+\frac {3 e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)}-\frac {2 e \left (a+b \log \left (c x^n\right )\right )^2}{b d^5 n}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)^2}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 (d+e x)^3}+\frac {4 e \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}-\frac {a+b \log \left (c x^n\right )}{d^4 x}+\frac {4 b e n}{3 d^4 (d+e x)}+\frac {b e n}{6 d^3 (d+e x)^2}+\frac {4 b e n \log (x)}{3 d^5}-\frac {13 b e n \log (d+e x)}{3 d^5}-\frac {b n}{d^4 x} \]
Antiderivative was successfully verified.
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Rule 31
Rule 44
Rule 2301
Rule 2304
Rule 2314
Rule 2317
Rule 2319
Rule 2351
Rule 2391
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^4} \, dx &=\int \left (\frac {a+b \log \left (c x^n\right )}{d^4 x^2}-\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{d^5 x}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)^4}+\frac {2 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)^3}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)^2}+\frac {4 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{d^4}-\frac {(4 e) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{d^5}+\frac {\left (4 e^2\right ) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^5}+\frac {\left (3 e^2\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d^4}+\frac {\left (2 e^2\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{d^3}+\frac {e^2 \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^4} \, dx}{d^2}\\ &=-\frac {b n}{d^4 x}-\frac {a+b \log \left (c x^n\right )}{d^4 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 (d+e x)^3}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)^2}+\frac {3 e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)}-\frac {2 e \left (a+b \log \left (c x^n\right )\right )^2}{b d^5 n}+\frac {4 e \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^5}-\frac {(4 b e n) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^5}+\frac {(b e n) \int \frac {1}{x (d+e x)^2} \, dx}{d^3}+\frac {(b e n) \int \frac {1}{x (d+e x)^3} \, dx}{3 d^2}-\frac {\left (3 b e^2 n\right ) \int \frac {1}{d+e x} \, dx}{d^5}\\ &=-\frac {b n}{d^4 x}-\frac {a+b \log \left (c x^n\right )}{d^4 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 (d+e x)^3}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)^2}+\frac {3 e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)}-\frac {2 e \left (a+b \log \left (c x^n\right )\right )^2}{b d^5 n}-\frac {3 b e n \log (d+e x)}{d^5}+\frac {4 e \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^5}+\frac {4 b e n \text {Li}_2\left (-\frac {e x}{d}\right )}{d^5}+\frac {(b e n) \int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx}{d^3}+\frac {(b e n) \int \left (\frac {1}{d^3 x}-\frac {e}{d (d+e x)^3}-\frac {e}{d^2 (d+e x)^2}-\frac {e}{d^3 (d+e x)}\right ) \, dx}{3 d^2}\\ &=-\frac {b n}{d^4 x}+\frac {b e n}{6 d^3 (d+e x)^2}+\frac {4 b e n}{3 d^4 (d+e x)}+\frac {4 b e n \log (x)}{3 d^5}-\frac {a+b \log \left (c x^n\right )}{d^4 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 (d+e x)^3}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)^2}+\frac {3 e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)}-\frac {2 e \left (a+b \log \left (c x^n\right )\right )^2}{b d^5 n}-\frac {13 b e n \log (d+e x)}{3 d^5}+\frac {4 e \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^5}+\frac {4 b e n \text {Li}_2\left (-\frac {e x}{d}\right )}{d^5}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 231, normalized size = 1.09 \[ \frac {-\frac {2 d^3 e \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3}-\frac {6 d^2 e \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}-\frac {18 d e \left (a+b \log \left (c x^n\right )\right )}{d+e x}+24 e \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 d \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {12 e \left (a+b \log \left (c x^n\right )\right )^2}{b n}+24 b e n \text {Li}_2\left (-\frac {e x}{d}\right )+b e n \left (\frac {d (3 d+2 e x)}{(d+e x)^2}-2 \log (d+e x)+2 \log (x)\right )+18 b e n (\log (x)-\log (d+e x))+6 b e n \left (\frac {d}{d+e x}-\log (d+e x)+\log (x)\right )-\frac {6 b d n}{x}}{6 d^5} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left (c x^{n}\right ) + a}{e^{4} x^{6} + 4 \, d e^{3} x^{5} + 6 \, d^{2} e^{2} x^{4} + 4 \, d^{3} e x^{3} + d^{4} x^{2}}, x\right ) \]
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 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{4} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 1083, normalized size = 5.13 \[ \text {result too large to display} \]
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 \[ -\frac {1}{3} \, a {\left (\frac {12 \, e^{3} x^{3} + 30 \, d e^{2} x^{2} + 22 \, d^{2} e x + 3 \, d^{3}}{d^{4} e^{3} x^{4} + 3 \, d^{5} e^{2} x^{3} + 3 \, d^{6} e x^{2} + d^{7} x} - \frac {12 \, e \log \left (e x + d\right )}{d^{5}} + \frac {12 \, e \log \relax (x)}{d^{5}}\right )} + b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{e^{4} x^{6} + 4 \, d e^{3} x^{5} + 6 \, d^{2} e^{2} x^{4} + 4 \, d^{3} e x^{3} + d^{4} x^{2}}\,{d x} \]
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 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^2\,{\left (d+e\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 141.37, size = 595, normalized size = 2.82 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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