Optimal. Leaf size=263 \[ -\frac {6 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^6 (d+e x)}-\frac {10 e^2 \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^6}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{d^5 x}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^4 (d+e x)^2}-\frac {a+b \log \left (c x^n\right )}{2 d^4 x^2}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^3}+\frac {10 b e^2 n \text {Li}_2\left (-\frac {d}{e x}\right )}{d^6}-\frac {11 b e^2 n \log (x)}{6 d^6}+\frac {47 b e^2 n \log (d+e x)}{6 d^6}-\frac {11 b e^2 n}{6 d^5 (d+e x)}+\frac {4 b e n}{d^5 x}-\frac {b e^2 n}{6 d^4 (d+e x)^2}-\frac {b n}{4 d^4 x^2} \]
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Rubi [A] time = 0.35, antiderivative size = 285, normalized size of antiderivative = 1.08, number of steps used = 15, 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} \[ -\frac {10 b e^2 n \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^6}-\frac {6 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^6 (d+e x)}+\frac {5 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b d^6 n}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^4 (d+e x)^2}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^3}-\frac {10 e^2 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^6}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{d^5 x}-\frac {a+b \log \left (c x^n\right )}{2 d^4 x^2}-\frac {11 b e^2 n}{6 d^5 (d+e x)}-\frac {b e^2 n}{6 d^4 (d+e x)^2}-\frac {11 b e^2 n \log (x)}{6 d^6}+\frac {47 b e^2 n \log (d+e x)}{6 d^6}+\frac {4 b e n}{d^5 x}-\frac {b n}{4 d^4 x^2} \]
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^3 (d+e x)^4} \, dx &=\int \left (\frac {a+b \log \left (c x^n\right )}{d^4 x^3}-\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{d^5 x^2}+\frac {10 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^6 x}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)^4}-\frac {3 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)^3}-\frac {6 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)^2}-\frac {10 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^6 (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx}{d^4}-\frac {(4 e) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{d^5}+\frac {\left (10 e^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{d^6}-\frac {\left (10 e^3\right ) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^6}-\frac {\left (6 e^3\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d^5}-\frac {\left (3 e^3\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{d^4}-\frac {e^3 \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^4} \, dx}{d^3}\\ &=-\frac {b n}{4 d^4 x^2}+\frac {4 b e n}{d^5 x}-\frac {a+b \log \left (c x^n\right )}{2 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 )}{3 d^3 (d+e x)^3}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^4 (d+e x)^2}-\frac {6 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^6 (d+e x)}+\frac {5 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b d^6 n}-\frac {10 e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^6}+\frac {\left (10 b e^2 n\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^6}-\frac {\left (3 b e^2 n\right ) \int \frac {1}{x (d+e x)^2} \, dx}{2 d^4}-\frac {\left (b e^2 n\right ) \int \frac {1}{x (d+e x)^3} \, dx}{3 d^3}+\frac {\left (6 b e^3 n\right ) \int \frac {1}{d+e x} \, dx}{d^6}\\ &=-\frac {b n}{4 d^4 x^2}+\frac {4 b e n}{d^5 x}-\frac {a+b \log \left (c x^n\right )}{2 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 )}{3 d^3 (d+e x)^3}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^4 (d+e x)^2}-\frac {6 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^6 (d+e x)}+\frac {5 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b d^6 n}+\frac {6 b e^2 n \log (d+e x)}{d^6}-\frac {10 e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^6}-\frac {10 b e^2 n \text {Li}_2\left (-\frac {e x}{d}\right )}{d^6}-\frac {\left (3 b e^2 n\right ) \int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx}{2 d^4}-\frac {\left (b e^2 n\right ) \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^3}\\ &=-\frac {b n}{4 d^4 x^2}+\frac {4 b e n}{d^5 x}-\frac {b e^2 n}{6 d^4 (d+e x)^2}-\frac {11 b e^2 n}{6 d^5 (d+e x)}-\frac {11 b e^2 n \log (x)}{6 d^6}-\frac {a+b \log \left (c x^n\right )}{2 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 )}{3 d^3 (d+e x)^3}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^4 (d+e x)^2}-\frac {6 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^6 (d+e x)}+\frac {5 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b d^6 n}+\frac {47 b e^2 n \log (d+e x)}{6 d^6}-\frac {10 e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^6}-\frac {10 b e^2 n \text {Li}_2\left (-\frac {e x}{d}\right )}{d^6}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 276, normalized size = 1.05 \[ \frac {\frac {4 d^3 e^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3}+\frac {18 d^2 e^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}-\frac {6 d^2 \left (a+b \log \left (c x^n\right )\right )}{x^2}+\frac {72 d e^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x}-120 e^2 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {48 d e \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {60 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b n}-\frac {3 b d^2 n}{x^2}-120 b e^2 n \text {Li}_2\left (-\frac {e x}{d}\right )-\frac {2 b d e^2 n (3 d+2 e x)}{(d+e x)^2}-\frac {18 b d e^2 n}{d+e x}-72 b e^2 n (\log (x)-\log (d+e x))+22 b e^2 n \log (d+e x)+\frac {48 b d e n}{x}-22 b e^2 n \log (x)}{12 d^6} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left (c x^{n}\right ) + a}{e^{4} x^{7} + 4 \, d e^{3} x^{6} + 6 \, d^{2} e^{2} x^{5} + 4 \, d^{3} e x^{4} + d^{4} x^{3}}, x\right ) \]
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 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{4} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 1324, normalized size = 5.03 \[ \text {result too large to display} \]
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 \[ \frac {1}{6} \, a {\left (\frac {60 \, e^{4} x^{4} + 150 \, d e^{3} x^{3} + 110 \, d^{2} e^{2} x^{2} + 15 \, d^{3} e x - 3 \, d^{4}}{d^{5} e^{3} x^{5} + 3 \, d^{6} e^{2} x^{4} + 3 \, d^{7} e x^{3} + d^{8} x^{2}} - \frac {60 \, e^{2} \log \left (e x + d\right )}{d^{6}} + \frac {60 \, e^{2} \log \relax (x)}{d^{6}}\right )} + b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{e^{4} x^{7} + 4 \, d e^{3} x^{6} + 6 \, d^{2} e^{2} x^{5} + 4 \, d^{3} e x^{4} + d^{4} x^{3}}\,{d x} \]
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 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^3\,{\left (d+e\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 148.35, size = 649, normalized size = 2.47 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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