Optimal. Leaf size=229 \[ \frac {d^2 \log \left (\frac {e x}{d}+1\right ) \left (60 a+60 b \log \left (c x^n\right )+47 b n\right )}{6 e^6}-\frac {x^3 \left (20 a+20 b \log \left (c x^n\right )+9 b n\right )}{6 e^3 (d+e x)}-\frac {x^4 \left (5 a+5 b \log \left (c x^n\right )+b n\right )}{6 e^2 (d+e x)^2}-\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}+\frac {x^2 \left (60 a+60 b \log \left (c x^n\right )+47 b n\right )}{12 e^4}-\frac {d x (60 a+47 b n)}{6 e^5}-\frac {10 b d x \log \left (c x^n\right )}{e^5}+\frac {10 b d^2 n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^6}+\frac {10 b d n x}{e^5}-\frac {5 b n x^2}{2 e^4} \]
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Rubi [A] time = 0.32, antiderivative size = 260, normalized size of antiderivative = 1.14, number of steps used = 15, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {43, 2351, 2295, 2304, 2319, 44, 2314, 31, 2317, 2391} \[ \frac {10 b d^2 n \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^6}+\frac {d^5 \left (a+b \log \left (c x^n\right )\right )}{3 e^6 (d+e x)^3}-\frac {5 d^4 \left (a+b \log \left (c x^n\right )\right )}{2 e^6 (d+e x)^2}-\frac {10 d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)}+\frac {10 d^2 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^6}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac {4 a d x}{e^5}-\frac {4 b d x \log \left (c x^n\right )}{e^5}-\frac {b d^4 n}{6 e^6 (d+e x)^2}+\frac {13 b d^3 n}{6 e^6 (d+e x)}+\frac {13 b d^2 n \log (x)}{6 e^6}+\frac {47 b d^2 n \log (d+e x)}{6 e^6}+\frac {4 b d n x}{e^5}-\frac {b n x^2}{4 e^4} \]
Antiderivative was successfully verified.
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Rule 31
Rule 43
Rule 44
Rule 2295
Rule 2304
Rule 2314
Rule 2317
Rule 2319
Rule 2351
Rule 2391
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx &=\int \left (-\frac {4 d \left (a+b \log \left (c x^n\right )\right )}{e^5}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac {d^5 \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)^4}+\frac {5 d^4 \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)^3}-\frac {10 d^3 \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)^2}+\frac {10 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)}\right ) \, dx\\ &=-\frac {(4 d) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^5}+\frac {\left (10 d^2\right ) \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^5}-\frac {\left (10 d^3\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^5}+\frac {\left (5 d^4\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{e^5}-\frac {d^5 \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^4} \, dx}{e^5}+\frac {\int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^4}\\ &=-\frac {4 a d x}{e^5}-\frac {b n x^2}{4 e^4}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^4}+\frac {d^5 \left (a+b \log \left (c x^n\right )\right )}{3 e^6 (d+e x)^3}-\frac {5 d^4 \left (a+b \log \left (c x^n\right )\right )}{2 e^6 (d+e x)^2}-\frac {10 d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)}+\frac {10 d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^6}-\frac {(4 b d) \int \log \left (c x^n\right ) \, dx}{e^5}-\frac {\left (10 b d^2 n\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^6}+\frac {\left (5 b d^4 n\right ) \int \frac {1}{x (d+e x)^2} \, dx}{2 e^6}-\frac {\left (b d^5 n\right ) \int \frac {1}{x (d+e x)^3} \, dx}{3 e^6}+\frac {\left (10 b d^2 n\right ) \int \frac {1}{d+e x} \, dx}{e^5}\\ &=-\frac {4 a d x}{e^5}+\frac {4 b d n x}{e^5}-\frac {b n x^2}{4 e^4}-\frac {4 b d x \log \left (c x^n\right )}{e^5}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^4}+\frac {d^5 \left (a+b \log \left (c x^n\right )\right )}{3 e^6 (d+e x)^3}-\frac {5 d^4 \left (a+b \log \left (c x^n\right )\right )}{2 e^6 (d+e x)^2}-\frac {10 d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)}+\frac {10 b d^2 n \log (d+e x)}{e^6}+\frac {10 d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^6}+\frac {10 b d^2 n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^6}+\frac {\left (5 b d^4 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 e^6}-\frac {\left (b d^5 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 e^6}\\ &=-\frac {4 a d x}{e^5}+\frac {4 b d n x}{e^5}-\frac {b n x^2}{4 e^4}-\frac {b d^4 n}{6 e^6 (d+e x)^2}+\frac {13 b d^3 n}{6 e^6 (d+e x)}+\frac {13 b d^2 n \log (x)}{6 e^6}-\frac {4 b d x \log \left (c x^n\right )}{e^5}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^4}+\frac {d^5 \left (a+b \log \left (c x^n\right )\right )}{3 e^6 (d+e x)^3}-\frac {5 d^4 \left (a+b \log \left (c x^n\right )\right )}{2 e^6 (d+e x)^2}-\frac {10 d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)}+\frac {47 b d^2 n \log (d+e x)}{6 e^6}+\frac {10 d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^6}+\frac {10 b d^2 n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^6}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 249, normalized size = 1.09 \[ \frac {\frac {4 d^5 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3}-\frac {30 d^4 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}+\frac {120 d^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x}+120 d^2 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )+6 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )-48 a d e x-48 b d e x \log \left (c x^n\right )+120 b d^2 n \text {Li}_2\left (-\frac {e x}{d}\right )-2 b d^2 n \left (\frac {d (3 d+2 e x)}{(d+e x)^2}-2 \log (d+e x)+2 \log (x)\right )-120 b d^2 n (\log (x)-\log (d+e x))+30 b d^2 n \left (\frac {d}{d+e x}-\log (d+e x)+\log (x)\right )+48 b d e n x-3 b e^2 n x^2}{12 e^6} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{5} \log \left (c x^{n}\right ) + a x^{5}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{5}}{{\left (e x + d\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.23, size = 1153, 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 \[ \frac {1}{6} \, a {\left (\frac {60 \, d^{3} e^{2} x^{2} + 105 \, d^{4} e x + 47 \, d^{5}}{e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}} + \frac {60 \, d^{2} \log \left (e x + d\right )}{e^{6}} + \frac {3 \, {\left (e x^{2} - 8 \, d x\right )}}{e^{5}}\right )} + b \int \frac {x^{5} \log \relax (c) + x^{5} \log \left (x^{n}\right )}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}\,{d x} \]
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 \[ \int \frac {x^5\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 133.88, size = 598, normalized size = 2.61 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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