Optimal. Leaf size=294 \[ -\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^7}-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{d^7 (d+e x)}+\frac {a+b \log \left (c x^n\right )}{2 d^5 (d+e x)^2}+\frac {a+b \log \left (c x^n\right )}{3 d^4 (d+e x)^3}+\frac {a+b \log \left (c x^n\right )}{4 d^3 (d+e x)^4}+\frac {a+b \log \left (c x^n\right )}{5 d^2 (d+e x)^5}+\frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}+\frac {b n \text {Li}_2\left (-\frac {d}{e x}\right )}{d^7}+\frac {49 b n \log (d+e x)}{20 d^7}-\frac {29 b n \log (x)}{20 d^7}-\frac {29 b n}{20 d^6 (d+e x)}-\frac {19 b n}{40 d^5 (d+e x)^2}-\frac {37 b n}{180 d^4 (d+e x)^3}-\frac {11 b n}{120 d^3 (d+e x)^4}-\frac {b n}{30 d^2 (d+e x)^5} \]
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Rubi [A] time = 0.73, antiderivative size = 316, normalized size of antiderivative = 1.07, number of steps used = 27, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2347, 2344, 2301, 2317, 2391, 2314, 31, 2319, 44} \[ -\frac {b n \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^7}-\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^7}-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{d^7 (d+e x)}+\frac {a+b \log \left (c x^n\right )}{2 d^5 (d+e x)^2}+\frac {a+b \log \left (c x^n\right )}{3 d^4 (d+e x)^3}+\frac {a+b \log \left (c x^n\right )}{4 d^3 (d+e x)^4}+\frac {a+b \log \left (c x^n\right )}{5 d^2 (d+e x)^5}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b d^7 n}+\frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}-\frac {29 b n}{20 d^6 (d+e x)}-\frac {19 b n}{40 d^5 (d+e x)^2}-\frac {37 b n}{180 d^4 (d+e x)^3}-\frac {11 b n}{120 d^3 (d+e x)^4}-\frac {b n}{30 d^2 (d+e x)^5}+\frac {49 b n \log (d+e x)}{20 d^7}-\frac {29 b n \log (x)}{20 d^7} \]
Antiderivative was successfully verified.
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Rule 31
Rule 44
Rule 2301
Rule 2314
Rule 2317
Rule 2319
Rule 2344
Rule 2347
Rule 2391
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^7} \, dx &=\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^6} \, dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^7} \, dx}{d}\\ &=\frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}+\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^5} \, dx}{d^2}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^6} \, dx}{d^2}-\frac {(b n) \int \frac {1}{x (d+e x)^6} \, dx}{6 d}\\ &=\frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}+\frac {a+b \log \left (c x^n\right )}{5 d^2 (d+e x)^5}+\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^4} \, dx}{d^3}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^5} \, dx}{d^3}-\frac {(b n) \int \frac {1}{x (d+e x)^5} \, dx}{5 d^2}-\frac {(b n) \int \left (\frac {1}{d^6 x}-\frac {e}{d (d+e x)^6}-\frac {e}{d^2 (d+e x)^5}-\frac {e}{d^3 (d+e x)^4}-\frac {e}{d^4 (d+e x)^3}-\frac {e}{d^5 (d+e x)^2}-\frac {e}{d^6 (d+e x)}\right ) \, dx}{6 d}\\ &=-\frac {b n}{30 d^2 (d+e x)^5}-\frac {b n}{24 d^3 (d+e x)^4}-\frac {b n}{18 d^4 (d+e x)^3}-\frac {b n}{12 d^5 (d+e x)^2}-\frac {b n}{6 d^6 (d+e x)}-\frac {b n \log (x)}{6 d^7}+\frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}+\frac {a+b \log \left (c x^n\right )}{5 d^2 (d+e x)^5}+\frac {a+b \log \left (c x^n\right )}{4 d^3 (d+e x)^4}+\frac {b n \log (d+e x)}{6 d^7}+\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^3} \, dx}{d^4}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^4} \, dx}{d^4}-\frac {(b n) \int \frac {1}{x (d+e x)^4} \, dx}{4 d^3}-\frac {(b n) \int \left (\frac {1}{d^5 x}-\frac {e}{d (d+e x)^5}-\frac {e}{d^2 (d+e x)^4}-\frac {e}{d^3 (d+e x)^3}-\frac {e}{d^4 (d+e x)^2}-\frac {e}{d^5 (d+e x)}\right ) \, dx}{5 d^2}\\ &=-\frac {b n}{30 d^2 (d+e x)^5}-\frac {11 b n}{120 d^3 (d+e x)^4}-\frac {11 b n}{90 d^4 (d+e x)^3}-\frac {11 b n}{60 d^5 (d+e x)^2}-\frac {11 b n}{30 d^6 (d+e x)}-\frac {11 b n \log (x)}{30 d^7}+\frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}+\frac {a+b \log \left (c x^n\right )}{5 d^2 (d+e x)^5}+\frac {a+b \log \left (c x^n\right )}{4 d^3 (d+e x)^4}+\frac {a+b \log \left (c x^n\right )}{3 d^4 (d+e x)^3}+\frac {11 b n \log (d+e x)}{30 d^7}+\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{d^5}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{d^5}-\frac {(b n) \int \frac {1}{x (d+e x)^3} \, dx}{3 d^4}-\frac {(b n) \int \left (\frac {1}{d^4 x}-\frac {e}{d (d+e x)^4}-\frac {e}{d^2 (d+e x)^3}-\frac {e}{d^3 (d+e x)^2}-\frac {e}{d^4 (d+e x)}\right ) \, dx}{4 d^3}\\ &=-\frac {b n}{30 d^2 (d+e x)^5}-\frac {11 b n}{120 d^3 (d+e x)^4}-\frac {37 b n}{180 d^4 (d+e x)^3}-\frac {37 b n}{120 d^5 (d+e x)^2}-\frac {37 b n}{60 d^6 (d+e x)}-\frac {37 b n \log (x)}{60 d^7}+\frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}+\frac {a+b \log \left (c x^n\right )}{5 d^2 (d+e x)^5}+\frac {a+b \log \left (c x^n\right )}{4 d^3 (d+e x)^4}+\frac {a+b \log \left (c x^n\right )}{3 d^4 (d+e x)^3}+\frac {a+b \log \left (c x^n\right )}{2 d^5 (d+e x)^2}+\frac {37 b n \log (d+e x)}{60 d^7}+\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{d^6}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d^6}-\frac {(b n) \int \frac {1}{x (d+e x)^2} \, dx}{2 d^5}-\frac {(b 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^4}\\ &=-\frac {b n}{30 d^2 (d+e x)^5}-\frac {11 b n}{120 d^3 (d+e x)^4}-\frac {37 b n}{180 d^4 (d+e x)^3}-\frac {19 b n}{40 d^5 (d+e x)^2}-\frac {19 b n}{20 d^6 (d+e x)}-\frac {19 b n \log (x)}{20 d^7}+\frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}+\frac {a+b \log \left (c x^n\right )}{5 d^2 (d+e x)^5}+\frac {a+b \log \left (c x^n\right )}{4 d^3 (d+e x)^4}+\frac {a+b \log \left (c x^n\right )}{3 d^4 (d+e x)^3}+\frac {a+b \log \left (c x^n\right )}{2 d^5 (d+e x)^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{d^7 (d+e x)}+\frac {19 b n \log (d+e x)}{20 d^7}+\frac {\int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{d^7}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^7}-\frac {(b n) \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^5}+\frac {(b e n) \int \frac {1}{d+e x} \, dx}{d^7}\\ &=-\frac {b n}{30 d^2 (d+e x)^5}-\frac {11 b n}{120 d^3 (d+e x)^4}-\frac {37 b n}{180 d^4 (d+e x)^3}-\frac {19 b n}{40 d^5 (d+e x)^2}-\frac {29 b n}{20 d^6 (d+e x)}-\frac {29 b n \log (x)}{20 d^7}+\frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}+\frac {a+b \log \left (c x^n\right )}{5 d^2 (d+e x)^5}+\frac {a+b \log \left (c x^n\right )}{4 d^3 (d+e x)^4}+\frac {a+b \log \left (c x^n\right )}{3 d^4 (d+e x)^3}+\frac {a+b \log \left (c x^n\right )}{2 d^5 (d+e x)^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{d^7 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b d^7 n}+\frac {49 b n \log (d+e x)}{20 d^7}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^7}+\frac {(b n) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^7}\\ &=-\frac {b n}{30 d^2 (d+e x)^5}-\frac {11 b n}{120 d^3 (d+e x)^4}-\frac {37 b n}{180 d^4 (d+e x)^3}-\frac {19 b n}{40 d^5 (d+e x)^2}-\frac {29 b n}{20 d^6 (d+e x)}-\frac {29 b n \log (x)}{20 d^7}+\frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}+\frac {a+b \log \left (c x^n\right )}{5 d^2 (d+e x)^5}+\frac {a+b \log \left (c x^n\right )}{4 d^3 (d+e x)^4}+\frac {a+b \log \left (c x^n\right )}{3 d^4 (d+e x)^3}+\frac {a+b \log \left (c x^n\right )}{2 d^5 (d+e x)^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{d^7 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b d^7 n}+\frac {49 b n \log (d+e x)}{20 d^7}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^7}-\frac {b n \text {Li}_2\left (-\frac {e x}{d}\right )}{d^7}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 349, normalized size = 1.19 \[ \frac {\frac {360 a \log \left (c x^n\right )}{n}+\frac {60 a d^6}{(d+e x)^6}+\frac {72 a d^5}{(d+e x)^5}+\frac {90 a d^4}{(d+e x)^4}+\frac {120 a d^3}{(d+e x)^3}+\frac {180 a d^2}{(d+e x)^2}+\frac {360 a d}{d+e x}-360 a \log \left (\frac {e x}{d}+1\right )+\frac {60 b d^6 \log \left (c x^n\right )}{(d+e x)^6}+\frac {72 b d^5 \log \left (c x^n\right )}{(d+e x)^5}+\frac {90 b d^4 \log \left (c x^n\right )}{(d+e x)^4}+\frac {120 b d^3 \log \left (c x^n\right )}{(d+e x)^3}+\frac {180 b d^2 \log \left (c x^n\right )}{(d+e x)^2}+\frac {360 b d \log \left (c x^n\right )}{d+e x}-360 b \log \left (c x^n\right ) \log \left (\frac {e x}{d}+1\right )+\frac {180 b \log ^2\left (c x^n\right )}{n}-\frac {12 b d^5 n}{(d+e x)^5}-\frac {33 b d^4 n}{(d+e x)^4}-\frac {74 b d^3 n}{(d+e x)^3}-\frac {171 b d^2 n}{(d+e x)^2}-360 b n \text {Li}_2\left (-\frac {e x}{d}\right )-\frac {522 b d n}{d+e x}+882 b n \log (d+e x)-882 b n \log (x)}{360 d^7} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left (c x^{n}\right ) + a}{e^{7} x^{8} + 7 \, d e^{6} x^{7} + 21 \, d^{2} e^{5} x^{6} + 35 \, d^{3} e^{4} x^{5} + 35 \, d^{4} e^{3} x^{4} + 21 \, d^{5} e^{2} x^{3} + 7 \, d^{6} e x^{2} + d^{7} x}, x\right ) \]
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 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{7} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.22, size = 1427, normalized size = 4.85 \[ \text {result too large to display} \]
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 \[ \frac {1}{60} \, a {\left (\frac {60 \, e^{5} x^{5} + 330 \, d e^{4} x^{4} + 740 \, d^{2} e^{3} x^{3} + 855 \, d^{3} e^{2} x^{2} + 522 \, d^{4} e x + 147 \, d^{5}}{d^{6} e^{6} x^{6} + 6 \, d^{7} e^{5} x^{5} + 15 \, d^{8} e^{4} x^{4} + 20 \, d^{9} e^{3} x^{3} + 15 \, d^{10} e^{2} x^{2} + 6 \, d^{11} e x + d^{12}} - \frac {60 \, \log \left (e x + d\right )}{d^{7}} + \frac {60 \, \log \relax (x)}{d^{7}}\right )} + b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{e^{7} x^{8} + 7 \, d e^{6} x^{7} + 21 \, d^{2} e^{5} x^{6} + 35 \, d^{3} e^{4} x^{5} + 35 \, d^{4} e^{3} x^{4} + 21 \, d^{5} e^{2} x^{3} + 7 \, d^{6} e x^{2} + d^{7} x}\,{d x} \]
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 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,{\left (d+e\,x\right )}^7} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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