Optimal. Leaf size=285 \[ -\frac{7 b d n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^8}-\frac{x^6 \left (7 a+7 b \log \left (c x^n\right )+b n\right )}{30 e^2 (d+e x)^5}-\frac{x^5 \left (42 a+42 b \log \left (c x^n\right )+13 b n\right )}{120 e^3 (d+e x)^4}-\frac{x^4 \left (210 a+210 b \log \left (c x^n\right )+107 b n\right )}{360 e^4 (d+e x)^3}-\frac{x^3 \left (420 a+420 b \log \left (c x^n\right )+319 b n\right )}{360 e^5 (d+e x)^2}-\frac{x^2 \left (140 a+140 b \log \left (c x^n\right )+153 b n\right )}{40 e^6 (d+e x)}-\frac{d \log \left (\frac{e x}{d}+1\right ) \left (140 a+140 b \log \left (c x^n\right )+223 b n\right )}{20 e^8}-\frac{x^7 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6}+\frac{x (140 a+223 b n)}{20 e^7}+\frac{7 b x \log \left (c x^n\right )}{e^7}-\frac{7 b n x}{e^7} \]
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Rubi [A] time = 0.571391, antiderivative size = 351, normalized size of antiderivative = 1.23, number of steps used = 23, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {43, 2351, 2295, 2319, 44, 2314, 31, 2317, 2391} \[ -\frac{7 b d n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^8}+\frac{d^7 \left (a+b \log \left (c x^n\right )\right )}{6 e^8 (d+e x)^6}-\frac{7 d^6 \left (a+b \log \left (c x^n\right )\right )}{5 e^8 (d+e x)^5}+\frac{21 d^5 \left (a+b \log \left (c x^n\right )\right )}{4 e^8 (d+e x)^4}-\frac{35 d^4 \left (a+b \log \left (c x^n\right )\right )}{3 e^8 (d+e x)^3}+\frac{35 d^3 \left (a+b \log \left (c x^n\right )\right )}{2 e^8 (d+e x)^2}+\frac{21 d x \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)}-\frac{7 d \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^8}+\frac{a x}{e^7}+\frac{b x \log \left (c x^n\right )}{e^7}-\frac{b d^6 n}{30 e^8 (d+e x)^5}+\frac{37 b d^5 n}{120 e^8 (d+e x)^4}-\frac{241 b d^4 n}{180 e^8 (d+e x)^3}+\frac{153 b d^3 n}{40 e^8 (d+e x)^2}-\frac{197 b d^2 n}{20 e^8 (d+e x)}-\frac{197 b d n \log (x)}{20 e^8}-\frac{223 b d n \log (d+e x)}{20 e^8}-\frac{b n x}{e^7} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2351
Rule 2295
Rule 2319
Rule 44
Rule 2314
Rule 31
Rule 2317
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^7 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx &=\int \left (\frac{a+b \log \left (c x^n\right )}{e^7}-\frac{d^7 \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)^7}+\frac{7 d^6 \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)^6}-\frac{21 d^5 \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)^5}+\frac{35 d^4 \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)^4}-\frac{35 d^3 \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)^3}+\frac{21 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)^2}-\frac{7 d \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)}\right ) \, dx\\ &=\frac{\int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^7}-\frac{(7 d) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^7}+\frac{\left (21 d^2\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^7}-\frac{\left (35 d^3\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{e^7}+\frac{\left (35 d^4\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^4} \, dx}{e^7}-\frac{\left (21 d^5\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^5} \, dx}{e^7}+\frac{\left (7 d^6\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^6} \, dx}{e^7}-\frac{d^7 \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^7} \, dx}{e^7}\\ &=\frac{a x}{e^7}+\frac{d^7 \left (a+b \log \left (c x^n\right )\right )}{6 e^8 (d+e x)^6}-\frac{7 d^6 \left (a+b \log \left (c x^n\right )\right )}{5 e^8 (d+e x)^5}+\frac{21 d^5 \left (a+b \log \left (c x^n\right )\right )}{4 e^8 (d+e x)^4}-\frac{35 d^4 \left (a+b \log \left (c x^n\right )\right )}{3 e^8 (d+e x)^3}+\frac{35 d^3 \left (a+b \log \left (c x^n\right )\right )}{2 e^8 (d+e x)^2}+\frac{21 d x \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)}-\frac{7 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^8}+\frac{b \int \log \left (c x^n\right ) \, dx}{e^7}+\frac{(7 b d n) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{e^8}-\frac{\left (35 b d^3 n\right ) \int \frac{1}{x (d+e x)^2} \, dx}{2 e^8}+\frac{\left (35 b d^4 n\right ) \int \frac{1}{x (d+e x)^3} \, dx}{3 e^8}-\frac{\left (21 b d^5 n\right ) \int \frac{1}{x (d+e x)^4} \, dx}{4 e^8}+\frac{\left (7 b d^6 n\right ) \int \frac{1}{x (d+e x)^5} \, dx}{5 e^8}-\frac{\left (b d^7 n\right ) \int \frac{1}{x (d+e x)^6} \, dx}{6 e^8}-\frac{(21 b d n) \int \frac{1}{d+e x} \, dx}{e^7}\\ &=\frac{a x}{e^7}-\frac{b n x}{e^7}+\frac{b x \log \left (c x^n\right )}{e^7}+\frac{d^7 \left (a+b \log \left (c x^n\right )\right )}{6 e^8 (d+e x)^6}-\frac{7 d^6 \left (a+b \log \left (c x^n\right )\right )}{5 e^8 (d+e x)^5}+\frac{21 d^5 \left (a+b \log \left (c x^n\right )\right )}{4 e^8 (d+e x)^4}-\frac{35 d^4 \left (a+b \log \left (c x^n\right )\right )}{3 e^8 (d+e x)^3}+\frac{35 d^3 \left (a+b \log \left (c x^n\right )\right )}{2 e^8 (d+e x)^2}+\frac{21 d x \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)}-\frac{21 b d n \log (d+e x)}{e^8}-\frac{7 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^8}-\frac{7 b d n \text{Li}_2\left (-\frac{e x}{d}\right )}{e^8}-\frac{\left (35 b d^3 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^8}+\frac{\left (35 b d^4 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^8}-\frac{\left (21 b d^5 n\right ) \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 e^8}+\frac{\left (7 b d^6 n\right ) \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 e^8}-\frac{\left (b d^7 n\right ) \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 e^8}\\ &=\frac{a x}{e^7}-\frac{b n x}{e^7}-\frac{b d^6 n}{30 e^8 (d+e x)^5}+\frac{37 b d^5 n}{120 e^8 (d+e x)^4}-\frac{241 b d^4 n}{180 e^8 (d+e x)^3}+\frac{153 b d^3 n}{40 e^8 (d+e x)^2}-\frac{197 b d^2 n}{20 e^8 (d+e x)}-\frac{197 b d n \log (x)}{20 e^8}+\frac{b x \log \left (c x^n\right )}{e^7}+\frac{d^7 \left (a+b \log \left (c x^n\right )\right )}{6 e^8 (d+e x)^6}-\frac{7 d^6 \left (a+b \log \left (c x^n\right )\right )}{5 e^8 (d+e x)^5}+\frac{21 d^5 \left (a+b \log \left (c x^n\right )\right )}{4 e^8 (d+e x)^4}-\frac{35 d^4 \left (a+b \log \left (c x^n\right )\right )}{3 e^8 (d+e x)^3}+\frac{35 d^3 \left (a+b \log \left (c x^n\right )\right )}{2 e^8 (d+e x)^2}+\frac{21 d x \left (a+b \log \left (c x^n\right )\right )}{e^7 (d+e x)}-\frac{223 b d n \log (d+e x)}{20 e^8}-\frac{7 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^8}-\frac{7 b d n \text{Li}_2\left (-\frac{e x}{d}\right )}{e^8}\\ \end{align*}
Mathematica [A] time = 0.566353, size = 356, normalized size = 1.25 \[ -\frac{2520 b d n \text{PolyLog}\left (2,-\frac{e x}{d}\right )-\frac{60 a d^7}{(d+e x)^6}+\frac{504 a d^6}{(d+e x)^5}-\frac{1890 a d^5}{(d+e x)^4}+\frac{4200 a d^4}{(d+e x)^3}-\frac{6300 a d^3}{(d+e x)^2}+\frac{7560 a d^2}{d+e x}+2520 a d \log \left (\frac{e x}{d}+1\right )-360 a e x-\frac{60 b d^7 \log \left (c x^n\right )}{(d+e x)^6}+\frac{504 b d^6 \log \left (c x^n\right )}{(d+e x)^5}-\frac{1890 b d^5 \log \left (c x^n\right )}{(d+e x)^4}+\frac{4200 b d^4 \log \left (c x^n\right )}{(d+e x)^3}-\frac{6300 b d^3 \log \left (c x^n\right )}{(d+e x)^2}+\frac{7560 b d^2 \log \left (c x^n\right )}{d+e x}+2520 b d \log \left (c x^n\right ) \log \left (\frac{e x}{d}+1\right )-360 b e x \log \left (c x^n\right )+\frac{12 b d^6 n}{(d+e x)^5}-\frac{111 b d^5 n}{(d+e x)^4}+\frac{482 b d^4 n}{(d+e x)^3}-\frac{1377 b d^3 n}{(d+e x)^2}+\frac{3546 b d^2 n}{d+e x}+4014 b d n \log (d+e x)-4014 b d n \log (x)+360 b e n x}{360 e^8} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.23, size = 1584, normalized size = 5.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{60} \, a{\left (\frac{1260 \, d^{2} e^{5} x^{5} + 5250 \, d^{3} e^{4} x^{4} + 9100 \, d^{4} e^{3} x^{3} + 8085 \, d^{5} e^{2} x^{2} + 3654 \, d^{6} e x + 669 \, d^{7}}{e^{14} x^{6} + 6 \, d e^{13} x^{5} + 15 \, d^{2} e^{12} x^{4} + 20 \, d^{3} e^{11} x^{3} + 15 \, d^{4} e^{10} x^{2} + 6 \, d^{5} e^{9} x + d^{6} e^{8}} - \frac{60 \, x}{e^{7}} + \frac{420 \, d \log \left (e x + d\right )}{e^{8}}\right )} + b \int \frac{x^{7} \log \left (c\right ) + x^{7} \log \left (x^{n}\right )}{e^{7} x^{7} + 7 \, d e^{6} x^{6} + 21 \, d^{2} e^{5} x^{5} + 35 \, d^{3} e^{4} x^{4} + 35 \, d^{4} e^{3} x^{3} + 21 \, d^{5} e^{2} x^{2} + 7 \, d^{6} e x + d^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{7} \log \left (c x^{n}\right ) + a x^{7}}{e^{7} x^{7} + 7 \, d e^{6} x^{6} + 21 \, d^{2} e^{5} x^{5} + 35 \, d^{3} e^{4} x^{4} + 35 \, d^{4} e^{3} x^{3} + 21 \, d^{5} e^{2} x^{2} + 7 \, d^{6} e x + d^{7}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} x^{7}}{{\left (e x + d\right )}^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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