Optimal. Leaf size=243 \[ \frac{b n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^7}-\frac{x^5 \left (6 a+6 b \log \left (c x^n\right )+b n\right )}{30 e^2 (d+e x)^5}-\frac{x^4 \left (30 a+30 b \log \left (c x^n\right )+11 b n\right )}{120 e^3 (d+e x)^4}-\frac{x^3 \left (60 a+60 b \log \left (c x^n\right )+37 b n\right )}{180 e^4 (d+e x)^3}-\frac{x^2 \left (20 a+20 b \log \left (c x^n\right )+19 b n\right )}{40 e^5 (d+e x)^2}-\frac{x \left (20 a+20 b \log \left (c x^n\right )+29 b n\right )}{20 e^6 (d+e x)}+\frac{\log \left (\frac{e x}{d}+1\right ) \left (20 a+20 b \log \left (c x^n\right )+49 b n\right )}{20 e^7}-\frac{x^6 \left (a+b \log \left (c x^n\right )\right )}{6 e (d+e x)^6} \]
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Rubi [A] time = 0.538136, antiderivative size = 316, normalized size of antiderivative = 1.3, number of steps used = 21, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {43, 2351, 2319, 44, 2314, 31, 2317, 2391} \[ \frac{b n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^7}-\frac{d^6 \left (a+b \log \left (c x^n\right )\right )}{6 e^7 (d+e x)^6}+\frac{6 d^5 \left (a+b \log \left (c x^n\right )\right )}{5 e^7 (d+e x)^5}-\frac{15 d^4 \left (a+b \log \left (c x^n\right )\right )}{4 e^7 (d+e x)^4}+\frac{20 d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^7 (d+e x)^3}-\frac{15 d^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^7 (d+e x)^2}-\frac{6 x \left (a+b \log \left (c x^n\right )\right )}{e^6 (d+e x)}+\frac{\log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^7}+\frac{b d^5 n}{30 e^7 (d+e x)^5}-\frac{31 b d^4 n}{120 e^7 (d+e x)^4}+\frac{163 b d^3 n}{180 e^7 (d+e x)^3}-\frac{79 b d^2 n}{40 e^7 (d+e x)^2}+\frac{71 b d n}{20 e^7 (d+e x)}+\frac{49 b n \log (d+e x)}{20 e^7}+\frac{71 b n \log (x)}{20 e^7} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2351
Rule 2319
Rule 44
Rule 2314
Rule 31
Rule 2317
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^6 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^7} \, dx &=\int \left (\frac{d^6 \left (a+b \log \left (c x^n\right )\right )}{e^6 (d+e x)^7}-\frac{6 d^5 \left (a+b \log \left (c x^n\right )\right )}{e^6 (d+e x)^6}+\frac{15 d^4 \left (a+b \log \left (c x^n\right )\right )}{e^6 (d+e x)^5}-\frac{20 d^3 \left (a+b \log \left (c x^n\right )\right )}{e^6 (d+e x)^4}+\frac{15 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^6 (d+e x)^3}-\frac{6 d \left (a+b \log \left (c x^n\right )\right )}{e^6 (d+e x)^2}+\frac{a+b \log \left (c x^n\right )}{e^6 (d+e x)}\right ) \, dx\\ &=\frac{\int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^6}-\frac{(6 d) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^6}+\frac{\left (15 d^2\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{e^6}-\frac{\left (20 d^3\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^4} \, dx}{e^6}+\frac{\left (15 d^4\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^5} \, dx}{e^6}-\frac{\left (6 d^5\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^6} \, dx}{e^6}+\frac{d^6 \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^7} \, dx}{e^6}\\ &=-\frac{d^6 \left (a+b \log \left (c x^n\right )\right )}{6 e^7 (d+e x)^6}+\frac{6 d^5 \left (a+b \log \left (c x^n\right )\right )}{5 e^7 (d+e x)^5}-\frac{15 d^4 \left (a+b \log \left (c x^n\right )\right )}{4 e^7 (d+e x)^4}+\frac{20 d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^7 (d+e x)^3}-\frac{15 d^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^7 (d+e x)^2}-\frac{6 x \left (a+b \log \left (c x^n\right )\right )}{e^6 (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^7}-\frac{(b n) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{e^7}+\frac{\left (15 b d^2 n\right ) \int \frac{1}{x (d+e x)^2} \, dx}{2 e^7}-\frac{\left (20 b d^3 n\right ) \int \frac{1}{x (d+e x)^3} \, dx}{3 e^7}+\frac{\left (15 b d^4 n\right ) \int \frac{1}{x (d+e x)^4} \, dx}{4 e^7}-\frac{\left (6 b d^5 n\right ) \int \frac{1}{x (d+e x)^5} \, dx}{5 e^7}+\frac{\left (b d^6 n\right ) \int \frac{1}{x (d+e x)^6} \, dx}{6 e^7}+\frac{(6 b n) \int \frac{1}{d+e x} \, dx}{e^6}\\ &=-\frac{d^6 \left (a+b \log \left (c x^n\right )\right )}{6 e^7 (d+e x)^6}+\frac{6 d^5 \left (a+b \log \left (c x^n\right )\right )}{5 e^7 (d+e x)^5}-\frac{15 d^4 \left (a+b \log \left (c x^n\right )\right )}{4 e^7 (d+e x)^4}+\frac{20 d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^7 (d+e x)^3}-\frac{15 d^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^7 (d+e x)^2}-\frac{6 x \left (a+b \log \left (c x^n\right )\right )}{e^6 (d+e x)}+\frac{6 b n \log (d+e x)}{e^7}+\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^7}+\frac{b n \text{Li}_2\left (-\frac{e x}{d}\right )}{e^7}+\frac{\left (15 b d^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 e^7}-\frac{\left (20 b d^3 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^7}+\frac{\left (15 b d^4 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^7}-\frac{\left (6 b d^5 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^7}+\frac{\left (b d^6 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^7}\\ &=\frac{b d^5 n}{30 e^7 (d+e x)^5}-\frac{31 b d^4 n}{120 e^7 (d+e x)^4}+\frac{163 b d^3 n}{180 e^7 (d+e x)^3}-\frac{79 b d^2 n}{40 e^7 (d+e x)^2}+\frac{71 b d n}{20 e^7 (d+e x)}+\frac{71 b n \log (x)}{20 e^7}-\frac{d^6 \left (a+b \log \left (c x^n\right )\right )}{6 e^7 (d+e x)^6}+\frac{6 d^5 \left (a+b \log \left (c x^n\right )\right )}{5 e^7 (d+e x)^5}-\frac{15 d^4 \left (a+b \log \left (c x^n\right )\right )}{4 e^7 (d+e x)^4}+\frac{20 d^3 \left (a+b \log \left (c x^n\right )\right )}{3 e^7 (d+e x)^3}-\frac{15 d^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^7 (d+e x)^2}-\frac{6 x \left (a+b \log \left (c x^n\right )\right )}{e^6 (d+e x)}+\frac{49 b n \log (d+e x)}{20 e^7}+\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^7}+\frac{b n \text{Li}_2\left (-\frac{e x}{d}\right )}{e^7}\\ \end{align*}
Mathematica [A] time = 0.447716, size = 333, normalized size = 1.37 \[ \frac{360 b n \text{PolyLog}\left (2,-\frac{e x}{d}\right )+\frac{432 a d^5 (d+e x)-1350 a d^4 (d+e x)^2+2400 a d^3 (d+e x)^3-2700 a d^2 (d+e x)^4-60 a d^6+2160 a d (d+e x)^5+360 a (d+e x)^6 \log \left (\frac{e x}{d}+1\right )+432 b d^5 (d+e x) \log \left (c x^n\right )-1350 b d^4 (d+e x)^2 \log \left (c x^n\right )+2400 b d^3 (d+e x)^3 \log \left (c x^n\right )-2700 b d^2 (d+e x)^4 \log \left (c x^n\right )-60 b d^6 \log \left (c x^n\right )+2160 b d (d+e x)^5 \log \left (c x^n\right )+360 b (d+e x)^6 \log \left (c x^n\right ) \log \left (\frac{e x}{d}+1\right )+12 b d^5 n (d+e x)-93 b d^4 n (d+e x)^2+326 b d^3 n (d+e x)^3-711 b d^2 n (d+e x)^4+1278 b d n (d+e x)^5+882 b n (d+e x)^6 \log (d+e x)}{(d+e x)^6}-882 b n \log (x)}{360 e^7} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.174, size = 1416, normalized size = 5.8 \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{360 \, d e^{5} x^{5} + 1350 \, d^{2} e^{4} x^{4} + 2200 \, d^{3} e^{3} x^{3} + 1875 \, d^{4} e^{2} x^{2} + 822 \, d^{5} e x + 147 \, d^{6}}{e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}} + \frac{60 \, \log \left (e x + d\right )}{e^{7}}\right )} + b \int \frac{x^{6} \log \left (c\right ) + x^{6} \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^{6} \log \left (c x^{n}\right ) + a x^{6}}{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^{6}}{{\left (e x + d\right )}^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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