Optimal. Leaf size=133 \[ -\frac{3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{6 x^6}-\frac{3 d e^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^4}-\frac{e^3 \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{3 b d^2 e n}{25 x^5}-\frac{b d^3 n}{36 x^6}-\frac{3 b d e^2 n}{16 x^4}-\frac{b e^3 n}{9 x^3} \]
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Rubi [A] time = 0.0959554, antiderivative size = 100, normalized size of antiderivative = 0.75, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {43, 2334, 12, 14} \[ -\frac{1}{60} \left (\frac{36 d^2 e}{x^5}+\frac{10 d^3}{x^6}+\frac{45 d e^2}{x^4}+\frac{20 e^3}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n}{25 x^5}-\frac{b d^3 n}{36 x^6}-\frac{3 b d e^2 n}{16 x^4}-\frac{b e^3 n}{9 x^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2334
Rule 12
Rule 14
Rubi steps
\begin{align*} \int \frac{(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^7} \, dx &=-\frac{1}{60} \left (\frac{10 d^3}{x^6}+\frac{36 d^2 e}{x^5}+\frac{45 d e^2}{x^4}+\frac{20 e^3}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{-10 d^3-36 d^2 e x-45 d e^2 x^2-20 e^3 x^3}{60 x^7} \, dx\\ &=-\frac{1}{60} \left (\frac{10 d^3}{x^6}+\frac{36 d^2 e}{x^5}+\frac{45 d e^2}{x^4}+\frac{20 e^3}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{60} (b n) \int \frac{-10 d^3-36 d^2 e x-45 d e^2 x^2-20 e^3 x^3}{x^7} \, dx\\ &=-\frac{1}{60} \left (\frac{10 d^3}{x^6}+\frac{36 d^2 e}{x^5}+\frac{45 d e^2}{x^4}+\frac{20 e^3}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{60} (b n) \int \left (-\frac{10 d^3}{x^7}-\frac{36 d^2 e}{x^6}-\frac{45 d e^2}{x^5}-\frac{20 e^3}{x^4}\right ) \, dx\\ &=-\frac{b d^3 n}{36 x^6}-\frac{3 b d^2 e n}{25 x^5}-\frac{3 b d e^2 n}{16 x^4}-\frac{b e^3 n}{9 x^3}-\frac{1}{60} \left (\frac{10 d^3}{x^6}+\frac{36 d^2 e}{x^5}+\frac{45 d e^2}{x^4}+\frac{20 e^3}{x^3}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0515982, size = 113, normalized size = 0.85 \[ -\frac{60 a \left (36 d^2 e x+10 d^3+45 d e^2 x^2+20 e^3 x^3\right )+60 b \left (36 d^2 e x+10 d^3+45 d e^2 x^2+20 e^3 x^3\right ) \log \left (c x^n\right )+b n \left (432 d^2 e x+100 d^3+675 d e^2 x^2+400 e^3 x^3\right )}{3600 x^6} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.138, size = 571, normalized size = 4.3 \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 [A] time = 1.10918, size = 193, normalized size = 1.45 \begin{align*} -\frac{b e^{3} n}{9 \, x^{3}} - \frac{b e^{3} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac{3 \, b d e^{2} n}{16 \, x^{4}} - \frac{a e^{3}}{3 \, x^{3}} - \frac{3 \, b d e^{2} \log \left (c x^{n}\right )}{4 \, x^{4}} - \frac{3 \, b d^{2} e n}{25 \, x^{5}} - \frac{3 \, a d e^{2}}{4 \, x^{4}} - \frac{3 \, b d^{2} e \log \left (c x^{n}\right )}{5 \, x^{5}} - \frac{b d^{3} n}{36 \, x^{6}} - \frac{3 \, a d^{2} e}{5 \, x^{5}} - \frac{b d^{3} \log \left (c x^{n}\right )}{6 \, x^{6}} - \frac{a d^{3}}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.082, size = 382, normalized size = 2.87 \begin{align*} -\frac{100 \, b d^{3} n + 600 \, a d^{3} + 400 \,{\left (b e^{3} n + 3 \, a e^{3}\right )} x^{3} + 675 \,{\left (b d e^{2} n + 4 \, a d e^{2}\right )} x^{2} + 432 \,{\left (b d^{2} e n + 5 \, a d^{2} e\right )} x + 60 \,{\left (20 \, b e^{3} x^{3} + 45 \, b d e^{2} x^{2} + 36 \, b d^{2} e x + 10 \, b d^{3}\right )} \log \left (c\right ) + 60 \,{\left (20 \, b e^{3} n x^{3} + 45 \, b d e^{2} n x^{2} + 36 \, b d^{2} e n x + 10 \, b d^{3} n\right )} \log \left (x\right )}{3600 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 17.4311, size = 231, normalized size = 1.74 \begin{align*} - \frac{a d^{3}}{6 x^{6}} - \frac{3 a d^{2} e}{5 x^{5}} - \frac{3 a d e^{2}}{4 x^{4}} - \frac{a e^{3}}{3 x^{3}} - \frac{b d^{3} n \log{\left (x \right )}}{6 x^{6}} - \frac{b d^{3} n}{36 x^{6}} - \frac{b d^{3} \log{\left (c \right )}}{6 x^{6}} - \frac{3 b d^{2} e n \log{\left (x \right )}}{5 x^{5}} - \frac{3 b d^{2} e n}{25 x^{5}} - \frac{3 b d^{2} e \log{\left (c \right )}}{5 x^{5}} - \frac{3 b d e^{2} n \log{\left (x \right )}}{4 x^{4}} - \frac{3 b d e^{2} n}{16 x^{4}} - \frac{3 b d e^{2} \log{\left (c \right )}}{4 x^{4}} - \frac{b e^{3} n \log{\left (x \right )}}{3 x^{3}} - \frac{b e^{3} n}{9 x^{3}} - \frac{b e^{3} \log{\left (c \right )}}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28759, size = 213, normalized size = 1.6 \begin{align*} -\frac{1200 \, b n x^{3} e^{3} \log \left (x\right ) + 2700 \, b d n x^{2} e^{2} \log \left (x\right ) + 2160 \, b d^{2} n x e \log \left (x\right ) + 400 \, b n x^{3} e^{3} + 675 \, b d n x^{2} e^{2} + 432 \, b d^{2} n x e + 1200 \, b x^{3} e^{3} \log \left (c\right ) + 2700 \, b d x^{2} e^{2} \log \left (c\right ) + 2160 \, b d^{2} x e \log \left (c\right ) + 600 \, b d^{3} n \log \left (x\right ) + 100 \, b d^{3} n + 1200 \, a x^{3} e^{3} + 2700 \, a d x^{2} e^{2} + 2160 \, a d^{2} x e + 600 \, b d^{3} \log \left (c\right ) + 600 \, a d^{3}}{3600 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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