Optimal. Leaf size=90 \[ -\frac{(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 d x^4}-\frac{b d^2 e n}{3 x^3}-\frac{b d^3 n}{16 x^4}-\frac{3 b d e^2 n}{4 x^2}+\frac{b e^4 n \log (x)}{4 d}-\frac{b e^3 n}{x} \]
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Rubi [A] time = 0.082418, antiderivative size = 90, normalized size of antiderivative = 1., 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 = {37, 2334, 12, 43} \[ -\frac{(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 d x^4}-\frac{b d^2 e n}{3 x^3}-\frac{b d^3 n}{16 x^4}-\frac{3 b d e^2 n}{4 x^2}+\frac{b e^4 n \log (x)}{4 d}-\frac{b e^3 n}{x} \]
Antiderivative was successfully verified.
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Rule 37
Rule 2334
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^5} \, dx &=-\frac{(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 d x^4}-(b n) \int -\frac{(d+e x)^4}{4 d x^5} \, dx\\ &=-\frac{(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 d x^4}+\frac{(b n) \int \frac{(d+e x)^4}{x^5} \, dx}{4 d}\\ &=-\frac{(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 d x^4}+\frac{(b n) \int \left (\frac{d^4}{x^5}+\frac{4 d^3 e}{x^4}+\frac{6 d^2 e^2}{x^3}+\frac{4 d e^3}{x^2}+\frac{e^4}{x}\right ) \, dx}{4 d}\\ &=-\frac{b d^3 n}{16 x^4}-\frac{b d^2 e n}{3 x^3}-\frac{3 b d e^2 n}{4 x^2}-\frac{b e^3 n}{x}+\frac{b e^4 n \log (x)}{4 d}-\frac{(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 d x^4}\\ \end{align*}
Mathematica [A] time = 0.0519544, size = 109, normalized size = 1.21 \[ -\frac{12 a \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )+12 b \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right ) \log \left (c x^n\right )+b n \left (16 d^2 e x+3 d^3+36 d e^2 x^2+48 e^3 x^3\right )}{48 x^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.14, size = 569, normalized size = 6.3 \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 = 1.13851, size = 193, normalized size = 2.14 \begin{align*} -\frac{b e^{3} n}{x} - \frac{b e^{3} \log \left (c x^{n}\right )}{x} - \frac{3 \, b d e^{2} n}{4 \, x^{2}} - \frac{a e^{3}}{x} - \frac{3 \, b d e^{2} \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac{b d^{2} e n}{3 \, x^{3}} - \frac{3 \, a d e^{2}}{2 \, x^{2}} - \frac{b d^{2} e \log \left (c x^{n}\right )}{x^{3}} - \frac{b d^{3} n}{16 \, x^{4}} - \frac{a d^{2} e}{x^{3}} - \frac{b d^{3} \log \left (c x^{n}\right )}{4 \, x^{4}} - \frac{a d^{3}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.0451, size = 352, normalized size = 3.91 \begin{align*} -\frac{3 \, b d^{3} n + 12 \, a d^{3} + 48 \,{\left (b e^{3} n + a e^{3}\right )} x^{3} + 36 \,{\left (b d e^{2} n + 2 \, a d e^{2}\right )} x^{2} + 16 \,{\left (b d^{2} e n + 3 \, a d^{2} e\right )} x + 12 \,{\left (4 \, b e^{3} x^{3} + 6 \, b d e^{2} x^{2} + 4 \, b d^{2} e x + b d^{3}\right )} \log \left (c\right ) + 12 \,{\left (4 \, b e^{3} n x^{3} + 6 \, b d e^{2} n x^{2} + 4 \, b d^{2} e n x + b d^{3} n\right )} \log \left (x\right )}{48 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 7.80881, size = 206, normalized size = 2.29 \begin{align*} - \frac{a d^{3}}{4 x^{4}} - \frac{a d^{2} e}{x^{3}} - \frac{3 a d e^{2}}{2 x^{2}} - \frac{a e^{3}}{x} - \frac{b d^{3} n \log{\left (x \right )}}{4 x^{4}} - \frac{b d^{3} n}{16 x^{4}} - \frac{b d^{3} \log{\left (c \right )}}{4 x^{4}} - \frac{b d^{2} e n \log{\left (x \right )}}{x^{3}} - \frac{b d^{2} e n}{3 x^{3}} - \frac{b d^{2} e \log{\left (c \right )}}{x^{3}} - \frac{3 b d e^{2} n \log{\left (x \right )}}{2 x^{2}} - \frac{3 b d e^{2} n}{4 x^{2}} - \frac{3 b d e^{2} \log{\left (c \right )}}{2 x^{2}} - \frac{b e^{3} n \log{\left (x \right )}}{x} - \frac{b e^{3} n}{x} - \frac{b e^{3} \log{\left (c \right )}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35221, size = 213, normalized size = 2.37 \begin{align*} -\frac{48 \, b n x^{3} e^{3} \log \left (x\right ) + 72 \, b d n x^{2} e^{2} \log \left (x\right ) + 48 \, b d^{2} n x e \log \left (x\right ) + 48 \, b n x^{3} e^{3} + 36 \, b d n x^{2} e^{2} + 16 \, b d^{2} n x e + 48 \, b x^{3} e^{3} \log \left (c\right ) + 72 \, b d x^{2} e^{2} \log \left (c\right ) + 48 \, b d^{2} x e \log \left (c\right ) + 12 \, b d^{3} n \log \left (x\right ) + 3 \, b d^{3} n + 48 \, a x^{3} e^{3} + 72 \, a d x^{2} e^{2} + 48 \, a d^{2} x e + 12 \, b d^{3} \log \left (c\right ) + 12 \, a d^{3}}{48 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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