Optimal. Leaf size=60 \[ -\frac{(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}+\frac{b e^2 n \log (x)}{2 d}-\frac{b d n}{4 x^2}-\frac{b e n}{x} \]
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Rubi [A] time = 0.0492517, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {37, 2334, 12, 43} \[ -\frac{(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}+\frac{b e^2 n \log (x)}{2 d}-\frac{b d n}{4 x^2}-\frac{b e n}{x} \]
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) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx &=-\frac{(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}-(b n) \int -\frac{(d+e x)^2}{2 d x^3} \, dx\\ &=-\frac{(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}+\frac{(b n) \int \frac{(d+e x)^2}{x^3} \, dx}{2 d}\\ &=-\frac{(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}+\frac{(b n) \int \left (\frac{d^2}{x^3}+\frac{2 d e}{x^2}+\frac{e^2}{x}\right ) \, dx}{2 d}\\ &=-\frac{b d n}{4 x^2}-\frac{b e n}{x}+\frac{b e^2 n \log (x)}{2 d}-\frac{(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}\\ \end{align*}
Mathematica [A] time = 0.0227524, size = 41, normalized size = 0.68 \[ -\frac{2 a (d+2 e x)+2 b (d+2 e x) \log \left (c x^n\right )+b n (d+4 e x)}{4 x^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.097, size = 232, normalized size = 3.9 \begin{align*} -{\frac{b \left ( 2\,ex+d \right ) \ln \left ({x}^{n} \right ) }{2\,{x}^{2}}}-{\frac{2\,i\pi \,bex{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-2\,i\pi \,bex{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -2\,i\pi \,bex \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,i\pi \,bex \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +4\,\ln \left ( c \right ) bex+4\,benx+4\,aex+i\pi \,bd{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,bd{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -i\pi \,bd \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \,bd \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +2\,\ln \left ( c \right ) bd+bdn+2\,ad}{4\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1398, size = 77, normalized size = 1.28 \begin{align*} -\frac{b e n}{x} - \frac{b e \log \left (c x^{n}\right )}{x} - \frac{b d n}{4 \, x^{2}} - \frac{a e}{x} - \frac{b d \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac{a d}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.975369, size = 140, normalized size = 2.33 \begin{align*} -\frac{b d n + 2 \, a d + 4 \,{\left (b e n + a e\right )} x + 2 \,{\left (2 \, b e x + b d\right )} \log \left (c\right ) + 2 \,{\left (2 \, b e n x + b d n\right )} \log \left (x\right )}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.74062, size = 75, normalized size = 1.25 \begin{align*} - \frac{a d}{2 x^{2}} - \frac{a e}{x} - \frac{b d n \log{\left (x \right )}}{2 x^{2}} - \frac{b d n}{4 x^{2}} - \frac{b d \log{\left (c \right )}}{2 x^{2}} - \frac{b e n \log{\left (x \right )}}{x} - \frac{b e n}{x} - \frac{b e \log{\left (c \right )}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31876, size = 77, normalized size = 1.28 \begin{align*} -\frac{4 \, b n x e \log \left (x\right ) + 4 \, b n x e + 4 \, b x e \log \left (c\right ) + 2 \, b d n \log \left (x\right ) + b d n + 4 \, a x e + 2 \, b d \log \left (c\right ) + 2 \, a d}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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