Optimal. Leaf size=48 \[ -\frac{d \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-\frac{b d n}{x} \]
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Rubi [A] time = 0.0497989, antiderivative size = 43, normalized size of antiderivative = 0.9, 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 = {43, 2334, 14, 2301} \[ -\left (\frac{d}{x}-e \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b d n}{x}-\frac{1}{2} b e n \log ^2(x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 2334
Rule 14
Rule 2301
Rubi steps
\begin{align*} \int \frac{(d+e x) \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx &=-\left (\frac{d}{x}-e \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{-d+e x \log (x)}{x^2} \, dx\\ &=-\left (\frac{d}{x}-e \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac{d}{x^2}+\frac{e \log (x)}{x}\right ) \, dx\\ &=-\frac{b d n}{x}-\left (\frac{d}{x}-e \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-(b e n) \int \frac{\log (x)}{x} \, dx\\ &=-\frac{b d n}{x}-\frac{1}{2} b e n \log ^2(x)-\left (\frac{d}{x}-e \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0254346, size = 48, normalized size = 1. \[ -\frac{d \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-\frac{b d n}{x} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.139, size = 250, normalized size = 5.2 \begin{align*} -{\frac{b \left ( -ex\ln \left ( x \right ) +d \right ) \ln \left ({x}^{n} \right ) }{x}}-{\frac{-i\ln \left ( x \right ) \pi \,be{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}x+i\ln \left ( x \right ) \pi \,be{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) x+i\ln \left ( x \right ) \pi \,be \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}x-i\ln \left ( x \right ) \pi \,be \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) x+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 ) +enb \left ( \ln \left ( x \right ) \right ) ^{2}x-2\,\ln \left ( x \right ) \ln \left ( c \right ) bex-2\,\ln \left ( x \right ) aex+2\,\ln \left ( c \right ) bd+2\,bdn+2\,ad}{2\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15116, size = 66, normalized size = 1.38 \begin{align*} \frac{b e \log \left (c x^{n}\right )^{2}}{2 \, n} + a e \log \left (x\right ) - \frac{b d n}{x} - \frac{b d \log \left (c x^{n}\right )}{x} - \frac{a d}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.02131, size = 136, normalized size = 2.83 \begin{align*} \frac{b e n x \log \left (x\right )^{2} - 2 \, b d n - 2 \, b d \log \left (c\right ) - 2 \, a d + 2 \,{\left (b e x \log \left (c\right ) - b d n + a e x\right )} \log \left (x\right )}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 16.6151, size = 53, normalized size = 1.1 \begin{align*} - \frac{a d}{x} + a e \log{\left (x \right )} + b d \left (- \frac{n}{x} - \frac{\log{\left (c x^{n} \right )}}{x}\right ) - b e \left (\begin{cases} - \log{\left (c \right )} \log{\left (x \right )} & \text{for}\: n = 0 \\- \frac{\log{\left (c x^{n} \right )}^{2}}{2 n} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26003, size = 76, normalized size = 1.58 \begin{align*} \frac{b n x e \log \left (x\right )^{2} + 2 \, b x e \log \left (c\right ) \log \left (x\right ) - 2 \, b d n \log \left (x\right ) + 2 \, a x e \log \left (x\right ) - 2 \, b d n - 2 \, b d \log \left (c\right ) - 2 \, a d}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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