Optimal. Leaf size=29 \[ a x+\frac{b (d+e x) \log \left (c (d+e x)^n\right )}{e}-b n x \]
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Rubi [A] time = 0.0158583, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2389, 2295} \[ a x+\frac{b (d+e x) \log \left (c (d+e x)^n\right )}{e}-b n x \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2295
Rubi steps
\begin{align*} \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx &=a x+b \int \log \left (c (d+e x)^n\right ) \, dx\\ &=a x+\frac{b \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}\\ &=a x-b n x+\frac{b (d+e x) \log \left (c (d+e x)^n\right )}{e}\\ \end{align*}
Mathematica [A] time = 0.0075962, size = 29, normalized size = 1. \[ a x+\frac{b (d+e x) \log \left (c (d+e x)^n\right )}{e}-b n x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 36, normalized size = 1.2 \begin{align*} ax+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) x-bnx+{\frac{\ln \left ( ex+d \right ) bdn}{e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11399, size = 54, normalized size = 1.86 \begin{align*} -b e n{\left (\frac{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} + b x \log \left ({\left (e x + d\right )}^{n} c\right ) + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.15658, size = 93, normalized size = 3.21 \begin{align*} \frac{b e x \log \left (c\right ) -{\left (b e n - a e\right )} x +{\left (b e n x + b d n\right )} \log \left (e x + d\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.546044, size = 42, normalized size = 1.45 \begin{align*} a x + b \left (\begin{cases} \frac{d n \log{\left (d + e x \right )}}{e} + n x \log{\left (d + e x \right )} - n x + x \log{\left (c \right )} & \text{for}\: e \neq 0 \\x \log{\left (c d^{n} \right )} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22442, size = 62, normalized size = 2.14 \begin{align*}{\left ({\left (x e + d\right )} n e^{\left (-1\right )} \log \left (x e + d\right ) -{\left (x e + d\right )} n e^{\left (-1\right )} +{\left (x e + d\right )} e^{\left (-1\right )} \log \left (c\right )\right )} b + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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