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 = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, 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 2295
Rule 2389
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*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 1.00 \[ a x+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e}-b n x \]
Antiderivative was successfully verified.
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fricas [A] time = 1.43, size = 40, normalized size = 1.38 \[ \frac {b e x \log \relax (c) - {\left (b e n - a e\right )} x + {\left (b e n x + b d n\right )} \log \left (e x + d\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 46, normalized size = 1.59 \[ {\left ({\left (x e + d\right )} n e^{\left (-1\right )} \log \left (x e + d\right ) - {\left (x e + d\right )} n e^{\left (-1\right )} + {\left (x e + d\right )} e^{\left (-1\right )} \log \relax (c)\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 36, normalized size = 1.24 \[ \frac {b d n \ln \left (e x +d \right )}{e}-b n x +b x \ln \left (c \left (e x +d \right )^{n}\right )+a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 40, normalized size = 1.38 \[ -b e n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} + b x \log \left ({\left (e x + d\right )}^{n} c\right ) + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 35, normalized size = 1.21 \[ x\,\left (a-b\,n\right )+b\,x\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )+\frac {b\,d\,n\,\ln \left (d+e\,x\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 42, normalized size = 1.45 \[ a x + b \left (\begin {cases} \frac {d n \log {\left (d + e x \right )}}{e} + n x \log {\left (d + e x \right )} - n x + x \log {\relax (c )} & \text {for}\: e \neq 0 \\x \log {\left (c d^{n} \right )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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