Optimal. Leaf size=33 \[ x \log \left (d \left (b x+c x^2\right )^n\right )+\frac {b n \log (b+c x)}{c}-2 n x \]
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Rubi [A] time = 0.02, antiderivative size = 33, 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 = {2523, 43} \[ x \log \left (d \left (b x+c x^2\right )^n\right )+\frac {b n \log (b+c x)}{c}-2 n x \]
Antiderivative was successfully verified.
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Rule 43
Rule 2523
Rubi steps
\begin {align*} \int \log \left (d \left (b x+c x^2\right )^n\right ) \, dx &=x \log \left (d \left (b x+c x^2\right )^n\right )-n \int \frac {b+2 c x}{b+c x} \, dx\\ &=x \log \left (d \left (b x+c x^2\right )^n\right )-n \int \left (2-\frac {b}{b+c x}\right ) \, dx\\ &=-2 n x+\frac {b n \log (b+c x)}{c}+x \log \left (d \left (b x+c x^2\right )^n\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 0.94 \[ x \log \left (d (x (b+c x))^n\right )+\frac {b n \log (b+c x)}{c}-2 n x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 38, normalized size = 1.15 \[ \frac {c n x \log \left (c x^{2} + b x\right ) - 2 \, c n x + b n \log \left (c x + b\right ) + c x \log \relax (d)}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 37, normalized size = 1.12 \[ n x \log \left (c x^{2} + b x\right ) - {\left (2 \, n - \log \relax (d)\right )} x + \frac {b n \log \left (c x + b\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 34, normalized size = 1.03 \[ \frac {b n \ln \left (c x +b \right )}{c}-2 n x +x \ln \left (d \left (c \,x^{2}+b x \right )^{n}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 36, normalized size = 1.09 \[ -n {\left (2 \, x - \frac {b \log \left (c x + b\right )}{c}\right )} + x \log \left ({\left (c x^{2} + b x\right )}^{n} d\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 33, normalized size = 1.00 \[ x\,\ln \left (d\,{\left (c\,x^2+b\,x\right )}^n\right )-2\,n\,x+\frac {b\,n\,\ln \left (b+c\,x\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.38, size = 56, normalized size = 1.70 \[ \begin {cases} \frac {b n \log {\left (b + c x \right )}}{c} + n x \log {\left (b x + c x^{2} \right )} - 2 n x + x \log {\relax (d )} & \text {for}\: c \neq 0 \\n x \log {\relax (b )} + n x \log {\relax (x )} - n x + x \log {\relax (d )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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