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 = 0.0163603, antiderivative size = 33, 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 = {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 2523
Rule 43
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*}
Mathematica [A] time = 0.006888, 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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Maple [A] time = 0.006, size = 34, normalized size = 1. \begin{align*} -2\,nx+{\frac{bn\ln \left ( cx+b \right ) }{c}}+x\ln \left ( d \left ( c{x}^{2}+bx \right ) ^{n} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15723, size = 49, normalized size = 1.48 \begin{align*} -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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52262, size = 95, normalized size = 2.88 \begin{align*} \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 \left (d\right )}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.21605, size = 56, normalized size = 1.7 \begin{align*} \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{\left (d \right )} & \text{for}\: c \neq 0 \\n x \log{\left (b \right )} + n x \log{\left (x \right )} - n x + x \log{\left (d \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22886, size = 50, normalized size = 1.52 \begin{align*} n x \log \left (c x^{2} + b x\right ) -{\left (2 \, n - \log \left (d\right )\right )} x + \frac{b n \log \left (c x + b\right )}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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