Optimal. Leaf size=47 \[ -\frac{\log \left (d \left (b x+c x^2\right )^n\right )}{x}+\frac{c n \log (x)}{b}-\frac{c n \log (b+c x)}{b}-\frac{n}{x} \]
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Rubi [A] time = 0.0438773, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2525, 77} \[ -\frac{\log \left (d \left (b x+c x^2\right )^n\right )}{x}+\frac{c n \log (x)}{b}-\frac{c n \log (b+c x)}{b}-\frac{n}{x} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 77
Rubi steps
\begin{align*} \int \frac{\log \left (d \left (b x+c x^2\right )^n\right )}{x^2} \, dx &=-\frac{\log \left (d \left (b x+c x^2\right )^n\right )}{x}+n \int \frac{b+2 c x}{x^2 (b+c x)} \, dx\\ &=-\frac{\log \left (d \left (b x+c x^2\right )^n\right )}{x}+n \int \left (\frac{1}{x^2}+\frac{c}{b x}-\frac{c^2}{b (b+c x)}\right ) \, dx\\ &=-\frac{n}{x}+\frac{c n \log (x)}{b}-\frac{c n \log (b+c x)}{b}-\frac{\log \left (d \left (b x+c x^2\right )^n\right )}{x}\\ \end{align*}
Mathematica [A] time = 0.0120372, size = 45, normalized size = 0.96 \[ -\frac{\log \left (d (x (b+c x))^n\right )}{x}+\frac{c n \log (x)}{b}-\frac{c n \log (b+c x)}{b}-\frac{n}{x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.016, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( d \left ( c{x}^{2}+bx \right ) ^{n} \right ) }{{x}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00471, size = 62, normalized size = 1.32 \begin{align*} -n{\left (\frac{c \log \left (c x + b\right )}{b} - \frac{c \log \left (x\right )}{b} + \frac{1}{x}\right )} - \frac{\log \left ({\left (c x^{2} + b x\right )}^{n} d\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92155, size = 113, normalized size = 2.4 \begin{align*} -\frac{c n x \log \left (c x + b\right ) - c n x \log \left (x\right ) + b n \log \left (c x^{2} + b x\right ) + b n + b \log \left (d\right )}{b x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.67932, size = 76, normalized size = 1.62 \begin{align*} \begin{cases} - \frac{n \log{\left (b x + c x^{2} \right )}}{x} - \frac{n}{x} - \frac{\log{\left (d \right )}}{x} - \frac{2 c n \log{\left (b + c x \right )}}{b} + \frac{c n \log{\left (b x + c x^{2} \right )}}{b} & \text{for}\: b \neq 0 \\- \frac{n \log{\left (c \right )}}{x} - \frac{2 n \log{\left (x \right )}}{x} - \frac{2 n}{x} - \frac{\log{\left (d \right )}}{x} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30503, size = 63, normalized size = 1.34 \begin{align*} -\frac{c n \log \left (c x + b\right )}{b} + \frac{c n \log \left (x\right )}{b} - \frac{n \log \left (c x^{2} + b x\right )}{x} - \frac{n + \log \left (d\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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