Optimal. Leaf size=53 \[ -n \text{PolyLog}\left (2,-\frac{c x}{b}\right )+\log (x) \log \left (d \left (b x+c x^2\right )^n\right )-n \log (x) \log \left (\frac{c x}{b}+1\right )-\frac{1}{2} n \log ^2(x) \]
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Rubi [A] time = 0.125706, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2524, 1593, 2357, 2301, 2317, 2391} \[ -n \text{PolyLog}\left (2,-\frac{c x}{b}\right )+\log (x) \log \left (d \left (b x+c x^2\right )^n\right )-n \log (x) \log \left (\frac{c x}{b}+1\right )-\frac{1}{2} n \log ^2(x) \]
Antiderivative was successfully verified.
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Rule 2524
Rule 1593
Rule 2357
Rule 2301
Rule 2317
Rule 2391
Rubi steps
\begin{align*} \int \frac{\log \left (d \left (b x+c x^2\right )^n\right )}{x} \, dx &=\log (x) \log \left (d \left (b x+c x^2\right )^n\right )-n \int \frac{(b+2 c x) \log (x)}{b x+c x^2} \, dx\\ &=\log (x) \log \left (d \left (b x+c x^2\right )^n\right )-n \int \frac{(b+2 c x) \log (x)}{x (b+c x)} \, dx\\ &=\log (x) \log \left (d \left (b x+c x^2\right )^n\right )-n \int \left (\frac{\log (x)}{x}+\frac{c \log (x)}{b+c x}\right ) \, dx\\ &=\log (x) \log \left (d \left (b x+c x^2\right )^n\right )-n \int \frac{\log (x)}{x} \, dx-(c n) \int \frac{\log (x)}{b+c x} \, dx\\ &=-\frac{1}{2} n \log ^2(x)-n \log (x) \log \left (1+\frac{c x}{b}\right )+\log (x) \log \left (d \left (b x+c x^2\right )^n\right )+n \int \frac{\log \left (1+\frac{c x}{b}\right )}{x} \, dx\\ &=-\frac{1}{2} n \log ^2(x)-n \log (x) \log \left (1+\frac{c x}{b}\right )+\log (x) \log \left (d \left (b x+c x^2\right )^n\right )-n \text{Li}_2\left (-\frac{c x}{b}\right )\\ \end{align*}
Mathematica [A] time = 0.0203252, size = 50, normalized size = 0.94 \[ \log (x) \log \left (d (x (b+c x))^n\right )-n \left (\text{PolyLog}\left (2,-\frac{c x}{b}\right )+\log (x) \log \left (\frac{b+c x}{b}\right )+\frac{\log ^2(x)}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( d \left ( c{x}^{2}+bx \right ) ^{n} \right ) }{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0547, size = 108, normalized size = 2.04 \begin{align*} -n \log \left (c x^{2} + b x\right ) \log \left (x\right ) + \frac{1}{2} \,{\left (2 \, \log \left (c x^{2} + b x\right ) \log \left (x\right ) - 2 \, \log \left (\frac{c x}{b} + 1\right ) \log \left (x\right ) - \log \left (x\right )^{2} - 2 \,{\rm Li}_2\left (-\frac{c x}{b}\right )\right )} n + \log \left ({\left (c x^{2} + b x\right )}^{n} d\right ) \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left ({\left (c x^{2} + b x\right )}^{n} d\right )}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (d \left (b x + c x^{2}\right )^{n} \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (c x^{2} + b x\right )}^{n} d\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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