Optimal. Leaf size=53 \[ \log (x) \log \left (d \left (b x+c x^2\right )^n\right )-n \text {Li}_2\left (-\frac {c x}{b}\right )-n \log (x) \log \left (\frac {c x}{b}+1\right )-\frac {1}{2} n \log ^2(x) \]
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Rubi [A] time = 0.13, antiderivative size = 53, normalized size of antiderivative = 1.00, 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 1593
Rule 2301
Rule 2317
Rule 2357
Rule 2391
Rule 2524
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*}
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Mathematica [A] time = 0.02, size = 50, normalized size = 0.94 \[ \log (x) \log \left (d (x (b+c x))^n\right )-n \left (\text {Li}_2\left (-\frac {c x}{b}\right )+\log (x) \log \left (\frac {b+c x}{b}\right )+\frac {\log ^2(x)}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (c x^{2} + b x\right )}^{n} d\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (c x^{2} + b x\right )}^{n} d\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (d \left (c \,x^{2}+b x \right )^{n}\right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 80, normalized size = 1.51 \[ -n \log \left (c x^{2} + b x\right ) \log \relax (x) + \frac {1}{2} \, {\left (2 \, \log \left (c x^{2} + b x\right ) \log \relax (x) - 2 \, \log \left (\frac {c x}{b} + 1\right ) \log \relax (x) - \log \relax (x)^{2} - 2 \, {\rm Li}_2\left (-\frac {c x}{b}\right )\right )} n + \log \left ({\left (c x^{2} + b x\right )}^{n} d\right ) \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\ln \left (d\,{\left (c\,x^2+b\,x\right )}^n\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (d \left (b x + c x^{2}\right )^{n} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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