Integrand size = 18, antiderivative size = 47 \[ \int \frac {\log \left (d \left (b x+c x^2\right )^n\right )}{x^2} \, dx=-\frac {n}{x}+\frac {c n \log (x)}{b}-\frac {c n \log (b+c x)}{b}-\frac {\log \left (d \left (b x+c x^2\right )^n\right )}{x} \]
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Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2605, 78} \[ \int \frac {\log \left (d \left (b x+c x^2\right )^n\right )}{x^2} \, dx=-\frac {\log \left (d \left (b x+c x^2\right )^n\right )}{x}+\frac {c n \log (x)}{b}-\frac {c n \log (b+c x)}{b}-\frac {n}{x} \]
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Rule 78
Rule 2605
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.96 \[ \int \frac {\log \left (d \left (b x+c x^2\right )^n\right )}{x^2} \, dx=-\frac {n}{x}+\frac {c n \log (x)}{b}-\frac {c n \log (b+c x)}{b}-\frac {\log \left (d (x (b+c x))^n\right )}{x} \]
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Time = 0.12 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.02
method | result | size |
parts | \(-\frac {\ln \left (d \left (c \,x^{2}+b x \right )^{n}\right )}{x}+n \left (-\frac {1}{x}+\frac {c \ln \left (x \right )}{b}-\frac {c \ln \left (x c +b \right )}{b}\right )\) | \(48\) |
parallelrisch | \(\frac {2 \ln \left (x \right ) x \,c^{2} n^{2}-x \ln \left (d \left (x \left (x c +b \right )\right )^{n}\right ) c^{2} n -\ln \left (d \left (x \left (x c +b \right )\right )^{n}\right ) b c n -b c \,n^{2}}{x b c n}\) | \(69\) |
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Time = 0.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.98 \[ \int \frac {\log \left (d \left (b x+c x^2\right )^n\right )}{x^2} \, dx=-\frac {c n x \log \left (c x + b\right ) - c n x \log \left (x\right ) + b n \log \left (c x^{2} + b x\right ) + b n + b \log \left (d\right )}{b x} \]
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Time = 0.98 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.40 \[ \int \frac {\log \left (d \left (b x+c x^2\right )^n\right )}{x^2} \, dx=\begin {cases} - \frac {n}{x} - \frac {\log {\left (d \left (b x + c x^{2}\right )^{n} \right )}}{x} - \frac {2 c n \log {\left (b + c x \right )}}{b} + \frac {c \log {\left (d \left (b x + c x^{2}\right )^{n} \right )}}{b} & \text {for}\: b \neq 0 \\- \frac {2 n}{x} - \frac {\log {\left (d \left (c x^{2}\right )^{n} \right )}}{x} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.98 \[ \int \frac {\log \left (d \left (b x+c x^2\right )^n\right )}{x^2} \, dx=-n {\left (\frac {c \log \left (c x + b\right )}{b} - \frac {c \log \left (x\right )}{b} + \frac {1}{x}\right )} - \frac {\log \left ({\left (c x^{2} + b x\right )}^{n} d\right )}{x} \]
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Time = 0.33 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (d \left (b x+c x^2\right )^n\right )}{x^2} \, dx=-\frac {c n \log \left (c x + b\right )}{b} + \frac {c n \log \left (x\right )}{b} - \frac {n \log \left (c x^{2} + b x\right )}{x} - \frac {n + \log \left (d\right )}{x} \]
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Time = 1.98 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.91 \[ \int \frac {\log \left (d \left (b x+c x^2\right )^n\right )}{x^2} \, dx=-\frac {\ln \left (d\,{\left (c\,x^2+b\,x\right )}^n\right )}{x}-\frac {n}{x}-\frac {2\,c\,n\,\mathrm {atanh}\left (\frac {2\,c\,x}{b}+1\right )}{b} \]
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