Optimal. Leaf size=47 \[ -\frac {n}{x}+\frac {c n \log (x)}{b}-\frac {c n \log (b+c x)}{b}-\frac {\log \left (d \left (b x+c x^2\right )^n\right )}{x} \]
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Rubi [A]
time = 0.03, antiderivative size = 47, normalized size of antiderivative = 1.00, 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 = {2605, 78}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 2605
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*}
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Mathematica [A]
time = 0.01, size = 45, normalized size = 0.96 \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (d \left (c \,x^{2}+b x \right )^{n}\right )}{x^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 46, normalized size = 0.98 \begin {gather*} -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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 46, normalized size = 0.98 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.10, size = 66, normalized size = 1.40 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.82, size = 47, normalized size = 1.00 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.80, size = 43, normalized size = 0.91 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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