Optimal. Leaf size=72 \[ -\frac {c^2 n \log (x)}{2 b^2}+\frac {c^2 n \log (b+c x)}{2 b^2}-\frac {\log \left (d \left (b x+c x^2\right )^n\right )}{2 x^2}-\frac {c n}{2 b x}-\frac {n}{4 x^2} \]
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Rubi [A] time = 0.05, antiderivative size = 72, 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 = {2525, 77} \[ -\frac {c^2 n \log (x)}{2 b^2}+\frac {c^2 n \log (b+c x)}{2 b^2}-\frac {\log \left (d \left (b x+c x^2\right )^n\right )}{2 x^2}-\frac {c n}{2 b x}-\frac {n}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 77
Rule 2525
Rubi steps
\begin {align*} \int \frac {\log \left (d \left (b x+c x^2\right )^n\right )}{x^3} \, dx &=-\frac {\log \left (d \left (b x+c x^2\right )^n\right )}{2 x^2}+\frac {1}{2} n \int \frac {b+2 c x}{x^3 (b+c x)} \, dx\\ &=-\frac {\log \left (d \left (b x+c x^2\right )^n\right )}{2 x^2}+\frac {1}{2} n \int \left (\frac {1}{x^3}+\frac {c}{b x^2}-\frac {c^2}{b^2 x}+\frac {c^3}{b^2 (b+c x)}\right ) \, dx\\ &=-\frac {n}{4 x^2}-\frac {c n}{2 b x}-\frac {c^2 n \log (x)}{2 b^2}+\frac {c^2 n \log (b+c x)}{2 b^2}-\frac {\log \left (d \left (b x+c x^2\right )^n\right )}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 65, normalized size = 0.90 \[ \frac {1}{2} n \left (-\frac {c^2 \log (x)}{b^2}+\frac {c^2 \log (b+c x)}{b^2}-\frac {c}{b x}-\frac {1}{2 x^2}\right )-\frac {\log \left (d (x (b+c x))^n\right )}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 70, normalized size = 0.97 \[ \frac {2 \, c^{2} n x^{2} \log \left (c x + b\right ) - 2 \, c^{2} n x^{2} \log \relax (x) - 2 \, b c n x - 2 \, b^{2} n \log \left (c x^{2} + b x\right ) - b^{2} n - 2 \, b^{2} \log \relax (d)}{4 \, b^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 65, normalized size = 0.90 \[ \frac {c^{2} n \log \left (c x + b\right )}{2 \, b^{2}} - \frac {c^{2} n \log \relax (x)}{2 \, b^{2}} - \frac {n \log \left (c x^{2} + b x\right )}{2 \, x^{2}} - \frac {2 \, c n x + b n + 2 \, b \log \relax (d)}{4 \, b x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (d \left (c \,x^{2}+b x \right )^{n}\right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 62, normalized size = 0.86 \[ \frac {1}{4} \, n {\left (\frac {2 \, c^{2} \log \left (c x + b\right )}{b^{2}} - \frac {2 \, c^{2} \log \relax (x)}{b^{2}} - \frac {2 \, c x + b}{b x^{2}}\right )} - \frac {\log \left ({\left (c x^{2} + b x\right )}^{n} d\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.48, size = 54, normalized size = 0.75 \[ \frac {c^2\,n\,\mathrm {atanh}\left (\frac {2\,c\,x}{b}+1\right )}{b^2}-\frac {\frac {n}{2}+\frac {c\,n\,x}{b}}{2\,x^2}-\frac {\ln \left (d\,{\left (c\,x^2+b\,x\right )}^n\right )}{2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.95, size = 110, normalized size = 1.53 \[ \begin {cases} - \frac {n \log {\left (b x + c x^{2} \right )}}{2 x^{2}} - \frac {n}{4 x^{2}} - \frac {\log {\relax (d )}}{2 x^{2}} - \frac {c n}{2 b x} + \frac {c^{2} n \log {\left (b + c x \right )}}{b^{2}} - \frac {c^{2} n \log {\left (b x + c x^{2} \right )}}{2 b^{2}} & \text {for}\: b \neq 0 \\- \frac {n \log {\relax (c )}}{2 x^{2}} - \frac {n \log {\relax (x )}}{x^{2}} - \frac {n}{2 x^{2}} - \frac {\log {\relax (d )}}{2 x^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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