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 = 0.0497263, antiderivative size = 72, normalized size of antiderivative = 1., 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 2525
Rule 77
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*}
Mathematica [A] time = 0.0320771, size = 65, normalized size = 0.9 \[ \frac{1}{2} n \left (-\frac{c^2 \log (x)}{b^2}+\frac{c^2 \log (b+c x)}{b^2}-\frac{c}{b x}-\frac{1}{2 x^2}\right )-\frac{\log \left (d (x (b+c x))^n\right )}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( d \left ( c{x}^{2}+bx \right ) ^{n} \right ) }{{x}^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03116, size = 84, normalized size = 1.17 \begin{align*} \frac{1}{4} \, n{\left (\frac{2 \, c^{2} \log \left (c x + b\right )}{b^{2}} - \frac{2 \, c^{2} \log \left (x\right )}{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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8428, size = 169, normalized size = 2.35 \begin{align*} \frac{2 \, c^{2} n x^{2} \log \left (c x + b\right ) - 2 \, c^{2} n x^{2} \log \left (x\right ) - 2 \, b c n x - 2 \, b^{2} n \log \left (c x^{2} + b x\right ) - b^{2} n - 2 \, b^{2} \log \left (d\right )}{4 \, b^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.9759, size = 110, normalized size = 1.53 \begin{align*} \begin{cases} - \frac{n \log{\left (b x + c x^{2} \right )}}{2 x^{2}} - \frac{n}{4 x^{2}} - \frac{\log{\left (d \right )}}{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{\left (c \right )}}{2 x^{2}} - \frac{n \log{\left (x \right )}}{x^{2}} - \frac{n}{2 x^{2}} - \frac{\log{\left (d \right )}}{2 x^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26964, size = 88, normalized size = 1.22 \begin{align*} \frac{c^{2} n \log \left (c x + b\right )}{2 \, b^{2}} - \frac{c^{2} n \log \left (x\right )}{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 \left (d\right )}{4 \, b x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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