Optimal. Leaf size=57 \[ -\frac{b^2 n \log (b+c x)}{2 c^2}+\frac{1}{2} x^2 \log \left (d \left (b x+c x^2\right )^n\right )+\frac{b n x}{2 c}-\frac{n x^2}{2} \]
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Rubi [A] time = 0.040766, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2525, 77} \[ -\frac{b^2 n \log (b+c x)}{2 c^2}+\frac{1}{2} x^2 \log \left (d \left (b x+c x^2\right )^n\right )+\frac{b n x}{2 c}-\frac{n x^2}{2} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 77
Rubi steps
\begin{align*} \int x \log \left (d \left (b x+c x^2\right )^n\right ) \, dx &=\frac{1}{2} x^2 \log \left (d \left (b x+c x^2\right )^n\right )-\frac{1}{2} n \int \frac{x (b+2 c x)}{b+c x} \, dx\\ &=\frac{1}{2} x^2 \log \left (d \left (b x+c x^2\right )^n\right )-\frac{1}{2} n \int \left (-\frac{b}{c}+2 x+\frac{b^2}{c (b+c x)}\right ) \, dx\\ &=\frac{b n x}{2 c}-\frac{n x^2}{2}-\frac{b^2 n \log (b+c x)}{2 c^2}+\frac{1}{2} x^2 \log \left (d \left (b x+c x^2\right )^n\right )\\ \end{align*}
Mathematica [A] time = 0.0202679, size = 49, normalized size = 0.86 \[ \frac{1}{2} x^2 \log \left (d (x (b+c x))^n\right )-\frac{1}{2} n \left (\frac{b^2 \log (b+c x)}{c^2}-\frac{b x}{c}+x^2\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.03, size = 0, normalized size = 0. \begin{align*} \int x\ln \left ( d \left ( c{x}^{2}+bx \right ) ^{n} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01231, size = 69, normalized size = 1.21 \begin{align*} \frac{1}{2} \, x^{2} \log \left ({\left (c x^{2} + b x\right )}^{n} d\right ) - \frac{1}{2} \, n{\left (\frac{b^{2} \log \left (c x + b\right )}{c^{2}} + \frac{c x^{2} - b x}{c}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55161, size = 132, normalized size = 2.32 \begin{align*} \frac{c^{2} n x^{2} \log \left (c x^{2} + b x\right ) - c^{2} n x^{2} + c^{2} x^{2} \log \left (d\right ) + b c n x - b^{2} n \log \left (c x + b\right )}{2 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.34266, size = 92, normalized size = 1.61 \begin{align*} \begin{cases} - \frac{b^{2} n \log{\left (b + c x \right )}}{2 c^{2}} + \frac{b n x}{2 c} + \frac{n x^{2} \log{\left (b x + c x^{2} \right )}}{2} - \frac{n x^{2}}{2} + \frac{x^{2} \log{\left (d \right )}}{2} & \text{for}\: c \neq 0 \\\frac{n x^{2} \log{\left (b \right )}}{2} + \frac{n x^{2} \log{\left (x \right )}}{2} - \frac{n x^{2}}{4} + \frac{x^{2} \log{\left (d \right )}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21367, size = 69, normalized size = 1.21 \begin{align*} \frac{1}{2} \, n x^{2} \log \left (c x^{2} + b x\right ) - \frac{1}{2} \,{\left (n - \log \left (d\right )\right )} x^{2} + \frac{b n x}{2 \, c} - \frac{b^{2} n \log \left (c x + b\right )}{2 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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