Optimal. Leaf size=71 \[ -\frac{b^2 n x}{3 c^2}+\frac{b^3 n \log (b+c x)}{3 c^3}+\frac{1}{3} x^3 \log \left (d \left (b x+c x^2\right )^n\right )+\frac{b n x^2}{6 c}-\frac{2 n x^3}{9} \]
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Rubi [A] time = 0.0526, antiderivative size = 71, 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} \[ -\frac{b^2 n x}{3 c^2}+\frac{b^3 n \log (b+c x)}{3 c^3}+\frac{1}{3} x^3 \log \left (d \left (b x+c x^2\right )^n\right )+\frac{b n x^2}{6 c}-\frac{2 n x^3}{9} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 77
Rubi steps
\begin{align*} \int x^2 \log \left (d \left (b x+c x^2\right )^n\right ) \, dx &=\frac{1}{3} x^3 \log \left (d \left (b x+c x^2\right )^n\right )-\frac{1}{3} n \int \frac{x^2 (b+2 c x)}{b+c x} \, dx\\ &=\frac{1}{3} x^3 \log \left (d \left (b x+c x^2\right )^n\right )-\frac{1}{3} n \int \left (\frac{b^2}{c^2}-\frac{b x}{c}+2 x^2-\frac{b^3}{c^2 (b+c x)}\right ) \, dx\\ &=-\frac{b^2 n x}{3 c^2}+\frac{b n x^2}{6 c}-\frac{2 n x^3}{9}+\frac{b^3 n \log (b+c x)}{3 c^3}+\frac{1}{3} x^3 \log \left (d \left (b x+c x^2\right )^n\right )\\ \end{align*}
Mathematica [A] time = 0.0288719, size = 63, normalized size = 0.89 \[ \frac{c n x \left (-6 b^2+3 b c x-4 c^2 x^2\right )+6 b^3 n \log (b+c x)+6 c^3 x^3 \log \left (d (x (b+c x))^n\right )}{18 c^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}\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 = 1.08318, size = 88, normalized size = 1.24 \begin{align*} \frac{1}{3} \, x^{3} \log \left ({\left (c x^{2} + b x\right )}^{n} d\right ) + \frac{1}{18} \, n{\left (\frac{6 \, b^{3} \log \left (c x + b\right )}{c^{3}} - \frac{4 \, c^{2} x^{3} - 3 \, b c x^{2} + 6 \, b^{2} x}{c^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50534, size = 171, normalized size = 2.41 \begin{align*} \frac{6 \, c^{3} n x^{3} \log \left (c x^{2} + b x\right ) - 4 \, c^{3} n x^{3} + 6 \, c^{3} x^{3} \log \left (d\right ) + 3 \, b c^{2} n x^{2} - 6 \, b^{2} c n x + 6 \, b^{3} n \log \left (c x + b\right )}{18 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.21884, size = 107, normalized size = 1.51 \begin{align*} \begin{cases} \frac{b^{3} n \log{\left (b + c x \right )}}{3 c^{3}} - \frac{b^{2} n x}{3 c^{2}} + \frac{b n x^{2}}{6 c} + \frac{n x^{3} \log{\left (b x + c x^{2} \right )}}{3} - \frac{2 n x^{3}}{9} + \frac{x^{3} \log{\left (d \right )}}{3} & \text{for}\: c \neq 0 \\\frac{n x^{3} \log{\left (b \right )}}{3} + \frac{n x^{3} \log{\left (x \right )}}{3} - \frac{n x^{3}}{9} + \frac{x^{3} \log{\left (d \right )}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1419, size = 88, normalized size = 1.24 \begin{align*} \frac{1}{3} \, n x^{3} \log \left (c x^{2} + b x\right ) - \frac{1}{9} \,{\left (2 \, n - 3 \, \log \left (d\right )\right )} x^{3} + \frac{b n x^{2}}{6 \, c} - \frac{b^{2} n x}{3 \, c^{2}} + \frac{b^{3} n \log \left (c x + b\right )}{3 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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