Optimal. Leaf size=136 \[ \frac{b n \left (b^2-3 a c\right ) \log \left (a+b x+c x^2\right )}{6 c^3}-\frac{n x \left (b^2-2 a c\right )}{3 c^2}+\frac{n \sqrt{b^2-4 a c} \left (b^2-a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{3 c^3}+\frac{1}{3} x^3 \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac{b n x^2}{6 c}-\frac{2 n x^3}{9} \]
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Rubi [A] time = 0.149037, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {2525, 800, 634, 618, 206, 628} \[ \frac{b n \left (b^2-3 a c\right ) \log \left (a+b x+c x^2\right )}{6 c^3}-\frac{n x \left (b^2-2 a c\right )}{3 c^2}+\frac{n \sqrt{b^2-4 a c} \left (b^2-a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{3 c^3}+\frac{1}{3} x^3 \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac{b n x^2}{6 c}-\frac{2 n x^3}{9} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 800
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int x^2 \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx &=\frac{1}{3} x^3 \log \left (d \left (a+b x+c x^2\right )^n\right )-\frac{1}{3} n \int \frac{x^3 (b+2 c x)}{a+b x+c x^2} \, dx\\ &=\frac{1}{3} x^3 \log \left (d \left (a+b x+c x^2\right )^n\right )-\frac{1}{3} n \int \left (\frac{b^2-2 a c}{c^2}-\frac{b x}{c}+2 x^2-\frac{a \left (b^2-2 a c\right )+b \left (b^2-3 a c\right ) x}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=-\frac{\left (b^2-2 a c\right ) n x}{3 c^2}+\frac{b n x^2}{6 c}-\frac{2 n x^3}{9}+\frac{1}{3} x^3 \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac{n \int \frac{a \left (b^2-2 a c\right )+b \left (b^2-3 a c\right ) x}{a+b x+c x^2} \, dx}{3 c^2}\\ &=-\frac{\left (b^2-2 a c\right ) n x}{3 c^2}+\frac{b n x^2}{6 c}-\frac{2 n x^3}{9}+\frac{1}{3} x^3 \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac{\left (b \left (b^2-3 a c\right ) n\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{6 c^3}-\frac{\left (\left (b^4-5 a b^2 c+4 a^2 c^2\right ) n\right ) \int \frac{1}{a+b x+c x^2} \, dx}{6 c^3}\\ &=-\frac{\left (b^2-2 a c\right ) n x}{3 c^2}+\frac{b n x^2}{6 c}-\frac{2 n x^3}{9}+\frac{b \left (b^2-3 a c\right ) n \log \left (a+b x+c x^2\right )}{6 c^3}+\frac{1}{3} x^3 \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac{\left (\left (b^4-5 a b^2 c+4 a^2 c^2\right ) n\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{3 c^3}\\ &=-\frac{\left (b^2-2 a c\right ) n x}{3 c^2}+\frac{b n x^2}{6 c}-\frac{2 n x^3}{9}+\frac{\sqrt{b^2-4 a c} \left (b^2-a c\right ) n \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{3 c^3}+\frac{b \left (b^2-3 a c\right ) n \log \left (a+b x+c x^2\right )}{6 c^3}+\frac{1}{3} x^3 \log \left (d \left (a+b x+c x^2\right )^n\right )\\ \end{align*}
Mathematica [A] time = 0.109442, size = 122, normalized size = 0.9 \[ \frac{c n x \left (-4 c \left (c x^2-3 a\right )-6 b^2+3 b c x\right )+3 b n \left (b^2-3 a c\right ) \log (a+x (b+c x))+6 n \sqrt{b^2-4 a c} \left (b^2-a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )+6 c^3 x^3 \log \left (d (a+x (b+c x))^n\right )}{18 c^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.089, size = 870, normalized size = 6.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9961, size = 691, normalized size = 5.08 \begin{align*} \left [-\frac{4 \, c^{3} n x^{3} - 6 \, c^{3} x^{3} \log \left (d\right ) - 3 \, b c^{2} n x^{2} + 3 \,{\left (b^{2} - a c\right )} \sqrt{b^{2} - 4 \, a c} n \log \left (\frac{2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt{b^{2} - 4 \, a c}{\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 6 \,{\left (b^{2} c - 2 \, a c^{2}\right )} n x - 3 \,{\left (2 \, c^{3} n x^{3} +{\left (b^{3} - 3 \, a b c\right )} n\right )} \log \left (c x^{2} + b x + a\right )}{18 \, c^{3}}, -\frac{4 \, c^{3} n x^{3} - 6 \, c^{3} x^{3} \log \left (d\right ) - 3 \, b c^{2} n x^{2} - 6 \,{\left (b^{2} - a c\right )} \sqrt{-b^{2} + 4 \, a c} n \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 6 \,{\left (b^{2} c - 2 \, a c^{2}\right )} n x - 3 \,{\left (2 \, c^{3} n x^{3} +{\left (b^{3} - 3 \, a b c\right )} n\right )} \log \left (c x^{2} + b x + a\right )}{18 \, c^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29281, size = 197, normalized size = 1.45 \begin{align*} \frac{1}{3} \, n x^{3} \log \left (c x^{2} + b x + a\right ) - \frac{1}{9} \,{\left (2 \, n - 3 \, \log \left (d\right )\right )} x^{3} + \frac{b n x^{2}}{6 \, c} - \frac{{\left (b^{2} n - 2 \, a c n\right )} x}{3 \, c^{2}} + \frac{{\left (b^{3} n - 3 \, a b c n\right )} \log \left (c x^{2} + b x + a\right )}{6 \, c^{3}} - \frac{{\left (b^{4} n - 5 \, a b^{2} c n + 4 \, a^{2} c^{2} n\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{3 \, \sqrt{-b^{2} + 4 \, a c} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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