Optimal. Leaf size=167 \[ -\frac{n \left (2 a^2 c^2-4 a b^2 c+b^4\right ) \log \left (a+b x+c x^2\right )}{8 c^4}-\frac{n x^2 \left (b^2-2 a c\right )}{8 c^2}+\frac{b n x \left (b^2-3 a c\right )}{4 c^3}-\frac{b n \sqrt{b^2-4 a c} \left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{4 c^4}+\frac{1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac{b n x^3}{12 c}-\frac{n x^4}{8} \]
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Rubi [A] time = 0.188672, antiderivative size = 167, 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} \[ -\frac{n \left (2 a^2 c^2-4 a b^2 c+b^4\right ) \log \left (a+b x+c x^2\right )}{8 c^4}-\frac{n x^2 \left (b^2-2 a c\right )}{8 c^2}+\frac{b n x \left (b^2-3 a c\right )}{4 c^3}-\frac{b n \sqrt{b^2-4 a c} \left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{4 c^4}+\frac{1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac{b n x^3}{12 c}-\frac{n x^4}{8} \]
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^3 \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx &=\frac{1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )-\frac{1}{4} n \int \frac{x^4 (b+2 c x)}{a+b x+c x^2} \, dx\\ &=\frac{1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )-\frac{1}{4} n \int \left (-\frac{b \left (b^2-3 a c\right )}{c^3}+\frac{\left (b^2-2 a c\right ) x}{c^2}-\frac{b x^2}{c}+2 x^3+\frac{a b \left (b^2-3 a c\right )+\left (b^4-4 a b^2 c+2 a^2 c^2\right ) x}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac{b \left (b^2-3 a c\right ) n x}{4 c^3}-\frac{\left (b^2-2 a c\right ) n x^2}{8 c^2}+\frac{b n x^3}{12 c}-\frac{n x^4}{8}+\frac{1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )-\frac{n \int \frac{a b \left (b^2-3 a c\right )+\left (b^4-4 a b^2 c+2 a^2 c^2\right ) x}{a+b x+c x^2} \, dx}{4 c^3}\\ &=\frac{b \left (b^2-3 a c\right ) n x}{4 c^3}-\frac{\left (b^2-2 a c\right ) n x^2}{8 c^2}+\frac{b n x^3}{12 c}-\frac{n x^4}{8}+\frac{1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac{\left (b \left (b^2-4 a c\right ) \left (b^2-2 a c\right ) n\right ) \int \frac{1}{a+b x+c x^2} \, dx}{8 c^4}-\frac{\left (\left (b^4-4 a b^2 c+2 a^2 c^2\right ) n\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{8 c^4}\\ &=\frac{b \left (b^2-3 a c\right ) n x}{4 c^3}-\frac{\left (b^2-2 a c\right ) n x^2}{8 c^2}+\frac{b n x^3}{12 c}-\frac{n x^4}{8}-\frac{\left (b^4-4 a b^2 c+2 a^2 c^2\right ) n \log \left (a+b x+c x^2\right )}{8 c^4}+\frac{1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )-\frac{\left (b \left (b^2-4 a c\right ) \left (b^2-2 a c\right ) n\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{4 c^4}\\ &=\frac{b \left (b^2-3 a c\right ) n x}{4 c^3}-\frac{\left (b^2-2 a c\right ) n x^2}{8 c^2}+\frac{b n x^3}{12 c}-\frac{n x^4}{8}-\frac{b \sqrt{b^2-4 a c} \left (b^2-2 a c\right ) n \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{4 c^4}-\frac{\left (b^4-4 a b^2 c+2 a^2 c^2\right ) n \log \left (a+b x+c x^2\right )}{8 c^4}+\frac{1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )\\ \end{align*}
Mathematica [A] time = 0.147535, size = 151, normalized size = 0.9 \[ \frac{-3 n \left (2 a^2 c^2-4 a b^2 c+b^4\right ) \log (a+x (b+c x))+c n x \left (2 b c \left (c x^2-9 a\right )-3 c^2 x \left (c x^2-2 a\right )-3 b^2 c x+6 b^3\right )-6 b n \sqrt{b^2-4 a c} \left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )+6 c^4 x^4 \log \left (d (a+x (b+c x))^n\right )}{24 c^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.1, size = 1146, normalized size = 6.9 \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 = 1.96504, size = 826, normalized size = 4.95 \begin{align*} \left [-\frac{3 \, c^{4} n x^{4} - 6 \, c^{4} x^{4} \log \left (d\right ) - 2 \, b c^{3} n x^{3} + 3 \,{\left (b^{2} c^{2} - 2 \, a c^{3}\right )} n x^{2} + 3 \,{\left (b^{3} - 2 \, a b 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^{3} c - 3 \, a b c^{2}\right )} n x - 3 \,{\left (2 \, c^{4} n x^{4} -{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} n\right )} \log \left (c x^{2} + b x + a\right )}{24 \, c^{4}}, -\frac{3 \, c^{4} n x^{4} - 6 \, c^{4} x^{4} \log \left (d\right ) - 2 \, b c^{3} n x^{3} + 3 \,{\left (b^{2} c^{2} - 2 \, a c^{3}\right )} n x^{2} + 6 \,{\left (b^{3} - 2 \, a b 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^{3} c - 3 \, a b c^{2}\right )} n x - 3 \,{\left (2 \, c^{4} n x^{4} -{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} n\right )} \log \left (c x^{2} + b x + a\right )}{24 \, c^{4}}\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 = 1.27076, size = 238, normalized size = 1.43 \begin{align*} \frac{1}{4} \, n x^{4} \log \left (c x^{2} + b x + a\right ) - \frac{1}{8} \,{\left (n - 2 \, \log \left (d\right )\right )} x^{4} + \frac{b n x^{3}}{12 \, c} - \frac{{\left (b^{2} n - 2 \, a c n\right )} x^{2}}{8 \, c^{2}} + \frac{{\left (b^{3} n - 3 \, a b c n\right )} x}{4 \, c^{3}} - \frac{{\left (b^{4} n - 4 \, a b^{2} c n + 2 \, a^{2} c^{2} n\right )} \log \left (c x^{2} + b x + a\right )}{8 \, c^{4}} + \frac{{\left (b^{5} n - 6 \, a b^{3} c n + 8 \, a^{2} b c^{2} n\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{4 \, \sqrt{-b^{2} + 4 \, a c} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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