Optimal. Leaf size=258 \[ -\frac{2 n \text{PolyLog}\left (2,-\frac{\frac{2 c x}{\sqrt{b^2-4 a c}}+\frac{b}{\sqrt{b^2-4 a c}}+1}{-\frac{2 c x}{\sqrt{b^2-4 a c}}-\frac{b}{\sqrt{b^2-4 a c}}+1}\right )}{e \sqrt{b^2-4 a c}}-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{e \sqrt{b^2-4 a c}}+\frac{2 n \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )^2}{e \sqrt{b^2-4 a c}}-\frac{4 n \log \left (\frac{2}{-\frac{2 c x}{\sqrt{b^2-4 a c}}-\frac{b}{\sqrt{b^2-4 a c}}+1}\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{e \sqrt{b^2-4 a c}} \]
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Rubi [A] time = 0.345819, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 9, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.281, Rules used = {618, 206, 2527, 12, 6121, 5984, 5918, 2402, 2315} \[ -\frac{2 n \text{PolyLog}\left (2,-\frac{\frac{2 c x}{\sqrt{b^2-4 a c}}+\frac{b}{\sqrt{b^2-4 a c}}+1}{-\frac{2 c x}{\sqrt{b^2-4 a c}}-\frac{b}{\sqrt{b^2-4 a c}}+1}\right )}{e \sqrt{b^2-4 a c}}-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{e \sqrt{b^2-4 a c}}+\frac{2 n \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )^2}{e \sqrt{b^2-4 a c}}-\frac{4 n \log \left (\frac{2}{-\frac{2 c x}{\sqrt{b^2-4 a c}}-\frac{b}{\sqrt{b^2-4 a c}}+1}\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{e \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
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Rule 618
Rule 206
Rule 2527
Rule 12
Rule 6121
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{a e+b e x+c e x^2} \, dx &=-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{\sqrt{b^2-4 a c} e}-n \int \frac{2 (-b-2 c x) \tanh ^{-1}\left (\frac{b}{\sqrt{b^2-4 a c}}+\frac{2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} e \left (a+b x+c x^2\right )} \, dx\\ &=-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{\sqrt{b^2-4 a c} e}-\frac{(2 n) \int \frac{(-b-2 c x) \tanh ^{-1}\left (\frac{b}{\sqrt{b^2-4 a c}}+\frac{2 c x}{\sqrt{b^2-4 a c}}\right )}{a+b x+c x^2} \, dx}{\sqrt{b^2-4 a c} e}\\ &=-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{\sqrt{b^2-4 a c} e}+\frac{n \operatorname{Subst}\left (\int \frac{\sqrt{b^2-4 a c} x \tanh ^{-1}(x)}{-\frac{b^2-4 a c}{4 c}+\frac{\left (b^2-4 a c\right ) x^2}{4 c}} \, dx,x,\frac{b}{\sqrt{b^2-4 a c}}+\frac{2 c x}{\sqrt{b^2-4 a c}}\right )}{c e}\\ &=-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{\sqrt{b^2-4 a c} e}+\frac{\left (\sqrt{b^2-4 a c} n\right ) \operatorname{Subst}\left (\int \frac{x \tanh ^{-1}(x)}{-\frac{b^2-4 a c}{4 c}+\frac{\left (b^2-4 a c\right ) x^2}{4 c}} \, dx,x,\frac{b}{\sqrt{b^2-4 a c}}+\frac{2 c x}{\sqrt{b^2-4 a c}}\right )}{c e}\\ &=\frac{2 n \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )^2}{\sqrt{b^2-4 a c} e}-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{\sqrt{b^2-4 a c} e}-\frac{(4 n) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}(x)}{1-x} \, dx,x,\frac{b}{\sqrt{b^2-4 a c}}+\frac{2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} e}\\ &=\frac{2 n \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )^2}{\sqrt{b^2-4 a c} e}-\frac{4 n \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \log \left (\frac{2}{1-\frac{b}{\sqrt{b^2-4 a c}}-\frac{2 c x}{\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} e}-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{\sqrt{b^2-4 a c} e}+\frac{(4 n) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-x}\right )}{1-x^2} \, dx,x,\frac{b}{\sqrt{b^2-4 a c}}+\frac{2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} e}\\ &=\frac{2 n \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )^2}{\sqrt{b^2-4 a c} e}-\frac{4 n \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \log \left (\frac{2}{1-\frac{b}{\sqrt{b^2-4 a c}}-\frac{2 c x}{\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} e}-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{\sqrt{b^2-4 a c} e}-\frac{(4 n) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\frac{b}{\sqrt{b^2-4 a c}}-\frac{2 c x}{\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} e}\\ &=\frac{2 n \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )^2}{\sqrt{b^2-4 a c} e}-\frac{4 n \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \log \left (\frac{2}{1-\frac{b}{\sqrt{b^2-4 a c}}-\frac{2 c x}{\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} e}-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{\sqrt{b^2-4 a c} e}-\frac{2 n \text{Li}_2\left (1-\frac{2}{1-\frac{b}{\sqrt{b^2-4 a c}}-\frac{2 c x}{\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} e}\\ \end{align*}
Mathematica [A] time = 0.162295, size = 339, normalized size = 1.31 \[ \frac{-2 n \text{PolyLog}\left (2,\frac{\sqrt{b^2-4 a c}-b-2 c x}{2 \sqrt{b^2-4 a c}}\right )+2 n \text{PolyLog}\left (2,\frac{\sqrt{b^2-4 a c}+b+2 c x}{2 \sqrt{b^2-4 a c}}\right )+2 \log \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \log \left (d (a+x (b+c x))^n\right )-2 \log \left (\sqrt{b^2-4 a c}+b+2 c x\right ) \log \left (d (a+x (b+c x))^n\right )-n \log ^2\left (-\sqrt{b^2-4 a c}+b+2 c x\right )+n \log ^2\left (\sqrt{b^2-4 a c}+b+2 c x\right )-2 n \log \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{2 \sqrt{b^2-4 a c}}\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x\right )+2 n \log \left (\frac{\sqrt{b^2-4 a c}-b-2 c x}{2 \sqrt{b^2-4 a c}}\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{2 e \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.229, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( d \left ( c{x}^{2}+bx+a \right ) ^{n} \right ) }{ce{x}^{2}+bex+ea}}\, dx \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left ({\left (c x^{2} + b x + a\right )}^{n} d\right )}{c e x^{2} + b e x + a e}, x\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (c x^{2} + b x + a\right )}^{n} d\right )}{c e x^{2} + b e x + a e}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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