Optimal. Leaf size=258 \[ -\frac {2 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{e \sqrt {b^2-4 a c}}-\frac {2 n \text {Li}_2\left (-\frac {\frac {b}{\sqrt {b^2-4 a c}}+\frac {2 c x}{\sqrt {b^2-4 a c}}+1}{-\frac {b}{\sqrt {b^2-4 a c}}-\frac {2 c x}{\sqrt {b^2-4 a c}}+1}\right )}{e \sqrt {b^2-4 a c}}+\frac {2 n \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )^2}{e \sqrt {b^2-4 a c}}-\frac {4 n \log \left (\frac {2}{-\frac {2 c x}{\sqrt {b^2-4 a c}}-\frac {b}{\sqrt {b^2-4 a c}}+1}\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{e \sqrt {b^2-4 a c}} \]
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Rubi [A] time = 0.35, antiderivative size = 258, normalized size of antiderivative = 1.00, 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 12
Rule 206
Rule 618
Rule 2315
Rule 2402
Rule 2527
Rule 5918
Rule 5984
Rule 6121
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*}
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Mathematica [A] time = 0.19, size = 339, normalized size = 1.31 \[ \frac {2 \log \left (-\sqrt {b^2-4 a c}+b+2 c x\right ) \log \left (d (a+x (b+c x))^n\right )-2 \log \left (\sqrt {b^2-4 a c}+b+2 c x\right ) \log \left (d (a+x (b+c x))^n\right )-2 n \text {Li}_2\left (\frac {-b-2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )+2 n \text {Li}_2\left (\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )-n \log ^2\left (-\sqrt {b^2-4 a c}+b+2 c x\right )+n \log ^2\left (\sqrt {b^2-4 a c}+b+2 c x\right )-2 n \log \left (\frac {\sqrt {b^2-4 a c}+b+2 c x}{2 \sqrt {b^2-4 a c}}\right ) \log \left (-\sqrt {b^2-4 a c}+b+2 c x\right )+2 n \log \left (\frac {\sqrt {b^2-4 a c}-b-2 c x}{2 \sqrt {b^2-4 a c}}\right ) \log \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{2 e \sqrt {b^2-4 a c}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.22, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (c x^{2} + b x + a\right )}^{n} d\right )}{c e x^{2} + b e x + a e}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (c x^{2} + b x + a\right )}^{n} d\right )}{c e x^{2} + b e x + a e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.71, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (d \left (c \,x^{2}+b x +a \right )^{n}\right )}{c e \,x^{2}+b e x +a e}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (d\,{\left (c\,x^2+b\,x+a\right )}^n\right )}{c\,e\,x^2+b\,e\,x+a\,e} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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