Optimal. Leaf size=372 \[ \frac {2 n \log \left (c (a+b x)^n\right ) \text {Li}_2\left (\frac {2 f (a+b x)}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {2 n \log \left (c (a+b x)^n\right ) \text {Li}_2\left (\frac {2 f (a+b x)}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (-\sqrt {e^2-4 d f}+e+2 f x\right )}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (\sqrt {e^2-4 d f}+e+2 f x\right )}{2 a f-b \left (\sqrt {e^2-4 d f}+e\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {2 n^2 \text {Li}_3\left (\frac {2 f (a+b x)}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {2 n^2 \text {Li}_3\left (\frac {2 f (a+b x)}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}} \]
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Rubi [A] time = 0.41, antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2418, 2396, 2433, 2374, 6589} \[ \frac {2 n \log \left (c (a+b x)^n\right ) \text {PolyLog}\left (2,\frac {2 f (a+b x)}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {2 n \log \left (c (a+b x)^n\right ) \text {PolyLog}\left (2,\frac {2 f (a+b x)}{2 a f-b \left (\sqrt {e^2-4 d f}+e\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {2 n^2 \text {PolyLog}\left (3,\frac {2 f (a+b x)}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {2 n^2 \text {PolyLog}\left (3,\frac {2 f (a+b x)}{2 a f-b \left (\sqrt {e^2-4 d f}+e\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (-\sqrt {e^2-4 d f}+e+2 f x\right )}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (\sqrt {e^2-4 d f}+e+2 f x\right )}{2 a f-b \left (\sqrt {e^2-4 d f}+e\right )}\right )}{\sqrt {e^2-4 d f}} \]
Antiderivative was successfully verified.
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Rule 2374
Rule 2396
Rule 2418
Rule 2433
Rule 6589
Rubi steps
\begin {align*} \int \frac {\log ^2\left (c (a+b x)^n\right )}{d+e x+f x^2} \, dx &=\int \left (\frac {2 f \log ^2\left (c (a+b x)^n\right )}{\sqrt {e^2-4 d f} \left (e-\sqrt {e^2-4 d f}+2 f x\right )}-\frac {2 f \log ^2\left (c (a+b x)^n\right )}{\sqrt {e^2-4 d f} \left (e+\sqrt {e^2-4 d f}+2 f x\right )}\right ) \, dx\\ &=\frac {(2 f) \int \frac {\log ^2\left (c (a+b x)^n\right )}{e-\sqrt {e^2-4 d f}+2 f x} \, dx}{\sqrt {e^2-4 d f}}-\frac {(2 f) \int \frac {\log ^2\left (c (a+b x)^n\right )}{e+\sqrt {e^2-4 d f}+2 f x} \, dx}{\sqrt {e^2-4 d f}}\\ &=\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {(2 b n) \int \frac {\log \left (c (a+b x)^n\right ) \log \left (\frac {b \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{-2 a f+b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{a+b x} \, dx}{\sqrt {e^2-4 d f}}+\frac {(2 b n) \int \frac {\log \left (c (a+b x)^n\right ) \log \left (\frac {b \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{-2 a f+b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{a+b x} \, dx}{\sqrt {e^2-4 d f}}\\ &=\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {(2 n) \operatorname {Subst}\left (\int \frac {\log \left (c x^n\right ) \log \left (\frac {b \left (\frac {-2 a f+b \left (e-\sqrt {e^2-4 d f}\right )}{b}+\frac {2 f x}{b}\right )}{-2 a f+b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{x} \, dx,x,a+b x\right )}{\sqrt {e^2-4 d f}}+\frac {(2 n) \operatorname {Subst}\left (\int \frac {\log \left (c x^n\right ) \log \left (\frac {b \left (\frac {-2 a f+b \left (e+\sqrt {e^2-4 d f}\right )}{b}+\frac {2 f x}{b}\right )}{-2 a f+b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{x} \, dx,x,a+b x\right )}{\sqrt {e^2-4 d f}}\\ &=\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {2 n \log \left (c (a+b x)^n\right ) \text {Li}_2\left (\frac {2 f (a+b x)}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {2 n \log \left (c (a+b x)^n\right ) \text {Li}_2\left (\frac {2 f (a+b x)}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {\left (2 n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {2 f x}{-2 a f+b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{x} \, dx,x,a+b x\right )}{\sqrt {e^2-4 d f}}+\frac {\left (2 n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {2 f x}{-2 a f+b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{x} \, dx,x,a+b x\right )}{\sqrt {e^2-4 d f}}\\ &=\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {2 n \log \left (c (a+b x)^n\right ) \text {Li}_2\left (\frac {2 f (a+b x)}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {2 n \log \left (c (a+b x)^n\right ) \text {Li}_2\left (\frac {2 f (a+b x)}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {2 n^2 \text {Li}_3\left (\frac {2 f (a+b x)}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {2 n^2 \text {Li}_3\left (\frac {2 f (a+b x)}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 655, normalized size = 1.76 \[ \frac {2 n \sqrt {4 d f-e^2} \log \left (c (a+b x)^n\right ) \text {Li}_2\left (\frac {2 f (a+b x)}{2 a f+b \left (\sqrt {e^2-4 d f}-e\right )}\right )-2 n \sqrt {4 d f-e^2} \log \left (c (a+b x)^n\right ) \text {Li}_2\left (\frac {2 f (a+b x)}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )+2 n \sqrt {4 d f-e^2} \log (a+b x) \log \left (c (a+b x)^n\right ) \log \left (1-\frac {2 f (a+b x)}{2 a f+b \sqrt {e^2-4 d f}+b (-e)}\right )-2 n \sqrt {4 d f-e^2} \log (a+b x) \log \left (c (a+b x)^n\right ) \log \left (\frac {2 f (a+b x)}{b \left (\sqrt {e^2-4 d f}+e\right )-2 a f}+1\right )+2 \sqrt {e^2-4 d f} \tan ^{-1}\left (\frac {e+2 f x}{\sqrt {4 d f-e^2}}\right ) \log ^2\left (c (a+b x)^n\right )-4 n \sqrt {e^2-4 d f} \log (a+b x) \tan ^{-1}\left (\frac {e+2 f x}{\sqrt {4 d f-e^2}}\right ) \log \left (c (a+b x)^n\right )-2 n^2 \sqrt {4 d f-e^2} \text {Li}_3\left (\frac {2 f (a+b x)}{-e b+\sqrt {e^2-4 d f} b+2 a f}\right )+2 n^2 \sqrt {4 d f-e^2} \text {Li}_3\left (\frac {2 f (a+b x)}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )-n^2 \sqrt {4 d f-e^2} \log ^2(a+b x) \log \left (1-\frac {2 f (a+b x)}{2 a f+b \sqrt {e^2-4 d f}+b (-e)}\right )+n^2 \sqrt {4 d f-e^2} \log ^2(a+b x) \log \left (\frac {2 f (a+b x)}{b \left (\sqrt {e^2-4 d f}+e\right )-2 a f}+1\right )+2 n^2 \sqrt {e^2-4 d f} \log ^2(a+b x) \tan ^{-1}\left (\frac {e+2 f x}{\sqrt {4 d f-e^2}}\right )}{\sqrt {-\left (e^2-4 d f\right )^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{2}}{f x^{2} + e x + d}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{2}}{f x^{2} + e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 20.05, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \left (b x +a \right )^{n}\right )^{2}}{f \,x^{2}+e x +d}\, dx \]
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 \[ \int \frac {{\ln \left (c\,{\left (a+b\,x\right )}^n\right )}^2}{f\,x^2+e\,x+d} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (c \left (a + b x\right )^{n} \right )}^{2}}{d + e x + f x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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