Integrand size = 15, antiderivative size = 67 \[ \int \log ^2\left (\frac {c x^2}{(b+a x)^2}\right ) \, dx=x \log ^2\left (\frac {c x^2}{(b+a x)^2}\right )+\frac {4 b \log \left (\frac {c x^2}{(b+a x)^2}\right ) \log \left (\frac {b}{b+a x}\right )}{a}+\frac {8 b \operatorname {PolyLog}\left (2,1-\frac {b}{b+a x}\right )}{a} \]
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Time = 0.10 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2536, 2542, 2458, 2378, 2370, 2352} \[ \int \log ^2\left (\frac {c x^2}{(b+a x)^2}\right ) \, dx=x \log ^2\left (\frac {c x^2}{(a x+b)^2}\right )+\frac {4 b \log \left (\frac {b}{a x+b}\right ) \log \left (\frac {c x^2}{(a x+b)^2}\right )}{a}+\frac {8 b \operatorname {PolyLog}\left (2,1-\frac {b}{b+a x}\right )}{a} \]
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Rule 2352
Rule 2370
Rule 2378
Rule 2458
Rule 2536
Rule 2542
Rubi steps \begin{align*} \text {integral}& = x \log ^2\left (\frac {c x^2}{(b+a x)^2}\right )-(4 b) \int \frac {\log \left (\frac {c x^2}{(b+a x)^2}\right )}{b+a x} \, dx \\ & = x \log ^2\left (\frac {c x^2}{(b+a x)^2}\right )+\frac {4 b \log \left (\frac {c x^2}{(b+a x)^2}\right ) \log \left (\frac {b}{b+a x}\right )}{a}-\frac {\left (8 b^2\right ) \int \frac {\log \left (\frac {b}{b+a x}\right )}{x (b+a x)} \, dx}{a} \\ & = x \log ^2\left (\frac {c x^2}{(b+a x)^2}\right )+\frac {4 b \log \left (\frac {c x^2}{(b+a x)^2}\right ) \log \left (\frac {b}{b+a x}\right )}{a}-\frac {\left (8 b^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {b}{x}\right )}{x \left (-\frac {b}{a}+\frac {x}{a}\right )} \, dx,x,b+a x\right )}{a^2} \\ & = x \log ^2\left (\frac {c x^2}{(b+a x)^2}\right )+\frac {4 b \log \left (\frac {c x^2}{(b+a x)^2}\right ) \log \left (\frac {b}{b+a x}\right )}{a}+\frac {\left (8 b^2\right ) \text {Subst}\left (\int \frac {\log (b x)}{\left (-\frac {b}{a}+\frac {1}{a x}\right ) x} \, dx,x,\frac {1}{b+a x}\right )}{a^2} \\ & = x \log ^2\left (\frac {c x^2}{(b+a x)^2}\right )+\frac {4 b \log \left (\frac {c x^2}{(b+a x)^2}\right ) \log \left (\frac {b}{b+a x}\right )}{a}+\frac {\left (8 b^2\right ) \text {Subst}\left (\int \frac {\log (b x)}{\frac {1}{a}-\frac {b x}{a}} \, dx,x,\frac {1}{b+a x}\right )}{a^2} \\ & = x \log ^2\left (\frac {c x^2}{(b+a x)^2}\right )+\frac {4 b \log \left (\frac {c x^2}{(b+a x)^2}\right ) \log \left (\frac {b}{b+a x}\right )}{a}+\frac {8 b \text {Li}_2\left (\frac {a x}{b+a x}\right )}{a} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.43 \[ \int \log ^2\left (\frac {c x^2}{(b+a x)^2}\right ) \, dx=\frac {a x \log ^2\left (\frac {c x^2}{(b+a x)^2}\right )+4 b \log \left (\frac {c x^2}{(b+a x)^2}\right ) \log \left (\frac {b}{b+a x}\right )-4 b \log \left (\frac {b}{b+a x}\right ) \left (2 \log \left (-\frac {a x}{b}\right )+\log \left (\frac {b}{b+a x}\right )\right )+8 b \operatorname {PolyLog}\left (2,1+\frac {a x}{b}\right )}{a} \]
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\[\int \ln \left (\frac {c \,x^{2}}{\left (a x +b \right )^{2}}\right )^{2}d x\]
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\[ \int \log ^2\left (\frac {c x^2}{(b+a x)^2}\right ) \, dx=\int { \log \left (\frac {c x^{2}}{{\left (a x + b\right )}^{2}}\right )^{2} \,d x } \]
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\[ \int \log ^2\left (\frac {c x^2}{(b+a x)^2}\right ) \, dx=- 4 b \int \frac {\log {\left (\frac {c x^{2}}{a^{2} x^{2} + 2 a b x + b^{2}} \right )}}{a x + b}\, dx + x \log {\left (\frac {c x^{2}}{\left (a x + b\right )^{2}} \right )}^{2} \]
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Time = 0.20 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.76 \[ \int \log ^2\left (\frac {c x^2}{(b+a x)^2}\right ) \, dx=x \log \left (\frac {c x^{2}}{{\left (a x + b\right )}^{2}}\right )^{2} - \frac {4 \, b \log \left (a x + b\right ) \log \left (\frac {c x^{2}}{{\left (a x + b\right )}^{2}}\right )}{a} + \frac {4 \, {\left ({\left (\frac {c \log \left (a x + b\right )^{2}}{a} - \frac {2 \, {\left (\log \left (\frac {a x}{b} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {a x}{b}\right )\right )} c}{a}\right )} b - \frac {2 \, {\left (c \log \left (a x + b\right ) - c \log \left (x\right )\right )} b \log \left (a x + b\right )}{a}\right )}}{c} \]
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\[ \int \log ^2\left (\frac {c x^2}{(b+a x)^2}\right ) \, dx=\int { \log \left (\frac {c x^{2}}{{\left (a x + b\right )}^{2}}\right )^{2} \,d x } \]
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Timed out. \[ \int \log ^2\left (\frac {c x^2}{(b+a x)^2}\right ) \, dx=\int {\ln \left (\frac {c\,x^2}{{\left (b+a\,x\right )}^2}\right )}^2 \,d x \]
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