Optimal. Leaf size=67 \[ x \log ^2\left (\frac {c x^2}{(a x+b)^2}\right )+\frac {4 b \log \left (\frac {b}{a x+b}\right ) \log \left (\frac {c x^2}{(a x+b)^2}\right )}{a}+\frac {8 b \text {Li}_2\left (1-\frac {b}{b+a x}\right )}{a} \]
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Rubi [A] time = 0.15, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2486, 2488, 2411, 2343, 2333, 2315} \[ \frac {8 b \text {PolyLog}\left (2,1-\frac {b}{a x+b}\right )}{a}+x \log ^2\left (\frac {c x^2}{(a x+b)^2}\right )+\frac {4 b \log \left (\frac {b}{a x+b}\right ) \log \left (\frac {c x^2}{(a x+b)^2}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2333
Rule 2343
Rule 2411
Rule 2486
Rule 2488
Rubi steps
\begin {align*} \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 )-(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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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*}
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Mathematica [A] time = 0.01, size = 106, normalized size = 1.58 \[ x \log ^2\left (\frac {c x^2}{(a x+b)^2}\right )+\frac {4 b \log \left (\frac {b}{a x+b}\right ) \log \left (\frac {c x^2}{(a x+b)^2}\right )}{a}+\frac {8 b \text {Li}_2\left (\frac {b+a x}{b}\right )}{a}-\frac {4 b \log ^2\left (\frac {b}{a x+b}\right )}{a}-\frac {8 b \log \left (-\frac {a x}{b}\right ) \log \left (\frac {b}{a x+b}\right )}{a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\log \left (\frac {c x^{2}}{a^{2} x^{2} + 2 \, a b x + b^{2}}\right )^{2}, x\right ) \]
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 \[ \int \log \left (\frac {c x^{2}}{{\left (a x + b\right )}^{2}}\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.57, size = 0, normalized size = 0.00 \[ \int \ln \left (\frac {c \,x^{2}}{\left (a x +b \right )^{2}}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 118, normalized size = 1.76 \[ x \log \left (\frac {c x^{2}}{{\left (a x + b\right )}^{2}}\right )^{2} - \frac {4 \, b \log \left (a x + b\right ) \log \left (\frac {c x^{2}}{{\left (a x + b\right )}^{2}}\right )}{a} + \frac {4 \, {\left ({\left (\frac {c \log \left (a x + b\right )^{2}}{a} - \frac {2 \, {\left (\log \left (\frac {a x}{b} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-\frac {a x}{b}\right )\right )} c}{a}\right )} b - \frac {2 \, {\left (c \log \left (a x + b\right ) - c \log \relax (x)\right )} b \log \left (a x + b\right )}{a}\right )}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\ln \left (\frac {c\,x^2}{{\left (b+a\,x\right )}^2}\right )}^2 \,d x \]
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 \[ - 4 b \int \frac {\log {\left (\frac {c x^{2}}{a^{2} x^{2} + 2 a b x + b^{2}} \right )}}{a x + b}\, dx + x \log {\left (\frac {c x^{2}}{\left (a x + b\right )^{2}} \right )}^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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