Optimal. Leaf size=98 \[ x \log ^3\left (\frac {c x^2}{(b+a x)^2}\right )+\frac {6 b \log ^2\left (\frac {c x^2}{(b+a x)^2}\right ) \log \left (\frac {b}{b+a x}\right )}{a}+\frac {24 b \log \left (\frac {c x^2}{(b+a x)^2}\right ) \text {Li}_2\left (\frac {a x}{b+a x}\right )}{a}-\frac {48 b \text {Li}_3\left (\frac {a x}{b+a x}\right )}{a} \]
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Rubi [A]
time = 0.05, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2536, 2552,
2354, 2421, 6724} \begin {gather*} \frac {24 b \text {PolyLog}\left (2,\frac {a x}{a x+b}\right ) \log \left (\frac {c x^2}{(a x+b)^2}\right )}{a}-\frac {48 b \text {PolyLog}\left (3,\frac {a x}{a x+b}\right )}{a}+x \log ^3\left (\frac {c x^2}{(a x+b)^2}\right )+\frac {6 b \log \left (\frac {b}{a x+b}\right ) \log ^2\left (\frac {c x^2}{(a x+b)^2}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 2354
Rule 2421
Rule 2536
Rule 2552
Rule 6724
Rubi steps
\begin {align*} \int \log ^3\left (\frac {c x^2}{(b+a x)^2}\right ) \, dx &=x \log ^3\left (\frac {c x^2}{(b+a x)^2}\right )-(6 b) \int \frac {\log ^2\left (\frac {c x^2}{(b+a x)^2}\right )}{b+a x} \, dx\\ &=x \log ^3\left (\frac {c x^2}{(b+a x)^2}\right )+\frac {6 b \log ^2\left (\frac {c x^2}{(b+a x)^2}\right ) \log \left (\frac {b}{b+a x}\right )}{a}-\frac {\left (24 b^2\right ) \int \frac {\log \left (\frac {c x^2}{(b+a x)^2}\right ) \log \left (\frac {b}{b+a x}\right )}{x (b+a x)} \, dx}{a}\\ &=x \log ^3\left (\frac {c x^2}{(b+a x)^2}\right )+\frac {6 b \log ^2\left (\frac {c x^2}{(b+a x)^2}\right ) \log \left (\frac {b}{b+a x}\right )}{a}+\frac {24 b \log \left (\frac {c x^2}{(b+a x)^2}\right ) \text {Li}_2\left (1-\frac {b}{b+a x}\right )}{a}-\frac {\left (48 b^2\right ) \int \frac {\text {Li}_2\left (1-\frac {b}{b+a x}\right )}{x (b+a x)} \, dx}{a}\\ &=x \log ^3\left (\frac {c x^2}{(b+a x)^2}\right )+\frac {6 b \log ^2\left (\frac {c x^2}{(b+a x)^2}\right ) \log \left (\frac {b}{b+a x}\right )}{a}+\frac {24 b \log \left (\frac {c x^2}{(b+a x)^2}\right ) \text {Li}_2\left (1-\frac {b}{b+a x}\right )}{a}-\frac {48 b \text {Li}_3\left (1-\frac {b}{b+a x}\right )}{a}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 98, normalized size = 1.00 \begin {gather*} x \log ^3\left (\frac {c x^2}{(b+a x)^2}\right )+\frac {6 b \log ^2\left (\frac {c x^2}{(b+a x)^2}\right ) \log \left (\frac {b}{b+a x}\right )}{a}+\frac {24 b \log \left (\frac {c x^2}{(b+a x)^2}\right ) \text {Li}_2\left (\frac {a x}{b+a x}\right )}{a}-\frac {48 b \text {Li}_3\left (\frac {a x}{b+a x}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \ln \left (\frac {c \,x^{2}}{\left (a x +b \right )^{2}}\right )^{3}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} - 6 b \int \frac {\log {\left (\frac {c x^{2}}{a^{2} x^{2} + 2 a b x + b^{2}} \right )}^{2}}{a x + b}\, dx + x \log {\left (\frac {c x^{2}}{\left (a x + b\right )^{2}} \right )}^{3} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\ln \left (\frac {c\,x^2}{{\left (b+a\,x\right )}^2}\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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