Optimal. Leaf size=106 \[ -3 p^2 \text {Li}_3\left (\frac {b x^2}{a}+1\right ) \log \left (c \left (a+b x^2\right )^p\right )+\frac {3}{2} p \text {Li}_2\left (\frac {b x^2}{a}+1\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {1}{2} \log \left (-\frac {b x^2}{a}\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )+3 p^3 \text {Li}_4\left (\frac {b x^2}{a}+1\right ) \]
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Rubi [A] time = 0.16, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2454, 2396, 2433, 2374, 2383, 6589} \[ -3 p^2 \text {PolyLog}\left (3,\frac {b x^2}{a}+1\right ) \log \left (c \left (a+b x^2\right )^p\right )+\frac {3}{2} p \text {PolyLog}\left (2,\frac {b x^2}{a}+1\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+3 p^3 \text {PolyLog}\left (4,\frac {b x^2}{a}+1\right )+\frac {1}{2} \log \left (-\frac {b x^2}{a}\right ) \log ^3\left (c \left (a+b x^2\right )^p\right ) \]
Antiderivative was successfully verified.
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Rule 2374
Rule 2383
Rule 2396
Rule 2433
Rule 2454
Rule 6589
Rubi steps
\begin {align*} \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\log ^3\left (c (a+b x)^p\right )}{x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \log \left (-\frac {b x^2}{a}\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac {1}{2} (3 b p) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right ) \log ^2\left (c (a+b x)^p\right )}{a+b x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \log \left (-\frac {b x^2}{a}\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )-\frac {1}{2} (3 p) \operatorname {Subst}\left (\int \frac {\log ^2\left (c x^p\right ) \log \left (-\frac {b \left (-\frac {a}{b}+\frac {x}{b}\right )}{a}\right )}{x} \, dx,x,a+b x^2\right )\\ &=\frac {1}{2} \log \left (-\frac {b x^2}{a}\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )+\frac {3}{2} p \log ^2\left (c \left (a+b x^2\right )^p\right ) \text {Li}_2\left (1+\frac {b x^2}{a}\right )-\left (3 p^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (c x^p\right ) \text {Li}_2\left (\frac {x}{a}\right )}{x} \, dx,x,a+b x^2\right )\\ &=\frac {1}{2} \log \left (-\frac {b x^2}{a}\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )+\frac {3}{2} p \log ^2\left (c \left (a+b x^2\right )^p\right ) \text {Li}_2\left (1+\frac {b x^2}{a}\right )-3 p^2 \log \left (c \left (a+b x^2\right )^p\right ) \text {Li}_3\left (1+\frac {b x^2}{a}\right )+\left (3 p^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {x}{a}\right )}{x} \, dx,x,a+b x^2\right )\\ &=\frac {1}{2} \log \left (-\frac {b x^2}{a}\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )+\frac {3}{2} p \log ^2\left (c \left (a+b x^2\right )^p\right ) \text {Li}_2\left (1+\frac {b x^2}{a}\right )-3 p^2 \log \left (c \left (a+b x^2\right )^p\right ) \text {Li}_3\left (1+\frac {b x^2}{a}\right )+3 p^3 \text {Li}_4\left (1+\frac {b x^2}{a}\right )\\ \end {align*}
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Mathematica [B] time = 0.10, size = 279, normalized size = 2.63 \[ -\frac {3}{2} p^2 \left (-2 \text {Li}_3\left (\frac {b x^2}{a}+1\right )+2 \text {Li}_2\left (\frac {b x^2}{a}+1\right ) \log \left (a+b x^2\right )+\log \left (-\frac {b x^2}{a}\right ) \log ^2\left (a+b x^2\right )\right ) \left (p \log \left (a+b x^2\right )-\log \left (c \left (a+b x^2\right )^p\right )\right )+3 p \left (\log (x) \left (\log \left (a+b x^2\right )-\log \left (\frac {b x^2}{a}+1\right )\right )-\frac {1}{2} \text {Li}_2\left (-\frac {b x^2}{a}\right )\right ) \left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right )^2+\log (x) \left (\log \left (c \left (a+b x^2\right )^p\right )-p \log \left (a+b x^2\right )\right )^3+\frac {1}{2} p^3 \left (6 \text {Li}_4\left (\frac {b x^2}{a}+1\right )+3 \text {Li}_2\left (\frac {b x^2}{a}+1\right ) \log ^2\left (a+b x^2\right )-6 \text {Li}_3\left (\frac {b x^2}{a}+1\right ) \log \left (a+b x^2\right )+\log \left (-\frac {b x^2}{a}\right ) \log ^3\left (a+b x^2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{x}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.66, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )^{3}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.65, size = 217, normalized size = 2.05 \[ \frac {1}{2} \, {\left (\log \left (b x^{2} + a\right )^{3} \log \left (-\frac {b x^{2} + a}{a} + 1\right ) + 3 \, {\rm Li}_2\left (\frac {b x^{2} + a}{a}\right ) \log \left (b x^{2} + a\right )^{2} - 6 \, \log \left (b x^{2} + a\right ) {\rm Li}_{3}(\frac {b x^{2} + a}{a}) + 6 \, {\rm Li}_{4}(\frac {b x^{2} + a}{a})\right )} p^{3} + \frac {3}{2} \, {\left (\log \left (b x^{2} + a\right )^{2} \log \left (-\frac {b x^{2} + a}{a} + 1\right ) + 2 \, {\rm Li}_2\left (\frac {b x^{2} + a}{a}\right ) \log \left (b x^{2} + a\right ) - 2 \, {\rm Li}_{3}(\frac {b x^{2} + a}{a})\right )} p^{2} \log \relax (c) + \frac {3}{2} \, {\left (\log \left (b x^{2} + a\right ) \log \left (-\frac {b x^{2} + a}{a} + 1\right ) + {\rm Li}_2\left (\frac {b x^{2} + a}{a}\right )\right )} p \log \relax (c)^{2} + \log \relax (c)^{3} \log \relax (x) \]
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 \[ \int \frac {{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3}{x} \,d x \]
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 \[ \int \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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