Optimal. Leaf size=119 \[ \frac {3 b p^2 \text {Li}_2\left (\frac {b x^2}{a}+1\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}-\frac {\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}+\frac {3 b p \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a}-\frac {3 b p^3 \text {Li}_3\left (\frac {b x^2}{a}+1\right )}{a} \]
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Rubi [A] time = 0.15, antiderivative size = 119, 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, 2397, 2396, 2433, 2374, 6589} \[ \frac {3 b p^2 \text {PolyLog}\left (2,\frac {b x^2}{a}+1\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}-\frac {3 b p^3 \text {PolyLog}\left (3,\frac {b x^2}{a}+1\right )}{a}+\frac {3 b p \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a}-\frac {\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 2374
Rule 2396
Rule 2397
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^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\log ^3\left (c (a+b x)^p\right )}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}+\frac {(3 b p) \operatorname {Subst}\left (\int \frac {\log ^2\left (c (a+b x)^p\right )}{x} \, dx,x,x^2\right )}{2 a}\\ &=\frac {3 b p \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a}-\frac {\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}-\frac {\left (3 b^2 p^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )}{a+b x} \, dx,x,x^2\right )}{a}\\ &=\frac {3 b p \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a}-\frac {\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}-\frac {\left (3 b p^2\right ) \operatorname {Subst}\left (\int \frac {\log \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 )}{a}\\ &=\frac {3 b p \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a}-\frac {\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}+\frac {3 b p^2 \log \left (c \left (a+b x^2\right )^p\right ) \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{a}-\frac {\left (3 b p^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{a}\right )}{x} \, dx,x,a+b x^2\right )}{a}\\ &=\frac {3 b p \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a}-\frac {\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}+\frac {3 b p^2 \log \left (c \left (a+b x^2\right )^p\right ) \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{a}-\frac {3 b p^3 \text {Li}_3\left (1+\frac {b x^2}{a}\right )}{a}\\ \end {align*}
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Mathematica [B] time = 0.30, size = 302, normalized size = 2.54 \[ -\frac {-6 b p^2 x^2 \text {Li}_2\left (\frac {b x^2}{a}+1\right ) \log \left (c \left (a+b x^2\right )^p\right )-3 b p^2 x^2 \log ^2\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )+12 b p^2 x^2 \log (x) \log \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )-6 b p^2 x^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )+a \log ^3\left (c \left (a+b x^2\right )^p\right )-6 b p x^2 \log (x) \log ^2\left (c \left (a+b x^2\right )^p\right )+3 b p x^2 \log \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+6 b p^3 x^2 \text {Li}_3\left (\frac {b x^2}{a}+1\right )+b p^3 x^2 \log ^3\left (a+b x^2\right )-6 b p^3 x^2 \log (x) \log ^2\left (a+b x^2\right )+3 b p^3 x^2 \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (a+b x^2\right )}{2 a x^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{x^{3}}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.19, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )^{3}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 202, normalized size = 1.70 \[ \frac {1}{2} \, {\left (\frac {3 \, {\left (\log \left (b x^{2} + a\right )^{2} \log \left (-\frac {b x^{2} + a}{a} + 1\right ) + 2 \, {\rm Li}_2\left (\frac {b x^{2} + a}{a}\right ) \log \left (b x^{2} + a\right ) - 2 \, {\rm Li}_{3}(\frac {b x^{2} + a}{a})\right )} p^{2}}{a} + \frac {6 \, {\left (\log \left (b x^{2} + a\right ) \log \left (-\frac {b x^{2} + a}{a} + 1\right ) + {\rm Li}_2\left (\frac {b x^{2} + a}{a}\right )\right )} p \log \relax (c)}{a} + \frac {6 \, \log \relax (c)^{2} \log \relax (x)}{a} - \frac {p^{2} \log \left (b x^{2} + a\right )^{3} + 3 \, p \log \left (b x^{2} + a\right )^{2} \log \relax (c) + 3 \, \log \left (b x^{2} + a\right ) \log \relax (c)^{2}}{a}\right )} b p - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{2 \, x^{2}} \]
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 \[ \int \frac {{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3}{x^3} \,d x \]
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 \[ \int \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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