Integrand size = 18, antiderivative size = 219 \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=\frac {3 b^2 p^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2}-\frac {3 b p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a^2 x^2}-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{4 x^4}-\frac {3 b^2 p \log ^2\left (c \left (a+b x^2\right )^p\right ) \log \left (1-\frac {a}{a+b x^2}\right )}{4 a^2}+\frac {3 b^2 p^2 \log \left (c \left (a+b x^2\right )^p\right ) \operatorname {PolyLog}\left (2,\frac {a}{a+b x^2}\right )}{2 a^2}+\frac {3 b^2 p^3 \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )}{2 a^2}+\frac {3 b^2 p^3 \operatorname {PolyLog}\left (3,\frac {a}{a+b x^2}\right )}{2 a^2} \]
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Time = 0.25 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {2504, 2445, 2458, 2389, 2379, 2421, 6724, 2355, 2354, 2438} \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=\frac {3 b^2 p^2 \operatorname {PolyLog}\left (2,\frac {a}{b x^2+a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2}+\frac {3 b^2 p^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2}-\frac {3 b^2 p \log \left (1-\frac {a}{a+b x^2}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a^2}+\frac {3 b^2 p^3 \operatorname {PolyLog}\left (2,\frac {b x^2}{a}+1\right )}{2 a^2}+\frac {3 b^2 p^3 \operatorname {PolyLog}\left (3,\frac {a}{b x^2+a}\right )}{2 a^2}-\frac {3 b p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a^2 x^2}-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{4 x^4} \]
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Rule 2354
Rule 2355
Rule 2379
Rule 2389
Rule 2421
Rule 2438
Rule 2445
Rule 2458
Rule 2504
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\log ^3\left (c (a+b x)^p\right )}{x^3} \, dx,x,x^2\right ) \\ & = -\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{4 x^4}+\frac {1}{4} (3 b p) \text {Subst}\left (\int \frac {\log ^2\left (c (a+b x)^p\right )}{x^2 (a+b x)} \, dx,x,x^2\right ) \\ & = -\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{4 x^4}+\frac {1}{4} (3 p) \text {Subst}\left (\int \frac {\log ^2\left (c x^p\right )}{x \left (-\frac {a}{b}+\frac {x}{b}\right )^2} \, dx,x,a+b x^2\right ) \\ & = -\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{4 x^4}+\frac {(3 p) \text {Subst}\left (\int \frac {\log ^2\left (c x^p\right )}{\left (-\frac {a}{b}+\frac {x}{b}\right )^2} \, dx,x,a+b x^2\right )}{4 a}-\frac {(3 b p) \text {Subst}\left (\int \frac {\log ^2\left (c x^p\right )}{x \left (-\frac {a}{b}+\frac {x}{b}\right )} \, dx,x,a+b x^2\right )}{4 a} \\ & = -\frac {3 b p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a^2 x^2}-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{4 x^4}-\frac {3 b^2 p \log ^2\left (c \left (a+b x^2\right )^p\right ) \log \left (1-\frac {a}{a+b x^2}\right )}{4 a^2}+\frac {\left (3 b p^2\right ) \text {Subst}\left (\int \frac {\log \left (c x^p\right )}{-\frac {a}{b}+\frac {x}{b}} \, dx,x,a+b x^2\right )}{2 a^2}+\frac {\left (3 b^2 p^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {a}{x}\right ) \log \left (c x^p\right )}{x} \, dx,x,a+b x^2\right )}{2 a^2} \\ & = \frac {3 b^2 p^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2}-\frac {3 b p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a^2 x^2}-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{4 x^4}-\frac {3 b^2 p \log ^2\left (c \left (a+b x^2\right )^p\right ) \log \left (1-\frac {a}{a+b x^2}\right )}{4 a^2}+\frac {3 b^2 p^2 \log \left (c \left (a+b x^2\right )^p\right ) \text {Li}_2\left (\frac {a}{a+b x^2}\right )}{2 a^2}-\frac {\left (3 b^2 p^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{a}\right )}{x} \, dx,x,a+b x^2\right )}{2 a^2}-\frac {\left (3 b^2 p^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {a}{x}\right )}{x} \, dx,x,a+b x^2\right )}{2 a^2} \\ & = \frac {3 b^2 p^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2}-\frac {3 b p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a^2 x^2}-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{4 x^4}-\frac {3 b^2 p \log ^2\left (c \left (a+b x^2\right )^p\right ) \log \left (1-\frac {a}{a+b x^2}\right )}{4 a^2}+\frac {3 b^2 p^2 \log \left (c \left (a+b x^2\right )^p\right ) \text {Li}_2\left (\frac {a}{a+b x^2}\right )}{2 a^2}+\frac {3 b^2 p^3 \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{2 a^2}+\frac {3 b^2 p^3 \text {Li}_3\left (\frac {a}{a+b x^2}\right )}{2 a^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(477\) vs. \(2(219)=438\).
Time = 0.28 (sec) , antiderivative size = 477, normalized size of antiderivative = 2.18 \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=-\frac {3 b p \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2}{4 a x^2}-\frac {3 b^2 p \log (x) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2}{2 a^2}+\frac {3 b^2 p \log \left (a+b x^2\right ) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2}{4 a^2}-\frac {3 p \log \left (a+b x^2\right ) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2}{4 x^4}-\frac {\left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^3}{4 x^4}+3 p^2 \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right ) \left (-\frac {\log ^2\left (a+b x^2\right )}{4 x^4}+\frac {b \left (4 b x^2 \log (x)+\log \left (a+b x^2\right ) \left (-2 \left (a+b x^2\right )-2 b x^2 \log \left (-\frac {b x^2}{a}\right )+b x^2 \log \left (a+b x^2\right )\right )-2 b x^2 \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )\right )}{4 a^2 x^2}\right )+\frac {b^2 p^3 \left (\frac {\left (a+b x^2\right ) \left (a \left (3-2 \log \left (a+b x^2\right )\right )+\left (a+b x^2\right ) \left (-3+\log \left (a+b x^2\right )\right )\right ) \log ^2\left (a+b x^2\right )}{b^2 x^4}-3 \left (-2+\log \left (a+b x^2\right )\right ) \log \left (a+b x^2\right ) \log \left (1-\frac {a+b x^2}{a}\right )-6 \left (-1+\log \left (a+b x^2\right )\right ) \operatorname {PolyLog}\left (2,\frac {a+b x^2}{a}\right )+6 \operatorname {PolyLog}\left (3,\frac {a+b x^2}{a}\right )\right )}{4 a^2} \]
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\[\int \frac {{\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{3}}{x^{5}}d x\]
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\[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{x^{5}} \,d x } \]
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\[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=\int \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}{x^{5}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.23 \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=-\frac {1}{4} \, {\left (\frac {3 \, {\left (\log \left (b x^{2} + a\right )^{2} \log \left (-\frac {b x^{2} + a}{a} + 1\right ) + 2 \, {\rm Li}_2\left (\frac {b x^{2} + a}{a}\right ) \log \left (b x^{2} + a\right ) - 2 \, {\rm Li}_{3}(\frac {b x^{2} + a}{a})\right )} b p^{2}}{a^{2}} - \frac {6 \, {\left (p^{2} - p \log \left (c\right )\right )} {\left (\log \left (b x^{2} + a\right ) \log \left (-\frac {b x^{2} + a}{a} + 1\right ) + {\rm Li}_2\left (\frac {b x^{2} + a}{a}\right )\right )} b}{a^{2}} - \frac {6 \, {\left (2 \, p \log \left (c\right ) - \log \left (c\right )^{2}\right )} b \log \left (x\right )}{a^{2}} - \frac {b p^{2} x^{2} \log \left (b x^{2} + a\right )^{3} - 3 \, {\left ({\left (p^{2} - p \log \left (c\right )\right )} b x^{2} + a p^{2}\right )} \log \left (b x^{2} + a\right )^{2} - 3 \, a \log \left (c\right )^{2} - 3 \, {\left ({\left (2 \, p \log \left (c\right ) - \log \left (c\right )^{2}\right )} b x^{2} + 2 \, a p \log \left (c\right )\right )} \log \left (b x^{2} + a\right )}{a^{2} x^{2}}\right )} b p - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{4 \, x^{4}} \]
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\[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{x^{5}} \,d x } \]
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Timed out. \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=\int \frac {{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3}{x^5} \,d x \]
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