Integrand size = 18, antiderivative size = 352 \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=\frac {b^3 p^3 \log (x)}{a^3}-\frac {b^2 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^3 x^2}-\frac {b^3 p^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a^3}-\frac {b p \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a x^4}+\frac {b^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a^3 x^2}-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{6 x^6}-\frac {b^3 p^2 \log \left (c \left (a+b x^2\right )^p\right ) \log \left (1-\frac {a}{a+b x^2}\right )}{2 a^3}+\frac {b^3 p \log ^2\left (c \left (a+b x^2\right )^p\right ) \log \left (1-\frac {a}{a+b x^2}\right )}{2 a^3}+\frac {b^3 p^3 \operatorname {PolyLog}\left (2,\frac {a}{a+b x^2}\right )}{2 a^3}-\frac {b^3 p^2 \log \left (c \left (a+b x^2\right )^p\right ) \operatorname {PolyLog}\left (2,\frac {a}{a+b x^2}\right )}{a^3}-\frac {b^3 p^3 \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )}{a^3}-\frac {b^3 p^3 \operatorname {PolyLog}\left (3,\frac {a}{a+b x^2}\right )}{a^3} \]
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Time = 0.44 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {2504, 2445, 2458, 2389, 2379, 2421, 6724, 2355, 2354, 2438, 2356, 2351, 31} \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=-\frac {b^3 p^2 \operatorname {PolyLog}\left (2,\frac {a}{b x^2+a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a^3}-\frac {b^3 p^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a^3}-\frac {b^3 p^2 \log \left (1-\frac {a}{a+b x^2}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^3}+\frac {b^3 p \log \left (1-\frac {a}{a+b x^2}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a^3}+\frac {b^3 p^3 \operatorname {PolyLog}\left (2,\frac {a}{b x^2+a}\right )}{2 a^3}-\frac {b^3 p^3 \operatorname {PolyLog}\left (2,\frac {b x^2}{a}+1\right )}{a^3}-\frac {b^3 p^3 \operatorname {PolyLog}\left (3,\frac {a}{b x^2+a}\right )}{a^3}+\frac {b^3 p^3 \log (x)}{a^3}-\frac {b^2 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^3 x^2}+\frac {b^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a^3 x^2}-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{6 x^6}-\frac {b p \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a x^4} \]
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Rule 31
Rule 2351
Rule 2354
Rule 2355
Rule 2356
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^4} \, dx,x,x^2\right ) \\ & = -\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac {1}{2} (b p) \text {Subst}\left (\int \frac {\log ^2\left (c (a+b x)^p\right )}{x^3 (a+b x)} \, dx,x,x^2\right ) \\ & = -\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac {1}{2} p \text {Subst}\left (\int \frac {\log ^2\left (c x^p\right )}{x \left (-\frac {a}{b}+\frac {x}{b}\right )^3} \, dx,x,a+b x^2\right ) \\ & = -\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac {p \text {Subst}\left (\int \frac {\log ^2\left (c x^p\right )}{\left (-\frac {a}{b}+\frac {x}{b}\right )^3} \, dx,x,a+b x^2\right )}{2 a}-\frac {(b 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 )}{2 a} \\ & = -\frac {b p \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a x^4}-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{6 x^6}-\frac {(b 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 )}{2 a^2}+\frac {\left (b^2 p\right ) \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 )}{2 a^2}+\frac {\left (b p^2\right ) \text {Subst}\left (\int \frac {\log \left (c x^p\right )}{x \left (-\frac {a}{b}+\frac {x}{b}\right )^2} \, dx,x,a+b x^2\right )}{2 a} \\ & = -\frac {b p \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a x^4}+\frac {b^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a^3 x^2}-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac {b^3 p \log ^2\left (c \left (a+b x^2\right )^p\right ) \log \left (1-\frac {a}{a+b x^2}\right )}{2 a^3}+\frac {\left (b p^2\right ) \text {Subst}\left (\int \frac {\log \left (c x^p\right )}{\left (-\frac {a}{b}+\frac {x}{b}\right )^2} \, dx,x,a+b x^2\right )}{2 a^2}-\frac {\left (b^2 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 )}{a^3}-\frac {\left (b^2 p^2\right ) \text {Subst}\left (\int \frac {\log \left (c x^p\right )}{x \left (-\frac {a}{b}+\frac {x}{b}\right )} \, dx,x,a+b x^2\right )}{2 a^2}-\frac {\left (b^3 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 )}{a^3} \\ & = -\frac {b^2 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^3 x^2}-\frac {b^3 p^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a^3}-\frac {b p \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a x^4}+\frac {b^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a^3 x^2}-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{6 x^6}-\frac {b^3 p^2 \log \left (c \left (a+b x^2\right )^p\right ) \log \left (1-\frac {a}{a+b x^2}\right )}{2 a^3}+\frac {b^3 p \log ^2\left (c \left (a+b x^2\right )^p\right ) \log \left (1-\frac {a}{a+b x^2}\right )}{2 a^3}-\frac {b^3 p^2 \log \left (c \left (a+b x^2\right )^p\right ) \text {Li}_2\left (\frac {a}{a+b x^2}\right )}{a^3}+\frac {\left (b^2 p^3\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x}{b}} \, dx,x,a+b x^2\right )}{2 a^3}+\frac {\left (b^3 p^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {a}{x}\right )}{x} \, dx,x,a+b x^2\right )}{2 a^3}+\frac {\left (b^3 p^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{a}\right )}{x} \, dx,x,a+b x^2\right )}{a^3}+\frac {\left (b^3 p^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {a}{x}\right )}{x} \, dx,x,a+b x^2\right )}{a^3} \\ & = \frac {b^3 p^3 \log (x)}{a^3}-\frac {b^2 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^3 x^2}-\frac {b^3 p^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a^3}-\frac {b p \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a x^4}+\frac {b^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a^3 x^2}-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{6 x^6}-\frac {b^3 p^2 \log \left (c \left (a+b x^2\right )^p\right ) \log \left (1-\frac {a}{a+b x^2}\right )}{2 a^3}+\frac {b^3 p \log ^2\left (c \left (a+b x^2\right )^p\right ) \log \left (1-\frac {a}{a+b x^2}\right )}{2 a^3}+\frac {b^3 p^3 \text {Li}_2\left (\frac {a}{a+b x^2}\right )}{2 a^3}-\frac {b^3 p^2 \log \left (c \left (a+b x^2\right )^p\right ) \text {Li}_2\left (\frac {a}{a+b x^2}\right )}{a^3}-\frac {b^3 p^3 \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{a^3}-\frac {b^3 p^3 \text {Li}_3\left (\frac {a}{a+b x^2}\right )}{a^3} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 571, normalized size of antiderivative = 1.62 \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=-\frac {-6 b^3 p^3 x^6 \log \left (-\frac {b x^2}{a}\right )+6 b^3 p^3 x^6 \log \left (a+b x^2\right )-36 b^3 p^3 x^6 \log (x) \log \left (a+b x^2\right )+18 b^3 p^3 x^6 \log \left (-\frac {b x^2}{a}\right ) \log \left (a+b x^2\right )+9 b^3 p^3 x^6 \log ^2\left (a+b x^2\right )-12 b^3 p^3 x^6 \log (x) \log ^2\left (a+b x^2\right )+6 b^3 p^3 x^6 \log \left (-\frac {b x^2}{a}\right ) \log ^2\left (a+b x^2\right )+2 b^3 p^3 x^6 \log ^3\left (a+b x^2\right )+6 a b^2 p^2 x^4 \log \left (c \left (a+b x^2\right )^p\right )+36 b^3 p^2 x^6 \log (x) \log \left (c \left (a+b x^2\right )^p\right )-18 b^3 p^2 x^6 \log \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )+24 b^3 p^2 x^6 \log (x) \log \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )-12 b^3 p^2 x^6 \log \left (-\frac {b x^2}{a}\right ) \log \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )-6 b^3 p^2 x^6 \log ^2\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )+3 a^2 b p x^2 \log ^2\left (c \left (a+b x^2\right )^p\right )-6 a b^2 p x^4 \log ^2\left (c \left (a+b x^2\right )^p\right )-12 b^3 p x^6 \log (x) \log ^2\left (c \left (a+b x^2\right )^p\right )+6 b^3 p x^6 \log \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+2 a^3 \log ^3\left (c \left (a+b x^2\right )^p\right )+6 b^3 p^2 x^6 \left (3 p-2 \log \left (c \left (a+b x^2\right )^p\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )+12 b^3 p^3 x^6 \operatorname {PolyLog}\left (3,1+\frac {b x^2}{a}\right )}{12 a^3 x^6} \]
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\[\int \frac {{\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{3}}{x^{7}}d x\]
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\[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{x^{7}} \,d x } \]
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\[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=\int \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}{x^{7}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.96 \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=\frac {1}{12} \, {\left (\frac {6 \, {\left (\log \left (b x^{2} + a\right )^{2} \log \left (-\frac {b x^{2} + a}{a} + 1\right ) + 2 \, {\rm Li}_2\left (\frac {b x^{2} + a}{a}\right ) \log \left (b x^{2} + a\right ) - 2 \, {\rm Li}_{3}(\frac {b x^{2} + a}{a})\right )} b^{2} p^{2}}{a^{3}} - \frac {6 \, {\left (3 \, p^{2} - 2 \, p \log \left (c\right )\right )} {\left (\log \left (b x^{2} + a\right ) \log \left (-\frac {b x^{2} + a}{a} + 1\right ) + {\rm Li}_2\left (\frac {b x^{2} + a}{a}\right )\right )} b^{2}}{a^{3}} + \frac {12 \, {\left (p^{2} - 3 \, p \log \left (c\right ) + \log \left (c\right )^{2}\right )} b^{2} \log \left (x\right )}{a^{3}} - \frac {2 \, b^{2} p^{2} x^{4} \log \left (b x^{2} + a\right )^{3} + 6 \, {\left (p \log \left (c\right ) - \log \left (c\right )^{2}\right )} a b x^{2} + 3 \, a^{2} \log \left (c\right )^{2} - 3 \, {\left ({\left (3 \, p^{2} - 2 \, p \log \left (c\right )\right )} b^{2} x^{4} + 2 \, a b p^{2} x^{2} - a^{2} p^{2}\right )} \log \left (b x^{2} + a\right )^{2} + 6 \, {\left ({\left (p^{2} - 3 \, p \log \left (c\right ) + \log \left (c\right )^{2}\right )} b^{2} x^{4} + {\left (p^{2} - 2 \, p \log \left (c\right )\right )} a b x^{2} + a^{2} p \log \left (c\right )\right )} \log \left (b x^{2} + a\right )}{a^{3} x^{4}}\right )} b p - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{6 \, x^{6}} \]
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\[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{x^{7}} \,d x } \]
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Timed out. \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=\int \frac {{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3}{x^7} \,d x \]
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