Optimal. Leaf size=129 \[ -\frac {b^2 p \log \left (1-\frac {a}{a+b x^2}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2}+\frac {b^2 p^2 \text {Li}_2\left (\frac {a}{b x^2+a}\right )}{2 a^2}+\frac {b^2 p^2 \log (x)}{a^2}-\frac {b p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2 x^2}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{4 x^4} \]
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Rubi [A] time = 0.27, antiderivative size = 147, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {2454, 2398, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31} \[ -\frac {b^2 p^2 \text {PolyLog}\left (2,\frac {b x^2}{a}+1\right )}{2 a^2}+\frac {b^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a^2}-\frac {b^2 p \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2}+\frac {b^2 p^2 \log (x)}{a^2}-\frac {b p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2 x^2}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2301
Rule 2314
Rule 2317
Rule 2344
Rule 2347
Rule 2391
Rule 2398
Rule 2411
Rule 2454
Rubi steps
\begin {align*} \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\log ^2\left (c (a+b x)^p\right )}{x^3} \, dx,x,x^2\right )\\ &=-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{4 x^4}+\frac {1}{2} (b p) \operatorname {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x^2 (a+b x)} \, dx,x,x^2\right )\\ &=-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{4 x^4}+\frac {1}{2} p \operatorname {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 )\\ &=-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{4 x^4}+\frac {p \operatorname {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}-\frac {(b p) \operatorname {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}\\ &=-\frac {b p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2 x^2}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{4 x^4}-\frac {(b p) \operatorname {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 (b^2 p\right ) \operatorname {Subst}\left (\int \frac {\log \left (c x^p\right )}{x} \, dx,x,a+b x^2\right )}{2 a^2}+\frac {\left (b p^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x}{b}} \, dx,x,a+b x^2\right )}{2 a^2}\\ &=\frac {b^2 p^2 \log (x)}{a^2}-\frac {b p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2 x^2}-\frac {b^2 p \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2}+\frac {b^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a^2}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{4 x^4}+\frac {\left (b^2 p^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{a}\right )}{x} \, dx,x,a+b x^2\right )}{2 a^2}\\ &=\frac {b^2 p^2 \log (x)}{a^2}-\frac {b p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2 x^2}-\frac {b^2 p \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2}+\frac {b^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a^2}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{4 x^4}-\frac {b^2 p^2 \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 137, normalized size = 1.06 \[ \frac {\frac {b x^2 \left (-2 b p x^2 \left (\log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )+p \text {Li}_2\left (\frac {b x^2}{a}+1\right )\right )+b x^2 \log ^2\left (c \left (a+b x^2\right )^p\right )-2 a p \log \left (c \left (a+b x^2\right )^p\right )+2 b p^2 x^2 \left (2 \log (x)-\log \left (a+b x^2\right )\right )\right )}{a^2}-\log ^2\left (c \left (a+b x^2\right )^p\right )}{4 x^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{x^{5}}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.39, size = 1080, normalized size = 8.37 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 142, normalized size = 1.10 \[ -\frac {1}{4} \, b^{2} p^{2} {\left (\frac {\log \left (b x^{2} + a\right )^{2}}{a^{2}} - \frac {2 \, {\left (2 \, \log \left (\frac {b x^{2}}{a} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-\frac {b x^{2}}{a}\right )\right )}}{a^{2}} + \frac {2 \, \log \left (b x^{2} + a\right )}{a^{2}} - \frac {4 \, \log \relax (x)}{a^{2}}\right )} + \frac {1}{2} \, b p {\left (\frac {b \log \left (b x^{2} + a\right )}{a^{2}} - \frac {b \log \left (x^{2}\right )}{a^{2}} - \frac {1}{a x^{2}}\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{4 \, x^{4}} \]
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 \[ \int \frac {{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2}{x^5} \,d x \]
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 \[ \int \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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